TPISC_IV:_Details

Preface

*TPISC ( pisque—silent “T”): The Pythagorean — Inverse Square Connection*

The world of numbers is yes!

Yes heaven, yes hell.

So where do they start?

Why not 0, 1, 2, 3,…

Now go square them — smart!

Place them on a grid,

Axis and diagonal.

Clever you! Done did.

The numbers between —

Subtract a squared from a squared —

Are there to be seen!

A row that ripples

Of squared numbers, gives to you:

Triangle “Triples!”

~ ~ ~

The story of the BIM (BBS-ISL Matrix) and the connection with the Pythagorean Theorem — TPISC (The Pythagorean - Inverse Square Connection) — is fundamentally simple!

In fact, keep the 3-Steps to Nirvana in mind as we parse out some of the details.

3-Steps to Nirvana:

1. Make the BIM:

• draw a square with evenly spaced grid lines
• place 0-1-2-3-… along top and side (left) and their squared values along diagonal (PD=Prime Diagonal)
• fill in the rest of the grid cells with the difference (∆) between the horizontal and vertical PD values.
2. Locate the PTs (Pythagorean Triples):

• from the PD, starting with 9, drop down vertically until a squared number on a Row is found
• that squared number will always point to its complementary squared number (found on both the PD and along that SAME Row)
• this PT Row — with the two a2 and b2 values — terminates with c2 at the interception with the PD.
3. Connect the dots to make the ToPPT (Tree of Primitive Pythagorean Triples):

• every PT — both parental Primitive Pythagorean Triples (PPTs) and their children non-Primitive Pythagorean Triples (nPPTs) — will occupy a unique Row (with some occasional overlaps)
• every PT will follow the detailed r-value based Template, differing in only its specific a2, b2, c2, 4A, 8A, p, ƒ and ƒ2... variables
• connecting the dots (values) between the Universal Template and the Specific variables gives the ToPPT.

SEE Page 50 (Art Theory101: White Papers) for spreadsheets: BIM: How to Make.

HERE IS THE BIG OUTLINE OF THE CONTENTS:

D. Rows (and symmetrical Columns)

1. EVENs (>4)

​                  a. EVENs ÷6—> BIM ÷24

​                  b. EVENs NEVER PPT or PRIME

2. ODDs (≥5)

​                 a. ODDs NOT ÷3 —> BIM ÷24

​                   1.) ALL PPTs

​                        a.) 4 PPT iterations/Row + Column

b.) 4A # ALWAYS on Row/Column

​                        c.) 8A # ALWAYS on "Sister Row" below

​                        d.) Tripartite Tree of Pythagorena Triples

e.) ALL PPTs linked to each other

​                        f.) Forms nodes of asymmetry

g.) Information is: ubiquitous, infinite, and ALWAYS KNOWN

h.) Informs DSEQEC and CaCoST, along with the BIM.

​                   2.) ALL PRIMES

​                   3.) Some Both

​                   4.) Some Neither

​                 b. ODDs ÷3 —> NOT BIM ÷24

​                   1.) NEVER PPT

​                   2.) NEVER PRIME

E. PD EVENs vs ODDs

1. EVEN PD (>4):

​                 a. NEVER PPT

​                 b. NEVER PRIME

​                 c. BIM ÷24 (Two alternating sets)

​                   1.) ES1

​                        a.) Outer

​                        b.) Middle

​                   2.) ES2

​                        a.) Outer

​                        b.) Middle

2. ODD PD (≥5)

​                 a. ODD NOT ÷3, ALWAYS, after ±1, ÷24 = ARs

​                   1.) ALL PPTs

​                   2.) ALL PRIMES

​                   3.) Some Both

​                   4.) Some Neither

​                   5.) ∆ ALWAYS ÷24

​                   6.) ∆ between AR & NON-AR NEVER ÷24

​                 b. ODD ÷3, NEVER, after ± 1, ÷24 = NON-ARs

​                   1.) NEVER PPTs

​                   2.) NEVER PRIMES

​                   3.) Form INTERMEDIARIES between ARs

4.) Form INTERMEDIARIES between ARs ÷24 Sets

​                   5.) ∆ ALWAYS ÷24

​                   6.) ∆ between AR & NON-AR NEVER ÷24

IV. APPENDIX

(10 Easy steps to making the symmetrical BIM)

Parsing out the details is also fundamentally simple — but admittedly can appear complex — because it is the nature of the BIM that very easy and simple to understand number differences are the basis for the rich complex content that emerges. The connections and correlations are so vast that one must maintain a high sense of order to corral these into a meaningful, and ultimately useful, pattern arrangement — what we call Number Pattern Sequence(s) or NPS.

The Dickson Method (DM) — an algebraic method for determining ALL PTs — has been found to be inherently a geometric method in that the r-values become STEPS (r-steps) between geometries on the BIM. Initially, this led to the Extended Dickson Method (EDM) — described in TPISC II: Advanced — and thereafter, the Fully Extended Dickson Method (FEDM)TPISC_III: Clarity — and the work herein: TPISC IV: Details.

What we have is some form of algebraic geometry. A highly visual geometry-based picture has evolved in which the various Universal Template and Specific variable point locations can all be algebraically determined. The two completely re-enforce each other. More than that, this process of seeing the geometry with the algebra, and, seeing the algebra with the geometry, has been extremely fruitful in seeing the larger and larger picture of the TPISC. A picture that just keeps on growing and gaining clarity!

Right now, at this stage, that clarity has resulted in the ToPPT. An infinitely large and expanding structure embedded within the infinitely large and expanding BIM. This has come about by connecting the dots. There are still many, many more dots to connect!

Something Very NEW! ➗24: PPTs and PRIMES

ORGANIZATIONAL NOTE: While it may have made more sense to divide the entire work into three parts — the ➗24 all by itself in the second part, the Double-Slit Experiment—Quantum Entanglement Conjecture (DSEQEC) , and, the Creation and Conservation of SpaceTime (CaCoST), in the third — preference was given to keeping it all together under "Details" in that each part informs the other and keeping a constant "eye" on the bigger picture outweighs over-focusing on either part alone. In Nature, these "parts" are NOT separated! As before, the "discovery" phase of this work follows a loose chronological order. The storyline actually helps to see how the BIM reveals itself, little by little, by following its leads — bits of information acting like bread crumbs pointing to a path of even greater clarity — via greater detail! The HTML Canvas feature has been extensively utilized to tell this story in a sequential series of interactive images and forms the outline basis of TPISC IV: Details. Along with additional images, animations, tables and charts, reference will be made to this interactive page: Page 42:    TPISC III: Clarity & Simplification and TPISC IV: Details:  Tree of Primitive Pythagorean Triples (ToPPT) (~ A MathspeedST Supplement~) Interactive Graphics on BIM & BIMtree AREAS.. which the reader may find directly on my website under Portfolio>New Media>Art Theory 101>Page 42. This is an interactive graphics page on the BIM and BIMtree_AREAS — bridging TPISC III, and III with TPISC IV: Details. Each canvas may be interactively deconstructed to see how the geometric components are related. Simply refresh the page to start over. A great learning and teaching resource.

Along the way, several new and very exciting findings have popped up! It's just like the BIM to just keep on giving!

• The simple — and some of the not so simple — geometries of the BIM and BIMtree have been laid out in a series of interactive canvases. Each canvas opens up on its own webpage. Once open, you just click/touch and drag the geometry around, apart, this way and that way, even rebuilding the initial geometry side by side until you can really see how the parts are so simply related. It's deconstruction!

The entire geometry on canvas series was produced initially without any supporting text to keep it as clean, simple and intuitive as possible. The accompanying text will certainly help fill in the gaps, but it should be noted that the geometry on canvas series sequentially builds from one canvas to another.

• The algebraic binomials and their square proofs are widely known. Here the very same proofs overlap with that of the BIM, BIMtree and the ToPPTs! This, too, can be readily seen.

• The number 24 — and its factors of 1-24, 2-12, 3-8, and 4-6 have shown, once again, to be integral to both the BIM in general (remember, the EVEN Inner Grid cell numbers are all evenly divisible by 4), and, the ToPPTs. HOW SO? Well, this is new, really new, so just the surface has been touched, but so far:

• Marking (YELLOW) all BIM cells evenly ➗ by 24 generates a striking diamond-grid, criss-crossing pattern with 4 additional YELLOW marked cells in the center of each diamond; (TPISC IV: Details)
• Every PPT is found to exist on —and ONLY on — those Rows whose 1st Column grid cell are ➗ by 24 (YELLOW, with 1st Column grid cell marked as 'Z'), though not every such YELLOW marked Row contains a PPT. nPPT are only present on such Rows if accompanied by a PPT; (TPISC IV: Details)
• The “step-sister” of any given PT Row is found r-steps down the Axis from that Row (with 1st Column grid cell marked as 'Zf') and it, too, always and only exists on a YELLOW marked Row (“r” is part of the FTP originally derived from the Dickson Method for algebraically calculating all PTs); (TPISC IV: Details)
• The PPT Row (and nPPT Row) always contains the 4A (A=area) value of that PT and the “step-sister” PPT Row always contains the 8A value, both landing exclusively on YELLOW marked grid cells, giving a striking visualization of ALL PPTs and their r-based “step-sisters;” (TPISC III: Clarity & TPISC IV: Details)
• The significance of the “step-sister” is that it becomes the mathematical link to the “NEXT” PPT within the ToPPT — like the Russian-Doll model; (TPISC III: Clarity)
• The significance of the expanding and increasingly inter-connected PTs, as the BIM itself expands, is one in which the perfect-symmetry geometry of regular shapes and solids — equal triangles, squares, circles,… of the BIM allows — at certain articulation nodes (i.e., Rows) — the introduction of the slightly less-perfect-symmetry geometry (i.e., bilateral symmetry) of the full rectangle and oval that the non-isosceles right triangle PTs represent, into the unfolding structural framework, working from the ground up, if you will. The roots of fractals-based self organization are first to grow here! (TPISC III: Clarity & TPISC IV: Details) more…
• ALL self organization of any sort — be it force field or particulate matter — must have an organizing mathematical layer below driving it!
• Besides intimately tying the ToPPTs to a natural fractal pattern within the BIM:

• Thereafter, ALL PTs — primitive “parents” (PPT) and non-primitive “children” (nPPT) —are found on select Rows of the BIM by simply following the squared numbers on the the Prime Diagonal down to intersecting Rows; ( TPISC I: Basics & TPISC II: Advanced) more…

• The mathematical basis relating the PPTs was found, as was the consistent Fractal-Template Profile (FTP) that every PT follows; (TPISC II: Advanced & TPISC III: Clarity) more…

• A yet to be explained tie-in with the prime numbers and 24: The difference (∆) between the squares of ALL *PRIMES is evenly ➗ by 24!!! (Pattern in Number: from Primes to DNA; GoDNA: The Geometry of DNA; SCoDNA: Structure and Chemistry of DNA; Butterfly Primes: ~let the beauty seep in~; Butterfly Prime Directive: ~metamorphosis ~ ; and, Butterfly Prime Determinate Number Array (DNA): ~conspicuous abstinence ~; and an interactive Butterfly Primes new media net.art project.

• The FTP allowed all the PPTs to be sorted out and organized into a definitive Tree of Primitive Pythagorean Triples (ToPPT) that co-extends infinitely throughout the infinitely expanding BIM —>BIMtree or BIM-ToPPT; (TPISC III: Clarity) more… and more…

• Sub-Matrix

“Note: This amazing sub-matrix grid pattern explains the entire parent grid matrix and the Inverse Square Law, ISL, relationship that unfolds. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers.

Here we find the simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a sub-matrix ... lying below the original grid. And every number on that original grid is predicated on this simple pattern of 1,2,3,... in both the horizontal and vertical arrays. In fact, we now have complete integration of the axis numbers with the Inner Grid numbers ... together forming the Prime Diagonal (PD) of the Inverse Square Law (ISL). We have come full circle. And it begs the question: Who is the parent and who is the child. It seems the parent has become the child. How many more fractal matrix layers are there?”

Henceforth, we will call this Sub-Matrix 2.

A new sub-matrix, Sub-Matrix 1, results in the Active Row Sets (ARS) grid formed by dividing ALL Inner Grid numbers by 24.

Together Sub-Matrix 1 and Sub-Matrix 2 will provide the visual-geometric and algebraic-geometric location of ALL PPTs and PRIMES.

One could simply say that: subtract one from the squared values of any natural, whole integer number (WIN) and if it is evenly ➗ by 24, it is a candidate for being a PPT or 'step-sister' if that same squared value - 25 is ALSO ➗ by 24. (And, most interestingly, a PRIME candidate, as well. ALL PRIMES follow this. see the PRIMES Section. *NOTE: ACTIVE Rows may contain NO PTs or PRIMES, PTs but NO PRIMES, PRIMES but NO PTS, or BOTH PTs and/or their 'step-sister' AND PRIMES — but ALL PTs, 'step-sisters' and ALL PRIMES occur — as a NECESSARY prerequisite — on an ACTIVE ➗24 Row!)

If the Row contains BOTH a Factor Pair Set (two squared values that = a2 and b2) and the 4A value ONLY, it is a PPT Row.

If it contains BOTH the 4A and 8A values, it is a 'step-sister' Row, i.e. (c2-1)/24 and (c2-25)/24, if evenly divisible, are PPT and/or PPT 'step-sister' Rows, e.i. Row 17 is the PPT Row of the 8-15-17 PPT, but is also the 'step-sister' Row of the 5-12-13 PPT.

(NOTE: The PPTs intersect with grid cells/24 at: the subtraction of 1,5,7,11,13,17,19,23,25,… from the squared Axis number (c2) and that gives a cell spacing of 4-2-4-2-4-2-4-2-… respectively, or a blank step ∆ between of 3,1,3,1,3,1,3,1,… These are the "ACTIVE" Rows. ACTIVE Rows may contain NO PTs or PRIMES, PTs but NO PRIMES, PRIMES but NO PTS, or BOTH PTs and/or their 'step-sister' AND PRIMES — but ALL PTs, 'step-sisters' and ALL PRIMES occur — as a NECESSARY prerequisite — on an ACTIVE ➗24 Row!)

• Take the BIM and divide all numbers evenly divisible by 24.

This gives you a criss-cross pattern based on 12, i.e. 12, 24, 36 ,48,… from Axis.

Halfway between, are the rows based on 6.

On either side of this 6-based and 12-based frequency, the rows just before and just after, are ACTIVE Rows. These are ALWAYS ODD # Rows. They form an Active Row Set (ARS). Later these were colored PURPLE.

Their Axis #s are NEVER ➗3.They ALWAYS have their 1st Col value ➗ by 24.

Adding 24 to ANY of the ODD # NOT ➗3 ACTIVE Row Axis values ALWAYS sums to a value NOT ➗3 and thus to another ACTIVE Row Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).

Another ODD Axis # Row lies before and after each pair of ACTIVE Rows, i.e. between EVERY set of two ACTIVE Rows, is an ODD non-ACTIVE Row and their Col 1value is NOT ➗by 24.

Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).

While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.

Together, two ACTIVEs + one non-ACTIVE form a repetitive pattern down the Axis, i.e. ARS + non-Active Row.

*While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…

It follows that:

1. ALL PTs (gray with small black dot) fall on an ACTIVE Row.

2. ALL PRIMES (RED with faint RED circle) fall on an ACTIVE Row.

3. The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.

4. The difference, ∆, in the SQUARED Axis #s on an non-ACTIVE ODD Row and an ACTIVE Row is NEVER divisible by 24.

5. The difference, ∆, in the SQUARED Axis #s on any non-ACTIVE ODD Row and another non-ACTIVE ODD Row is ALWAYS divisible by 24.

6. Going sequentially down the Axis, every ODD number in the series follows this pattern:

nA—A-A—nA—A-A—nA—A-A—

7. #3—5-7—9—11-13—15—17-19—21... Every 3rd ODD # (starting with 3) is ➗by 3 = nA .

8. #5-7—9—11-13—15—17-19 Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A.

In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.

9. #3,4,5 re-stated: let A = ACTIVE Row Axis #, nA = non-ACTIVE Row Axis #

A22-A12= ➗ 24 and A ≠ ➗by 3

nA2-A2≠ ➗ 24

nA22-nA12= ➗ 24 and nA = ➗by 3

10. 6,7,8 re-stated:

ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME

ODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS ACTIVE Row members and candidates for being PT and/or PRIME.

In brief:

An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # :

A12+ 24x = A22 and √A22 = A2 = a PT and/or PRIME candidate;

ODD12+ 24x = ODD22 and √ODD22 = ODD2 = a PT and/or PRIME candidate, if and only if, its 1st Col. value is ➗ by 24.

The difference in the squared values of any two PTs/PRIME #s is ALWAYS a multiple of 24!

On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7)

BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under Why?

((((Table VI series is really the whole evolution of the BIMrow1-1000+sheets/Primes_sheets+PF, etc here and appendix.)))

referenced as Table VI b

Table VII.referenced as Table VI b with ACTIVE Rows in VioletTable VII.

Table VII close-up

DSEQEC: Double-Slit Experiment — Quantum Entanglement Conjecture

We conclude with the possible tie-in of the BIM-ToPPT with the central question of Quantum Mechanics (Quantum Field Theory): how do you explain the Double-Slit Experiment and the Quantum Entanglement phenomena — other than simply describing the results? The DSEQEC proposes an explanation.

Intro

The original Dickson Method (DM) showed that for r=EVEN numbers, such that r2=st was satisfied, ALL Pythagorean Triples (PTs) could be found. This was a strictly algebraic calculation method.

Previously, we found that ALL PTs could be uniquely located on specific Rows on the BBS-ISL Matrix (BIM) by finding the a2, b2, c2 values.

We also found that our Extended Dickson Method (EDM) provided a Template series — one for each r-set value — that more completely characterized a PT and its unique placement on specific Row on the BIM.

The Template upon a Row had a number of key focus points — values — upon, above, and below each PT Row. ALL PTs within a given r-set value, strictly follow the respective template for that r-set.

The EDM added d, e, f, f2, F, g, h, i, J, √J, k, l, m, n, o, p, q, U, v, W, √W, and d/p to the a, a2, b, b2, c, c2, r, r2, s, and t values found in the original DM.

In this work, we have Fully Extended the EDMFEDM — to better characterize important profiling focus point values of any given PT. There are now some 40+ focus point values.

The discovery of the Area (A) and Perimeter (P) values within a PT Row Template has provided new insights into the very foundation of the Pythagorean Theorem and its intimate connection with the Inverse Square Law, as fully visual algebraic geometry specifically and unequivocally shown on the BIM.

In this journey, we will methodically lay out how these 40+ values provide a simple, yet unique profile. The expanded proof of the Pythagorean Theorem directly on the BIM is in and of itself proof of The Pythagorean - Inverse Square Connection (TPISC)! Along the way we hope to share some great insights!

List of 40 characters/focus points:

• Every PT Row contains at least 8 key values directly;

• and another 8 values indirectly by counting STEPS;

• r/4r/2ratpsb
• giving another 8 values, by calculation/STEPS or a mixture of both;

• APUeff2Fr2

• 8A@Ƀd/pghiJ — √Jklmnp2qvVW√W¥.

I. Who?

PTs — Pythagorean Triples — and the squares and rectangles they represent.

A. The PT's sides, when squared, become SQUARES and have Squared Areas.

B. The PTs, themselves, when combined with their mirror opposite, form RECTANGLES and Areas.

0

What?

1

1a

1b

1c

C. As ALL Inner Grid (IG) numbers are simply differences (∆) in PD values, so too, ALL IG numbers are Areas.

D. The key is that those IG Areas that are SQUARES equal the Squared Area sides of Pythagorean Triples!

II. What?

What?

2

PTs are non-isosceles, 90°-right-triangles composed of Whole Integer Numbers (WINS).

A. PTs are composed of only WINs.

B. PTs come in two flavors:

1. Primitive, or Parent, PTs (PPTs) are irreducible;

2. non-Primitive, or Child, PTs (nPPTs) are reducible;

a. 6-8-10

b. 9-12-15

c. 10-24-26.

C. PTs are infinite — they grow infinitely large in size as WINs to infinity.

D. PT's sides, designated a & b for the short sides, c for the long, hypotenuse such that:

• a2 + b2 =c2.

E. PTs Area (A) times 4 + the difference () in the short sides squared(2) — (b-a)2 = f2 .

• 4A + f2 = c2 = a2 + b2.

TIP: PPTs always have 1-EVEN and 1-ODD short side.

TIP: nPPTs can have either EVEN & ODD, or, Both EVEN short sides (and long side, too).

>

III. Where?

Where?

3a

3b

3c

3d

3e

PTs — and their proofs — are ubiquitously located, in five (5) easy steps, throughout the infinitely expandable BIM. (see Chapter V: How for details)

A. All proofs are visually present right on the BIM, starting at the origin.

B. PTs and the entire BIM are both composed solely of WINs.

C. PTs and the entire BIM are both ultimately related to the ODD-number summation series:

• 1 + 2 = 3, 2 + 3 = 5, 3 + 4 = 7,… ——> 1,3,5,7,… become the 1st Diagonal Parallel to the Prime Diagonal (PD);
• 12 = 1, 22 = 4, 32 = 9, 42 =16,… ——>1,4,9,16,… the PD.

D. In that every IG WIN is an Area difference (∆) between two (2) PD perfect SQUARES Areas, it is ONLY those Rows (or Columns) containing the SAME perfect SQUARES Areas as in the PD, that are PT Rows. (see IV next)

E. The building up of sets of two (2) perfect, but asymmetric sized SQUARE Area sides — short sides 1 and 2 —the non-SQUARE RECTANGLE is generated, i.e. by combining the two (2) non-isosceles right-triangle PTs.

TIP: The Tree of Primitive Pythagorean Triples (ToPPT) is also relateable by its SQUARE Areas (TPISC III).

TIP: The Exponentials are also relateable by its SQUARE Areas (TPISC V).

TIP: Here is the BBS-ISL Matrix (BIM) 10x10 in 5 Easy Steps:

1. Make 10x10 Template:

1. Draw 10 x 10 grid + add 1 additional Row @top and Column @left side;
2. Label TOP AXIS Row 0–1-2-3-4-5-…. with BOLD numbers;
3. Label LEFT AXIS Column (0)-1-2-3-4-5-… with BOLD numbers;
4. Fill-in the Prime Diagonal (PD) numbers with BOLD numbers, as the squared value of the AXIS numbers as: 1-4-9-16-25-...
5. Note: the Matrix is symmetrical about the PD with a 90° mirror reflection symmetry, lower-upper;
6. (You only need, initially, to fill in the lower triangle values, and finish up by copying them above.)
2. Fill-in the 1st Parallel Diagonal that runs parallel just below the PD with the ODD WINS: 3-5-7-9…with a difference (∆) of two (2) between subsequent numbers.

3. Fill-in the 2nd Parallel Diagonal that runs parallel just below the PD with the EVEN WINS: 8-12-16-20…with a difference (∆) of four (4) between subsequent numbers.

4. Fill-in the 3rd Parallel Diagonal that runs parallel just below the PD with the ODD WINS: 15-21-27-33…with a difference (∆) of six (6) between subsequent numbers.

5. Continue filling in the rest of the Parallel Diagonals by adding (2)-(4)-(6)-8-10-12-…to each number to give the next number with the ∆ appropriate for that Parallel Diagonal:

1. The ∆ is simply the Axis number x 2;

2. The first, initial number in the IG is always its PD-1 — Prime Diagonal value minus one:

• PD4-1=3 the first number on that Row;
• PD9-1=8 the first number on that Row;
• PD16-1=15 the first number on that Row.
3. You can double-check your work by noticing that every IG number:

• Is the ∆ between its two PD number values — where the Row and Column intersects the PD;
• Is the product of its two Left-Vertical Axis numbers, set at 90° diagonals from the IG number;
• Is the product o the sum (∑) of its Row Asix + Column Axis numbers x the number of IG cell STEPS down from the PD.

IV. When?

When?

4

PTs are uniquely located on any Row (or Column) containing Paired Sets — 2 Squared Areas — that represent a2 and b2, with c2 at the endpoint on the PD.

A. PTs may be easily found by dropping down from the PD number (WIN>4) to intersect the Row with Paired Sets — a.k.a. Square Paired Sets (SPS)

1. PD 9, drop down to intersect 9 on Row Axis 5, PD 16, drop down to intersect 16 on same PT Row.
2. PD 64, drop down to intersect 64 on Row Axis 17, PD 226 drop down to intersect 225 on same PT Row.

B. PTs may also be easily found as being on the Row whose PD number has matching numbers on its Row and Column.

1. Row 5, with PD 25, has 9 and 16 on both the Row and on the PD above its Row Paired Sets values.
2. Row 17, with PD 289, has 64 and 225 on both the Row and on the PD respectively, above.

C. PTs may also be calculated algebraically and then located on the BIM following the Dickson Method (DM), Extended Dickson Method (EDM) or the Fully Extended Dickson Method (FEDM) as shown in Chapter V.

V. How?

How?

5

PTs have been eloquently and succinctly defined by algebraic geometry.

A. First, it was found that the strictly algebraic Dickson Method (DM) for calculating all PTs coincided with those found on the BIM as presented in TPISC I: Basics:

1. Let r=EVEN number, such that r2=2st is satisfied;

• r=2, r2 4=2st
2. r2/2=st, where s,t are the Factor Pairs (FPs) of r2/2;

• r2/2=4/2=st, and s=1, t=2 as FPs
3. Then, side a=r+s, side b=r+t, and side c=r+s+t;

• a=2+1=3, b=2+2=4, and c=2+1+2=5, for the 3-4-5 PT.

B. Second, it was found that the parameters described in the DM were actually the grid cell STEPS from one parameter — or focus point character — value to another as this algebra-geometry overlay become the Expanded or Extended Dickson Method (EDM), the subject of much of TPISC II: Advanced:

1. PTs with the EDM, were found to be parsed into r- and s-sets;

r-sets (examples, see Table 1c)

• r=2 set includes only the 3-4-5 PT;
• r=4 set includes 6-8-10 and 5-12-13 PTs;
• r=6 set includes the 9-12-15, 8-15-17, and 7-24-25 PTs.

s-sets (examples, see Tables 2, 3, 5,...)

• s=1 set includes 3-4-5, 5-12-13, 7-24-25,… PTs;
• s=2 set includes 8-15-17, 12-35-37, 16-63-65,… PTs;
• s=8 set includes 20-21-29, 28-45-53, 32-60-69,… PTs.
1. PT sets were found to form linked Generations (see Appendix and especially TPISC II).

C. Third, the PT r-sets formed a common Template of EDM relationships and the geometry becomes very visual in the Fully Extended Dickson Method (FEDM). The Tree of Primitive Pythagorean Triples (ToPPT), known in part since Pythagoras' time, became fully realized, and thus essentially defined, by its full presence on the BIM. It provides a large structural framework of forms based on the non-SQUARE RECTANGLES (and Ovals) that are formed from the co-joining of the non-isosceles right-triangle PTs. A new Clarification and Simplification of the TPISC as a whole becomes the subject of TPISC III:.

1. Within a given r-set, the Template was fixed for ALL PTs in that r-set;
2. The Template form remains constant, only its size scales up as one moves up into larger r-set;
3. The Template reveals some 40+ profiling parameters — focus points — for ANY PT. (Also see Appendix A/Notes2016-2018/Section 4C/Pages 41-49 for more.)

D. The BBS-ISL Matrix (BIM) is itself an infinitely expandable — yet highly patterned —Template of perfect SQUARE Areas and perfect CIRCLES into which, the PTs are embedded. Here are a few things to know:

PTs: 40+ FEDM derived profiling focus point parameters:

1. While the actual Area (A) and Perimeter (P) of the PT in question can be shown directly on the Matrix as grid cell areas — as can the PT proof — it is the 4x the A, and, the difference (∆) in the length of the sides, squared, that present the true Pythagorean - Inverse Square Connection:

4A + (b-a)2 = c2 = a2 + b2;

​ where, (b-a)2 = (t-s)2 = ƒ2, giving 4A + ƒ2 = c2 = a2 + b2.

2. Finding the A and P on the Matrix necessarily introduced some additional key focus point values — giving rise to the FEDM.

3. The 40+ key focus points are found consistently in each and every r-set Template — defining the PTs.

4. Every PT Row contains at least 8 key values directly;

5. And another 8 values indirectly by counting STEPS;

• r/4 — r/2 — ratpsb.
6. Giving another 8 values, by calculation/STEPS or a mixture of both;

• APUeƒƒ2Fr2.

• 8A@Ƀd/pghiJ — √Jklmnp2qvVW√W¥.
8. If you plot these out along their r-set STEPS (r-steps) spacings, the ∆ between values and their PD for any given Column follows 1r2 — 4r2 — 9r2 — 16r2 — 25r2 —…;

c2-o=1r2 169-153=16=1r2, c2-d=4r2 169-105=64=4r2, c2-g=9r2 169-25=144=9r2.

9. Naturally, the PD sequence, up from the Row values, follows the same 1—4—9—16—25—...

10. The "downward" Diagonal, perpendicular from the PD, back to the Row Axis gives the ɃV¥ and c-ƒ points, and, 8A is always r-steps down the grid from the 4A location on that PT Row.

(Also see Appendix A/Notes2016-2018/Section 4C/Pages 32-36 for more.)

PTs: 10 ways to approach the BIM

1. The PT Axis Row: ALL PTs have cXfob24Ada2c2 :

• c is on the Left SIDE Axis;
• Xƒ is r/2 steps in from Axis;
• o is r-steps in from Axis;
• b2 is a-steps in from Axis;
• 4A = c-ƒ steps coming back from the PD;
• d is 2r-steps in from the Axis;
• a2 is b-steps in from the Axis;
• c2 is on the PD at the intersect of the PT Row.
2. The Main Diagonals (Template) for ALL PTs:

​ PD — O (origin)—>c2 as oƒ2r2a2b2c2;

​ ⊥PD — c2—>2c (Axis) follows 4c8c12c16c sequence.

3. The Secondary Diagonals:

• pUd this important diagonal defines the Common Diagonal of the Golden Diamond ToPPTs;
• ded/p (on Axis).
4. The Tertiary Diagonals:

• √Wo;
• o√J (on Axis).
5. The Horizontal Axis (TOP) of Columns:

• 0—ALL of the above—from c —>c2;
• r/4—r/2—r—2r—3r—4r—.. and b-a=t-s.
6. The Vertical Axis (Left SIDE) of Rows:

• 0—ALL of the above—from c —>2c;
• sp√Wc√Jd/ptn—2cr—2r—3r—4r—....
7. The 4A8Ac2ƒ2 Rectangle Connection. (Also see Appendix A/Notes2016-2018/Section 4C/Pages 22-27 for more.)

8. The Proof a2 + b2 = c2 = 4A + ƒ2.

9. The Complementary Pair (Square Pairs) Sets of ANY PT.

10. ALL PTs have matching 4A values on both the PD Row itself and r-steps down that PD Column.

10 Basic, fundamental rules of the symmetrical BBS-ISL Matrix

1. Basic BBS-ISL Rule 1: All numbers (#s) related by the 1-4-9-...PD sequence.

2. Basic BBS-ISL Rule 2: Every # in the PD sequence is the square of an Axial #.

3. Basic BBS-ISL Rule 3: The Odd-Number Summation sequence forms the PD sequence.

4. Basic BBS-ISL Rule 4: Every EVEN Inner Grid (IG) # is divisible by 4 & all are present.

5. Basic BBS-ISL Rule 5: Every IG# is:

A: The difference (∆) between its two PD-sequence #s; (Note: A=B=C=D, and, E.)

• Ex: PD 25 - PD9 = 16.

B: The sum (∑) of the ∆s of each of its PD#s between its two PD-sequence #s (as above);

• Ex: (PD 25 - PD16) + (PD16 - PD9) = 16.

C: The ∆ between the squares of the two Axial #s forming that IG# (as above);

• Ex: 5^2 - 3^2 = 16.

D: The product of the Addition & Subtraction of the two Axial #s forming that IG# (as above);

• Ex: (5 + 3) x (5 - 3) = 16.

E: Also, the product of its 2 Axial #s intersected by that IG#'s 90° diagonals;

• Ex: 2 x 8 = 16.
6. Basic BBS-ISL Rule 6: Every ODD IG# is NOT PRIME & all are present;

• Corollary: NO EVEN NOT divisible by 4 #s are present on the IG.
7. Basic BBS-ISL Rule 7: The ODD-Number sequence, and the 1-4-9-...PD sequence, forms the sequential ∆ between ALL IG#s.

8. Basic BBS-ISL Rule 8: The ∆ between #s within the Parallel Diagonals is a constant 2 x its Axial #.

9. Basic BBS-ISL Rule 9: The ∆ between #s in the Perpendicular Diagonals follow:

A: From EVEN PD#s, √PD x 4 starts the sequence & follows x1-x2-x3-x4....

B: From ODD PD#s, √PD x 4 starts the sequence & follows x1-x2-x3-x4....

C: From ODD Perpendicular Diagonals between the EVEN-ODD diagonals (above), the sequence starts with the same value as the Axis number ending the diagonal, the sequence following x1-x3-x5-x7..

10. Basic BBS-ISL Rule 10: Every #, especially the #s in the ONEs Column, informs both smaller and larger Sub-set symmetries (much larger grids required to demonstrate).

5 Basic, fundamental rules of the symmetrical BBS-ISL Matrix Inner Grid

1. IGGR 1: The IG is formed of two equal & symmetrical 90°-right, isosceles triangles that are bilaterally symmetrical about the PD — and, infinitely expandable.
2. IGGR 2: The 90°-right-triangle — inherent to ALL squares and rectangles by definition — both forms the alternating EVEN-ODD square grid cells within the Matrix, and, is responsible for all major patterns and sequences, thereupon.
3. IGGR 3: Subtraction-Addition: Every IG# is simply the ∆ between its two PD#s (subtraction), and, the sum (∑) of any IG# + its PD# above = the PD# on the end of that Row (or, Column).
4. IGGR 4: Multiplication-Division: Every IG# is simply the product of the two AXIAL #s intersected by the two diagonals — of that said IG# — pointing back to the Axis at a 90° angle (multiplication), and, the dividend of the Axial divisor and quotient (division).
5. IGGR 5: The actual # of grid-cell steps — i.e., the actual # of STEPS from a given IG# to another by a strictly horizontal, vertical, or 45° diagonal path — forms a simple, yet often fundamental descriptor to the pattern-sequence templates that inform the more advanced patterns, e.i., Exponentials and especially the Pythagorean Triples (PTs). STEPS are particularly important in the geometric visualizations within the BBS-ISL Matrix (as alluded to in IGGR 2, above).

Pythagorean Triples and BBS-ISL Fundamentals (TPISC: The Pythagorean-Inverse Square Connection)

3 Basic, fundamental rules of the symmetrical BBS-ISL Matrix Inner Grid that encompass the PTs.

1. TPISC-BBS-ISL Rule 1: Every IG EVEN Squared # is part of a Square Paired- Set (SPS) that:
• A: Has reciprocal SPS members on the PD vertically above;
• B: Both SPS members reside on the SAME Row;
• C: They represent the a2 and b2 values of a PT, whose c2 value is on the PD intersection.
1. TPISC-BBS-ISL Rule 2: Every PT is found on the BBS-ISL Matrix and can be located by this intersection of EVERY PD (9>) and a Row with SPSs.
2. TPISC-BBS-ISL Rule 3: Every PT — including its sides, perimeter, area and proof — can also be found and fully profiled (and, predicted) as r-set, s-,t-set members of the Dickson Method (DM), Expanded Dickson Method (EDM), and the Fully Expanded Dickson Method (FEDM), shown herein. (ADD the content pg 1-27-30 here)

Matrix Flow:

1. ALL WINs.
2. Axis (vertical, horizontal) +IG + PD = BIM.
3. BIM - (Axes & PD) = Inner Grid (IG).
4. IG - 1st Parallel Diagonal (P∥D)= Strict Inner Grid (SIG).
5. Axes = ALL WINs (X) sequentially on both vertical (Left SIDE) and horizontal (TOP) Axis.
6. PD = Axis2 = X2 = a2b2c2 of ALL Pythagorean Triples (PTs).
7. Axis Rows & Columns define the PD, IG, ALL PTs and ALL Exponentials Xn.
8. Parallel Diagonals (to the PD) define ALL Exponentials Xn (Exp Xn).
9. ALL PTs and ALL Exp Xn are on the BIM.
10. ALL Squared #s, by definition, are on the PD.
11. ALL Squared #s are ALSO on the IG.
12. ALL Squared Sides (a2, b2, c2) of ALL PTs (across Rows) & ALL Exp X>2 (down Parallel Diagonals) are also on the IG.
13. Therefore, ANY Squared # on the IG is PART OF A PT & some, but NOT ALL, are also Exp X>2.
14. ALL Exp Xn are found on the IG, specifically along their Parallel Diagonals (P∥Ds).
15. ALL Exp Xn values, like ALL IG #s, are simply the ∆ between their two PD Area values.

Exponentials Summary (see TPISC V: Exponentials):(Also see Appendix A/Notes2016-2018/Section 3/Page LF2 for more.)

1. For any given # X, located on the Axis, its respective x2 is, of course, located on the PD, while X3, X4, X5,.. (Xn), are ALL found on that X Diagonal Parallel ( P∥D) to the PD.

2. The distance — # of steps diagonally — between successive Exponentials X1,2,3... for a given X, follows a Number Sequence Pattern (NPS) equal to is Xn sequence value.

3. The Sum () of the Axis Column & Row # values x that Diagonal Axis # X, equals the Xn value:

• ∑(AxisCol + AxisRow) x X = Xn as does the AxisCol x AxisRow product;
• ∑(6+10) x 4 = 64 = 43, where X=4, and, AxCol x AxRow = 4 x 16 =64 = 43.
4. The Sums (s) of the ∆s between the PD #s of a given X35=X3 and is simply an expression of the IGGR:

• 23 = 1-4-9 with ∆s of 3 & 5, where 3 + 5 =8 = 23.
5. The Area (# of grid cells) of a given Xx = Area Xx~uo~ - Area Xx~LD~, and flollows the same NPS progression sequence as X3, X4, X5,… in Area and in # of PD steps:

• 1st P∥D = # ∆ 2, ODD #s (Prime & Not-Prime [NP]);
• 2nd P∥D = # ∆ 4, EVEN #s ➗ 4 = ALL Exp 2n;
• 3rd P∥D = # ∆ 6, ODD #s NP= ALL Exp 3n;
• 4th P∥D = # ∆ 8, EVEN #s ➗ 4 = ALL Exp 4n;
• 5th P∥D = # ∆ 10, ODD #s NP= ALL Exp 5n;
• 6th P∥D = # ∆ 12, EVEN #s ➗ 4 = ALL Exp 6n;
• 7th P∥D = # ∆ 14, ODD #s NP= ALL Exp 7n;
• 8th P∥D = # ∆ 16, EVEN #s ➗ 4 = ALL Exp 8n.

VI. Why?

Everything you need to know is covered here and in the Summary and Conclusion. The Appendix gives you the source of the original Figures,Notes and Tables.

Why?

6a

6b

B. Areas, Perimeters, Proofs.(Also see Appendix A/Notes2016-2018/Section 4c/Pages 4-29, 40 for more.)

D+S

7

8a

8b

9a

9b

9c

9d

10a

10b

11b

11c

11d

13b

13c

14b

14c

14d

14e

15b

16b

16c

16d

16e

16f

16g

16h

16i

16j
1. (Also see Appendix A/Notes2016-2018/Section 4a/Pages 1-19; Appendix A/Notes2016-2018/Section 4B/Pages 0000-10; Appendix A/Notes2016-2018/Section 4C/Pages 0000-50c, and thereafter: see Appendix A/Notes2016-2018/Sections 1 and 2/ ALL Pages for more.)

2. (complete mapping-Template of A, P, 4A, 8A, ƒ ƒ2,...)

3. Because of the bilateral symmetry of the BIM, each PT is represented 4 times (4x) =4A:

Row;

ver1

mirror of ver 1 = ver 2

Column;

Symmetry of ver 1 on Row = ver 3

Mirror of ver 3 = ver 4.

Remember: ALL IG #s, just like ALL PD #s, are Areas!

17b

17c

17d

17e

17f

17g

17h

17i

17j

17k

17l

17m

Sub-Matrix

(Also see Appendix Figures and Tables for BIM/24, PPTs and PRIMES, as well as the DSEQEC.)

“Note: This amazing sub-matrix grid pattern explains the entire parent grid matrix and the Inverse Square Law, ISL, relationship that unfolds. See an animated example by CLICKING IMAGE above. Patience. Follow the bold numbers.

Here we find the simplest and most basic pattern of the simple whole number sequence ... 1,2,3,... forming a truly fundamental base layer ... a sub-matrix ... lying below the original grid. And every number on that original grid is predicated on this simple pattern of 1,2,3,... in both the horizontal and vertical arrays. In fact, we now have complete integration of the axis numbers with the Inner Grid numbers ... together forming the Prime Diagonal (PD) of the Inverse Square Law (ISL). We have come full circle. And it begs the question: Who is the parent and who is the child. It seems the parent has become the child. How many more fractal matrix layers are there?”

Henceforth, we will call this Sub-Matrix 2.

A new sub-matrix, Sub-Matrix 1, results in the Active Row Sets (ARS) grid formed by dividing ALL Inner Grid numbers by 24.

First Sub-Matrix 1:

1. The simple — and some of the not so simple — geometries of the BIM and BIMtree have been laid out in a series of interactive canvases. Each canvas opens up on its own webpage. Once open, you just click/touch and drag the geometry around, apart, this way and that way, even rebuilding the initial geometry side by side until you can really see how the parts are so simply related. It's deconstruction!

The entire geometry on canvas series was produced initially without any supporting text to keep it as clean, simple and intuitive as possible. The accompanying text will certainly help fill in the gaps, but it should be noted that the geometry on canvas series sequentially builds from one canvas to another.

2. The algebraic binomials and their square proofs are widely known. Here the very same proofs overlap with that of the BIM, BIMtree and the ToPPTs! This, too, can be readily seen.

3. The number 24 — and its factors of 1-24, 2-12, 3-8, and 4-6 have shown, once again, to be integral to both the BIM in general (remember, the EVEN Inner Grid cell numbers are all evenly divisible by 4), and, the ToPPTs. HOW SO? Well, this is new, really new, so just the surface has been touched, but so far:

• Marking (YELLOW) all BIM cells evenly ➗ by 24 generates a striking diamond-grid, criss-crossing pattern with 4 additional YELLOW marked cells in the center of each diamond; (TPISC IV: Details)
• Every PPT is found to exist on —and ONLY on — those Rows whose 1st Column grid cell are ➗ by 24 (YELLOW, with 1st Column grid cell marked as 'Z'), though not every such YELLOW marked Row contains a PPT. nPPT are only present on such Rows if accompanied by a PPT; (TPISC IV: Details)
• The “step-sister” of any given PT Row is found r-steps down the Axis from that Row (with 1st Column grid cell marked as 'Zf') and it, too, always and only exists on a YELLOW marked Row (“r” is part of the FTP originally derived from the Dickson Method for algebraically calculating all PTs); (TPISC IV: Details)
• The PPT Row (and nPPT Row) always contains the 4A (A=area) value of that PT and the “step-sister” PPT Row always contains the 8A value, both landing exclusively on YELLOW marked grid cells, giving a striking visualization of ALL PPTs and their r-based “step-sisters;” (TPISC III: Clarity & TPISC IV: Details)
• The significance of the “step-sister” is that it becomes the mathematical link to the “NEXT” PPT within the ToPPT — like the Russian-Doll model; (TPISC III: Clarity)
• The significance of the expanding and increasingly inter-connected PTs, as the BIM itself expands, is one in which the perfect-symmetry geometry of regular shapes and solids — equal triangles, squares, circles,… of the BIM allows — at certain articulation nodes (i.e., Rows) — the introduction of the slightly less-perfect-symmetry geometry (i.e., bilateral symmetry) of the full rectangle and oval that the non-isosceles right triangle PTs represent, into the unfolding structural framework, working from the ground up, if you will. The roots of fractals-based self organization are first to grow here! (TPISC III: Clarity & TPISC IV: Details) more…
• ALL self organization of any sort — be it force field or particulate matter — must have an organizing mathematical layer below driving it!
• Besides intimately tying the ToPPTs to a natural fractal pattern within the BIM:

One could simply say that: subtract one from the squared values of any natural, whole integer number (WIN) and if it is evenly ➗ by 24, it is a candidate for being a PPT or 'step-sister' if that same squared value - 25 is ALSO ➗ by 24.

If the Row contains BOTH a Factor Pair Set (two squared values that = a2 and b2) and the 4A value ONLY, it is a PPT Row.

If it contains BOTH the 4A and 8A values, it is a 'step-sister' Row, i.e. (c2-1)/24 and (c2-25)/24, if evenly divisible, are PPT and/or PPT 'step-sister' Rows, e.i. Row 17 is the PPT Row of the 8-15-17 PPT, but is also the 'step-sister' Row of the 5-12-13 PPT.

(NOTE: The PPTs intersect with grid cells/24 at: the subtraction of 1,5,7,11,13,17,19,23,25,… from the squared Axis number (c2) and that gives a cell spacing of 4-2-4-2-4-2-4-2-… respectively, or a blank step ∆ between of 3,1,3,1,3,1,3,1,... )

• Take the BIM and divide all numbers evenly divisible by 24.

This gives you a criss-cross pattern based on 12, i.e. 12, 24, 36 ,48,… from Axis.

Halfway between, are the rows based on 6.

On either side of this 6-based and 12-based frequency, the rows just before and just after, are ACTIVE Rows. These are ALWAYS ODD # Rows. They form an Active Row Set (ARS). Later colored in PURPLE.

Their Axis #s are NEVER ➗3.They ALWAYS have their 1st Col value ➗ by 24.

Adding 24 to ANY of the ODD # NOT ➗3 ACTIVE Row Axis values ALWAYS sums to a value NOT ➗3 and thus to another ACTIVE Row Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).

Another ODD Axis # Row lies before and after each pair of ACTIVE Rows, i.e. between EVERY set of two ACTIVE Rows, is an ODD non-ACTIVE Row and their Col 1value is NOT ➗by 24.

Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).

While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.

Together, two ACTIVEs + one non-ACTIVE form a repetitive pattern down the Axis, i.e. ARS + non-Active Row.

*While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…

It follows that:

1. ALL PTs (gray with small black dot) fall on an ACTIVE Row.

2. ALL PRIMES (RED with faint RED circle) fall on an ACTIVE Row.

3. The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.

4. The difference, ∆, in the SQUARED Axis #s on an non-ACTIVE ODD Row and an ACTIVE Row is NEVER divisible by 24.

5. The difference, ∆, in the SQUARED Axis #s on any non-ACTIVE ODD Row and another non-ACTIVE ODD Row is ALWAYS divisible by 24.

6. Going sequentially down the Axis, every ODD number in the series follows this pattern:

nA—A-A—nA—A-A—nA—A-A—

7. #3—5-7—9—11-13—15—17-19—21... Every 3rd ODD # (starting with 3) is ➗by 3 = nA .

8. #5-7—9—11-13—15—17-19 Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A.

In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.

9. #3,4,5 re-stated: let A = ACTIVE Row Axis #, nA = non-ACTIVE Row Axis #

A22-A12= ➗ 24 and A ≠ ➗by 3

nA2-A2≠ ➗ 24

nA22-nA12= ➗ 24 and nA = ➗by 3

10. 6,7,8 re-stated:

ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME

ODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS ACTIVE Row members and candidates for being PT and/or PRIME.

• In brief:

An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # :

A12+ 24x = A22 and √A22 = A2 = a PT and/or PRIME candidate;

ODD12+ 24x = ODD22 and √ODD22 = ODD2 = a PT and/or PRIME candidate, if and only if, its 1st Col. value is ➗ by 24.

The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!

On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7).

BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under Why?

Now that have overloaded the field with details, let's simply (again)!:

• ALL PTs and ALL PRIMES are exclusively on Rows that:

are referred to as “ACTIVE."

bookend the Rows that are multiples (evenly ➗ by) 6: i.e. 6,12,18,24,30,36…

some of these bookend on 6-based Rows do NOT have a PT and/or Prime, e.i. Rows 35, 49, 55, 77, 91, 95,… (Certainly, those ➗ by factors other than themselves and one, are NOT Prime.)

as one progresses across the Matrix Row, the YELLOW and YELLOW-ORANGE cells on “ACTIVERows follow a pattern under Column #: 1–5—7—11—13—17—19—23—25…i.e., + - - - + - + - - - + - + - - - + - + - - - and so on (see Gray-Violet & White on the Table below).

the cell value numbers of the YELLOW and YELLOW-ORANGE cells on “ACTIVERows follow exactly the numbers on Col C of the Table VII: Axis_Sqd_Diff_24x.numbers.

the Table view gives the number values and relationships in tabular form, the Matrix in a more visual, geometric form.

remember: ALL candidates for PTs and/or Primes MUST have Col 1/24 = (x2 - 1)/24 as TRUE.

• Distilling the BIM/24 into the underlying "sub-matrix" of ➗24 Actives, as shown in Figure below. ___ , reveals exactly why the BIM/24 pattern is what it is.

NOTES: Sections 1-10

Section 1: 2018 (29pgs)

• ~~ ~~ ~~

In that the discovery that ALL PRIMES land only on ARS — exactly as do the PPTs — yet most emphatically do NOT follow the exact same distribution pattern leads to some open questions:

• what exactly is the relationship between PPTs and PRIMES?
• why do they BOTH land on ARS?
• why do some ARS have both, neither, or one or the other?
• can the PRIMES be used to predict the PPTs?
• can the PPTs be used to predict the PRIMES?

While a great deal in the way of proof of the PRIMES dependency on the ARS for their distribution will be presented in a number of tables (Tables 1-23, mostly redirected to the APPENDIX), we must keep these questions in mind as we pursue the PPTs and their relationship to the BIM (Tables 24-28).

Now, we introduce

Sub-Matrix 2:

• Table 24 re-introduces the Sub-Matrix 1 and how the PPTs (and PRIMES) are distributed strictly on the ARS.
• Table 25 introduces the full Sub-Matrix 2, with the Columns 1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47, and 49 values shown in the colored inset boxes across the BIM.
• Table 26 reduces Table 25 down to a 5x1000 BIM to focus primarily on the Col 1 Sub-Matrix 2 values (shown in the colored box insets).
• Table 27 reduces this further to a 5x100 BIM.
• Table 28 reduces this further to a 5x50 BIM.

The details are the same:

• These (colored inset boxes) Sub-Matrix 2 values:

• ALL PPTs have Col 1 ➗4;
• NO PPTs have Col 1 NOT ➗4;
• For any given Active Rows Set, only 1 Row is a ➗4 Row PPT, never both;
• SOME Col 1 ➗4 Rows are NOT PPTs ( starred );
• The NOT PPTs ( starred ) Axis #s are ➗Prime Factors*.

Fool-proof Steps to Find ALL PPTs:

1. Axis# must be ODD, NOT ➗3 = Active Row Set (ARS) member;

2. Only 1 of the 2 ARS can be a PPT;

3. Sub-Matrix Col 1 # MUST be ➗4;

4. SOME may NOT be PPT if ➗Prime Factor (>5);

5. Remaining Axis # is a PPT. Exceptions:

• Squared #s that are PPTs, remain PPTs when x2 or √x:

• ALL Squared #s that are PPTs, remain PPTs. ANY PPT #(x) times itself, times its square (x2) and/or times it serial products = NEW PPT;
• Example1: 5x5=25, 5x25=125, 5x125=625, 5x625=3125, 5x3125=15625=1252, 5x15625=78125, 5x78125=390625=6252,… products are ALL PPTs;
• Example 2: 97x97=9409, 97x9409=912,673, 97x912,673=88,529,281=94092=ALL PPTs.
• Squared #s that are NOT, remain NOT when x2 or √x, as above.

Sub-Matrix 2 Sidebar: Exponentials of the PPTs

• Table 29 Exponentials of the first 10 PPTs c-values to be used in Tables 30a-g.

• Tables 30a-g The Sub-Matrix 2, when ➗4, and the difference (∆) between this and the next exponential PPT treated this way, is subsequently ➗ by its Sub-Matrix 2 variable, the PREVIOUS exponential within the series is revealed. Restated as an example: When one subtracts 1 from the exponential values of c (the c-value of the PPT) you get the Sub-Matrix 2 value. Divide that by 4 and take the difference (∆) between it and the next. Divide that by 3 to give the PREVIOUS PPT c-value in the series.

The Sub-Matrix 2 variable divisor = 3 = Sub-Matrix 2 value/4 = 12/4. These variables run: 1,3,4,6,7,9,10,...

As to answers to the open questions called above:

• what exactly is the relationship between PPTs and PRIMES?

• a loosely threaded connection is quite apparent;
• why do they BOTH land on ARS?

• they both must be ODD #s, not ➗3, whose (x2-1)/24 is true;
• why do some ARS have both, neither, or one or the other?

• some clouds, some clarity, at least for the PPTs;
• can the PRIMES be used to predict the PPTs?

• yes, in the sense that if the PPT candidate is ➗Prime Factor (>5), it will not be a PPT;
• can the PPTs be used to predict the PRIMES?

• currently, NO, yet the threaded connections are so great that the pattern will eventually emerge!

BIM÷24: SubMatrix Sidebar: What is the role of 24 in the underlying structure?

SEE: Tables: 33a, 33b and 33c. towards the end of Appendix B for some very NEW INFO on the BIM÷24.

The underlying geometry of the BIM÷24 PRE-SELECTS the Axis Rows into TWO Groups: ARs and NON-ARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NON-ARs. While both Groups follow (PD2 - PD2)÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.

The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.

The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The Pythagorean-Inverse Square Connection, and the PRIMES.

Open in separate browser tab/window to see all.

BIM➗PPTs and PRIMES

BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found. SEE: Tables 31-32 in Appendix B for a great deal more info and proofs!)

A dovetailing of PPTs and PRIMES on the BIM

The discovery of the Active Row Sets (ARS) — the direct result of the BIM ➗24 — in which it has been found that ALL PPTs and ALL PRIMES are exclusively found on, was in and of itself, a slow an arduous journey.

Once found, it has added a great deal of visual graphic clarity! In simple terms, it simply marks out the obvious. Both the PPTs and the PRIMES can not be on Axis Row #s that are EVEN, nor ➗3. This leaves ONLY Rows that are ODD #s and not ➗3.

The BIM ➗24 marked those Active Row Sets indirectly, by being on either side — i.e., +/- 1 — of the Axis Row # intercepted by the ➗24. Directly, the ARS was shown to be picked out by Sub-matrix 1 and 2 values of the 1st cell Column of those Rows.

So we have the PPTs and the PRIMES occupying the same footprint rows, the ARS Rows. Both as a necessary, but not sufficient requirement, i.e., some ARS Rows do NOT have a PPT or PRIME, or both. ALL PPTs and ALL PRIMES are ALWAYS found on an ARS Row, NEVER on a non-ARS Row. Some ARS Rows may have none, either a PPT or a PRIME, or both.

Nevertheless, on this vast matrix array of ISL whole integer numbers, that the PPTs and PRIMES exclusively occupy the same ➗24-based footprint points to an underlying connection!

The 1st connection was found and written about in the three white papers of 2005-6 on PRIMES:

The 2nd connection, as referenced below, has been the latest discovery that Euler’s 6n+1 and 6n-1 pick out, as a necessary — but not sufficient for primality — condition ALL the PRIMES.

When you look at the BIM24, you can readily see how this theorem simply picks out the very same ARS Rows! (For ARS 5 and above.)

The BIM24 becomes a DIRECT GRAPHIC VISUALIZATION of EULER’s PRIMES = 6n+1 and 6n-1, where n=1,2,3,..

The same holds true for Fermat's (Fermat-Euler) 4n + 1 = Sum of Two Squares Theorem, where 4n + 3 ≠ Sum of Two Squares. The 4n + 1 = Sum of Two Squares = Pythagorean Primes (PTs where c = Prime #).

These are simply those ARs that contain BOTH a PRIME (red circle) AND a PPT (black dot) in the figures.

Note: In any given ARS, only one of the two ARs may be a PPT, while both, neither, or one or the other ARs may be PRIME.

Black dot in a Red circle = Pythagorean Triple = 4n + 1 PRIME candidate = Sum of Two Squares.

The 3rd connection is that for those ARS Rows that do NOT contain PRIMES — e.i., 25, 35, 49, … and have been shown to negate the possibility of the # being prime because it is itself prime factorable — divisible by another set of primes — is ALSO DIRECTLY VISUALIZABLE ON THE BIM24 AS THE INTERSECTING PRIME COLUMNS!!!

The 4th connection reveals that the BIM Prime Diagonal (PD) — the simple squares of the Axis #s — defines:

• The ISL itself, as every BIM Inner Grid cell value is simply the difference between its horizontal and vertical PD values;
• The Pythagorean Triples, as every PD cell value points to a PT when one drops down from it to its ARS intercept;
• The PRIMES, as the difference in the squares of any two PRIMES (≧5) — every PD value of a PRIME Axis # — is evenly ➗24.

The 5th connection is that for those ARS Rows— that may or may not contain PPTs and/or PRIMES — their 1,5,7,11,13,17,19,23,25,.. ODD intersecting Columns NOT ➗3, ARE ALL➗24, and, this is ALSO DIRECTLY VISUALIZABLE ON THE BIM24 AS THE INTERSECTING COLUMNS (usually depicted graphically in YELLOW-ORANGE boxes/cells on the BIM as part of the diamond with centers pattern) !!!

￼ This is revealed in the Sub-Matrix 1 figures and tables: the BIM ÷24.

Additionally, Sub-Matrix 2 also selects for ALL ARS as Column 1 ALWAYS ÷4. (See figures.)

In ALL cases, the PRIMES (≧5) are necessary — but not sufficient to insure primality — located on ODD # Rows NOT ÷3. It is as simple as that!

The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL ARs. Euler's 6n +/-1 is the most direct, Fermat's 4n + 1 gets the Sums of Two Squares = Pythagorean Primes (while 4n + 3 gets the rest).

Fermat's Little Theorem (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.

But now there is a dead simple way to test for primality:

The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.

For example, take any random ODD # - 25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)

• 741

7412-52/24 = n= 22877.3 NOT PRIME

• 189

1892-52/24 = n = 1487.3 NOT PRIME

• 289

2892-52/24 = n = 3479 PRIME

￼ ￼ If P = X, evenly ➗24, then P = a PRIME Candidate where, X=evenly➗24, P=Prime, P22=larger Prime Squared, P21=smaller PRIME Squared, n=1,2,3,…

Reference:

"A New Kind of Prime

The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want."

Quote is from Quanta Mag 9/26/19 article: Big Question About Primes Proved in Small Number Systems

by Kevin Hartnett

~~ ~~ ~~ PRIME GAPS — Goldbach Conjecture ~~ ~~ ~~

• Twin Primes
• Primes separated by other EVEN numbers
• Euler's 6n+1 and 6n-1
• Fermat-Euler's 4n + 1= Sum of Two Squares Theorem (Pythagorean Primes)
• Dickson's Conjectures: Sophie Germain Primes
• Goldbach Conjecture (Euler's "strong" form)
• Primes - BIM - Pythagorean Triples

~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~

A new and easy visualization!

PRIME GAPS

Game Board

There is a new game board in town. It has actually been around for awhile, but few know of it. It's basically a matrix grid of natural numbers that defines the Inverse Square Law (ISL). It's called the BIM, short for the BBS-ISL Matrix. Every grid cell is uniquely occupied by a given number that is simply the difference between the horizontal and vertical intercept values of the main "Prime Diagonal" (not Prime number diagonal) that mirror-divides the whole matrix.

If you show all the matrix values that are evenly divisible by 24, a criss-crossing pattern of diamond-diagonal lines will appear and this ends up giving us a unique visual on the distribuition of ALL PRIMES!

Basically, all ODDs (3 and greater for this presentation) fall into a repetitive pattern of:

ODD ÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3...

The matrix ÷24 described above, nicely picks out these sets of —Not÷3—Not÷3 that we call Active Row Sets (ARS). Each ODD member of the ARS is referred to as an Active Row (AR).

The ÷3 ODD Rows that lie between are called Non-Active Rows, or NA.

It is a necessary, but not sufficient condition, that ALL PRIMES are strictly located on ARs. NO exceptions, except for some ARs have NO PRIMES.

Another striking visualization is that ALL Primitive Pythagorean Triples (PPTs) also lie only on the ARs, following the same necessary, but not sufficient, condition. NO exceptions here as well, except for some ARs also have NO PPTs. In addition, only one of the ARs within an ARS can have a PPT.

So any AR within an ARS can have 0/1 PRIMES and/or 0/1 PPTs in any combination, only with the one caveat: that only 1 PPT/ARS is allowed.

Now, not to belabor this BIM game board, let's talk PRIME GAPS!

Prime Gaps

Let's refer to the lower number value for the ARS as "Lower" and the higher as "Upper." All are ODDs. As will become apparent, you can always tell if your ODD AR is "Lower/Upper" simply by adding 2 and ÷3: if the result evenly ÷3 it is "Upper" and if not, "Lower."

The Even Gap between primes strictly follows this pattern: (all are necessary but not sufficient conditions for primality)

For any given set of twin primes, or any single prime, that is the smaller "Lower" of an AR set:

• Even Gap must be 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50,…
• Example: P=5: 7, 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55)
• P + 2, P + 2+4, P + 2+6, P + 2+10, P + 2+12, P + 2+16, P + 2+18, P + 2+22, P + 2+24,…
• P+21, P+2+22, P+23, P+22+23, P+2+22+23, P+2+24, P+22+24, P+23+24, P+2+2324
• The Even Gap difference pattern follows: 2-4-2-4-2-4-….

For any given set of twin primes, or any single prime, that is the larger "Upper" of an AR set:

• Even Gap must be 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54.…
• Example: P=7: 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55), 59, 61
• P + 4, P + 2+4, P + 2+8, P + 2+10, P + 2+14, P + 2+16, P + 2+20, P + 2+22, P + 2+26, P + 2+28,
• P+22, P+2+22, P+2+23, P+22+23, P+24, P+2+24, P+2+22+24, P+23+24, P+22+23+24, P+2+22+23+24,
• The Even Gap difference pattern follows: 4-2-4-2-4-2-…. and is identical in the middle, not the start!

One can readily see that all this is simply the natural result of two ARs alternating with a NA row:

AR—AR—NA—AR—AR—NA—AR—AR—NA—AR—AR—NA—…

“Lower” AR number

AR—5

AR—7

NA—9

AR—11

AR—13

NA—15

AR—17

AR—19

NA—21

AR—23

AR—25

NA—27…

It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.

To any “Lower” Active Axis Row ODD:

• Add 2, 4, or 6,…
• If sum is ÷3, it is a NA
• If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.

“Upper” AR number

AR—7

NA—9

AR—11

AR—13

NA—15

AR—17

AR—19

NA—21

AR—23

AR—25

NA—27…

To any “Upper” Active Axis Row ODD:

• Add 2, 4, or 6,…
• If sum is ÷3, it is a NA
• If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.

Again, It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.

The ARS pattern on the BIM clearly shows the above patterns and may be extrapolated to infinity:

PRIME Conjectures

A number of PRIME conjectures have been shown to be easily visualized on the BIM:

Reference: TPISC IV: Details: BIM + PTs + PRIMES

The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL ARs. Euler's 6n +/-1 is the most direct, Fermat's 4n + 1 gets the Sums of Two Squares = Pythagorean Primes(while 4n + 3 gets the rest).

Fermat's Little Theorem (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.

But now there is a dead simple way to test for primality:

The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.

For example, take any random ODD # - 25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)

• 741

7412-52/24 = n= 22877.3 NOT PRIME

• 189

1892-52/24 = n = 1487.3 NOT PRIME

• 289

2892-52/24 = n = 3479 PRIME

PRIMES vs NO-PRIMES

I would be remiss if I did not mention that throughout this long journey that began with the Inverse Square Law and the Primtive Pythagorean Triples that the PRIMES kinda of just fell out. They just kept popping up. Not the least, but certainly not without effort, the BIM actually directly visualizes ALL of the NO-PRIMES. In doing so, one is left with information that is simply the inverse of the PRIMES. Subtract the NO-PRIMES information from the list of ODDS (disregarding 2) and what remains are ALL the PRIMES! A simple algebraic expression falls out from that:

NP = 6yx +/- y

letting x=1,2,3,… and y=ODDs 3,5,7,… with the only caveat is that if you don't first eliminate all the ÷3 ODDS, you must include exponentials of 3 (3x) in the NP tally.

PRIMES vs NO-PRIMES

Goldbach Conjecture (Euler's "strong" form)

In 2009-10, a solution to Euler's "strong" form of the Goldbach Conjecture "that every even positive integer greater than or equal to 4 can be written as a sum of two primes" was presented as the BBS-ISL Matrix Rule 169 and 170. This work generated a Periodic Table of Primes (PTOP) in which Prime Pair Sets (PPsets) that sequentially formed the EVEN numbers were laid out. It is highly patterned table. A recently annotated version is included with the original below. (See above link for details.)

It turned out this PTOP was actually embedded — albeit hidden — within the BIM itself as shown in Rule 170 and here, too, a recently annotated version is included below. (See above link for details.)

Rule 169: Periodic Table of Primes.

Rule 169: annotated

Rule 170: Periodic Table of Primes (PTOP): embedded within Brooks (Base) Square.

Rule 170: annotated

With that review of the ongoing work, the full presentation of this work starts below. It will cover everything above plus the NEW work leading to the findings and proof of the Goldbach Conjecture. This will be the basis for the ebook: PTOP: Periodic Table of PRIMES & the Goldbach Conjecture

PTOP: Periodic Table Of PRIMES & the Proof of the Goldbach Conjecture

PTOP Goldbach Conjecture from Reginald Brooks on Vimeo.

PTOP: Periodic Table Of PRIMES & the

INTRODUCTION and SYNOPSIS

Periodic Table Of PRIMES (PTOP) and the Goldbach Conjecture. 2019 Reginald Brooks

Here’s the thing. Amongst a myriad of other connections, there exists an intimate connection between three number systems on the BBS-ISL Matrix (BIM):
1 The Inverse Square Law (ISL) as laid out in the BIM;
2 PTs — and most especially PPTs — as laid out on the BIM;
3 The PRIME numbers — PRIMES — as laid out on the BIM.
The BIM is the FIXED GRID numerical array of the ISL.
PTs are the Pythagorean Triples and PPTs are the Primitive Pythagorean Triples. They have been extensively covered in the TPISC (The Pythagorean-Inverse Square Connection) series as: TPISC I Basics, TPISC II Advanced, TPISC III Clarity, TPISC IV Details. (See links at bottom.)
The PRIMES vs NO-PRIMES (2019) was covered earlier.
The original MathspeedST (2009-14) work, Brooks (Base) Square (2009-11), that started this journey, was divided in to two sections:
I. TAOST (The Architecture Of SpaceTime);
II. TCAOP (The Conspicuous Absence Of PRIMES).
You see, other than the natural whole numbers that form the BIM Axis’ and the standard 1st Parallel Diagonal (containing ALL the ODDs (≥3), there are NO PRIMES on the BIM.
This is the basis of the PRIMES vs NO-PRIMES work.
Yet, convert that same 1st Parallel Diagonal to ALL EVENS (≥4, by adding 1 to each former ODD), and now the BIM reveals the stealthily hidden PRIMES relationship in forming symmetrical pairs of PRIMES that are ALWAYS equal distance (STEPS) from ANY given EVEN # divided by two.
All this occurs on the BIM Axis. The apex of the 90° Right-angled, isosceles triangle so formed lies on a straight line path from the EVEN/2 to the given EVEN # located on that converted 1st Parallel Diagonal. This relationship is geometrically true and easily seen on the BIM.
Extracting those PRIME-Pair sets (PPsets) for each given EVEN forms the basis for the PTOP — Periodic Table Of PRIMES.
While “hidden” on the BIM, it clearly forms a definitive pattern on the PTOP: for EVERY 3+PRIME PPset that forms the 2nd column on the PTOP — acting like a bifurcation point — a “Trail” of PPsets forms a zig-zagging diagonal pointing down and to the right.
These are ALWAYS — much like a fractal — added with the SAME PRIME Sequence (3,5,7,11,13,17,19,23,29,31,37,...).
When you read a given horizontal line from left to right across the PTOP, you see that each given PPset that was contributed by a PPset Trail, adds up — composes — its respective EVEN. The sum (∑) number of PPsets that form its EVEN is totaled in the last column.
The fractal-like addition of one new, additional PRIME Sequence PPset to each subsequent “Trail” formed results in the overlapping trails growing at a rate that far, far exceeds the growth and incidence of the PRIME Gaps. This ensures that there will always be at least one PPset that will be present to compose ANY EVEN.
The Goldbach Conjecture has been satisfied and a new PRIME pattern has been found.
MathspeedST-TPISC Resource Media Center Intro
MathspeedST-TPISC Resource Media Center
PTOP - Goldbach Conjecture Video

INTRODUCTIONSTATEMENT: Layout & EssentialsThere are five sets of data to consider:FINDINGS & PROOFSUMMARY and CONCLUSIONDATA: Images and TablesDATA: Images and TablesREFERENCES

Goldbach Conjecture

INTRODUCTION

While the (strong) Goldbach Conjecture has been verified up to 4x1018, it remains unproven.

A number of attempts have demonstrated substantial, provocative and often beautiful patterns and graphics, none have proven the conjecture.

Proof of the conjecture must not rely solely on the notion that extension of a pattern to infinity will automatically remain valid.

No, instead, a proof must, in its very nature, reveal something new about the distribution and behavior of PRIMES that it is absolutely inevitable that such pattern extension will automatically remain valid. The proof is in the pudding!

Proof offered herein is just such a proof. It offers very new insights, graphical tables and algebraic geometry visualizations into the distribution and behavior of PRIMES.

In doing so, the Proof of the Euler Strong form of the Goldbach Conjecture becomes a natural outcome of revealing the stealthy hidden Number Pattern Sequence (NPS) of the PRIMES.

STATEMENT: Layout & Essentials

Proof of the Goldbach Conjecture (strong form, ≥6)

1. Natural (n), Whole Integer Numbers (WIN) — 0,1,2,3,…infinity — form horizontal and vertical Axis of a simple matrix grid.

2. The squares of such WINs — n2=12=1, 22=4, 32=9,…infinity — forms the central Diagonal of said grid — dividing it into two bilaterally symmetric triangular halves.

3. Every Inner Grid (IG) cell within is simply the difference (∆) between its horizontal and vertical Diagonal intercept values. They extend to infinity. The Diagonal WINs form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex.

4. Every IG cell within is also the product of two Axis WINs (Either horizontal or vertical, not both), that form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex.

5. The complete matrix grid extends to infinity and is referred to as the BIM (BBS-ISL Matrix).The BIM forms — and informs — a ubiquitous map (algebraic geometry) to:

​ The Inverse Square Law (ISL);

• The Pythagorean Triples (PT);
• The PRIMES (stealthily hidden, but revealed by NPS.
6. The 1st Diagonal that runs parallel to either side of the main Prime Diagonal (PD, not of PRIME numbers, but primary), is composed of the ODD WINs: 1,3,5,…infinity.

7. If we add +1 to each value, that 1st Diagonal now becomes a sequence of ALL the EVEN WINs (≥4): 4, 6, 8,…infinity. NOTE: this is why the PTOP is hidden, in the normal, base BIM these remain ODDs.

8. Select ANY EVEN WIN and plot a line straight back to its Axis WIN — that Axis WIN = EVEN/2 = core Axis #.

9. Upon that same Axis, PRIME Pair sets (PPsets) — whose sum (∑) equals the EVEN WIN (on the 1st Diagonal) — will be found that form the base of 90° R-angled isosceles triangle(s) whose apex lie(s) on that straight line between the EVEN and its 1/2 Axis WINs. PPsets with identical PRIMES = 1/2 Axis value.

10. The proof that every EVEN WIN has ≥1 PPsets can be seen in the Periodic Table Of PRIMES (PTOP) that stealthily informs the BIM of how each and every EVEN WIN is geometrically related to one or more PPsets.

11. These PPsets are NOT randomly contributing their ∑s to equal the EVEN WINs, rather they come about as the consequence of a strict NPS: the sequential — combining, linking, concatenation — addition of the PRIMES Sequence (PS) — 3,5,7,11,13,17,19,…—to a base PS — 3,5,7,11,13,17,19,….

12. The NPS of this addition forms the PTOP: for each vertical PS — the 1st PRIME (P1) remains constant (3), the 2nd PRIME (P2) sequentially advances one (1) PS WIN — is matched diagonally with the 2nd PS, but now the 1st PRIME sequentially advances, while the 2nd PRIME remains constant within a given PPset.

13. This matching addition of the 2nd PS at the bifurcation point of the common 2nd PRIMES, forms the zig-zag diagonal PPset Trails that are the hallmark of the PTOP.

14. For every subsequent vertical PPset match, the Trail increases by one (1) PPset.

15. The rate of such PPset Trail growth far exceeds the PRIME Gap rate.

16. The zig-zag diagonal PPset Trails combine horizontally on the PTOP to give the ∑# of PPsets whose ∑s = The EVEN WIN.

17. More than simply proving the Goldbach Conjecture, the PTOP hidden within the BIM reveals a new NPS connection of the PRIMES: PRIMES + PRIMES = 90° R-angle isosceles triangles.

18. The entire BIM, including the ISL—Pythagorean Triples—and, PRIMES, is based on 90° R-triangles!

19. Similar to how the grid cell values of the Axis, PD and IG of the Pythagorean Triples reveal additional, intimate connections within the BIM, so too do the PPsets: the 1st PRIME values of each set points to the # of STEPS from the PD that intersects the given EVEN WIN (Axis2), on a straight line path back to its Axis, at the apex where its other PPset member intersects — this is no mere coincidence — and that apex is, of course, the 90° R-angle isosceles triangle that results. For example: EVEN = 24, Axis = 12, PD = 144, # of STEPS from PD towards Axis = 5 for the 5+19 PPset and 7 for the 7+17 PPset, and 11 for the 11+13 PPset that each forms the EVEN 24.

20. As the bifurcation concatenation of the PS — 3,5,7,11,13,17,19,..— with the same base PS — 3,5,7,11,13,17,19,…— of those EVEN WIN — that when “3” is subtracted, the remainder is a next-in-the-sequence PRIME — remains one of a similar split with the Pythagorean Triples: for every “Primitive” parent PT, there are multiple “Non-Primitive” child PTs and it is the PPTs (Primitive Pythagorean Triples) that ultimately form the inter-connectedness of ALL PTs back to the original PPT — the 3-4-5. With the PRIMES, one raises the question: why are these “EVENS” preselected to be the “parent” EVENS forming the “beginning” or “start” of every PPset Trail with all other EVENS hitching on to that Trail further down the sequence?

21. Another set of STEPS (S) from the core Axis value , directly on the Axis, identifies each symmetrical pair of a given PPset that forms that given EVEN. These STEPS may by Universally calculated from the EVENS, P1 and P2 values. Examples are given further down.

As every new discovery unlocks many more questions, it follows that the details of the PTOP and BIM visualizations should both satisfy the proof offered and, more importantly, provide provocative data that will advance the field for the next researcher!

There are five sets of data to consider:

1. BIM (BBS-ISL Matrix): grid visualizations that overview the entire work

• Fig. 1-PTOP: Periodic Table Of PRIMES (100, original)
• Fig. 2-PTOP: Periodic Table Of PRIMES (100, annotated)
• Fig. 3-PTOP: Periodic Table Of PRIMES (100, upgraded)
• Fig. 4-PTOP: Periodic Table Of PRIMES (200, annotated)
• Fig. 5-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (original)
• Fig. 6-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (annotated)
• Fig. 7-Table 46: Symmetrical STEPS of the PPsets for EVEN 128 annotated snapshot
• Fig. 8-BIM: Symmetrical STEPS of the PPsets for EVEN 126 snapshot
• Fig. 9-BIM: Symmetrical STEPS of the PPsets for EVEN 126
• Fig. 10-BIM: Symmetrical STEPS of the PPsets for EVEN 128 snapshot
• Fig. 11-BIM: Symmetrical STEPS of the PPsets for EVEN 128
• Fig. 12- animated gif of video (below)
• Video: PTOP rule 169-170 Annotated
2. PTOP: the actual Table

• Table 34: Original PTOP from 2009 (EVENS 6-100)
• Table 35: Upgraded PTOP (EVENS 6-200)
• Table 36: Upgraded PTOP (EVENS 6-400)
• Table 37: Upgraded PTOP (work in progress, EVENS 6-1000)
3. PTOP: Analysis

• Table 38: Distribution and NPS of the PPset Trails (EVENS 6-404)
• Table 39: Distribution and NPS of the PPset Trails (EVENS 6-914)
• Table 40: Distribution and NPS of the PPset Trails (EVENS 6-2360 and up and up)
• Table 41: Bifurcation Addition and PPset ∑s
4. Reference

• Table 42: PRIME Gaps
• Table 43: PRIME Partitions = PPsets per EVENS (4-2000)
• Table 44: Summaries of Table 43 (EVENS 4-2000)
• Table 45: Equations for PTOP Tables 38-41
5. PRIME PPset Trails

• Table 46: PRIME PPset Trails (3-10007 and up)
• Table 47: PRIME PPset Trails simplified and extended Table 46 (3-568201)
• Table 48: Working example: EVEN 8872, core Axis 4436 with 93 PPsets.
• Table 49: PRIME PPset Trails & EVENS divisible by 6,12 or 24 (3-10007 and up).

The DATA has been grouped in its own section down below. It is highly recommended that you, the reader, preview first — review, thereafter.

~~~~

FINDINGS & PROOF

The definitive proof depends on demonstrating that the PPset Trails grow and extend to cover the “next” EVENS at a rate that exceeds the Prime Gap rate. PRIMES Sequence, PS is the key.

The PS is well established. How many primes are there?

As every beginning PPset — of 3,P2 — forms from successively increasing the 2nd PRIME by the next number in the PS (3,5,7,11,13,17,…), the Trail formed by bifurcating off from that point increases the total number sum (∑) of PPsets by one, i.e. Trail, Trail+1, (Trail+1)+1, ((Trail+1)+1)+1,…

A necessary, and sufficient, condition is that the Trail lengths — i.e. the total number of PPsets, in their overlapping aggregate, always exceed both the number of PRIMES and their Gaps for any and all numbers.

Specifically from Table 46 and Fig.__, locate a P2 PRIME with a large Gap, e.i. P2= 23, Gap=6.

line#———PRIMES, P2—∑#set/Trail—PRIME Gap—∆Trail-Gap— EVEN:∑#sets—EC——Eending——EVEN, E

82386226: 3114626

We see that 23 has 8 PPsets in its Trail, it has a Gap of 6 to the next PRIME (for a ∆=2). It’s EVEN=26 and there are 3 PPsets (going horizontally across the PTOP) that will make 26. The 11 is the number of EVENS Covered (EC) as the PPsets zig-zag diagonally down the PTOP. If we add double this EC - 1 to the EVEN 26, we get the EVEN Ending of 46 as: 2(11-1) + 26 = 46. So this Trail alone inclusively covers the EVENS 26-46, although their will be some holes. To fill the holes, we look at the Trails that started up above this 23 Trail.

For the Specific set of equations exclusively for the 3,P2 sets, where ∆ = P2 - 3 (see Table 45: Equations):

2EC = P2 -1

Ending EVEN Covered = Ee = 2(EC - 1) + EVEN

51354116: 262616
61762420: 283420
71974322: 393822

We can readily see that the 17 and 19 Trails, when their EVENS are added to their 2EC-1, respectively, will equal or exceed the EVEN of the NEXT PRIME up by the GAP=6.

Take Trail 17: EVEN = 20 with EC = 8. 2(8-1) + 20 = 34.

Take Trail 19: EVEN = 22 with EC = 9. 2(9-1) + 22 = 38.

And these Trails 17 and 19 will overlap the Trail 23 EC span, filling in any and all holes between it and the next Trail 29.

92992732: 2145832

For completeness, we can see that Trail 13, while reaching Trail 23 at it’s start, does not overlap any further.

Take Trail 13: EVEN = 16 with EC = 6. 2(6-1) + 16 = 26. We also can see that 13 + 13 = 26.

~~~

The PPset Trails for EVEN 26 gives 3 PPsets via the overlapping Trails 17, 19 and 23, AND, their overlapping Trails extend past the next 3 EVENS (28, 30 and 32), ensuring that they are “covered” with PPsets in the Gap jump to the next PRIME (29). All very neat and clean.

• The actual PPsets ∑s equal the EVEN 26:

• 3 + 23 = 26
• 7 + 19 = 26
• 13 + 13 = 26
• which comes about more simply as the core Axis value ± steps away:

• 13 ± 10 = 3 and 23
• 13 ± 6 = 7 and 19
• 13 ± 0 = 13 and 13
• where the core Axis value = 1/2 the EVEN, or 26/2 = 13 and we locate it on Column 2, PRIMES (line 5) for reference. It is not actually the PRIME, as we will see. Here it happens to land directly on the PRIME 13 as it is one of the PPsets. Other times it may well be somewhere between two PRIMES in this column. We will clarify this further on, but for now it is a reference to numbers on the Axis of the BIM that, being core Axis values, are key in showing that the PPsets will ALWAYS have their Pair members symmetrically located on either side of this reference marker.

But what about a larger Gap? How about Trail 113, EVEN = 116 and a Gap of 14 to the next Trail 127? (See image from Table 36.)

Trail 113 is overlapped by Trails 73, 79, 97, 103, and 109 that account for the 6 PPsets that will form the EVEN = 116 (the 3,113 PPset is included in the tally and you can follow it in Table 43). Again, from Table 46:

1312-16: 1166
252208: 12108
3734-110: 231410
41142214: 252214
51354116: 262616
61762420: 283420
71974322: 393822
82386226: 3114626
92992732: 2145832
1031106434: 4156234
1137114740: 3187440
12411221044: 3208244
1343134946: 4218646
1447146850: 4239450
1553156956: 32610656
16591621462: 32911862
17611761164: 53012264
18671841470: 53313470
19711921774: 53514274
20732061476: 53614676
21792141782: 53915882
22832261686: 54116686
23892381592: 44417892
249724420100: 648194100
2510125223104: 550202104
2610326422106: 651206106
2710727225110: 653214110
2810928424112: 754218112
29113291415116: 656226116
3012730426130: 763254130

What about the Gap = 14 going to the Trail 127, EVEN = 130? Overlapping Trails 71, 83, 89, 101,107 and 113 combine with the 127 to account for the 7 PPsets that will form EVEN = 130.

Of course to account for all the EVENS between Trail 113 and 127 — EVENS 116 to 130, we have Trails 59 to and including 127 in various combinations to account for that. We can easily see this on Table 46, lines 16-30, where each has an Ee (118–254) that meets or exceeds those EVENS (116–130). Altogether, there are some 80 PPsets that account for these 15 EVENS 116-130. Easily done.

What if you want to know specifically who covers, say EVEN 128? Looking at the full table, line 18, we see that Trail 67 and above have Ee > 128. How can we determine that the 19+109, 31+97 and 61+67 PPsets are the ones we are looking for?

Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick:

• on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 128 = 64

• see that PRIMES 61 & 67 on either side = 128 when added together and both are PRIMES

• knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 64 slot like the 61 & 67

• the next pair out would be 31 & 97, followed by 19 & 109

• this is easily calculated as the core Axis value ± steps away:

• 64 ± 3 = 61 and 67
• 64 ± 33 = 31 and 97
• 64 ± 45 = 19 and 109
• where the steps away are literally the number of steps along the Axis to either side of the core Axis value that goes to a PPset
• their ∑s equal the EVEN 128:

• 61 + 67 = 128

• 31 + 97 = 128

• 19 + 109 = 128.

• We can confirm this on the PTOP (Table 35), and, of course, directly on the BIM. Tables 43 and 44 can also confirm ALL EVENS up to 2000. One can also use an online calculator to get the results.

• it can be quite helpful to follow these examples directly on the BIM. The visualizations of how the symmetrical PPsets contributed from the separate, individual PRIME Trails line up their 90° R-angled isosceles triangles over the central core with each of their respective apexes inline and pointing towards the EVEN that they are forming. On the Axis, one can easily see the symmetrical steps from the core to each PPset.

• What if you want to know specifically who covers, say EVEN 126? Looking at the full table, line 18, we see that Trail 67 and above have Ee > 126. How can we determine that the

• 13+113

• 17+109

• 19+107

• 23+103

• 29+97

• 37+89

• 43+83

• 47+79

• 53+73

• 59+67

PPsets are the ones we are looking for?

• Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick:

• on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 126 = 63

• see that PRIMES 59 & 67 on either side = 126 when added together and both are PRIMES

• knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 63 slot like the 59 & 67

• the next pair out would be 53 & 73, followed by 47 & 79.

• this is easily calculated as the core Axis value ± steps away:

• 63 ± 4 = 59 and 67

• 63 ± 10 = 53 and 73

• 63 ± 16 = 47 and 79

• 63 ± 20 = 43 and 83

• 63 ± 26 = 37 and 89

• 63 ± 34 = 29 and 97

• 63 ± 40 = 23 and 103

• 63 ± 44 = 19 and 107

• 63 ± 46 = 17 and 109

• 63 ± 50 = 13 and 113

• their ∑s equal the EVEN 126:

• 59 + 67 = 126
• 53 + 73 = 126
• 47 + 79 = 126
• 43 + 83 = 126
• 37 + 89 = 126
• 29 + 97 = 126
• 23 + 103 = 126
• 19 + 107 = 126
• 17 + 109 = 126
• 13 + 113 = 126
• The key is seeing that the PPsets are symmetrical about the core Axis value of the EVEN and applying these sequential, linear # of steps out from the core in identifying the PPsets that form that EVEN! EVERY PPset that forms an EVEN is part of a symmetrical pair of PPs, that together forms the EVEN. This is a critical and paramount finding in the behavior of the PRIMES and in the PROOF of the strong form of the Goldbach Conjecture.
• The ONLY EVENS (>4) with less than 2 PPsets are the EVENS 6, 8 and 12. EVEN 6 has only 1 PPset of 3,3 and EVEN 8 has only 1 PPset of 3,5. The EVEN 8 is the beginning of the 3 + 5 Trail, while the EVEN 6 is BOTH the start and end of the 3 + 3 Trail that begins the PTOP. The EVEN 12 is the one isolated example where the middle of a PRIME Trail — here the 3 + 7 Trail — is the only PPset to form EVEN 12, as the 3 + 5 Trail is — being the early start of Trails — not yet sufficient enough in length to cover the EVEN 12 slot. All 3 EVENS — 6,8,12 — are the exceptions precisely because they are the very beginning of the PTOP where the PRIME Trails are first being established.
• So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically flanked on either side by ODD WINs, and, as we have shown here on the PTOP and the PTOP as hidden within the BIM, will have two (2) or more ODD PPs pairs within that symmetrical flanking that will, as PPsets, form said EVEN.
• The symmetry simply falls out of the inherent inter-connection between the number values of each cell within the BIM. Every 1st Parallel EVEN has a straight line path back to its Axis and along that path the cell values must reflect the product of the two Axis values located symmetrically from the core Axis value outward, such that at their 90° R-angle, isosceles intersection on that path, one finds that product.
• The real wonder — and part of this amazing new finding — is that when such Axis numbers are found that are PRIME, they ALWAYS have a symmetrical ODD # counterpart on the other side of their common, core Axis number: and at the minimum, two or more candidates will have their ODDs be PRIMES, thus completing two or more PPsets!
• To remember, the Axis core number may itself be either EVEN or ODD as it is simply 1/2 of the EVEN that the path points to from its 1st Parallel Diagonal location. Also, the above rule does NOT mean that ALL ODD #s found on the Axis are part of the PPsets that form the EVEN. Even though symmetrical, and forming the same products on the path, BOTH PPset MEMBERS FORMING THE PAIR MUST BE PRIMES, e.i. an Axis 9, 15, 21,.. are NOT PRIME and NOT part of a PPset forming the EVEN. The geometry defines the algebra = algebraic geometry.
• The growth rate of the staggered, zig-zagging diagonal Trails of the PS ensures that the number of overlapping Trails will always exceed the growth rate of the PRIME Gaps by a sizable margin.

Towards the bottom of Table 46, lines 1092 - 1107, there is a good run of PRIME Gaps in close proximity. Below is the table in plain form with the same column headings:

line #PRIMES, P**2≥3**∑# of 3+P**2PPsets/Trail**Prime Gap∆ **Trail-Gap**EVEN: ∑# of**PPsets**EC=# of EVENs coveredE**e = Ending EVEN covered**EVEN
10808681108081072944340173628684
108186891081410771004344173788692
10828693108261076894346173868696
10838699108381075964349173988702
108487071084610781324353174148710
10858713108561079924356174268716
1086871910861210741124359174388722
108787311087610811014365174628734
108887371088410841364368174748740
10898741108961083934370174828744
109087471090610841384373174948750
109187531091810831044376175068756
1092876110921810741114380175228764
10938779109341089914389175588782
109487831094201074954391175668786
109588031095410911254401176068806
1096880710961210841244403176148810
109788191097210951024409176388822
109888211098101088904410176428824
10998831109961093934415176628834
110088371100210981414418176748840
110188391101101091934419176788842
110288491102121090914424176988852
11038861110321101934430177228864
110488631104411001184431177268866
1105886711052010851214433177348870
110688871106611001484443177748890
110788931107301077944446177868896
11088923110861102954461178468926
110989291109411051254464178588932
11108933111081102954466178668936
1111894111111011011014470178828944
1112895111121211001044475179028954
11138963111361107964481179268966
11148969111421112844484179388972
1115897111152810871044485179428974
111689991116211141104499179989002
11179001111761111954500180029004
111890071118411141334503180149010
11199011111921117964505180229014
1120901311201611041194506180269016

Without even knowing the values for the Column 6 (Table 46) which have been inserted here: EVEN, ∑# of PPsets, one can take EVEN 8872:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick: (see *below for a simple Universal calculation method as shown in Table 45: Equations.)

• on Table 46, find where EVEN 8872 would be located as the core Axis value between lines 601 and 602 as 1/2 of 8872 = 4436

• see that PRIMES 4423 & 4441 on either side = 8864 when added together and both are PRIMES we are close

• knowing from the BIM that ALL PPsets are symmetrical about the center core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 4436 slot like the 93 PPsets that equal EVEN 8872:

• 5+8867
• 11+8861
• 23+8849
• 41+8831
• 53+8819
• 89+8783
• 131+8741
• 173+8699
• 179+8693
• 191+8681
• 263+8609
• 359+8513
• 443+8429
• 449+8423
• 503+8369
• 509+8363
• 599+8273
• 641+8231
• 653+8219
• 701+8171
• 761+8111
• 863+8009
• 953+7919
• 971+7901
• 1019+7853
• 1031+7841
• 1049+7823
• 1181+7691
• 1223+7649
• 1229+7643
• 1283+7589
• 1289+7583
• 1373+7499
• 1439+7433
• 1523+7349
• 1619+7253
• 1721+7151
• 1871+7001
• 1889+6983
• 1901+6971
• 1913+6959
• 1973+6899
• 2003+6869
• 2039+6833
• 2069+6803
• 2081+6791
• 2111+6761
• 2153+6719
• 2213+6659
• 2273+6599
• 2309+6563
• 2351+6521
• 2381+6491
• 2399+6473
• 2423+6449
• 2543+6329
• 2549+6323
• 2609+6263
• 2699+6173
• 2729+6143
• 2741+6131
• 2819+6053
• 2843+6029
• 2861+6011
• 2969+5903
• 3011+5861
• 3023+5849
• 3089+5783
• 3203+5669
• 3221+5651
• 3299+5573
• 3371+5501
• 3389+5483
• 3491+5381
• 3539+5333
• 3593+5279
• 3701+5171
• 3719+5153
• 3821+5051
• 3833+5039
• 3851+5021
• 3863+5009
• 3929+4943
• 4001+4871
• 4073+4799
• 4079+4793
• 4139+4733
• 4229+4643
• 4289+4583
• 4349+4523
• 4391+4481
• 4409+4463
• 4421+4451
• as before, these can be simplified as:

• 4436 ± 5 = 4431 and 4441. Together, 4431 + 4441 = 8872. See Table 48 for the complete list.
• 4436 ± 11
• 4436 ± 23
• 4436 ± 41
• 4436 ± 53
• 4436 ± 89
• 4436 ± 131
• 4436 ± 173
• 4436 ± 179
• 4436 ± 191
• 4436 ± 263
• 4436 ± 359
• 4436 ± 443
• 4436 ± 449
• 4436 ± 503
• 4436 ± 509
• 4436 ± 599
• 4436 ± 641
• 4436 ± 653
• 4436 ± 701
• 4436 ± 761
• 4436 ± 863
• 4436 ± 953
• 4436 ± 1019
• 4436 ± 1031
• 4436 ± 1049
• 4436 ± 1181
• 4436 ± 1223
• 4436 ± 1229
• 4436 ± 1283
• 4436 ± 1289
• 4436 ± 1373
• 4436 ± 1439
• 4436 ± 1523
• 4436 ± 1619
• 4436 ± 1721
• 4436 ± 1871
• 4436 ± 1889
• 4436 ± 1901
• 4436 ± 1913
• 4436 ± 1973
• 4436 ± 2003
• 4436 ± 2039
• 4436 ± 2069
• 4436 ± 2081
• 4436 ± 2111
• 4436 ± 2153
• 4436 ± 2213
• 4436 ± 2273
• 4436 ± 2309
• 4436 ± 2351
• 4436 ± 2381
• 4436 ± 2399
• 4436 ± 2423
• 4436 ± 2543
• 4436 ± 2549
• 4436 ± 2609
• 4436 ± 2699
• 4436 ± 2729
• 4436 ± 2741
• 4436 ± 2819
• 4436 ± 2843
• 4436 ± 2861
• 4436 ± 2969
• 4436 ± 3011
• 4436 ± 3023
• 4436 ± 3089
• 4436 ± 3203
• 4436 ± 3221
• 4436 ± 3299
• 4436 ± 3371
• 4436 ± 3389
• 4436 ± 3491
• 4436 ± 3539
• 4436 ± 3593
• 4436 ± 3701
• 4436 ± 3719
• 4436 ± 3821
• 4436 ± 3833
• 4436 ± 3851
• 4436 ± 3863
• 4436 ± 3929
• 4436 ± 4001
• 4436 ± 4073
• 4436 ± 4079
• 4436 ± 4139
• 4436 ± 4229
• 4436 ± 4289
• 4436 ± 4349
• 4436 ± 4391
• 4436 ± 4409
• 4436 ± 4421

We can further simplify by applying these Universal (vs Specific, see Table 45:) equations:

• By definition, EVEN = P1 + P2 = PPset
• let S = steps, E = EVEN = 2(core Axis value) = 2(Ax), as E/2 = Ax

• S = P2 - E/2 = P2 - Ax

• the PPset for a given EVEN:

• P2 = S + E/2 = S + Ax

• P1 = P2 - (2S)

• 2S = P2 - P1

• S = AX - P1

• E = P1 + P2

Example: 3,5 PPset for EVEN = 8: S = P2 - E/2 = P2 - Ax 1 = 5 -4

P2 = S + E/2 = S + Ax

5 = 1 + 4

P1 = P2 - (2S) 3 = 5 - (2*1)

And, as P1 = E - P2, or E = P1 + P2

P1 = P2 - (2S) = E - P2

2P2 - E = 2S 2(5) - 8 = 2(1) S = 1

As S = (2P2 - E)/2

S = (2P2 - E)/2 = P2 - E/2

2P2/2 - E/2 = P2 - E/2

P2 - E/2 = P2 - E/2 = S

P2 - Ax = P2 - Ax = S

The built-in symmetry of the PPsets around the core Axis value is easily calculated as these four examples show:

• EVEN 24 with core Axis value (Ax) = 24 / 2 = 12 with 3 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 11,13 13-12 = 1 13 - (2x1) = 11

• 7,17 17-12 = 5 17 - (2x5) = 7

• 5,19 19-12 = 7 19 - (2x7) = 5

• EVEN 26 with core Axis value = 26 / 2 = 13 with 3 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 13,13 13-13 = 0 13 - (2x0) = 13

• 7,19 19-13 = 6 19 - (2x6) = 7

• 3,23 23-13 = 10 23 - (2x10) = 3

• EVEN 22 with core Axis value = 22/2 = 11 with 3 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 11,11 11-11 = 0 11 - (2x0) = 11

• 5,17 17-11 = 6 17 - (2x6) = 5

• 3,19 19-11 = 8 19 - (2x8) = 3

• EVEN 100 with core Axis value = 100 / 2 = 50 with 6 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 47,53 53-50 = 3 53 - (2x3) = 47

• 43,57 57-50 = 7 57 - (2x7) = 43

• 41,59 59-50 = 9 59 - (2x9) = 41

• 29,71 71-50 = 21 71 - (2x21) = 29

• 17,83 83-50 = 33 83 - (2x33) = 17

• 11,89 89-50 = 39 89 - (2x39) = 11

~~~~~~~~

As one moves successively along the EVENS, the Trails get longer and longer, adding one new member for each successive PS (See Tables 44-46.). This rate of increase far exceeds the size and incidence rate of the PRIME Gaps (See Tables 42 and 46.), ensuring that for every EVEN ≥6, there is at least one PPset of ODD PRIMES that will form it. Actually, as we have shown, there are always a minimum of 2 sets of PPsets that form the EVENS (≥14, including 10).

Table 44 shows how ∑s of the number of PPsets/ EVEN grows. It increases such that for every 60 successive EVENS on average, the ∑ increases by 3 (as calculated from 4-1080).

For example:

Columns generally have 60 entries ending in multiples of 120.

∑of PPsets per column Ave/Col

Col A 262 / 59 = 4.4 = 4

Col B 516 / 60 = 8.6 = 9

Col C 718 / 60 = 12 = 12

Col D 930 / 60 = 15.5 = 16

Col E 1076 / 60 = 17.9 = 18

Col F 1267 / 60 = 21.1 = 21

Col G 1302 / 60 = 21.7 = 22

Col H 1534 / 60 = 25.6 = 26

Col I 1687 / 60 = 28.1 = 28

SUMMARY and CONCLUSION

It is worth repeating here from above:

• So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically flanked on either side by ODD WINs, and, as we have shown here on the PTOP and the PTOP as hidden within the BIM, will have two (2) or more ODD PPs pairs within that symmetrical flanking that will, as PPsets, form said EVEN.
• The symmetry simply falls out of the inherent inter-connection between the number values of each cell within the BIM. Every 1st Parallel EVEN has a straight line path back to its Axis and along that path the cell values must reflect the product of the two Axis values located symmetrically from the core Axis value outward, such that at their 90° R-angle, isosceles intersection on that path, one finds that product.
• The real wonder — and part of this amazing new finding — is that when such Axis numbers are found that are PRIME, they ALWAYS have a symmetrical ODD # counterpart on the other side of their common, core Axis number: and at the minimum, two or more candidates will have their ODDs be PRIMES, thus completing two or more PPsets!
• To remember, the Axis core number may itself be either EVEN or ODD as it is simply 1/2 of the EVEN that the path points to from its 1st Parallel Diagonal location. Also, the above rule does NOT mean that ALL ODD #s found on the Axis are part of the PPsets that form the EVEN. Even though symmetrical, and forming the same products on the path, BOTH PPset MEMBERS FORMING THE PAIR MUST BE PRIMES, e.i. an Axis 9, 15, 21,.. are NOT PRIME and NOT part of a PPset forming the EVEN. The geometry defines the algebra = algebraic geometry.
• The growth rate of the staggered, zig-zagging diagonal Trails of the PS ensures that the number of overlapping Trails will always exceed the growth rate of the PRIME Gaps by a sizable margin.

As seen in Table 44, EVEN 68 it the last EVEN to only have 2 PPsets, EVEN 128 is the last to have 3, 332 the last 6, 398 the last 7, 488 the last 9, 632 the last 10, 878 the last 14,…as the table shows the progression steadily goes forth never regressing by more than 1 or 2 to a previous lower value. The number of PPsets for:

• 100000 = 810,
• 1000000 = 5402,
• 10000000 = 38807,
• 100000000 = 291400
• 1000000000 = 2274205.

If we take a sampling (from Table 46), the ∑# of PPsets and and divide by their respective EVENS, we get the percentage incidence of:

• 1 / 6 = 0.167
• 2 / 10 = 0.2
• 3 / 26 = 0.115
• 4 / 50 = 0.08
• 6 / 100 = 0.06
• 9 / 200 = 0.045
• 8 / 296 = 0.027
• 14 / 400 = 0.035
• 15 / 502 = 0.029
• 28 / 1000 = 0.028
• 27 / 1502 = 0.018
• 37 / 2000 = 0.018
• 46 / 2480 = 0.018
• 49 / 2974 = 0.016
• 80 / 4300 = 0.019
• 67 / 5594 = 0.012
• 104 / 9976 = 0.010
• 191 / 10010 = 0.0191
• 810 / 100000 = 0.008
• 5402 / 1000000 = 0.005
• 38807 / 10000000 = 0.004
• 291400 / 100000000 = 0.003
• 2274205 / 1000000000 = 0.002
• Clearly the percentage drops with the size of EVENS. Naturally, the percentage is highest at the beginning of the EVEN sequence.
• We know from the Prime Number Theorem (PNT) that the PRIME Gaps increase in number as the PRIMES occur less often as their size increases.
• From Tables 38-40, we know that the size of the PPset Trails = (EVEN/2)-2. For example EVEN 400 that would be: (400/2)-2 = 198. EVEN 400 has 14 PPsets (see above) yet is the start of a PPset Trail that has 198 PPs that zig-zag diagonally down (see Tables 34-36) to inform the next 198 EVENS in the EVENS sequence. The overlapping, zig-zag Trails and the inherent PPset symmetry about the core Axis value together ensures that all EVENS ≥6 are composed of two or more PRIMES.
• The PPsets Trails increase sequentially with each step in the PRIME Sequence (PS), effectively counteractively covering the larger EVENS as their size increases with a decreasing density of PRIMES. This may be the insight needed in solving the Riemann Hypothesis — and specifically the Riemann zeta function — as the “critical line” of “trivial zeros” may be the very counterbalancing symmetry between the actual numbers of PPsets solving the Goldbach Conjecture, and, the play between the ever-lengthening Trails vs the PRIMES density falloff as the EVENS increase in size. (See Tables 34-42.)

This built-in geometry provides the necessary coverage that ALL EVENS (≥14) will easily be covered by two or more PPsets. The geometry defines the algebra = algebraic geometry.

The Goldbach Conjecture is proven. With the EVENS having been actually tested for compliance with the conjecture up to 4x10**18, one can say that it is statistically impossible for any of the holes not to be covered by the PPset Trails no matter what size EVEN is considered! The largest confirmed PRIME Gap of 1550 would have —————

DATA: Images and Tables

Here is the index of the five sets of data to consider (click links to go):

1. BIM (BBS-ISL Matrix): grid visualizations that overview the entire work

• Fig. 1-PTOP: Periodic Table Of PRIMES (100, original)
• Fig. 2-PTOP: Periodic Table Of PRIMES (100, annotated)
• Fig. 3-PTOP: Periodic Table Of PRIMES (100, upgraded)
• Fig. 4-PTOP: Periodic Table Of PRIMES (200, annotated)
• Fig. 5-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (original)
• Fig. 6-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (annotated)
• Fig. 7-Table 46: Symmetrical STEPS of the PPsets for EVEN 128 annotated snapshot
• Fig. 8-BIM: Symmetrical STEPS of the PPsets for EVEN 126 snapshot
• Fig. 9-BIM: Symmetrical STEPS of the PPsets for EVEN 126
• Fig. 10-BIM: Symmetrical STEPS of the PPsets for EVEN 128 snapshot
• Fig. 11-BIM: Symmetrical STEPS of the PPsets for EVEN 128
• Fig. 12- animated gif of video (below)
• Video: PTOP rule 169-170 Annotated
2. PTOP: the actual Table

• Table 34: Original PTOP from 2009 (EVENS 6-100)
• Table 35: Upgraded PTOP (EVENS 6-200)
• Table 36: Upgraded PTOP (EVENS 6-400)
• Table 37: Upgraded PTOP (work in progress, EVENS 6-1000)
3. PTOP: Analysis

• Table 38: Distribution and NPS of the PPset Trails (EVENS 6-404)
• Table 39: Distribution and NPS of the PPset Trails (EVENS 6-914)
• Table 40: Distribution and NPS of the PPset Trails (EVENS 6-2360 and up and up)
• Table 41: Bifurcation Addition and PPset ∑s
4. Reference

• Table 42: PRIME Gaps
• Table 43: PRIME Partitions = PPsets per EVENS (4-2000)
• Table 44: Summaries of Table 43 (EVENS 4-2000)
• Table 45: Equations for PTOP Tables 38-41
5. PRIME PPset Trails

• Table 46: PRIME PPset Trails (3-10007 and up)
• Table 47: PRIME PPset Trails simplified and extended Table 46 (3-568201)
• Table 48: Working example: EVEN 8872, core Axis 4436 with 93 PPsets.
• Table 49: PRIME PPset Trails & EVENS divisible by 6,12 or 24 (3-10007 and up).

DATA: Images and Tables

1. BIM (BBS-ISL Matrix): grid visualizations that overview the entire work

• Fig. 1-PTOP: Periodic Table Of PRIMES (100, original)

• Fig. 2-PTOP: Periodic Table Of PRIMES (100, annotated)

• Fig. 3-PTOP: Periodic Table Of PRIMES (100, upgraded)

• Fig. 4-PTOP: Periodic Table Of PRIMES (200, annotated)

• Fig. 5-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (original)

• Fig. 6-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (annotated)

• Fig. 7-Table 46: Symmetrical STEPS of the PPsets for EVEN 128 annotated snapshot

• Fig. 8-BIM: Symmetrical STEPS of the PPsets for EVEN 126 snapshot

• Fig. 9-BIM: Symmetrical STEPS of the PPsets for EVEN 126

• Fig. 10-BIM: Symmetrical STEPS of the PPsets for EVEN 128 snapshot

• Fig. 11-BIM: Symmetrical STEPS of the PPsets for EVEN 128

• Fig. 12- animated gif of video (below)

• Video: PTOPrule170Annotated.mov
• (https://vimeo.com/372243091)

PTOP & Goldbach Conjecture from Reginald Brooks on Vimeo.

2. PTOP: the actual Table

• Table 34: Original PTOP from 2009 (EVENS 6-100)
• Table 35: Upgraded PTOP (EVENS 6-200)
• Table 36: Upgraded PTOP (EVENS 6-400)
• Table 37: Upgraded PTOP (work in progress, EVENS 6-1000)

3. PTOP: Analysis

• Table 38: Distribution and NPS of the PPset Trails (EVENS 6-404)
• Table 39: Distribution and NPS of the PPset Trails (EVENS 6-914)
• Table 39_page-2
• Table 40: Distribution and NPS of the PPset Trails (EVENS 6-2360 and up and up)
• Table 40_page-2
• Please see ebook for remaining pages.
• Table 40_page-3
• Please see ebook for remaining pages.
• Table 40_page-4
• Please see ebook for remaining pages.
• Table 40_page-5
• Please see ebook for remaining pages.
• Table 40_page-6
• Please see ebook for remaining pages.
• Table 40_page-7
• Please see ebook for remaining pages.
• Table 40_page-8
• Please see ebook for remaining pages.
• Table 40_page-9
• Please see ebook for remaining pages.
• Table 40_page-10
• Please see ebook for remaining pages.

• Table 41: Bifurcation Addition and PPset ∑s>

4. Reference

• Table 42: PRIME Gaps
• Table 43: PRIME Partitions = PPsets per EVENS (4-2000)
• Table 43_page-2
• Please see ebook for remaining pages.
• Table 43_page-3
• Please see ebook for remaining pages.
• Table 43_page-4
• Please see ebook for remaining pages.
• Table 43_page-5
• Please see ebook for remaining pages.
• Table 43_page-6
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• Table 43_page-7
• Please see ebook for remaining pages.
• Table 43_page-8
• Please see ebook for remaining pages.
• Table 43_page-9
• Please see ebook for remaining pages.
• Table 43_page-10
• Please see ebook for remaining pages.
• Table 43_page-11
• Please see ebook for remaining pages.

• Table 44: Summaries of Table 43 (EVENS 4-2000)

• Table 44_page-2
• Table 45: Equations for PTOP Tables 38-41

5. PRIME PPset Trails

• Table 46: PRIME PPset Trails (3-10007 and up)
• Table 46_page-2
• Please see ebook for remaining pages.
• Table 46_page-3
• Please see ebook for remaining pages.
• Table 46_page-4
• Please see ebook for remaining pages.
• Table 46_page-5
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• Table 46_page-6
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• Table 46_page-7
• Please see ebook for remaining pages.
• Table 46_page-8
• Please see ebook for remaining pages.
• Table 46_page-9
• Please see ebook for remaining pages.
• Table 46_page-10

• Table 47: PRIME PPset Trails simplified and extended Table 46 (3-568201)
• Table 47_page-2
• Please see ebook for remaining pages.
• Table 47_page-3
• Please see ebook for remaining pages.
• Table 47_page-4
• Please see ebook for remaining pages.
• Table 47_page-5
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• Table 47_page-6
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• Table 47_page-7
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• Table 47_page-8
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• Table 47_page-9
• Please see ebook for remaining pages.
• Table 47_page-10
• Please see ebook for remaining pages.
• Table 47_page-11
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• Table 47_page-12
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• Table 47_page-13
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• Table 47_page-14
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• Table 47_page-15
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• Table 47_page-16
• Please see ebook for remaining pages.
• Table 47_page-17
• Please see ebook for remaining pages.
• Table 47_page-18
• Please see ebook for remaining pages.

• Table 48: Working example: EVEN 8872, core Axis 4436 with 93 PPsets.
• Table 49: PRIME PPset Trails & EVENS divisible by 6,12 or 24 (3-10007 and up)
• Table 49_page-2
• Please see ebook for remaining pages.
• Table 49_page-3
• Please see ebook for remaining pages.
• Table 49_page-4
• Please see ebook for remaining pages.
• Table 49_page-5
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• Table 49_page-6
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• Table 49_page-7
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• Table 49_page-8
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• Table 49_page-9
• Please see ebook for remaining pages.
• Table 49_page-10

New PTOP on the BIM

PREFACE

Fractal — Symmetry — Inverse Square Law (ISL). These are not the usual descriptive terms associated with the Prime numbers (PRIMES) Yet that is exactly what best describes the PRIMES!

The PRIMES, when brought together as P1 & P2 members of PRIME Pair Sets (PPsets) demonstrate a robust symmetry and an intimate relationship with the ISL when shown on the BIM (BBS-ISL Matrix grid of the Inverse Square Law). This symmetry is brought out in the geometric relationship between the PPsets and the EVEN numbers (EVENS) that they inform consistent with Euler’s Strong Form of the Goldbach Conjecture.

This come about as the well established PRIMES Sequence (PS) — 3-5-7-11-13-17-19-23-29-31-... acts in a fractal-like manner, i.e., it demonstrates redundant, repetitive and re-iterative behavior in presenting self-similar reflection of itself as it constructs PPsets and PPset “TRAILS.”

The TRAILS are formed from the concatenation of PS’s progressively onto each successive PRIME of a given PS — forming a series of PPsets along the way. The TRAILS can also be seen to be formed directly as individual PPsets on the BIM. Here, each PPset is easily seen to be intersection of 1 PPset member from the Horizontal AXIS and 1 PPset member from the Vertical AXIS, together forming the P1, P2 members. The TRAIL is simply all those PPsets thus formed across a Row — or down a Column — on the bilaterally symmetrical BIM. All this is easily visible on the BIM.

That leads us to our story about the PRIMES—Fractals—Symmetry—and the intimate relationship they have with the ISL as seen on the BIM.

ABSTRACT

The “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” (2019), also referred to as BIM: Part I, updated and clarified the PTOP, PPsets and their PPset TRAILS that were originally presented in MathspeedST (2010). BIM = BBS-ISL Matrix.

In this work, a major update and refinement has been made. Each of the three parts — BIM: II, III and IV presents new findings that visibly demonstrate the PRIMES on the BIM.

Each and every PRIME, when treated as part of a PPset (P1, P2), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.

Plotting all the Lower Diagonal P2 PRIMES on a table and then re-plotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P2 PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place — BIM: Part III.

In BIM: Part I-III, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.

In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.

Treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found. As the PS-Fractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…

And while this intimate relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.

While Euler’s Strong Form of the Goldbach Conjecture is proved along the way, the real significance is what we have seen unfolding in BIM: Parts I-IV. The role of Symmetry and Fractal underlie everything about the PRIMES.

The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!

What is marvelous, incredible, mind-blowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.

The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.

DESCRIPTION

SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM (BBS-ISL Matrix).

In MathspeedST, the PRIMES were found to be stealthily hidden within the BIM, revealing their presence ONLY when the BIM itself had its ODD #s 1st Parallel Diagonal modified to EVENS by the addition of 1 to each ODD.

This resulted in the original PTOP (Periodic Table Of PRIMES) on the BIM. It was refined by pulling the values off the BIM and making a separate PTOP. All this became the subject of “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” ebook and white paper (2019). It is also referred to as BIM: Part I. The PPsets and their TRAILS were introduced.

Each and every PRIME, when treated as part of a PPset, can be found and individually profiled DIRECTLY on the BIM. BIM: Part II was formed.

Plotting all the Lower Diagonal P2 PRIMES on a table and then re-plotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P2 PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place.

This is a direct consequence of locating the PPsets as the grid space resulting from the intersection of the PRIMES Sequence (PS) of the Horizontal AXIS with the PS of the Vertical AXIS. This becomes the basis for the BIM: Part III.

In BIM: Part I-III, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.

In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.

What we find is that by treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found. Yes, as the PS-Fractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…

And while this relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.

While Euler’s Strong Form of the Goldbach Conjecture is proved along the way on this journey, the real significance is what we have seen unfolding in BIM: Parts I-IV. The role of Symmetry and Fractal underlie everything about the PRIMES.

The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!

What is marvelous, incredible, mind-blowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.

The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.

Introduction

The PRIMES have been found on the BIM despite there being NO PRIMES in the Strict Inner Grid (SIG). The SIG is simply the BIM without the AXIS, Prime Diagonal (PD) that runs diagonally down the middle, dividing the grid into two bilaterally symmetrical triangular halves, and, the 1st Parallel Diagonal of ALL ODD numbers.

So where are they?

BIM: Part I and the original PTOP

Initially, they were found in a quasi-stealth mode on the BIM. Why quasi-stealth? The PRIMES were naturally found on the AXIS (n=1,2,3,…). It was only when they were paired symmetrically to either side of another AXIS #—a number that turned out to be an EVEN ÷ 2— that the lines of the triangular sides (diagonals) meet on a line from said EVEN ÷2 # on the AXIS, back to the 1st Parallel Diagonal. Yes, here we had to use the entire BIM, and, that 1st Parallel Diagonal had to have ALL its ODD #s advanced (+1) to make them EVEN.

From this, it was found that every EVEN # on the adjusted 1st Parallel Diagonal, had, indeed, one or more sets, or pairs, of PRIMES on either side of the EVEN/2, found on the AXIS.

Visually, this demonstrated that every EVEN was flanked by one or more pair sets of PRIMES. We call these PPsets and each contains a P1 and P2 value that resides symmetrically on either side of the EVEN/2 AXIS #. The only exception being when P1 = P2, then the symmetry is one, i.e., EVEN = 10 has EVEN/2=5 and there are two PPsets that inform EVEN = 10 are 3,7 and 5,5.

Visually, this also demonstrated both the symmetrical nature of the PPsets and the isosceles, right-triangle each made pointing back to the line connecting the EVEN with its EVEN/2 AXIS #.

This early (discovered in 2009–10) PRIMES on the BIM” will be referred to as BIM: Part I.

A simplification of BIM: Part I formed the basis of the PTOP (Periodic Table Of PRIMES, as first described in MathspeedST, 2010).

The PTOP extracts the PPsets and EVENS from the BIM and relates them to each other in a straightforward and graphical manner.

A distinct PPset “TRAIL” is seen for each PPset starting with the P1=3 and P2 = each successive number in the natural PRIMES sequence—(2),3,5,7,11,13,17,19,23,29,31,37,…

In fact, this natural PRIMES Sequence (PS) acts in every way just like a fractal, whereby a fractal set consisting of the PS is “added” —i.e. concatenated—to each successive P1 value down the Column 3 on the PTOP.

This fractal PS property ensures that there are more overlapping “TRAILS” of PPsets than PRIME Gaps, ensuring that ALL EVENS = the sum of two PRIMES. Euler’s Strong Form of the Goldbach Conjecture is proven. More importantly, a definitive relationship between PRIMES and the Inverse Square Law is defined. This joins the ISL, Primitive Pythagorean Triples and PRIMES together under the umbrella of the Inverse Square Law as presented on the BIM. Part I and the original PTOP is fully described in both the white paper and ebook: PTOP (Periodic Table Of PRIMES) and the Goldbach Conjecture (2019). PRIMES Index.

BIM: Part II

In BIM: Part II, the focus was shifted to looking at individual PPsets on the BIM—those making one EVEN number at a time. It is quite interesting how once again those stealthy PRIMES only appear on the AXIS and 1st Parallel Diagonal, yet make a clear, distinctive Number Pattern Sequence (NPS) by their interactions.

In this case, for every EVEN/2 on the AXIS, an isosceles triangle is formed with the base along the AXIS, one side along the PD to the apex of the (EVEN/2)2 and the other side back to the base on the AXIS.

The PPset P1s are found on the Upper Diagonal PD as the √ of where the PS # on the AXIS intersects the PD—there the square root of that PD #=P1, or more easily, just take its Axial value.

The Lower Diagonal (LD) ALWAYS points back to the EVEN on the AXIS. Along the way, it intercepts with the PS values on the AXIS above, to give the P2 values. Dividing the various grid values along the LD by 4x(Even/2) will select out and confirm which values house the P2 PRIMES.

Incredibly, it turns out the number of STEPS from one apex to another around each of the triangles formed, will show them to ALWAYS be equal, thus equilateral. The larger base:PD:LD isosceles triangle is actually made of two equilateral triangles positioned back-to-back with the centerline being the line from the EVEN/2 back to it squared value on the PD.

Table 51 was formed by plotting all the LD values for each EVEN as seen in BIM: Part II. These values were then plotted back on to the BIM and BIM: Part III was formed.

BIM: Part III

In Part III, we combine and simplify everything from the Part I & II including:

SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM.
~~~~~~

These PPsets are formed from the Horizontal and Vertical AXIS numbers that intercept those BLUE circle values. The PRIMES Sequence (PS)—3,5,7,11,13,17,19,23,…— determines the pattern. Notice that the PPsets ALSO fall exactly on the same BIM/24 Active ROWS (PURPLE Bands) as do the PPTs and PRIMES

Those BLUE values from Table 51 are now BLUE CIRCLE values on the BIM. And when those values are overlaid with the PPsets it now becomes Table 52 and the NEW PTOP (Periodic Table Of PRIMES) on the BIM! Here they are plain and simple. Let the story begin.

Like shining a light at a sculpture in the dark, each angle reveals a new “look,” yet it is just one sculpture. That’s what we are doing here..

BIM: Part III becomes significant when one finds that the NPS of those PPsets falls on exactly the SAME Active Rows that previously ALL PRIMES and ALL PPTs were found to occupy on the BIM! No small coincidence and quite easily seen when, as in Table 52, the actual PPset values are overlaid on those LD values. The PS fractals along the Horizontal and Vertical AXIS intercept at these points on the BIM. They form the NPS of the PTOP directly on the BIM! It is no longer hidden!

The PRIMES Sequence Fractal—3–5–7–11–13–17–19–23–29–31-…—found along each AXIS, together forms the PPsets = EVENS = Goldbach Conjecture Proof.

The PPset TRAILS (Rows) ALWAYS INCREASE FASTER than the PRIME Gaps ensuring that the PPsets = EVENS = Goldbach Conjecture Proof.

One could say, that by definition, ALL EVENS are indeed formed from the OVERLAPPING PS Fractal TRAILS of PPsets = EVENS = Goldbach Conjecture Proof.

~~~

EVEN = 30
EVEN/2 = 15

7, 23 = 7 + 23 = 30
23 - 7 = 16
16/2 = 8
8 = STEPS to either side of 15 as:
8 + 15 = 23
15 - 8 = 7

11,19 = 11 + 19 = 30
19 –11 = 8
8/2 = 4
4 = STEPS to either side of 15 as:
4 + 15 = 19
15 - 4 = 11

13, 17 = 13 + 17 = 30
17 - 13 = 4
4/2 = 2
2 = STEPS to either side of 15 as:
2 + 15 = 17
15 - 2 = 13

EVEN = 30 = 3 PPsets = (7, 23), (11, 19), and (13, 17)

The 3 PPsets—(7, 23), (11, 19), and (13, 17)—result from the overlap of
3 PPset TRAILS: starting at (3, 17), (3, 19) and (3, 23) PS Rows.

The TRAIL overlap occurs under the
7, 11 and 13 PS Columns, respectively.

One can see that combining the P2 PS Row values with the P1 PS Column values gives the
PPsets = (7, 23), (11, 19), and (13, 17).

And, of course, the symmetrical STEPS between the P2 - P1 values follows as shown.

It also follows that taking half of the difference in sums of the P2 - P1 = the sums of STEPS.

[(17 + 19 + 23) - (7 + 11 + 13)] ÷ 2 = STEPS = 14 = 2 + 4 + 8 STEPS.

These STEPS are BOTH the number of steps from the PD to each P1 or P2, and,
the number of steps from the P1 or P2 to the center Core Axis Value (EVEN/2) line — the line, or ROW, that connects the EVEN/2 on the AXIS to the square of the EVEN/2 on the PD.

EVEN/2 = 15 and Row 15 line intersects and terminates on the PD at (EVEN/2)2.

The NPS holds true for ALL PPsets that together define any given EVEN!

Summary of the math:
let E = EVENS, V= Value of grid cell

Like shining a light at a sculpture in the dark, each angle reveals a new “look,” yet it is just one sculpture. That’s what we are doing here. Hopefully, the BIM: Part III look will give us the simplest, most direct and informative view of just how EVERY EVEN IS MADE OF SYMMETRY AND PS FRACTAL ENDOWED PPsets IN THE FORM OF THEIR OVERLAID PPset TRAILS. Euler’s Strong Form of the Goldbach Conjecture simply falls out from this builtin relationship between the ISL, the PPTs and PRIMES as seen on the BIM.

BIM: Part IV

In closing out the “PTOP (Periodic Table Of PRIMES) on the BIM: Parts I, II and III,” one may ask if that is it it?

The hidden PTOP (Part I) was visibly profiled as individual PPsets (Part II), and the two combined to be fully revealed in plain sight in Part III. But is that it?

Besides showing that the Primitive Pythagorean Triangles, PRIMES and the PPsets and their PPset TRAILS were ALL similarly located on specific Active Rows on the BIM—Rows identified by dividing the BIM Inner Grid cell values evenly by 24—all the EVENS were shown to be informed by the PPsets. These PPsets were themselves informed geometrically by the symmetry and PRIMES Sequence (PS)-fractal nature of the PRIMES patterning in their very formation. The PS is a fractal that interacts dynamically with other PS-fractal iterations that together form the PTOP.

But is that it? Is that all? Is there something more these PRIMES can tell us about the basic numbers that inform the Universe?

Most of the graphics on the PTOP on the BIM have focused on looking at just one side of the bilaterally symmetric BIM square—and often that “square” has been stretched into a rectangular format to accommodate both the ever-increasing larger numbers and some just plain, simplified space-shapes to present info and comments on.

But what about looking at the whole BIM? In doing so, we find that the AREAS enclosed by looking at both of the bilaterally symmetric sides of the BIM gives us some NEW INSIGHTS into just what is the relationship of the PRIMES to the ISL (Inverse Square Law) ?

In PTOP (Periodic Table Of PRIMES) on the BIM: Part IV” we will be doing just that.

Here is an older graphic, now overlayed with the NEW PRIMES-PPsets. Take all the Inner Grid cell values on the BIM that divide evenly by 24, color them YELLOW-ORANGE, and this is what you get. It showed that the Primitive Pythagorean Triangles (PPTs) always and exclusively fell on the Active ROWs—where the 1st Col. values were either YELLOW or ORANGE. Turns out the SAME is true for the PRIMES. They, too, only land on the Active ROWs. Note: some Active ROWs have neither PPT or PRIME, some have both, and some one but not the other.

Notice the pattern unfolding when the PRIMES Sequence (PS)-Fractals—3–5–7–11–13–17–19–23–29-…—from the horizontal and vertical AXIS combine on the BIM. Soon we will be looking at this pattern across the entire BIM grid.

Clarifications: The BIM is NOT a traditional multiplication table. NO WAY. While the central Prime Diagonal (PD) does represent the squares of the AXIS #s, ALL the Inner Grid (IG) values are determined by the difference (∆) in their horizontal and vertical PD intercept values, ei., 16 is PD 25 - PD 9 (25–9=16).

That said, in the PPsets as presented in the PTOP on the BIM, we will go from this standard approach, to overlaying key matrix values with the coordinates of their AXIS, i.e., 9 is overlaid with AXIS 3, AXIS 3 or simply 3,3 and 16 is overlaid with 3,5 and so on. This forms a “Sub-Matrix” on the BIM of overlaid values that will be quite useful. Once you see how simple this overlay is, a great simplification overall will reveal itself.

One other note: the PPsets as 3,3–3,5–5,5–… form the EVENS by adding the set members, i.e. 3,3 = 3 + 3 = 6, and, a diagonal line from a given PPset back to the AXIS will reveal that EVEN.

Object AREAS:

AREAS — as Object Areas — are introduced here. Object Areas simply are the # of objects within an “area.” Here that AREA is a square on the BIM. Counting the # of objects within each AREA will give the number. More surprises await.

The PPset Areas are NOT traditional areas. Instead, they represent the number (#) of PPsets captured within the boundaries of the squared AXIS #s, i.e., the squared values of the PS-fractal 3–5–7–11–13–17–19–23–29-… are 9–25–49–121–169–289–361–529–841-… but they are represented, as a “Sub-Matrix” on the BIM, as 3,3–5,5–7,7–11,11–13,13–17,17–19,19–23,23–29,29—…PPsets.

It is the # of these PPsets within the squared PS-fractal AXIS value treated as area. One is really capturing the # of “objects” or Object AREAS.

This IS a different way of capturing the way the PRIMES as PPsets are part of the BIM ISL 1–4–9–16–25-… pattern!

This way of capturing the # of objects is NOT limited to the PPsets, but rather is a natural characteristic of the BIM. Other non-prime or prime with non-prime mixtures of objects can be shown to also demonstrate this ISL pattern. It is simply the result of progressively adding 1,2,3 or more objects — in any interval — regularly to the squared AREA. It will naturally generate an ISL pattern. This points, again, to the larger role that the ISL relationship of numbers—as shown in the BIM—encompasses the expression of “other” number systems like the PPTs, PRIMES, and, any regular, progressive enlargement of the number of objects within a class of objects whose initial size = 1, singularly or as a group acting as one.

Object AREAS: PPset TRAILS

From the above, it may seem all too easy to just disregard the PPsets as simple “objects” not unlike those of any other random selection. But there is more here, than just that.

We have already seen in the BIM: Parts I-III that the PPsets are intimately related to and described by the ISL as seen on the BIM.

In Parts II & III, we saw directly how each and every PPset – set as factors on the BIM – related diagonally back to the AXIS EVEN number.

We also saw how the PPset TRAILS – that were extensively examined in Part I – are now easily seen as the PPset values occupying any Active Row on the BIM.

Now, in Part IV, we open the BIM up to see both symmetrical sides. In doing so, we see that underlining those PPset TRAILS Rows gives us the progressively larger squares: the Object AREAS of PPset TRAIL numbers.

That’s all in well, but significantly, those PPset TRAIL “objects” are not just any objects. They are a very specific collection of objects – specific PPsets that are part of specific PPset TRAILS, or groups, that ONLY exist and occupy their specific location on the BIM. A location that both allows them to individually relate back diagonally to the EVEN # on the AXIS, and, to relate ACROSS the Row as the collective set, or group, of ALL PPsets whose members consist of one constant PRIME and one PRIME Sequence PRIME, e.i., 3,11–5,11–7,11–11,11. The latter become the PPset TRAILS.

Ultimately, it is the PPset TRAILS on the Rows that interact and inform the diagonally-related EVENS. Their rate of growth – (increasing by 2 for each PS Object AREA) – far exceeds the Prime Gaps growth. This is built into the geometry of the BIM. It ensures that Euler’s Strong Form of the Goldbach Conjecture is fully satisfied!

The 1–4–9–16–25-… ISL relationship of the Object AREAS: PPset TRAILS provides a definitive and intimate connection of the PRIMES to the ISL as seen on the BIM.

*Each square AREA = sum of the ODD#s leading to it, i.e. the ODD number summation series. Thus AREA = 9 = the sum of the ODDs 1 – 3 – 5. The next AREA = 16 = the sum of the ODDs 1 – 3 – 5 – 7. The next AREA = 25 = the sum of the ODDS 1 – 3 – 5 – 7 – 9, and so on. One can calculate the tally of any given square AREA by subtracting the two previous square AREAS to get their ODDs summation value, then add 2 to that and increase the larger square AREA by its sum, e.i., 25 - 16 = 9, add 9 + 2 = ll, add 11 + 25 = 36. The next square AREA is 36 and it contains 11 PPsets in its TRAIL.

As the TRAILS follow the ISL NPS — 1—4—9—–16—25—36—49—64—81—100—…—the actual number of unique PPsets within a trail = √(square), e.i., √36 = 6. There are 6 unique PPsets: Since the TRAIL falls on Row 17, those PPsets are: 3,17—5,17—7,17—11,17—13,17—17,17.

We know Object AREA 36 = 6*6, i.e., it is the 6th in the ISL NPS. Likewise, we can take any whole √ of a number and know where it is at in the ISL NPS. Another example, √100 = 10, so on Row 31, we have 19 PPsets, with 10 of them unique: 3,31—5,31—7,31—11,31—13,31—17,31—19,31—23,31—29,31—31,31. How do we know there are 19 PPsets? The ISL NPS is, of course, based on the squares of the numbers 1,2,3,…. The difference (∆) in the AREAS being the ODD # summation series: 1+3+5+7+…difference, or simply adding 2 to the previous ODD. Adding the √ of the current and previous AREAS will give the ∆, e.i. √100 + √ 81 = 10 + 9 = 17.

So once any Row in the ISL NPS is known, we can readily determine:
1. AXIS and squared AXIS — PD — numbers;
2. The ∆ between the current ISL NPS AREA and the one previous;
3. The total number of PPsets within the current AREA;
4. The net number of unique PPsets forming the PPset TRAIL;
5. The numbers, both total and unique, of ALL other AREAS before and after.

For example, Row 31:
1. AXIS = 31 and squared AXIS = 961 on AREA = 100;
2. ∆ 100 - 81 = 19;
3. PPsets total = 19;
4. Net PPsets = (19–1)/2 = 10, or simply the √100;
5. Previous AREA = 100–19 = 81, 81 –17 = 64, 64 –15 = 49, … and Next AREA = 100 + 21 = 121, 121 + 23 = 144, 144 + 25 = 169,… or simply follow the 12—22—32—42—52—62—72—82—92—102—112—122—132—…

The question of how do we know Row 31 is where the ISL NPS AREA = 100 lands? Or any other PPset TRAIL Object AREA?

The rigidly regular ISL NPS — 1—4—9—16—25—36—49—64—81—100—…, — itself built on 12—22—32—42—52—62—72—82—92—102 —…, the ∆ being the ODD # summation series of 1+3+5+7+9 +11+.. with a constant ∆ of 2 — must be reconciled with the PRIMES Sequence (PS) with its variable Prime Gap.

What Row does ISL NPS AREA, say, PS 41, occupy?

We could think, well, it must be AREA 121, up 21 from AREA 100, but on what Row?

Wrong we would be, as the NEXT PS after 31 is 37, followed by 41. So Row 37 is AREA 121 and Row 41 is actually AREA 144, i.e., 102—112—122, respectively, for Rows 31—37—41.

Currently, one must know one to get the other and that one must be the PS Row. From that, one can calculate the ISL NPS AREAS and the number and configuration of ALL the PPsets within the PPset TRAIL.

If we know Row 41 — as PS 41 — we can calculate back to a known Row-AREA like Row 31—AREA 100, count the number of PS jumps from 31 to 41 — jump one to 37, jump two to 41 — and know that we have moved two positions up the ISL NPS — 100 to 121 to 144 — giving 144 as the respective AREA for Row 41.

From (122 = 144) - (112 = 121) we see there is a ∆ of 23 between the AREA 144 and the previous AREA 121.

Since we know from the PS, the previous to 41 is 37, thus AREA 121 is on Row 37. The total number of PPsets equals the AREA difference from the previous AREA, thus for 144 it is 23 and for 121 it is 21. The net unique number of 12 PPsets within the TRAIL for Row 41—AREA 144 = √144 = 12. For the Row 37—AREA 121 it equals √121 = 11.

Let’s go back to Row 41—Area 144. What lies ahead on ISL NPS AREA 441. What Row is it on? Aside from simply looking at the BIM:

• The √441 = 21.
• There are 21 unique PPsets in the PPset TRAIL.
• There are 9 jump steps from AREA 144 to AREA 441 — 144—>169—>196—>225—>256—>289—>324—>361—>400—>441, corresponding to 122—>132—>142—>152—>162—>172—>182—>192—>202—>212.
• There must be 9 corresponding jump steps for the PS: 41—>43—>47—>53—>59—>61—>67—>71—>73—>79.
• AREA 441 = Row 79
• Take 441 - 400 = 41.
• There are 41 total PPsets in the AREA 441.
• Another way to get the total 41 is to add 20 + 21.
• All the Row—AREAS between Row 41—AREA 144 and Row 79—AREA 441 are easily determined in a like manner.

One more example: Starting with PS 113, what ISL NPS AREA does it occupy and what are the numbers?

• We could start from scratch with PS 3 and count the jump steps to 113 = 28.
• Now match that with 28 jump steps from ISL NPS 1 + 28 = 29.
• 292 = 841.
• There are 29 unique PPsets in the PPset TRAIL on Row 113—AREA 841.
• Take (292 = 841) - (282 = 784) = 57.
• There 57 total PPsets in the AREA 841.
• Another way to get the total 57 is to add 28 + 29.
• All the Row—AREAS between Row 3—AREA 1 and Row 113—AREA 841 are easily determined in a like manner.

There are several ways to calculate the PPsets and their associated ISL-NPS AREAS. A tip to keep in mind: as the Object AREAS increase as 12,22,32,...the ∆ between them being the ODD # Summation series — 3--5--7--9... — the ∆ of the ∆, if you will, is 2. ALL total PPsets within an AREA will be ODD. Another visual way to look at this: as each AREA progressively enlarges, one new PPset is added to the Horizontal and one new PPset is added to the Vertical Row/Col, the sum equals 2, so starting with 1, 1 + 2 = 3, 3 + 2 = 5, 5 + 2 = 7,... The unique PPsets may be either.

With optimum computing power, one could determine ALL PS Row—ISL NPS AREAs up to and including the largest PRIME.

One can use the NO-PRIMES (NP) = 6yx ± y (covered extensively in PRIMES vs NO-PRIMES, 2019) to determine ALL the non-PRIMES between any run of numbers.
Example: what are the NPs from 3–59?
Consulting NP=6yx+-y_with3^x.jpg, you could simply pick out the NPs.
Calculating the NPs:
NP = 6yx ± y. Let x = 1,2,3,… and y = ODD # ≥3, and add 3 exponential, 3x:
1. let x = 1, y = 3
2. NP = 6yx ± y = 6(3)(1)±3 = 18 ± 3 = 15, 21
3. let x =1, y = 5
4. NP = 6yx ± y = 6(5)(1)±5 = 30 ± 5 = 25, 35
5. let x =1, y = 7
6. NP = 6yx ± y = 6(7)(1)±7 = 42 ± 7 = 35, 49
7. let x =1, y = 9
8. NP = 6yx ± y = 6(9)(1)±9 = 54 ± 9 = 45, 63
9. let x =1, y = 11
10. NP = 6yx ± y = 6(11)(1)±1 = 66 ± 11 = 55, 77
11. let x =2 y = 3
12. NP = 6yx ± y = 6(3)(2)±3 = 36 ± 3 = 33, 39
13. let x =3 y = 3
14. NP = 6yx ± y = 6(3)(3)±3 = 54 ± 3 = 51, 57
15. add 3 exponential, 3x = 32 = 9, 33 = 27
16. NPs 3—59 = 9,15,21,25,27,33,35,39,45,49,51,55, and 57.
Knowing the NPs allows one to make the jumps from one known PS and its associated ISL NPS AREA to another.

The value of determining the net unique number of PPsets within a TRAIL is that each TRAIL member will sequentially contribute to the pool of PPsets whose members add up to form a given EVEN. This constant and regular growth of the PPsets outpaces the rate of growth of the Prime Gaps ensuring the Goldbach Conjecture is always fulfilled.

Another valuable outcome of determining ALL the unique PPsets within ALL the encompassing AREAS is that ALL PRIMES are accounted for. Should a new and/or larger PRIME be discovered, its PS Row—ISL NPS AREA can be calculated and the net unique PPsets determined. There should NOT be any holes!

~~~~

From PRIMES vs NO-PRIMES we know that we can test-validate that any ODD (≥5) is PRIME by seeing if the difference between its squared value and that of any other squared PRIME value is an even multiple of 24, e.i. 72 - 52 = 24, 312 - 112 = 840 and 840/24=35 and so on.

Note that it is the PS PRIMES from the AXIS on the BIM that is squared on the PD that follows this PRIMES validation, not the ISL NPS Object AREAS, per se, i.e., the 312 - 112 example above works as expected, but the overlaid ISL NPS Object AREAS of 100 and 16 values, respectively, do not work this way.

The latter only works when the values are from squared PS values like 121 and 49, i.e., 121 = 112 and 49 = 72, the ∆ being 121 - 49 = 72 and 72/24 = 3.

Closing remarks:

SYMMETRY, STEPS, EVENS, EVENS/2, PPsets, PRIME SEQUENCE FRACTALS, ISOSCELES & EQUILATERAL TRIANGLES, INVERSE SQUARE LAW, PYTHAGOREAN TRIPLES; ALL ON THE BIM.

In the first publication of the BIM (BBS-ISL Matrix — originally referred to as the Brooks (Base) Square—Inverse Square Law Matrix) in MathspeedST (2010), the work was divided in to two main sections:

• TAOST: The Architecture Of SpaceTime, for the simple reason that all considerations of the actual architecture of spacetime MUST revolve around the ISL as it underlies all fundamental expressions of energy — as matter and fields — dispersion.
• TCAOP: The Conspicuous Absence Of PRIMES. How can these fundamental, “atomic building blocks” of numbers actually be absent from a matrix grid that so fundamentally informs all of spacetime?

In MathspeedST, the PRIMES were found to be stealthily hidden within the BIM, revealing there presence ONLY when the BIM itself had its ODD #s 1st Parallel Diagonal modified to EVENS by the addition of 1 to each ODD.

This resulted in the original PTOP (Periodic Table Of PRIMES) on the BIM. It was refined by pulling the values off the BIM and making a separate PTOP. All this became the subject of “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” ebook and white paper (2019). It is also referred to as BIM: Part I. The PPsets and their TRAILS were introduced.

Not good enough just by itself, further research found that, indeed, each and every PRIME, when treated as part of a PPset, could be found and individually profiled DIRECTLY on the BIM. BIM: Part II was formed.

Still not good enough, plotting all the Lower Diagonal P2 PRIMES on a table and then re-plotting those results back onto the BIM opened up a new vista. By substituting the AXIS values for each of those P2 PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place.

One immediately sees that they occupy EXACTLY the same footprint on the BIM as do the individual PRIMES (and the PPTs): namely, falling only on the BIM/24 Active Rows.

In the case of the PPsets, one can also see that this is a direct consequence of locating the PPsets as the grid space resulting from the intersection of the PRIMES Sequence (PS) of the Horizontal AXIS with the PS of the Vertical AXIS. This becomes the basis for the BIM: Part III.

In BIM: Part I-III, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.

In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.

What we find is that by treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL NPS is found. Yes, as the PS-Fractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS:
1—4—9—16—25–… And while this relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.

While Euler’s Strong Form of the Goldbach Conjecture is proved along the way on this journey, the real significance is what we have seen unfolding in BIM: Parts I-IV. The role of Symmetry and Fractal underlie everything about the PRIMES.

The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!

What is marvelous, incredible, mind-blowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.

The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.

ISL—> BIM

BIM—> PPT
BIM—> PRIMES
BIM—> PPset Object AREAS
BIM—> other Object AREAS

As in showing that ALL EVENS (≥6) can be formed from the sum of two ODD PRIMES (Euler’s Strong Form of the Goldbach Conjecture), one would like to see — in the context of the PRIMES being the fundamental “atoms” — the PRIMES being the source of the ISL and everything thereafter!

PRIMES—> ISL—>BIM—>PPT & PRIMES patterns on the BIM, etc, etc.

We are not there yet.

PRIMES Index

PRIMES on the BIM Part IV

MathspeedST: TPISC Media Center

MathspeedST: eBook (free)

Apple Books

Artist Link in iTunes Apple Books Store: Reginald Brooks

NEWLY ADDED (after TPISC IV published):

Back to Part III of the BIM-Goldbach_Conjecture.

BACK: ---> PRIMES Index on a separate White Paper

BACK: ---> Periodic Table Of PRIMES (PTOP) and the Goldbach Conjecture on a separate White Paper (REFERENCES found here.)

BACK: ---> Periodic Table Of PRIMES (PTOP) - Goldbach Conjecture ebook on a separate White Paper

BACK: ---> Simple Path BIM to PRIMES on a separate White Paper

BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper

BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper

BACK: ---> PRIME GAPS on a separate White Paper

Reginald Brooks

Brooks Design

Portland, OR

brooksdesign-ps.net

REFERENCES

Back to Part III of the BIM-Goldbach_Conjecture.

BACK: ---> Simple Path BIM to PRIMES on a separate White Paper

BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper

BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper

BACK: ---> PRIME GAPS on a separate White Paper (This work.)

BACK: ---> PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper

BACK: ---> Make the PTOP with Fractals on a separate White Paper

TPISC, The Pythagorean-Inverse Square Connection, has evolved into a connection with the PRIMES (TPISC-P).

(See Appendix B: BIM ➗24 PPTs and PRIMES for ALL the Figures and Tables relating to BIM ➗24. Note: open the PDFs into a separate tab/page to see the full content. The png image files give a quick-look only.)

REFERENCES:

https://primes.utm.edu/notes/faq/six.html

“Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

"Perhaps the most rediscovered result about primes numbers is the following:

"I found that every prime number over 3 lies next to a number divisible by six. Using Matlab with the help of a friend, we wrote a program to test this theory and found that at least within the first 1,000,000 primes this holds true.

"Checking a million primes is certainly energetic, but it is not necessary (and just looking at examples can be misleading in mathematics). Here is how to prove your observation: take any integer n greater than 3, and divide it by 6. That is, write

n = 6q + r

where q is a non-negative integer and the remainder r is one of 0, 1, 2, 3, 4, or 5.

• If the remainder is 0, 2 or 4, then the number n is divisible by 2, and can not be prime.
• If the remainder is 3, then the number n is divisible by 3, and can not be prime.

"So if n is prime, then the remainder r is either

• 1 (and n = 6q + 1 is one more than a multiple of six), or
• 5 (and n = 6q + 5 = 6(q+1) - 1 is one less than a multiple of six).

"Remember that being one more or less than a multiple of six does not make a number prime. We have only shown that all primes other than 2 and 3 (which divide 6) have this form.”

From Another prime page by Chris K. Caldwell caldwell@utm.edu

~~~

"Euler's 6n+1 Theorem

"Every Prime of the form can be written in the form ."

https://archive.lib.msu.edu/crcmath/math/math/e/e282.htm

~~~

https://www.quora.com/Why-do-prime-numbers-always-satisfy-the-6n+1-and-6n-1-conditions-Is-there-mathematical-logic-behind-it

~~~

~~~

https://math.stackexchange.com/questions/1896636/eulers-theorem-to-validate-prime-numbers-shows-non-primes-as-valid

~~~

https://en.m.wikipedia.org/wiki/Formula_for_primes

~~

http://www2.mae.ufl.edu/~uhk/NUMBER- FRACTION.pdf

~~~

http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html

~~

http://eulerarchive.maa.org See Number Theory Section

http://eulerarchive.maa.org/docs/translations/E744en.pdf

NOTE: These article links from the Euler Archive/Number Theory/ INDEX are IMPORTANT READS for PRIMES & Squares. You can link directly from the Index as well.

http://eulerarchive.maa.org/pages/E026.html

http://eulerarchive.maa.org/pages/E054.html

http://eulerarchive.maa.org/pages/E134.html

http://eulerarchive.maa.org/pages/E175.html

http://eulerarchive.maa.org/pages/E191.html

http://eulerarchive.maa.org/pages/E241.html

http://eulerarchive.maa.org/pages/E242.html

http://eulerarchive.maa.org/pages/E243.html

http://eulerarchive.maa.org/pages/E244.html

http://eulerarchive.maa.org/pages/E256.html

http://eulerarchive.maa.org/pages/E262.html

http://eulerarchive.maa.org/pages/E270.html

http://eulerarchive.maa.org/pages/E279.html

http://eulerarchive.maa.org/pages/E283.html

http://eulerarchive.maa.org/pages/E369.html

http://eulerarchive.maa.org/pages/E394.html

http://eulerarchive.maa.org/pages/E405.html

http://eulerarchive.maa.org/pages/E445.html

http://eulerarchive.maa.org/pages/E449.html

http://eulerarchive.maa.org/pages/E467.html

http://eulerarchive.maa.org/pages/E523.html

http://eulerarchive.maa.org/pages/E542.html

http://eulerarchive.maa.org/pages/E552.html

http://eulerarchive.maa.org/pages/E554.html

http://eulerarchive.maa.org/pages/E564.html

http://eulerarchive.maa.org/pages/E596.html

http://eulerarchive.maa.org/pages/E610.html

http://eulerarchive.maa.org/pages/E699.html

http://eulerarchive.maa.org/pages/E708.html

http://eulerarchive.maa.org/pages/E715.html

http://eulerarchive.maa.org/pages/E718.html

http://eulerarchive.maa.org/pages/E719.html

http://eulerarchive.maa.org original 744 Euler article

https://www.britannica.com/science/Fermats-theorem

http://mathworld.wolfram.com/FermatsLittleTheorem.html

http://mathworld.wolfram.com/EulersTotientTheorem.html

http://mathworld.wolfram.com/TotientFunction.html

https://www.geeksforgeeks.org/fermats-little-theorem/

https://www.geeksforgeeks.org/eulers-totient-function/

http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html

https://brilliant.org/wiki/fermats-little-theorem/

https://brilliant.org/wiki/eulers-theorem/

https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html

https://primes.utm.edu/notes/conjectures/ Conjectures

https://primes.utm.edu/notes/faq/ FAQ Index

https://primes.utm.edu/notes/faq/six.html Six —>Important Page

http://unsolvedproblems.org/index.htm Unsolved Problems Index

https://en.m.wikipedia.org/wiki/Proofs_of_Fermat%27s_little_theorem

https://en.m.wikipedia.org/wiki/Fermat%27s_little_theorem

https://en.m.wikipedia.org/wiki/Euler%27s_theorem

~~NEW REFERENCE LIST 11-08-2018~Fermat’s Little Theorem ++++Euler

https://web.math.princeton.edu/swim/SWIM%202010/Shi-Xie%20Presentation%20SWIM%202010.pdf

http://mathworld.wolfram.com/Fermats4nPlus1Theorem.html

http://nonagon.org/ExLibris/euler-proves-fermats-theorem-sum-two-squares

https://storyofmathematics.com/17th_fermat.html

https://en.wikipedia.org/wiki/Prime_number

https://www.britannica.com/science/prime-number-theorem

https://going-postal.com/2018/02/fermats-little-theorem/

https://ibmathsresources.com/2014/03/15/fermats-theorem-on-the-sum-of-two-squares/

https://oeis.org/A002144 Pythagorean Primes of the form 4n + 1

https://oeis.org/A002144/list List of Pythagorean Primes of the form 4n + 1

https://www.britannica.com/science/Mersenne-prime Mersenne Primes

https://primes.utm.edu/mersenne/

http://mathworld.wolfram.com/MersenneNumber.html

http://mathworld.wolfram.com/MersennePrime.html

https://www.encyclopediaofmath.org/index.php/Mersenne_number

https://www.mersenne.org/primes/

https://primes.utm.edu/lists/small/millions/ Primes LIST by section

http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php Primes LIST by section

http://primerecords.dk/primegaps/maximal.htm

https://en.wikipedia.org/wiki/Prime_gap

http://www.trnicely.net

http://www.primegaps.com

https://primes.utm.edu/notes/GapsTable.html

https://en.wikipedia.org/wiki/Prime_k-tuple

https://primes.utm.edu/glossary/page.php?sort=ktuple

https://en.wikipedia.org/wiki/Twin_prime

http://mathworld.wolfram.com/TwinPrimes.html

https://primes.utm.edu/notes/faq/

~~

https://oeis.org/wiki/Pythagorean_primes Pythagorean vs non-Pythagorean Primes

http://oeis.org/A002145 non-Pythagorean Primes of the form 4n + 3

http://oeis.org/A002145/list

http://oeis.org/A002144 Pythagorean Primes of the form 4n + 1

http://oeis.org/A002144/list

https://en.wikipedia.org/wiki/Double_Mersenne_number Double Mersenne Primes

https://primes.utm.edu/mersenne/index.html#unknown Questions remain

~~~~~~~~~~NEW REFERENCE LIST 12-13-2018~~~videos——must watch~~~~

See: Animated math: videos by 3Blue1Brown (Grant Sanderson, et al.) https://www.3blue1brown.com/videos/

Feynman's Lost Lecture (ft. 3Blue1Brown) https://www.youtube.com/watch?v=xdIjYBtnvZU

But WHY is a sphere's surface area four times its shadow?: video by 3Blue1Brown https://youtu.be/GNcFjFmqEc8

Why is pi here? And why is it squared? A geometric answer to the Basel problem: video by 3Blue1Brown https://www.youtube.com/watch?v=d-o3eB9sfls&frags=pl%2Cwn

Pi hiding in prime regularities: video by 3Blue1Brown https://youtu.be/NaL_Cb42WyY

Visualizing the Riemann hypothesis and analytic continuation: video by 3Blue1Brown https://youtu.be/sD0NjbwqlYw

All possible pythagorean triples, visualized: video by 3Blue1Brown https://www.youtube.com/watch?v=QJYmyhnaaek

((((Table VI series is really the whole evolution of the BIMrow1-1000+sheets/Primes_sheets+PF, etc here and appendix.)))

referenced as Table VI a

referenced as Table VI b

referenced as Table VI b

Table VII.

From TPISC III: Clarity Conclusion:

• Thomas Young established the wave nature of light in the Young Double-Slit Experiment.

• Max Planck established the quantum nature of light energy mathematically (E=hⱱ).

• Albert Einstein established that ALL energy is quantized (E=hⱱ), energy and matter are inter-convertible ((E=mc2), and that the speed of light is always constant (c=ƴⱱ) — and later, the curvature of ST as the driving dynamic informing gravitation (General Relativity).

• Niels Bohr established the quantization of the atomic orbitals within the atom (Bohr Model of the Atom). Along with Heisenberg, the leading voices in the Copenhagen Interpretation of Quantum Mechanics.

• Louis de Broglie established that all objects express the wave-particle duality as defined by its momentum being its energy per wavelength (p=h/ƴ).

• Werner Heisenberg established the Uncertainty Principle that reveals the limitation of precisely and simultaneously defining both an objects position and its momentum (∆x∆p≥h/2𝛑).

• Arnold Schrödinger established the time-dependent Schrödinger (Quantum) Wave Equation.

• Max Born established the statistical basis of the same Quantum Wave function.

• Wolfgang Pauli established the Pauli Exclusion Principle that disallows any quantum particle to occupy the same quantum state (parameters) within the same ST.

• Paul Dirac established the Dirac Equation, a relativistic generalization of the Schrödinger equation

wave equation for fermionic spin ½ particles, bringing together the two pillars of special relativity (3 dimensions of space+1 dimension of time, and the constancy of the speed of light, c=ƴⱱ) and quantum mechanics (h/2𝛑=E/v) leading to Quantum Field Theory and the Standard Model.

• Quantum Entanglement , initially raised in opposition in 1935 in the Einstein, Podolsky, Rosen paper (EPD Paradox) and papers by Schrödinger, was brought to the forefront of debate in 1964 when John Stewart Bell's Inequality Theorem ruled out the "local hidden variables" of the opposition. Since then the quantum entanglement — i.e., particles bound in entanglement cannot be factored out into separate states that would explain their entanglement, the particles act together as a whole, not the sum or product of their parts — has been successfully tested and appears to be profoundly consistent with the Quantum Field theory. It is in direct opposition to Einstein's nothing — no information — can travel faster than the speed of light. And yet, here we have proofs stacking up that show, for example, two entangled particles with equal but opposite spin will simultaneously settle into one of the other spin directions opposite its entangled partner instantly once its partner's spin is determined — faster than the speed of light. Einstein's "local hidden variables" may not be completely ruled out if, indeed, one reveals a different sort of underlying geometric order that sidesteps being ruled out by the Inequality Theorem — a geometric order that precedes the very formation of ST itself (BIM-BIMtree).

~~ ~~ ~~

BACK: ---> Part I of II CaCoST-DSEQEC on a separate White Paper ~~ ~~ ~~

~~ ~~ ~~

BACK: ---> Part II of II CaCoST-DSEQEC on a separate White Paper ~~ ~~ ~~

Double-Slit Experiment:

In essence, even a single photon or electron passing through a double-slit will display interference, collapsing upon any interacting measurement upon the initiating wave particle.

Fig.73. Double-Slit Experiment

Quantum Entanglement:

In essence, any two (and possibly more) wave-particle ST units born of and thus assigned the same, equal but opposite quantum number state, i.e. spin angular momentum, will maintain and function as a single wave-particle system having a single quantum number state regardless of distance or time separation, in the absence of external negating influences. Transference is possible.

If c, the speed of light, is fundamentally buit into the formation of ST itself — moving ƴ distance with each pulse-propagating ⱱ frequency — as described in LightspeedST — and as combined with MathspeedST, i.e. through the BIM informing that same ST pulse-propagating unit — all potential ST information, as to density and disposition of space per time, is inherently known!

The BIM itself is defined as a Universal Relationship. Its numerical value of the quantity of parts (the value in each grid cell), in relation to its whole (the square values of the PD — that, by the way, directly inform the parts) is instantly and ubiquitously known for every value throughout BIM as extended to infinity.

There is no propagation of information faster than c because all its information is simultaneously known. We thus have a basis for Quantum Entanglement: two or more particles born of the same quantum number state wholy remain connected regardless of distance of separation precisely because they are informed by the SAME BIM-derived ST pulse. No information is exchanged — no violation of faster-than-light signals because both (all) particles have the SAME BIM information map informing their quantum state parameters at any and all ST unit pulses ().

The Quantum Entanglement example most referenced is that of spin, a vector momentum parameter.

Two entangled particles (e.i., two photons or an electron-positron pair) are simultaneously born, each existing with both possible equal but opposite spin states.

Upon any distance (or time) separation, determining the spin quantum state of one particle — collapsing its wave potential of both into that of one polemic — immediately determines the other as equal but opposite.

Equal and opposite what?

The spin quantum state is a vector whose magnitude and expression is dependent on orientation (direction).

So, for whatever context that vector exists within, a polemic is established, i.e., +/-, ↑/↓, yes/no, that inherently establishes a positive and negative state (Dirac Equation) — e.i., matter/anti-matter, +charge/-charge, spin+/spin-.

The Quantum Entanglement State (QES) is just such a context — a container, if you will — holding the full potential of either/or vector states within its potential wave form.

Measurement (Heisenberg Uncertainty Principle) —that is interacting with such potential state — necessarily collapses the potential wave-state into a definitive, particulate-like state — call it +/-, ↑/↓, yes/no, or Even/Odd.

In effect, one has induced the vector state. In doing so, the other entangled member has its vector state induced to that equal, but opposite polemic value to its partner.

Now, let the QES=BIM. Let collapsing the wave-potential (the entire BIM) = √BIM. Since BIM is built on squaring the 1,2,3… Axis numbers to give the PD — and all the Inner Grid cell values as its children — taking the √BIM is akin to collapsing BIM back to the equal and symmetrical opposite Axial numbers along the horizontal and vertical Axises.

Alternatively, the BIM may indeed project into the imaginary number space, i, giving the equal but opposite value a more hidden meaning.

Yet another alternative — and within the context of the current mathematical space the BIM occupies — one may induce the ubiquitous and infinity expanding Pythagorean Triples to provide the necessary polemical expression of a vector from the multi-potential of its wave-form. In this case — and exactly in tune with the BIM layout — there is a 90°-right angle relationship between the sides. In the BIM it is within a square, within the PTs it is within a rectangle (formed from the two orientations of the triangle). Both have a symmetry involved. The BIM has an absolute, bilateral mirror-reflection symmetry. The latter an algebraic symmetry in that various whole integer combinations of a2 + b2 =c2.

In fact, there are, as we mentioned in the Intro, 4 geometric manifestations of each algebraically described PT. Two for each side of the bilaterally symmetric BIM, and two orientations (180° flips along the row/column). Perhaps the polemic we are looking for is simply the wave-collapse selection of one of the 4 possible orientations of the PT, with its opposite simply its mirror reflection on the other side of the BIM.

Fig 74. The Quantum Entanglement Conjecture. Every PT has 4 potential presentations: each has a mirror symmetry, and because BIM has itself reflective, mirror bilateral symmetry, there are two more.

The Quantum Entanglement Conjecture:

Every PT has 4 potential presentations: each has a mirror symmetry, and because BIM is internally itself reflective, mirror bilateral symmetry, there are two more.

For simplicity, the TPISC painting image (Fig.74.) has been overlaid with three sets of PTs — each having 4 potential presentations. Exactly the same is true for the strictly Primitive PTs within the ToPPT.

As ST formation is informed most certainly by the ISL (BIM), the PTs, and most especially the Primitive PTs within the ToPPT, are viable candidates for laying down a bit of slightly asymmetrical bilateral symmetry framework embedded within the perfect rotational and bilateral symmetry the perfect squares and circles generated by the BIM.

While the BIM itself is composed of alternating ODD and EVEN whole integer numbers, the PTs, especially the PPTs, pick these ODD and EVEN numbers out as part of their natural structure. Every PPT has one short side made of an ODD number and the other is always the opposite — an EVEN number.

If one pictures the 4 potential presentations of the PT as existing as the wave-potential state of a forming particle, the naturally built-in polemics of these ubiquitous triangles comes into play: selecting to "measure" any one parameter as say, the spin ½ momentum vector, could collapse the wave function such that the other, opposite side of the polemic — which is always known as part of the Entangled Quantum State — could only express that opposite value.

It could be something as simple as solving the Pythagorean Theorem. If you measure (solve) for one short side — an let's say it is an EVEN whole integer number — you instantaneously know the other side is ODD. You also can know its value if you know the hypotenuse.

If you measure (solve) for one short side — an let's say it is an ODD whole integer number — you instantaneously know the other side is EVEN. You also can know its value if you know the hypotenuse.

The geometry of ST formation generates descriptive parameters referred to as quantum numbers or their quantum number state — unique identifiers for each and every subatomic particle. These wave-particle ST units exhibit unusual and non-intuitive behaviors that reveal a connection to each other below the physical reality we observe. The Conjecture is that that underlying connection is the BIM and BIMtree geometry that informs the creation and disposition of ALL ST — including their quantum number state identifiers.

Back to the Double-Slit Experiment Conjecture:

In the simplest terms, the ST formation of the wave-particle pulse-propagates into and out of existence according to its energy.

Each pulse expands form its own singularity outward via the BIM.

The ST-BIM expands outward in all directions as a sphere, despite its forward motion as a vector. Thus it travels through both slits and interferes with itself on the other side.

It is the BIM geometry information that is traveling through both slits, i.e. it doesn’t actually travel, rather it is known.

If you block one of the slits or otherwise collapse the geometry wave by measuring or interacting with it, no interference occurs.

How does it work?

There are several possibilities.

It may be as simple as positive interference patterns may appear when the same Square/Circle WIN values of the BIM resonate and negative when they do not — giving the classical light-dark-light-dark pattern.

Or, it may be the same resonance interference positive/negative superpositions for the PTs. As ST expands, a positive interference pattern would result when two (or more) 3-4-5 PPTs harmoniously arrive at the same ST location on the screen. The same for the 5-12-13, 8-15-17, etc. In either case, if you block one slit or collapse the expanding BIM geometry before entering the slit, interference will also collapse.

A third, and perhaps more relevant possibility includes the interaction with the slit material itself. Remember, ALL ST pulse-propagates into and out of existence. All matter and all energy, including the material that the slits are made of, pulse. When only a single slit is open, the BIM geometry of the incoming wave-particle interacting with the slit ST does NOT generate an interference pattern.

However, when two slits are open, the BIM-ST geometry of the slit material favorably interacts with the BIM-ST geometry of the wave-particle to generate interference!

One cannot ignore the contribution of the slit BIM-ST geometry in forming the net pattern. Having more than a single wave-particle ST unit passing through the slit(s) is simply magnifying the fundamental interaction between the slit material and wave-particle BIM-ST geometries!

As Heisenberg so profoundly pointed out, all contributions to any observation-measurement scenario must be accounted for.

Together, the Double-Slit Experiment and Quantum Entanglement Conjecture (DSEQEC) are but two sides of the same coin. Details to follow in TPISC IV and V.

The absolut key notion in the DSEQEC is that the geometry of the BIM and BIMtree both simultaneously and automatically precede and inform the subsequent ST unit expression, and, that this geometry is ubiquitous — expanding to infinity!

Since the initial DSEQEC Conjecture, presented as quoted above in TPISC III: Clarity (2017), a barrage of new articles confirming Quantum Entanglement have appeared. Perhaps the most intriguing, and relevant, has to be the series of new articles relating back to earlier work by Leonard Susskind, Juan Maldacena and Mark Van Raamsdonk — some of which are referenced below. In a brief, open letter to their ideas, I wrote:

Open letter addressing the "ER=EPR" works of Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …

Open letter addressing the "ER=EPR" works of Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …

The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.

What is needed is the other, complementary side that seals the deal: any entanglement of spacetime (ST) and ST units/particles, i.e. photons, gravitons, Higgs Bosons,… must reckon with:

1. ST itself is formed in such a way that it obeys the Inverse Square Law (ISL), is always consistent with the Conservation of Energy, and its very formation must be one and the same with the formation of its progeny ST units and the built-in velocity of light, c;
2. ST itself, and thus its ST progeny, must be conserved;
3. ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the Double-Slit Experiment.

One can satisfy all three constraints by allowing ST to be pulse-propagated from its singularity (S) into and back out of our ordinary view, forming the photon with its built-in velocity, c, with each pulse. A graviton is nothing more than the positive interference of two similar-spin photons, while the Higgs Boson, in the pure state, is two photons with opposite-spins interfering. The Higgs, of course, gains mass as it decouples into ST units with 1/2 spin, be it quarks or electrons-neutrinos, or their composites. Note that one could start with the Higgs and decouple, first to the photon and thereafter to its progeny. (LightspeedST)

Key to the ordered formation of ST is the ISL. A fully, self-consistent matrix grid array numerically depicting the ISL in all its glory is easily formed by filling in all the Inner Grid cell values as the difference between squares of the x- and y-axis whole integer numbers (MathspeedST). It’s called the BIM, or BBS-ISL Matrix.

Here’s the thing. Amongst a myriad of other connections, there exist an intimate connection between three number systems:

1. The ISL as laid out in the BIM;
2. PTs — and most especially PPTs — as laid out on the BIM;
3. The PRIME numbers — PRIMES — as laid out on the BIM.

The BIM is the FIXED GRID numerical array of the ISL.

The PTs are the Pythagorean Triples, and more specifically, the PPTs are the Primitive Pythagorean Triples (parent, non-reproducible). All PTs — primitive and non-primitive — are found on the BIM ([TPISC I: Basics, TPISC II: Advanced, TPISC III: Clarity [Tree of Primitive Pythagorean Triples, ToPPT], and TPISC IV: Details](http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/MSST-TPISC_resources/MSST-TPISC_resources.html)).

Amongst its vast array of inter-connecting Number Pattern Sequences (NPS) — i.e., number systems — two such systems stick out and do so in such an overtly visual — as well as mathematical — way that their connection to each other is more than implied.

You see, both the PPTs and PRIMES strictly align themselves on the SAME paths within the BIM.

But we are not here to discuss the PRIMES, just the BIM and the PPTs. And why is that? How do they shed light on the Double-Slit Experiment, Quantum Entanglement and the whole Geometry of Spacetime scenario that, I believe, underlies ER = EPR?

Briefly — very briefly — the PTs introduce rectangles, ovals and non-isosceles right triangles as articulation nodes within the architectural framework of the pure squares, circles and isosceles triangles that the pure ISL builds, as clearly revealed — through a highly visible algebraic geometry — on the BIM.

Visually, right on the BIM grid, one can see — literally draw — the PTs on each active Row (or Column since the BIM is bilaterally symmetric).

Now, two points of emphasis before carrying on: both the BIM and the embedded PTs are fixed, non-variable mathematical entities, and, they both extend to infinity. Every ST unit pulse is informed by this grid information at the start (from its S), and is known throughout its pulse-propagation history. It is “information” that exists with no need to travel at or faster than the speed of light. If one knows “any” value and location on the grid one inherently knows “all” the information that defines that history. Not only that, but the BIM and PTs are so intimately linked throughout the grid that one can be used to define the other.

Because the BIM is bilaterally symmetrical, and, because every PT can be inherently represented on the BIM in both “left” and “right” hand depictions — e.i. on an active Row, a 3-4-5 PPT can be depicted equally with the hypotenuse along the axis Row (lower left BIM) and its short side to the “left” and to the “right” and if you turn to the Columns side of the BIM (upper right), again the “left” and “right” versions of the same 3-4-5 PPT are depicted. What is even more amazing, the Area of the PPT x 4 is always present on that same Row (or Column) and is integral to defining the relationships to the next PPT within the ToPPTs. There is so much inter-connectedness between the BIM, the PTs and the next PPTs built right in to the grid that one feels a blue-print for the fractal-based holographic ST universe is right before one’s eyes!

Back to the point: let the 4 iterations of the PT — any PT — structurally represent the quantum state parameter of spin (spin angular momentum). When one measures the vector spin value and orientation, one is measuring one of the 4 PT iterations. Knowing one value, one always instantaneously knows the value of the “other” entangled ST unit coherently existing as part of a complementary pair sharing the same total quantum state. If your axis of measurement defines “left” (or -) on one ST unit moving in one direction, you will automatically register the equal but opposite “right” (or +) measurement on the other entangled ST unit moving in the opposite direction. No information is transmitted or signaled. It is inherently known and the very process of which axis is used to make the measurement pre-selects the two opposite spin orientations. Sorry, but the hidden information is neither hidden nor transferred at faster than lightspeed — it is simply known! This is the Quantum Entanglement Conjecture.

The Double-Slit Experiment Conjecture suggests that the very same known information is what is ultimately revealed on the screen as an interference pattern of even a single photon. It appears that the photon splits itself into two (or more) parts, each going through one of the double-slits and thereafter re-combining as positive and negative interference patterns, and while I do believe this is very possibly part of the story here, what is really being exposed — most emphatically not transferred — is the known information of the BIM/ISL as the collapse of the all possible wave probabilities of a given photon traveling through one or the other of the two slits. Information can demonstrate positive and negative interference!

Together, the Double-Slit Experiment Conjecture and the Quantum Entanglement Conjecture are the result of the one and same information expression: DSEC = QEC, or DSEQEC for short (Double-Slit Experiment Quantum Entanglement Conjecture).

The ER = EPR conjecture brilliantly ties the fractal, holographic Universe(s) into a self-consistent whole by turning the geometry of spacetime in on itself. It implies that connections between disparate “space” and disparate “time” realizations may be fruitfully manifested by the re-joining the one with the other by tunneling (ER wormhole). The DSEQEC provides for the very “mechanics” or “architecture” of that possibility. The statistical, probability of the all-possible paths quantum wave is simply this: every “possible” path has embedded within it the known information of the BIM — including the PPTs and PRIMES — and the collapse of that wave-form of possibilities into a manifest reality selects for that known information set of that particular quantum state identity at the time and place of collapse. Perhaps the most-positive interference of the known information leans that selection to the most statistically likely “choice.”

The information briefly presented here is freely available in its full form on my website and is included in several of my free iBooks. A few of the more advanced iBooks have a nominal charge, but I have free redeemable coupons available upon request.

NOTE: The decoupling of strict symmetry of the Higgs Boson generates mass expression in forming ST units with mass, i.e. fermions, baryons and mass+ bosons.

NOTE: Every PT is on the BIM, along with its proof. In addition, the Axis Row containing the PT always contains the squared value of its sides and hypotenuse, as well as its Area value x 4.

NOTE: Every PT is represented on the BIM 4x, a "left" and "right" version on both the Axis Row and its symmetrical Axis Column.

The ramifications of Susskind, Maldacena and Van Raamsdonk's works are most profound, offering a fresh, imaginative and far-reaching overview of the Geometry of Spacetime. The DSEQEC fully resonates with the ER = EPR conjecture — providing the complementary geometry from the "ground floor" — if you will — on up. The very formation of ST must crucially contain the constraining parameters of the large scale Universe(s) from the get go. These were alluded to in TPISC III: Clarity and re-affirmed here, now, in TPISC IV: Details.

DSEQEC-3 from Reginald Brooks on Vimeo.

Note: The hypothetical photon is a model of my own making (see LightspeedST, LUFE) and has no experimental confirmation. Despite its highly visualized nature within the proposed quantum, pulse-propagating context in which it is described, it still remains just a model. The proposed connection between the two Left- and two Right-handed PPTs — as shown above — as the fundamental, underlying ST source of the phenomena of "spin", as depicted on the BIM, remains! It is this information as to the vector parameter of spin — magnitude and direction — that exists for each ST unit particle. Information that provides equal, yet opposite, spin vector direction simultaneously. It is known, does not need to be transported, and becomes the in situ basis of entanglement.

DSEQEC: Double-Slit Experiment Quantum Entanglement Conjecture

On the BIM (BBS-ISL Matrix), ALL Pythagorean Triples — parent, Primitive Pythagorean Triples (PPTs) and child, non-Primitive PTs (nPPTs) — are found.

Because the BIM extends all Inverse Square Law (ISL) information to infinity, and, knowing any one matrix grid cell value and location, one (Nature) ubiquitously knows ALL the BIM values at ALL grid cell locations, and because the PTs, as part of the BIM, are also ubiquitously known, NO FASTER THAN LIGHT signal is required to extend the information to its ENTANGLED cohort particle regardless of the degree of space or time separation, i.e. SpaceTime (ST) separation.

Entangled ST unit particles, like the photon, coherently share the same quantum state, only with equal but opposite values. The Conservation Laws (energy, charge, spin angular momentum) require the combined coherent state of the entangled ST unit particles to be the net value of its individual units when combined. Spin is often the quantum state value measured to label two equal but opposite values contributed. Spin may be designated as up-down, left-right, + or -, …

It is proposed in the DSEQEC that the PTs represent the quantum state values of spin, with the direction of travel of the spin vector that is measured being one of the sides of the PT.

Since the BIM itself is bilaterally symmetrical, and, because every PT on the BIM is represented on a Row (or Column), and can be depicted in either “left-” or “right-“handed iterations, there are 4 total iterations of any given PT.

Ones particular axial frame of reference when measuring the spin vector as up-down, left-right, + or -,…. ALWAYS results in the measurement reading of the opposite, but coherently entangled particle spin vector to be INSTANTLY equal but opposite precisely because the spin information is ALWAYS known. The movement in opposite direction ensures that, given the same axial frame of reference, the opposite will always be chosen.

One may argue that the measurement is collapsing the wave-probability function in which both spin vector directions simultaneously exist and it is the act of measuring that collapses the wave function to select one or the other spin vector direction. And in one sense, that is true. But NOT in the sense that some hidden information traveling faster than light is then communicating, via a signal, to the other particle to register the opposite spin vector direction.

NO. There are is NO hidden variable information and there is NO faster than light signal pushing information to the other entangled particle. What is happening is that the choice of measurement to utilize a specific and consistent axial frame of reference to register the spin vector angular momentum direction crucially pre-selects the COMPLETELY, SIMULTANEOUSLY KNOWN INFORMATION of the other entangled ST unit particle, giving the equal, but opposite spin vector direction when measured with the same axial frame of reference.

The DSEQEC is consistent with and provides a complementary theoretical basis for the ER = EPR work of Leonard Susskind, Juan Maldacena and Mark Van Raamsdonk. The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.

But what of the Double-Slit Experiment? The same KNOWN INFORMATION of the PTs on the BIM means that even a single ST unit particle — be it a photon or an electron — can interfere with itself in going through the two slits because the INFORMATION of BOTH possible paths is KNOWN from the start. Now, it is the process of detecting (measuring) which slit the ST particle went through that destroys the heretofore entangled INFORMATION, resulting in the loss of the interference pattern.

A tip of the hat to Jacob Kafka’s “Rough Animator” app.

Original soundtrack.

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/netarti5.htm

Thanks for viewing!

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The quantum universe begs the question: what is the quantum?

Perhaps we can say the basic pulse-propagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Law-based ST, and they have distinct vector parameters inherently built into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtracted interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractal-like structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulse-propagation, and we we now have a fractal-based, holographic quantum universe(s)!

Closing thoughts

The Conservation Laws, and specifically the Conservation of Energy — the Conservation of Lightspeed — the Conservation of SpaceTime — has been at the center of the discovery of the BIM. The ISL is at the heart of ST formation and its expression as matter and fields. At the “end of the day” it must be conserved!

And while discovering how intimately and profoundly the Pythagorean Triples — especially in the form of the ToPPTs — are so incredibly Number Pattern Sequence inter-related to each other and their placement on the BIM — as also referred to as the BIMtree — is a giant step forward, it is its structural implications that give it meaning. The PTs give structural nodes and forms to the generalized BIM structure in the unfolding of ST.

As ST forms, i.e. expresses itself as, matter and waves, it must address the underlying imperative that in total, its Conservation Laws must be upheld.

If the ST parameters that give identity to wave-particle forms of matter and/or to wave-particle forms of an energy field — parameters such as energy, momentum, charge, baryon number, and lepton number — are a manifestation of their underlying quantum number state, then these too must be intimately involved in the Conservation Laws.

When a wave-particle ST unit seems to either separate into two parts that later recombine as they interfere with each other (Double-Slit Experiment), and/or, when they are part of a system of two (or more) ST units entangled (Quantum Entanglement) via sharing the same quantum number state, they BOTH MUST DO SO IN A WAY THAT SATISFIES THE CONSERVATION LAWS!

The total angular spin momentum of any system must be such that it does NOT violate the Conservation Laws. Naturally, if the entangled ST units have spin measured UP on one axis of one of the entangled pairs, it MUST measure DOWN on the same axis of measurement on the other entangled unit. This is the Conservation Law in action.

If the ST that informs the wave-particle and its parameters ALWAYS obeys the Conservation Laws, then the geometry that informs that ST formation must in and of itself ALWAYS obey the same laws.

The BIM and BIMtree, being the geometry of the BIM and BIMtree expression,both simultaneously and automatically precede and inform any and all subsequent ST unit expression, and, that this geometry is ubiquitous — expanding to infinity!

​ ~~ ~~ ~~

From the introduction to The Pythagorean—Inverse Square Connection (TPISC I: Basics) to its proof and distribution (TPISC II: Advanced) on the BBS-ISL Matrix (BIM), we have now come full circle to TPISC III: Clarity & Simplification: Tree of Primitive Pythagorean Triples (ToPPT).

Here we “see” the pattern: the arrangement of PPTs into triplet branches (Cluster of Tertiary Branches/Tiers) as organized into the ToPPT — with each and every PPT related within a Cluster, Cluster to Cluster, Cluster to Branch and Branch to Branch, back to the initiating 3-4-5 PPT Trunk.

The devilish details proving the linkage nevertheless — in the end — serve the clarity and simplification!

That a SpaceTime (ST)-forming Matrix patterned on simple Squares (& Circles) — a.k.a. the ISL — should pair-up their Squares to become and reveal in their sum, a larger Square — i.e. a2+b2=c2 — forming fractal iterations of Pythagorean Triples (PTs), themselves patterned throughout the BIM, is a sublime beauty to be realized with a small effort.

The fractal within a fractal!

Such profound harmony of form naturally begs the question: to what effect?

Several conjectures come to mind:

1. Would not the fractal-nodes of the PTs lead to structural articulation points along the expanding ST that would allow the geometry of the Squares to be enhanced by the embedded geometry of the non-Square Rectangles holding these PTs?
2. Would not these same fractal-nodes of the PTs be internal, integral and identifying structural parameters (Quantum State Numbers) of any wave-particle ST unit formed?
3. Would not the immensely inter-connecting linkage of the PTs, between the PTs and between the PTs and the BIM itself, provide the very basis of the wave-particle duality and entanglement phenomenon we see in the Double-Slit Experiment and the Quantum Entanglement (DSEQEConjecture) that lies at the core of our understanding the Quantum Universe?

Form follows function — function follows Form!

~~ ~~ ~~

BACK: ---> Part I of II CaCoST-DSEQEC on a separate White Paper ~~ ~~ ~~

~~ ~~ ~~

BACK: ---> Part II of II CaCoST-DSEQEC on a separate White Paper ~~ ~~ ~~

CaCoST — Creation and Conservation of SpaceTime

DSEQEC (Part I) SUMMARY or *"Where did all the antimatter go?"

What? How are these two even related?

(Note: Based on earlier work as fully referenced at the bottom of this Section under "My References.")

CaCoST-DSEQEC from Reginald Brooks on Vimeo.Q. Where is all the antimatter?

A. Right next to all the matter.

Q. Why can't we see it?

A. Ordinary matter exists (primarily) in Our View (OV). We can see and measure it directly. Antimatter exists (primarily) in the Alternate View (AV). We can not see it, only infer it by indirect measurements like the DSE and QE.

Q. What is the difference (∆) between the two?

A. Just as we've been taught: antimatter is matter with Equal And Opposite (EAO) charge and spin (spin angular momentum) direction.

Q. How does one characterize the two in their respective OV and AV SpaceTime (ST)?

A. Same form with EAO charge and EAO spin direction. Each quantum pulse-propagation (p-p) from the Singularity () — that dividing OV/AV — out into its view at and forming light speed (c)-based ST extension, and back to the .

All ST is formed from p-p of the virtual-pairs of ST units: one matter in OV and one antimatter in AV, most likely in an alternating sequence.

(Note: when you have an actual antimatter ST unit in OV, its virtual-pair partner is the EAO matter ST unit in the AV.)

Q. Does this virtual-pair p-p apply to all ST unit particles, and if so, can you explain?

A. Yes. Picture a line whose forward direction represents time. This is the . (The Planck Event Horizon [PEH] is the border interface of that line.) It actually contains ALL the energy, mass, momentum, charge and angular momentum of the Universe. The line is contiguous. It is a constant: neither created or destroyed.

The energy is the total field. The ST field.

All the other are parameters that define the expression of that ST energy field as it manifests itslef in quantum p-p as virtual-pairs, alternating EAO values first in OV, the AV, OV, AV,...

They do so in a manner in which whole discrete units of specific parameter expression — e.i., charge and spin — do so in discrete, quantum units of whole integer amounts (directly) or indirectly as simple ½, ⅓, ⅔ fractional amounts that add up to a net whole integer amount.

For example, electric charge is 1 WIN (Whole Integer Number), 2 WIN,... or, ⅓ + ⅔ = 1, or ⅔ - ⅓ - ⅓ = 0 WIN.

For example, spin, by mathematical convention, is in ½ "integral" amounts and may be +/- ½, or ½ + ½ = 1, or ½ + -½ = 0, or ½ + ½ + ½ + ½ = 2, or ½ + ½ + -½ + -½ = 0, the fermions having a net fractional spin ½ integral and the bosons a net 0, 1, 2 whole integer number.

The point is, that whatever the matter ST unit has, its antimatter virtual-pair partner will always have exactly the EAO entangled values.

(Note: think of charge as so much up/down relative to the line and spin as so much rotation at the 90° horizontal position to the charge vector.)

Q. Can you give a little more of an actual example, please?

A. The Higgs Boson (), in its purest form, is really the massless, chargeless, and spin-less ST unit/field that is parent to ALL ST units. It is composed of the two EAO virtual-pairs in OV/AV.

The first child is the photon (𝛾). Two EAO photons together make the .

(Note: interestingly — for future reference — two Not Equal And Opposite photons, i.e., two Equal And Same-spin photons in OV, will temporarily combine as gravitons[𝓖] as the two constructively interfere with each other. Temporary, in the sense that their point of interference is not static, but travels as in any ordinary wave-interference pattern, away from its source, following the Inverse Square Law [ISL].

The 𝓖 is actually the epitome of the wave-particle duality in that it really is a cross-wave interference expression that can collapse to its 2-photon wave-particle ST unit constituents.

The 𝓖 is also directly related — an even derived from — the in that 2-Higgs, composed of four photon spin-1 states, can reconfigure such that the 4 sets of spin recombine into two sets of EQUAL spin, i.e, spin 2 gravitons: as c+cc and cc+c reconfigure to c+c and cc+cc as shown in the image below.)

The photon (𝛾), itself a boson, gives birth to ALL other ST units — as can the — by deconstructing (i.e., re-arranging) its charge and spin vectors to form ST units with mass. Mass being a parameter expression in which some form of asymmetry results in the ST unit now unable to generate full lightspeed ST extension with each p-p.

In other words, it partly spins on itself and in doing so it is less than lightspeed extension gives what we measure as mass, e.i., the 𝛾 forms an electron/positron or positron/electron (e+/e-) virtual-pair with EAO charge (+1/-1), and, EAO spin vector (e+~R~/e-L or e+~L~/e-R). Each having lost the near perfect symmetry of the 𝛾 (spin 1, charge 0, mass 0), now express mass.

Q. Where does the ISL come in to play?

A. Everywhere! ALL energy — all influence — dissipates from its source inverse squarely with distance.

Those gravitons (𝓖) mentioned earlier, dissipate as an expanding sphere of ST curvature — a diminishing ST curvature as the area (surface) of constructive interference quadruples in size for every doubling of the distance from the source, while the actual number of interfering nodes remains constant.

Because the Conservation of Energy ultimately dictates the virtual-pair expression, we effectively have a Conservation of SpaceTime (ST) in effect. The must ultimately know and balance ALL ST curvature back to its constant — be it "0" or "1" depending on how you like to look at it.

Therefore, it is precluded that the ISL drives the amount and balance of ST curvature.

ST curvature so dilute that it effectively scores as negative will have an expansive, anti-gravity expression, a.k.a. Dark Energy.

Intense, concentrated ST curvature the opposite: dense constructive interference will express as Dark Matter upon the surrounding masses.

Thus Dark Energy = Dark Matter/the inverse of. The Conservation of ST is maintained.

(Note: The Sun's corona represents a local high-ST curvature-Dark Matter effect. The intense concentration of constructive interference results in a temperature expression of millions of degrees versus that of the actual surface of the sun at 5-6000 degrees C. See the final Summary.)

(Note: In 2008, "The Conservation of SpaceTime~ The role of the Higgs, graviton and photon bosons in defining Dark Matter = Dark Energy (the inverse of) in the fractal Universe ~" white paper presented in great detail much of what we are presenting here.

This was built on the 2003 "Dark Matter = Dark Energy (the inverse of): The Conservation of Spacetime by The Conservation of Force.

The History of the Universe in Scalar Graphics and The History of the Universe - update - The Big Void round out this early series, presenting a consolidated image and summary view. Well worth a read!

The BIM was discovered and published in 2009, adding the underlying mathematical foundation and validation of the ST concepts put forth then, and now! Dynamic ST has wave-particle interference manifestations that directly accounts for the current "mystery" of Dark Energy and Dark Matter. One simply must look in the right spot(s) to see the ripples!)

Here is an exerpt from "Dark-Dark-Light: Dark Matter = Dark Energy (the inverse of:"

Similar to the real world irony, here too, smaller size (area) represents a greater focus of energy. Thus the smaller the λ = the greater the energy, as shown earlier,

E = pc = hc/λ .

Therefore, smaller areas in the rectangle represent greater concentration of energy. And because all energy is gravitationally active, in that higher energy area (higher than boring spacetime energy of 1 · 1 units) the gravitational acceleration is relatively enhanced. Opposite to this are the lower energy areas (again lower than the ideal, boring spacetime energy), where the spacetime energy density is so relatively unattractive that it appears to be repulsive when viewed from the higher energy position. Those smaller rectangles will be progressively more attractive the smaller they get and vice versa.

Effectively, this describes a force in general, and here, the gravitational force in particular. Geometrically, the higher spacetime curvature of the higher mass-energy density gravitationally attracts all other energy to itself. The concentration of h units here, at higher frequencies than their surrounding spacetime, ST, leads to a net attractive force here.

Because dividing up the symmetrical 25 unit2 Total Energy, ET, Rectangle "B" (or "A") into an asymmetrical, weighted Rectangle "C", still at E T = 25 units2, the lighter, airy portion in the upper right of Rectangle "C" is equal and opposite to the heavy, denser portion in the lower left.

Equal and opposite to what? Equal and opposite in force, F. Conservation of Force. The degree, or amount of F that is gravitationally attractive, relative to boring old ideal ST of 1 · 1 = 1 unit2, is equal, and opposite, to the amount of F that is less attractive (i.e. repulsive) relative to that same boring ST.

Fattraction = Frepulsion (Conservation of Force)

Picture a cross-section through the ST continuum where the curvature of ST is represented by vertical lines. In the middle, which also would represent ideal, boring ST the lines would be perfectly vertical and evenly spaced. Towards the increasing mass-energy density on our left, the lines would increasingly curve towards the left as the spacing between the lines also decreases in progression, depicting what effectively becomes the gravitationally positive "missing mass", or dark matter. Towards the right where the mass-energy density progressively thins out, the vertical lines become spaced further and further apart, and, become increasingly curved to the right, depicting what effectively becomes the negative or antigravity of the dark energy. The image is not far removed from the common image of iron filings in the presence of the opposite poles of a magnet, i.e. a magnetic field.

Expansion and contraction are relative concepts…because of the relativity of ST. If ST consists of all the Planck h units and all the Planck h units times all their frequencies of presentation equals a constant Total Energy, ET, then there can be no net expansion or contraction of the universe…only that of one occurs at the expense of the other. The more contraction force is expressed here, the more the expansion force is expressed there. E = will not allow it any other way. The universe…and ST…is expanding at an accelerated rate if viewed from a relatively high ST contraction position.

It is very helpful to think of ST as space/time, or simply as distance (displacement) per unit of time (or frequency):

λ /t = λ ν .

All ST manifests itself as mass or mass-energy (matter or fields). All ST exists as pulses, at a given frequency related to their energy, of Planck's unit of h, and the energy of any such ST existence is

E = .

Mass, as matter, represents an asymmetry in a given ST parameter (an attribute which generally limits its mobility, but also provides for net expressions of the charge vector, whether +, -, or 0; and, the spin, or intrinsic angular momentum, vector).

Mass-energy, as a field, represents the relatively symmetrical expression of the ST (an attribute which allows for its lack of mass, lightspeed mobility, neutrality of the charge vector, and gauge boson behavior, i.e. the spin vector is always paired, resulting in only net integral spins of 0, 1 or 2). It is relatively symmetrical, and not absolutely symmetrical. The Higgs Boson represents the absolute symmetry, the progenitor of the ST field. Breaking the symmetry of the Higgs Boson generates the relatively symmetrical ST field mentioned above, which can itself form, through further symmetry breaking, the matter field, a.k.a. the mass expression.

This relatively symmetrical ST field mimics in every way the massless, chargeless, spin = 1 photon, whose λ and ν are always reciprocally related to insure that

c = λ ν ,

the velocity of light, equals the fundamental constant of nature.

In 2012-13, "LightspeedST" was published, bringing together in a single work all of the previous 30 years. Both as a freely available web project and as an ebook. Here is a movie from a page excerpt: "LightSpeed = ST: The Movie Script" (navigate to The Movie Script>Plot>Simplification>6) or this Vimeo summary movie below.

LightspeedST: The Movie (short) from Reginald Brooks on Vimeo. A full-length version is also on Vimeo.

Q. What is the BIM?

A. The BIM (BBS-ISL Matrix = Brooks Base Square-Inverse Square Law Matrix) describes the ISL in a simple fashion using nothing but WINs.

With the Axis of 0, 1, 2, 3,... and the central Prime Diagonal (PD) of their squares, every BIM cell is simply the ∆ between the horizontal (Row) and vertical (Column) PD numbers.

Every BIM cell value is known, fixed, and expands to infinity.

The BIM enjoys a 90° bilateral symmetry. Combining both triangular sides gives one the perfect square areas (and their circles).

Looking at just the Rows, one finds that dividing all cell values by 24 selects out ALL Active Rows: Rows of ODD numbers, not ➗3, that contain ALL the Primitive Pythagorean Triples (PPTs), as well as ALL (>2) PRIMES.

Q. What is the significance of ➗the BIM by 24?

A. There are 3 main number theory systems contained within the BIM. Each extends to infinity:

1. ISL is synonymous with the BIM.
2. The PTs are distributed throughout the BIM, the PPTs are ALWAYS on Active Rows.
3. The PRIMES — while not on the strict Inner Grid — ALWAYS have Active Axis Rows, occupying the same, but not necessarily identical Active Rows.

(Note: TPISC_IV:_Details presents the BIM, Pythagorean Triples and Primes in great — wait for it — detail.)

Q. What is the significance of the PTs, especially the PPTs?

A. As the PTs are both ubiquitously and infinitely embedded within the BIM, they provide key structural framework nodes for the expanding, p-p ST unit.

Key in that the non-isosceles, 90° right-triangle PTs are present with 4 iterations for each PT on the BIM.

Not only do these asymmetrical triangles form rectangles (and ovals) versus squares (and circles) when combined in pairs, individually each of their triangular iterations may actually be representative of a quantum state number (parameter).

Horizontally, on a Row, the PT has a Left and Right (mirror) iteration, and the same — when rotated 90° — on its bilaterally symmetrical Vertical Column.

If we assign the spin vector to an iteration, we have 2 sets of EAO spin vectors for every PT. As the iterations are known — ubiquitously and infinitely — we have the basis of the DSEQEC.

(*From the Big Bang genesis, "inflation" was both the result of and the solution to potential annihilation — by separating and sequencing matter and antimatter — giving the initial birth of ST.)

• Big Bang
• *INFLATION
• Matter = Antimatter
• Dark Matter = 1/Dark Energy
• DSE = QEC
• ER = EPR

The DSEQEC ~ One more time

The Quantum Mechanics that ultimately explains the Double-Slit Experiment (DSE, a.k.a. the wave-particle duality conundrum) must, ultimately, also explain and inform Quantum Entanglement (QE).

$Thus: DSE = QE —> DSEQE Conjecture$

DSE —> QE

1st Level: The photon (𝛾) consists of EAO ST units whose mass, charge and spin vectors give a net mass 0, charge 0 and spin 1.

On a simplistic level, the 𝛾 splits into its EAO parts, each going through, in a synchronous fashion, one of the slits.

2nd Level: The photon — and ALL other ST units — is but a child of the Higgs Boson () and exists as part of a virtual-pair — part time in Our View (OV) and part in the Alternate View (AV), not entirely unlike an AC electric current, yet we only perceive OV (½ the time it is in OV< the other ½ in the AV with EAO parameters).

The , in its ideal, perfect field state, is really 2 EAO photons, so this is not that difficult to envision, only in this scenario, its EAO photon STs occupy two separate views — OV and AV — and most likely in a serial, time-separated manner, not unlike a sine wave.

3rd Level: The Higgs=photon+photon ST unit forms ST — and its accompanying velocity of light (c) parameters — with each pulse-propagation (p-p) from its Singularity () out past its Planck Event Horizon (PEH) into OV (and sequentially, alternatively and in syncronicity in AV).

This all occurs under the constraints and directives of the Inverse Square Law (ISL), as portrayed on the BIM (BBS-ISL Matrix).

The BIM is bilaterally symmetric with ½ mirror reflected 90° to the other, divided by the Prime Diagonal (PD).

4th Level: Within the BIM, ALL possible Pythagorean Triples (PTs) are present and manifest their non-isosceles, 90° right-triangle direction-vector-specific presence — and automatically, the mirror of that and the 90° mirror symmetry of both — giving 4 iterations for each PT.

graphic)

5th Level: If we let the vector-specific-direction of the 4 PT triangle presentations present the respective horizontal and vertical spin axis of the individual 𝛾 ST, we can account for ALL possible manifestations at any moment in time over any extension in space as the 𝛾 ST unit is itself the unfolding, expansion of the BIM.

For example using the 3-4-5 PPT and letting the short side "a" — or, if you prefer, simply the "outward" area projection from the hypotenuse — represent the spin angular momentum vector (spin) on the horizontal plane. Turn it 90° to represent it — with the other two PT representations — on the vertical plane.

Thus 2-L's and 2 R's.

6th Level: The superposition of both spin states — spin ½L and spin ½R — is present simultaneously as two virtual states, but only one state is revealed upon measurement in OV — instantaneously, simultaneously and without transferring any information — the other EAO spin state is revealed in the AV.

Likewise, two entangled photons — with EAO spin states — do exactly the same thing, in effect 4 spin states are effected anytime any 1 spin state is measured.

7th Level: Important to notice is the maintenance of symmetry in service to the overall Conservation Laws: Conservation of Spin Angular Momentum, Conservation of Charge, Conservation of Energy and, of course, Conservation of ST.

8th Level: The Higgs Boson () represents the perfect symmetry — and Conservation — of ST. composed of 2 EAO entangled photons.

Let's say that again: The Higgs Boson = the state of 2 EAO entangled photons (𝛾E).

The Higgs Boson, in its ideal, perfect field state, is simply two EAO entangled photons.

(The Higgs in the transformative state of composing other ST units, will, of course, have various masses. )

When deconstructed into its 2 "independent" photons in OV, it simultaneously reflects as and EAO photon if AV.

9th Level: In other words, every photon (𝛾) comes as a virtual-pair, entangled set — ½ in OV, ½ in AV — together forming the Higgs ().

When two real OV photons are entangled (EAO) in real time OV, there are, indeed, 2 sets = 4 entangled photons = 2 Higgs Bosons. One in OV, the other in AV.

(Note: not to be confused with the graviton (𝓖), that is simply the constructive interference of two similar — Not EAO — photons, i.e.. 2 clockwise-R photons or 2-counterclockwise-L photons: 𝛾R + 𝛾R = 𝓖R and 𝛾L + 𝛾L = 𝓖L as spin 1 + spin 1 = spin 2 of the 𝓖.

Naturally, two EAO gravitons will form 2 Higgs Bosons: 𝓖R + 𝓖L = as spin 2c + spin 2cc = spin 0

)

Thus, even a single, real OV photon can demonstrate interference in the DSE.

10th Level: Everything about the —>𝛾 + 𝛾 virtual-pair scenario is true for ALL ST units from leptons - baryons, fermions - bosons, matter - antimatter.

—>𝛾 + 𝛾 —>Matter + Antimatter as described in the 1st Level.

The electron (e-)—positron (e+) are the virtual-pair components of the photon, directly, and the , indirectly.

The quark-based hadrons — mesons and baryons — are likewise virtual-pair components of the 𝛾 and/or .

When you "see" (measure) a real OV e-, it acts like a hole — a void —in ST awaiting to be filled with a positive charge ST unit. The e- represents so much integral *"down-ness" (↓) as opposed to the *"up-ness" (↑) of the + ST unit.

(*Relative to their Singularity and PEH. Diagrammatic convention places ↑ at the top, ↓ at the bottom.)

As a virtual pair, both ↓ and ↑ are always together — only one is expressed in OV, the other to balance it in the AV.

)

The exact same superposition, synchronous entanglement and wave-particle double-slit expression found in the photon is present in all the children ST units.

So where has all the missing antimatter gone?

Well, it never left. It's where it has always been: as part of the inflation-driven virtual-pair — matter:antimatter — that ALL ST units express!

The key point is that Nature embodies the Universe with the harmony of the one by her Conservation Laws — Laws that by combining simple mathematical rules of counting (a.k.a. the ISL), her balance sheet ALWAYS — in the end — reflects symmetry.

The unfolding apparent asymmetry:

• Big Bang
• Matter
• Antimatter
• Inflation
• Dark Matter
• Dark Energy
• DSE
• QE

is given a resultant symmetry by the entangled generation of SpaceTime(ST). A ST that is conserved.

Thus:

• Big Bang-Inflation

sequentially, over ST Inflation, separates the entangled

• Matter-Antimatter

into:

EAO, alternating entangled virtual-pairs whose internal frameworks of asymmetrical (contributed by the PTs), slows down and concentrates parts of the ST field into coalescing, entangled particles that pull part of the expansion back home to the common Ⓢ that balances the accounting ledger sheet, giving

• Dark Matter-1/Dark Energy
• DSE-QE.

We just happen to only "see" one side of the OV/AV coin!

DSEQEC: One last look at just the photon.

The time component will always ensure that the EAO spin is accounted for!

(Note: the "Inflation" expansion of the Big Bang — separating Matter - Antimatter — generates an EAO chiral spin of sequential, interlocking, double-helical spiral. This large scale structure is itself but a fractal re-iteration of the ST units.)

SUMMARY

*Have you ever wondered why that area between the sun's surface photosphere (~5000° C) and its corona (~1-10 M° C) is not hotter, indeed it is actually cooler than the surface?

Think of the corona as "Soft Hair," a term coined in Stephen Hawking's, et al, last published paper:* Black Hole Entropy and Soft Hair, as an accumulation of photons that are gradually released (along with their ST information) away from the outer surface of a black hole event horizon. Only here there is no black hole as the mass of the sun is too small. Yet there is this huge accumulation of photons at some distance from the sun's surface. Much less so in the chromosphere area between. How so?

The broad spectrum photons radiating out from the thermonuclear core (15 M° C) of the sun generate a harmonic wave band that forms the corona. Here the massive battle of constructive vs. destructive wave interference favors the former. A dense band of constructively interfering photons forms this 1st harmonic. Many of the photons will have matching coherent spin1 vectors, forming gravitons in the process.

The gravitons — representing local high curvature ST, i.e., Dark Matter — will gravitationally attract and curve and effectively "slow down" other photons within this "temporary" photon "soft hair" accretion disk.

Some of the interfering photons — and some of the ensuing interfering graviton waves — will destructively interact — opposite spin 1 vectors — to generate Higgs Bosons. The Higgs Boson field (spin 0) inherently favors ST expansion: witness the singularity of the Big Bang and the multiple singularities of the individual p-p ST units.

One may visualize a number of harmonic interference bands radiating out from a "point source"— accumulated mass — giving rise to "structural nodes" along the way. The corona is one. The formation and distribution of the planets, et al, within the Solar System is yet another.

The accretion and consolidation of smaller bits of mass-energy into preferential orbits about a central "point source" would be a natural consequence of these photon-graviton-Higgs resonance bands.

After firmly establishing a deep connection between Pythagorean Triples (and PRIMES) with the BIM-depicted Inverse Square Law, we have ventured back out to the world of quantum physics — or should we say quantum relativity physics. What have we put forward that might meaningfully inform our world?

In LightspeedST, we showed that nothing can physically travel faster than the speed of light precisely because the photon — the basic ST unit that p-p into and out of existence forming a single quantum of ST — is itself composed of and generates 1 unit of lightspeed ST with each pulse.

Decoupling the ideal Higgs Boson field ST unit gives the photon, and from further decoupling, all the other ST units downstream.

Re-coupling two coherent, like-minded, similar spin 1 photons generates the graviton (spin 2). This ellusive ST unit — that must be absolutely inundating the Universe — is really just the temporary constructive interference of two coherent superimposed photon ST unit waves at the moving point of their crossing.

As in all wave interferences, opposite spins interacting will neutralize the gravitational contracting action in favor of the ST expansion of the newly formed Higgs.

This +, -, 0 wave interference of photon ST units has many, many consequences, and dare we say, that — along with the accompanying BIM — gives insights, if not downright clarifications of a number of fascinating, yet heretofore, inexplicable physical phenomena.

To be complete, we must re-introduce (Dark Matter = Dark Energy

(the inverse of); The History of the Universe in Scalar Graphics; The History of the Universe - update - The Big Void; and, The Conservation of SpaceTime) the Our View (OV), Alternative View (AV), and the role of how symmetry, matter-antimatter, inflation, singularity, Planck Event Horizon, etc., fit into the p-p of the ST field and units.

We thus have:

1. BIM — the information
1. P-P — the structure
2. Symmetry — the mechanism.

1. The BIM — along with the PTs and PRIMES — provides the information on how ALL ST will be informed: the rate of and relationship to all expansion of ST. Key structural nodes — representing quantum number states — are ubiquitously embedded within the expanding BIM.

2. The underlying quantum is the photon ST unit, created and always expressing the constant, ultimate and maximum velocity of light expression (gauge bosons) that every subsequently derived ST unit with mass will necessarily be moving at less than light speed (i.e., fermions and other bosons).

The +, -, 0 wave interference effects are precisely the guiding dictates — along with the dictates of 1. and 2. — informing the physical manifestations of both the seen and unseen (Dark Matter, Dark Energy, as well as the phenomena underlying the wave-particle duality of the Double-Slit Experiment and Quantum Entanglement, DSEQEC) Universe.

3. The symmetry — that which informs the Conservation of Energy, the Conservation of Momentum and Angular Momentum, the Conservation of Charge, the Conservation of the Velocity of Light, and ultimately, all in service to the Conservation of ST — divides the Universe — or should we say Cosmos — into polemic parts whose equal part ratios ultimately demand a separation — a far from equilibrium separation — precisely such that the dynamics may unfold at these edges (OV-AV) in the drive to re-establish the equilibrium and a return to the ultimate neutral, perfect symmetry state of the ideal Higgs Boson ST field/particle.

This model addresses several outstanding issues:

Big Bang : The Singularity, or 0 or 1 (depending on your intial state reference preference) = ideal Higgs Boson ST field/particle of spin 0, mass 0, charge 0. Because ST expansion of the Singularity — Higgs — is inherently built into this perfect symmetry, we must conclude and invoke that such a spin 0, mass 0, charge 0 state is, however temporary, expansive. The same holds true for the individual Singularities of the individual p-p ST units.

Quantum : The Big Bang is the 1st quantum. It quickly decomposes into its fractal quantum units, each p-p into and out of existence at lightspeed from the Higgs Boson field. c=𝛾ⱱ

ST Units : The ST units — all derived from the photon (itself derived from the Higgs) — all p-p into and out of existence as aggregates of quarks and leptons. A photonic ST field — like a ripple in a pond — is generated with each pulse and continues to expansively pulse-propagate outward until it is neutralized by interference effects along the way.

The builtin symmetry of the massless gauge boson ST units allows them to travel 1 full quantum of spatial extension with each pulse frequency expression (lightspeed), while the other less-symmetrical ST units spend part of their ST p-p within their own spatial expression with each pulse (sub-lightspeed).

The ubiquitous information of the BIM/PTs/PRIMES is present and guides every p-p from its birth at the Singularity (Higgs) out past its Planck Event Horizon and on out to every possible ST extension throughout its possible expansion paths throughout the Universe. This information is encoded within the ST p-p and it never needs to be communicated or propagated — as per any carrier, particle or pilot-wave — to any such expansion as the information is "always known" — from the beginning to the end of each and every ST unit p-p. This becomes critical to our understanding of everything!

Inflation : Both the 1st quantum-Big Bang and all subsequent, subset quantum ST units p-p, expand outward at c=𝛾ⱱ. This builtin fractal expansion relationship defines the very existence of our Universe. Without so much spatial extension over so much temporal periodicity we have no ST and thus no Cosmos.

Inflation separates, i.e., pushes the ST units into far-from equilibrium status such that a re-organization into seen-unseen, OV-AV, matter-antimatter domains both prevents the senseless, chaotic annihilation of opposites — even if they are to repeat their expansion — and allows a very measured, controlled, ordered and fully accountable return to their neutral, perfect symmetry Singularity-Higgs state while maintaining the balance sheets of equal and opposites the Conservation Laws require. Inflation allows and drives the Creation and Conservation of SpaceTime (CaCoST) to exist.

As noted above, the inherent ST expansion force of the Singularity-Higgs underlies this inflation.

Entanglement : Without a doubt, in some sense, the entire Universe — every ST unit and field — is entangled. The BIM ensures that. Every ST unit — whether generated in OV as two real, Equal And Opposite (EAO) entangled ST units or as part of the virtual-pair — one member in OV, the other in AV — that every ST unit has upon its p-p, is entangled in several or more ways. This has to be! How else would Nature be able to account for and maintain the symmetry of the Conservation Laws?

Interference : (wave-particle duality, Double-Slit Experiment, Dark Matter, Dark Energy, gravitons, quantum gravity,...)

• wave-particle duality : Every ST unit p-p as a ST wave and its ST unit particle expression is simply a perceptual (measurement) of it within a local interpretation of its field. This is consistent with the "Copenhagen Interpretation" of Quantum Mechanics.

• Double-Slit Experiment (DSE) : Getting in the way — i.e., interfering with the information of the wave-state — the BIM information —of a p-p ST wave collapses its DSE interference precisely because the perception/measuring process blocks the BIM information. The information that informs ALL possible slit paths — thereby collapsing any possibility that even a single photon could interfere with itself. ALL ST units carry ALL possible path information within the embedded BIM information.

• Dark Matter — Dark Energy : Together, these two "mysteries" are really just part — albeit opposite ends — of the same interference of ST unit wave-field effects (+/- curved ST via gravitons).

Two or more p-p ST units will have some expression of each others presence as to energy density. The overlapping of the ST unit wave fronts will generate interference effects. A massive body of high energy ST unit photons will output dense bands of constructive interference. The constructive overlap of two similar, coherent photons will generate a node we call — when thinking or perceiving it as a particle ST unit — a graviton (spin 2).

It acts to *curve the ST locally.

Like any interfering wave-crest, it expands and moves outward from its "point source"— following the dictates of the ISL (BIM). It is "quantum" in the sense that its component parts are quantum p-p photon ST units whose c=𝛾ⱱ wavelength frequency properties match enough (coherence) to form a "temporary" curvature of ST by their constructive interference (spin 1 + spin 1 = spin 2), giving rise to our mysterious graviton. Again, the graviton ST unit as a particle is really only our designation of this moving interference node. It does NOT actually exist as a "particle" except for our convenience in defining it superposition wave state. Both the photon and graviton node at ST unit wave packets travel at the velocity of light.

The graviton as a vector-gauge exchange boson — passing the curvature of space information between bodies — needs to be looked at as the interfering field dynamic that it is rather than a mechanical "particle" exchange picture commonly held. Every corner of the Universe is flooded with these gravitons — i.e., flooded with +, -, 0 interference effects of ALL the other p-p ST units.

(*The spin 1 photon ST unit may itself be "sticky"— i.e., it contributes to the gravitational curvature of ST because it — as the creator of ST (via the Higgs) — inherently wants its spin 1 vector to act as an attractor, a ST curver while its ST p-p is pushing to expand outward with each pulse. The graviton node is simply double the spin vector without doubling that expansive push.)

• In Dark Matter, the concentration — and importantly — the resonance banding of waves of constructively interfering coherent photons generates the graviton ST curvature we know as gravity. When densely concentrated, it gives rise to the global effects of Dark Matter. Resonance plays a key role in that harmonics of interference wave/frequency expressions play out in the expanding bands of graviton-ST curvature.

• Sun's Corona is a simple expression of this phenomena. The corona's temperature is a reflection of its energy and its high energy is a reflection of the resonant concentration — the Dark Matter accumulation — of gravitons — constructively interfering photon ST unit pulses of the enormously dense photon-energy at its core.

• Black Hole Event Horizon, a case of increasing the energy density of the core will increase the Dark Matter energy density of the surrounding "corona" up to a point whereby its density is so great that the photons can no longer and/or barely escape, effectively forming an Event Horizon — and the now "unseen" core inside as a Black Hole. (Hawking, et al, "Soft Hair" referenced above.)

• Dark Energy, not to be, by any means, outdone, is simply the reflection of the opposite. The Conservation Laws require the total ST expression to be constant. *ST=energy. Concentration of ST in one part must necessarily generate dilution of ST in another part. The dilution of ST reflecta a curvature of ST property that is expansive — opposite to the contractive ST curvature property of the graviton. So if the same expanding photon + photon = graviton constructive interference node gets away from its contractive curvature, how does it later become expansive? Remember, that in any interfering wave phenomenal, one has +, -, and 0 expressions. The + and - of the interfering photon ST units represent the superposition addition — spin 1R + spin 1R and spin 1L + spin 1L — of two similar-spin photons giving rise to the graviton (spin 2). Combining two EAO photons — spin 1R + spin 1L — gives rise through superposition subtraction to our expansion-loving Higgs Boson ST unit field (spin 0). Dilute ST unit density equals the formation of the expansive Higgs field = Dark Energy. Dark Matter = Dark Energy (the inverse of).

A ST field can be divided between:

• positive (+) curvature (contraction via constructive interference)

• negative (-) curvature (expansion via destructive interference)

(* c = 𝛾ⱱ is the first step in ST creation: one dimension of space/one dimension of time;

c2 = (𝛾ⱱ)2 = potential difference=2 spatial/2 temporal dimensions;

c3 = (𝛾ⱱ)3 = current-amperage=3 spatial/3 temporal dimensions;

c4 = (𝛾ⱱ)4 = force = 4 spatial/4 temporal dimensions;

c5 = (𝛾ⱱ)5 = energy/time = 5 spatial/5 temporal dimensions; see The LUFE Matrix:The distillation of System International (SI) units into more fundamentally base units of Space-Time (ST) dimensions)

The alternating bands of contraction and expansion may be visualized as:

Everywhere within the Universe, one may look out and "see" expansion and contraction — i.e., Dark Energy and Dark Matter.

Just as the Higgs-expansion drives the Big Bang — as well as each and every individual ST unit p-p — within its expansion are the seeds of its contraction — via the superposition of coherent photons.

Matter - Antimatter : In quantum #1, the Big Bang, inflation — a natural consequence of the HIggs Boson ST field expansion — obviated annihilation by the separation of matter and antimatter ST into EAO domains. The former as Our View (OV) that take our privileged view as "reality," and the Alternate View (AV), were all the EAO antimatter hangs out in the "unseen" view.

Between the two, and the source and balancing factor of both, is the Singularity-Higgs field. Upon collapse it is the "one," the only one, the "zero" from which all ST expression — all energy — all fields and particles — arises. Each individual p-p ST unit is but a local fractal component of the larger global hologram.

The individual, fractal ST unit itself is composed ot two domains as well — OV and AV. EAO. The L-shaped and I-shaped quarks are all consistently modeled giving rise to their net quantum state expressions of spin, mass and charge. Every ST unit ("particle") is accounted for. (Please see LightspeedST.)

Symmetry : A=B. A is not B. A=EAO B.

Break and re-combine symmetries informs it all.

The tension — the asymmetrical imbalance — of far from equilibrium (pure symmetry) that gives rise to the dynamics of ST expression only exists to resolve to the apparently neutral, equilibrium threshold of perfect symmetry. The real mystery of the Cosmos is whether pure, perfect symmetry is anything other than an ideal, a portal, a perfect interference — and a temporary one at that!

EP = EPR :

Open letter addressing the "ER=EPR" work of Leonard Susskind, Juan Maldacena, and Mark Van Raamsdonk, …

The ER = EPR conjecture is a very interesting insight into the joining of quantum mechanics with quantum gravity through the geometry of spacetime.

What is needed is the other, complementary side that seals the deal: any entanglement of spacetime (ST) and ST units/particles, i.e. photons, gravitons, Higgs Bosons,… must reckon with:

1. ST itself is formed in such a way that it obeys the Inverse Square Law (ISL), is always consistent with the Conservation of Energy, and its very formation must be one and the same with the formation of its progeny ST units and the built-in velocity of light, c;
2. ST itself, and thus its ST progeny, must be conserved;
3. ST itself, and thus it ST progeny, must be fully accountable for all intricacies of the Double-Slit Experiment.

One can satisfy all three constraints by allowing ST to be pulse-propagated from its singularity (S) into and back out of our ordinary view, forming the photon with its built-in velocity, c, with each pulse. A graviton is nothing more than the positive interference of two similar-spin photons, while the Higgs Boson, in the pure state, is two photons with opposite-spins interfering. The Higgs, of course, gains mass as it decouples into ST units with 1/2 spin, be it quarks or electrons-neutrinos, or their composites. Note that one could start with the Higgs and decouple, first to the photon and thereafter to its progeny. (LightspeedST)

Key to the ordered formation of ST is the ISL....

Experimental verification : Despite our knowing how incredibly weak the gravitational force is, we nevertheless should be able to devise experimental verification of the CaCoST model.

Knowing that photons can transfer momentum while at the same time testing for the gravitational effects of constructively and destructively interfering photons, might one devise a setup up to null out the former while testing for the latter.

• A series of high density, high energy coherent, polarized (spin 1 vectors all pointing in the same direction) converged into a superposition state — a focus point —upon which lies a "target." The object is to show flicking the laser lights on locally curves the ST immediately surrounding the "target," causing it to deflect towards the light source (spin 1R + spin 1R or spin 1L + spin 1L — of two similar-spin photons giving rise to the graviton (spin 2)).

• A matched set of the same laser setup — only wtih 1/2 the photons polarized one way, the other 1/2 the other (Combining two EAO photons — spin 1R + spin 1L — gives rise through superposition subtraction to our expansion-loving Higgs Boson ST unit field (spin 0)). The "target" should be deflected away from the light sources.

• A hybrid of the two: move the later Higgs-generating laser setup to the opposite side of the "target." Repeat experiment with one, then the other, then both laser setups turned on. The "target" does not deflect. Some fine tuning will undoubtedly be required.

~~ ~~ ~~

BACK: ---> Part I of II CaCoST-DSEQEC ~~ ~~ ~~

References (specific to ER=EPR and some of the surrounding issues):

ER=EPR

https://en.wikipedia.org/wiki/Quantum_entanglement Quantum Entanglement

https://en.wikipedia.org/wiki/Wormhole = Einstein-Rosen Bridge (EP)

https://en.wikipedia.org/wiki/Black_hole Black Hole

https://en.wikipedia.org/wiki/Event_horizon Event Horizon

https://en.wikipedia.org/wiki/Einstein%27s_thought_experiments Einstein's Thought Experiments

https://en.wikipedia.org/wiki/Firewall_(physics) AMPS firewall

Mark Van Raamsdonk

https://en.wikipedia.org/wiki/Mark_Van_Raamsdonk

https://arxiv.org/pdf/1005.3035.pdf Building up spacetime with quantum entanglement

https://sitp.stanford.edu/topic/quantum-gravity-and-black-holes Gravity and Entanglement

https://www.nature.com/news/the-quantum-source-of-space-time-1.18797 The quantum source of space-time

https://www.sciencenews.org/blog/context/new-einstein-equation-wormholes-quantum-gravity A new ‘Einstein’ equation suggests wormholes hold key to quantum gravity

Leonard Susskind

https://en.wikipedia.org/wiki/Leonard_Susskind Leonard Susskind

http://theoreticalminimum.com/home

https://sitp.stanford.edu/people/leonard-susskind

https://sitp.stanford.edu/topic/quantum-gravity-and-black-holes ER = EPR" or "What's Behind the Horizons of Black Holes?" (Lecture 1 & 2)

https://arxiv.org/pdf/hep-th/9306069v2.pdf The Stretched Horizon and Black Hole Complementarity

Leonard Susskind, La ́rus Thorlacius, and John Uglum, Department of Physics

Stanford University, Stanford, CA 94305-4060

https://arxiv.org/pdf/1606.08444.pdf Space from Hilbert Space: Recovering Geometry from Bulk Entanglement

ChunJun Cao,1, ∗ Sean M. Carroll,1, † and Spyridon Michalakis1, 2, ‡

https://arxiv.org/pdf/1604.02589.pdf Copenhagen vs Everett, Teleportation, and ER=EPR

https://arxiv.org/pdf/1306.0533.pdf Cool horizons for entangled black holes Juan Maldacena and Leonard Susskind

Juan Martîn Maldacena

https://www.ias.edu/scholars/maldacena

http://www.sns.ias.edu/malda

https://en.wikipedia.org/wiki/AdS/CFT_correspondence Anti-de Sitter/Conformal Field Theory correspondence, sometimes called Maldacena duality or gauge/gravity duality

https://www.ias.edu/ideas/2013/maldacena-entanglement Entanglement and the Geometry of Spacetime

https://www.ias.edu/ideas/2015/general-relativity-at-100-conference General Relativity at 100

https://www.ias.edu/ideas/2011/maldacena-black-holes-string-theory Black Holes and the Information Paradox in String Theory

Patrick Hayden

https://sitp.stanford.edu/topic/quantum-information Decoding Spacetime

https://quantumfrontiers.com/tag/patrick-hayden/ Here’s one way to get out of a black hole!

http://iopscience.iop.org/article/10.1088/1126-6708/2007/09/120/meta;jsessionid=3A92BE27C55F2B4A3D8CFC70E14618CB.c3.iopscience.cld.iop.org Black holes as mirrors: quantum information in random subsystems

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https://www.nature.com/articles/s41586-018-0200-5 Deterministic delivery of remote entanglement on a quantum network

Peter C. Humphreys, Norbert Kalb, Jaco P. J. Morits, Raymond N. Schouten, Raymond F. L. Vermeulen, Daniel J. Twitchen, Matthew Markham & Ronald Hanson

Nature, volume 558, pages 268–273 (2018)

http://www.theory.caltech.edu/∼preskill/ph229

http://www.theory.caltech.edu/people/preskill/ph229/notes/chap4.pdf Chapter 4 Quantum Entanglement

https://plato.stanford.edu/entries/qt-entangle/ Quantum Entanglement and Information

https://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac

https://arxiv.org/pdf/1503.06237.pdf Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

Fernando Pastawski,a Beni Yoshidaa Daniel Harlow,b John Preskill,a

https://journals.aps.org/pr/abstract/10.1103/PhysRev.48.73 The Particle Problem in the General Theory of Relativity

A. Einstein and N. Rosen, Phys. Rev. 48, 73 – Published 1 July 1935

https://journals.aps.org/pr/abstract/10.1103/PhysRev.47.777 Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?

A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 – Published 15 May 1935

https://www.sciencenews.org/article/entanglement-gravitys-long-distance-connection
Entanglement: Gravity's long-distance connection

https://arxiv.org/pdf/0709.0390.pdf* Entanglement, EPR-correlations, Bell-nonlocality, and Steering

S. J. Jones, H. M. Wiseman, and A. C. Doherty, (Dated: May 28, 2018)

http://science.sciencemag.org/content/360/6387/40* Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates

Matteo Fadel, Tilman Zibold, Boris Décamps, Philipp Treutlein, Science 27 Apr 2018:

Vol. 360, Issue 6387, pp. 409-413 DOI: 10.1126/science.aao1850

https://www.quantamagazine.org/closed-loophole-confirms-the-unreality-of-the-quantum-world-20180725/

https://phys.org/news/2018-03-gravity-quantum-mechanics.html

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.240401 Spin Entanglement Witness for Quantum Gravity

Sougato Bose, Anupam Mazumdar, Gavin W. Morley, Hendrik Ulbricht, Marko Toroš, Mauro Paternostro, Andrew A. Geraci, Peter F. Barker, M. S. Kim, and Gerard Milburn, Phys. Rev. Lett. 119, 240401 – Published 13 December 2017

https://phys.org/news/2018-02-two-way-quantum-particle.html?utm_source=nwletter&utm_medium=email&utm_campaign=weekly-nwletter 'Two-way signaling' possible with a single quantum particle

Lisa Zyga, Phys.org, February 26, 2018

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.120.060503

Two-Way Communication with a Single Quantum Particle

Flavio Del Santo and Borivoje Dakić

Phys. Rev. Lett. 120, 060503 – Published 8 February 2018

https://en.wikipedia.org/wiki/ER%3DEPR

​ ~~ ~~ ~~

ER=EPR_quantumEntanglement Double-Slit DELAYED

https://www.physicsoftheuniverse.com/scientists_wheeler.html

https://en.wikipedia.org/wiki/Double-slit_experiment

https://en.wikipedia.org/wiki/Wheeler%27s_delayed_choice_experiment

https://www.physicsoftheuniverse.com/scientists_feynman.html

Tangled Up in Entanglement

Lawrence M. Krauss

https://www.newyorker.com/tech/elements/tangled-up-in-entanglement-quantum-mechanics

Strange Numbers Found in Particle Collisions | Quanta Magazine

https://www.quantamagazine.org/strange-numbers-found-in-particle-collisions-20161115/

Closed Loophole Confirms the Unreality of the Quantum World | Quanta Magazine

https://www.quantamagazine.org/closed-loophole-confirms-the-unreality-of-the-quantum-world-20180725/

Hyperuniformity Found in Birds, Math and Physics | Quanta Magazine

https://www.quantamagazine.org/hyperuniformity-found-in-birds-math-and-physics-20160712/

Physicists Hunt for the Big Bang’s Triangles | Quanta Magazine

https://www.quantamagazine.org/physicists-hunt-for-the-big-bangs-triangles-20160419/

Matter - Antimatter

Antimatter - Wikipedia

https://en.wikipedia.org/wiki/Antimatter

What is antimatter? - Scientific American

https://www.scientificamerican.com/article/what-is-antimatter-2002-01-24/

Antimatter | symmetry magazine

more on antimatter

10/19/17

Scientists make rare achievement in study of antimatter

Scientists on the BASE experiment vastly improved their measurement of a property of protons and antiprotons.

04/11/17

What’s left to learn about antimatter?

Experiments at CERN investigate antiparticles.

02/28/17

How to build a universe

Our universe should be a formless fog of energy. Why isn’t it?

01/30/17

Sign of a long-sought asymmetry

A result from the LHCb experiment shows what could be the first evidence of matter and antimatter baryons behaving differently.

01/19/17

Matter-antimatter mystery remains unsolved

Measuring with high precision, physicists at CERN found a property of antiprotons perfectly mirrored that of protons.

11/24/15

Charge-parity violation

Matter and antimatter behave differently. Scientists hope that investigating how might someday explain why we exist.

11/20/15

Physicists get a supercomputing boost

Scientists have made the first-ever calculation of a prediction involving the decay of certain matter and antimatter particles.

04/23/15

Scientists have proven the concept of the CUORE experiment, which will study neutrinos with the world’s coldest detector and ancient lead.

Where did all the antimatter go? | The United States at the LHC

The five greatest mysteries of antimatter | New Scientist

Antimatter mysteries 1: Where is all the antimatter? | New Scientist

Q: Where is all the anti-matter? | Ask a Mathematician / Ask a Physicist

Where did all the antimatter go? - Quora

Explainer: What is antimatter?

The Higgs Boson Simplified Through Animation - YouTube

Is the Higgs boson really the Higgs boson? - YouTube

Brian Greene explains some math behind the Higgs Boson - YouTube

https://www.youtube.com/watch?v=KWj00MCqSxs Brian Greene explains some of the math behind the Higgs Boson

The Higgs Mechanism Explained | Space Time | PBS Digital Studios

https://www.youtube.com/channel/UC7_gcs09iThXybpVgjHZ_7g * Space Time : PBS Studios

~

Particle Data Group

http://pdg.lbl.gov

Wikipedia Particle Physic, Standard Model, Quantum Field Theory

https://en.wikipedia.org/wiki/Quantum_field_theory

https://en.wikipedia.org/wiki/Standard_Model

https://en.wikipedia.org/wiki/Standard_Model#Fermions

https://en.wikipedia.org/wiki/Standard_Model#Gauge_bosons

https://en.wikipedia.org/wiki/Standard_Model#Higgs_boson

https://en.wikipedia.org/wiki/Lepton

https://en.wikipedia.org/wiki/Quark

https://en.wikipedia.org/wiki/Baryon

https://en.wikipedia.org/wiki/Meson

https://en.wikipedia.org/wiki/Gluon

https://en.wikipedia.org/wiki/W_and_Z_bosons

https://en.wikipedia.org/wiki/Graviton

https://en.wikipedia.org/wiki/Photon

https://en.wikipedia.org/wiki/List_of_particles

https://en.wikipedia.org/wiki/Wave–particle_duality

https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

https://en.wikipedia.org/wiki/Quantum_gravity

Cosmology

http://abyss.uoregon.edu/~js/cosmo/index.html

Early Universe

http://abyss.uoregon.edu/~js/cosmo/lectures/lec20.html

Javascript Wave Superposition Model

http://iwant2study.org/lookangejss/04waves_11superposition/ejss_model_wave1d01/wave1d01_Simulation.xhtml

http://www.physics.ohio-state.edu/~mathur/sissa.html

https://www.sciencedaily.com/terms/holographic_principle.htm

List of unsolved problems in physics

https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_physics

The 18 Biggest Unsolved Mysteries in Physics

https://www.livescience.com/34052-unsolved-mysteries-physics.html

Crunch time for physics: What's next?

https://www.newscientist.com/round-up/physics-crunch/

The 11 Greatest Unanswered Questions of Physics

http://discovermagazine.com/2002/feb/cover/

The Greatest Unsolved Problem in Theoretical Physics

http://scienceblogs.com/startswithabang/2012/09/19/the-greatest-unsolved-problem-in-theoretical-physics/

Five Great Problems in Theoretical Physics

https://www.thoughtco.com/five-great-problems-in-theoretical-physics-2699065

Physics: What We Do and Don’t Know

https://www.nybooks.com/articles/2013/11/07/physics-what-we-do-and-dont-know/

The 10 Biggest Unsolved Problems in Physics

http://files.aiscience.org/journal/article/html/70310003.html

Open Problems In Mathematics And Physics

http://www.openproblems.net

FUNDAMENTAL UNSOLVED PROBLEMS IN PHYSICS AND ASTROPHYSICS

http://www.calphysics.org/problems.pdf

5 Unanswered Questions that Will Keep Physicists Awake at Night

The Secret History of Gravitational Waves

​x​

My References:

Page 55:

PRIMES vs NO-PRIMES ebook in Apple Books.

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VII. Summary

Brief Synopsis:

• Invention/discovery that the Inverse Square Law (ISL) could be both visualized and mathematically consistent within a simple matrix grid: Brooks Base Square-Inverse Square Law Matrix —>BBS-ISL Matrix —>BIM; (MathspeedST) more…
• Within the BIM, it was found that the “children” of the 3-4-5 Primitive Pythagorean Triple (PPT) triangles were ubiquitous in their distribution; (MathspeedST & TPISC I: Basics, The Pythagorean-Inverse Square Connection) more…
• Thereafter, ALL PTs — primitive “parents” (PPT) and non-primitive “children” (nPPT) —are found on select Rows of the BIM by simply following the squared numbers on the the Prime Diagonal down to intersecting Rows; ( TPISC I: Basics & TPISC II: Advanced) more…
• The FTP allowed all the PPTs to be sorted out and organized into a definitive Tree of Primitive Pythagorean Triples (ToPPT) that co-extends infinitely throughout the infinitely expanding BIM —>BIMtree or BIM-ToPPT; (TPISC III: Clarity) more… and more…
• Marking (YELLOW) all BIM cells evenly ➗ by 24 generates a striking diamond-grid, criss-crossing pattern with 4 additional YELLOW marked cells in the center of each diamond; (TPISC IV: Details) See 24 Summary.
• Every PPT is found to exist on —and ONLY on — those Rows whose 1st Column grid cell are ➗ by 24 (YELLOW), though not every such YELLOW marked Row contains a PPT. nPPT are only present on such Rows if accompanied by a PPT; (TPISC IV: Details) See 24 Summary.
• The “step-sister” of any given PT Row is found r-steps down the Axis from that Row and it, too, always and only exists on a YELLOW marked Row (“r” is part of the FTP originally derived from the Dickson Method for algebraically calculating all PTs); (TPISC IV: Details)
• The PPT Row (and nPPT Row) always contains the 4A (A=area) value of that PT and the “step-sister” PPT Row always contains the 8A value, both landing exclusively on YELLOW marked grid cells, giving a striking visualization of ALL PPTs and their r-based “step-sisters;” (TPISC III: Clarity & TPISC IV: Details)
• The significance of the “step-sister” is that it becomes the mathematical link to the “NEXT” PPT within the ToPPT — like the Russian-Doll model; (TPISC III: Clarity)
• The significance of the expanding and increasingly inter-connected PTs, as the BIM itself expands, is one in which the perfect-symmetry geometry of regular shapes and solids — equal triangles, squares, circles,… of the BIM allows — at certain articulation nodes (i.e., Rows) — the introduction of the slightly less-perfect-symmetry geometry (i.e., bilateral symmetry) of the full rectangle and oval that the non-isosceles right triangle PTs represent, into the unfolding structural framework, working from the ground up, if you will. The roots of fractals-based self organization are first to grow here! (TPISC III: Clarity & TPISC IV: Details) more…
• ALL self organization of any sort — be it force field or particulate matter — must have an organizing mathematical layer below driving it!

24 SUMMARY

SUMMARY:
• The BIM is Symmetrical down the Diagonal.

• ALL Natural Whole Integer Numbers (WIN) on the Axis.

• ALL WINs ➗ by 24 in YELLOW (24Y).

• ALL PPTs and PRIMES (>2) are ODD #s.

• ALL PPTs and PRIMES (>2) are NEVER ➗3.

• ALL PTs & ALL PRIMES on 24Y Rows/Columns in PURPLE (= “ACTIVERows).

• Some PURPLE 24Y Rows/Columns have neither PT/PRIME.

• ODD #s ➗3 are NEVER PPTs or PRIMES, thus NEVER on ACTIVE (PURPLE) Rows.

• ODD #s ➗3 ALWAYS follow two ACTIVE Rows forming a repetitive set.

• Addition of 24 to ANY ACTIVE Row ODD # = another ACTIVE Row ODD #, while addition to an non-ACTIVE ODD # = another non-ACTIVE ODD # ➗3.

• ALL Squared #s that are PPTs, remain PPTs. ANY PPT #(x) times itself, times its square (x2) and/or times it serial products = NEW PPT

Example1: 5x5=25, 5x25=125, 5x125=625, 5x625=3125, 5x3125=15625=1252, 5x15625=78125, 5x78125=390625=6252,… products are ALL PPTs.

Example 2: 97x97=9409, 97x9409=912,673, 97x912,673=88,529,281=94092=ALL PPTs.

• ALL Squared #s that are NOT PPTs, remain NOT PPTs when x2 or √x as above.

DETAILS:
• Take BIM and divide all numbers evenly divisible by 24.
• This gives you a criss-cross pattern based on 12, i.e. 12,24,36,48,… from Axis.
• Halfway between, are the EVEN # rows based on 6.
• On either side of this 6-based and 12-based frequency, the rows just before and just after, are ACTIVE Rows. These are ALWAYS ODD # Rows. They form an Active Row Set (ARS).
• Their Axis #s are NEVER ➗3. They ALWAYS have their 1st Col value ➗ by 24.
• Adding 24 to ANY of the ODD # NOT ➗3 ACTIVE Row Axis values ALWAYS sums to a value NOT ➗3 and thus to another ACTIVE Row Axis value (as adding 2 + 4 = 6, ➗3 added to a value NOT ➗3 = NOT ➗3 sum*).
• Another ODD Axis # Row lies before and after each pair of ACTIVE Rows, i.e. between EVERY set of two ACTIVE Rows, is an ODD non-ACTIVE Row and their Col 1 value is NOT ➗by 24.
• Adding 24 to ANY of these ODD # ➗3 Axis values ALWAYS sums to a value also ➗3 (as 2 + 4 = 6, ➗3 added to a value already ➗3 = ➗3 sum *).
• While not an exclusive condition, it is a necessary condition, that ALL PPTs and ALL Primes have Col 1 evenly ➗ by 24.
• Together, two ACTIVEs + one non-ACTIVE form a repetitive pattern down the Axis, i.e. ARS + non-Active Row.
• *While 24 seems to define this relationship, any EVEN # ➗3 will pick out much if not all of this pattern, e.i., 6, 12, 18,…
• It follows that:
1. ALL PTs (gray with small black dot) fall on an ACTIVE Row.

2. ALL PRIMES (red with faint RED circle) fall on an ACTIVE Row.

3. The difference, ∆, in the SQUARED Axis #s on any two ACTIVE Rows is ALWAYS divisible by 24.

4. The difference, ∆, in the SQUARED Axis #s on an non-ACTIVE ODD Row and an ACTIVE Row is NEVER divisible by 24.

5. The difference, ∆, in the SQUARED Axis #s on any non-ACTIVE ODD Row and another non-ACTIVE ODD Row is ALWAYS divisible by 24.

6. Going sequentially down the Axis, every ODD number in the series follows this pattern:

nA—A-A—nA—A-A—nA—A-A—

7. #3—5-7—9—11-13—15—17-19—21… Every 3rd ODD # (starting with 3) is ➗by 3 = nA .

8. #5-7—9—11-13—15—17-19 Every 1st & 2nd, 4th & 5th, 7th & 8th,… ODD # is NOT ➗ by 3 = A. In other words, the two consecutive ODD #s, between the the nA ODD #s, are A ODD #s and are NOT ➗ by 3.

9. 3,4,5 re-stated: let A = ACTIVE Row Axis #, nA = non-ACTIVE Row Axis

A22-A12= ➗ 24 and A ≠ ➗by 3

nA2-A2≠ ➗ 24

nA22-nA12= ➗ 24 and nA = ➗by 3

1. 6,7,8 re-stated:

ODD Axis #s ➗by 3 (every 3rd ODD #) are NEVER ACTIVE Row members — thus never PT/PRIME

ODD Axis #s NOT ➗by 3 (every 1,2 — 4,5 — 7,8….ODD #s ) are ALWAYS ACTIVE Row members and candidates for being PT and/or PRIME.

In brief:

An ACTIVE Row ODD Axis # squared + a multiple of 24 (as 24x) = Another ACTIVE Row ODD Axis # squared , and the Square Root = a PT and/or a PRIME # candidate :

A12+ 24x = A22 and √A22 = A2 = a PT and/or PRIME candidate;

ODD12+ 24x = ODD22 and √ODD22 = ODD2 = a PT and/or PRIME candidate, if and only if, its 1st Col. value is ➗ by 24.

The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!

On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7)

BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found.) See below under Why?

Sub-Matrix 2:

Once again:

• These (colored inset boxes) Sub-Matrix 2 values:

• ALL PPTs have Col 1 ➗4
• NO PPTs have Col 1 NOT ➗4
• For any given Active Rows Set, only 1 Row is a ➗4 Row
• SOME Col 1 ➗4 Rows are NOT PPTs ( starred )
• The NOT PPTs ( starred ) Axis #s are ➗Prime Factors*.

Fool-proof Steps to Find ALL PPTs

1. Axis# must be ODD, NOT ➗3 = Active Row Set (ARS) member

2. Only 1 of the 2 ARS can be a PPT

3. Sub-Matrix Col 1 # MUST be ➗4

4. SOME may NOT be PPT if ➗Prime Factor (>5)

5. Remaining Axis # is a PPT. Exceptions:

• Squared #s that are PPTs, remain PPTs when x2 or √x,

• ALL Squared #s that are PPTs, remain PPTs. ANY PPT #(x) times itself, times its square (x2) and/or times it serial products = NEW PPT
• Example1: 5x5=25, 5x25=125, 5x125=625, 5x625=3125, 5x3125=15625=1252, 5x15625=78125, 5x78125=390625=6252,… products are ALL PPTs.
• Example 2: 97x97=9409, 97x9409=912,673, 97x912,673=88,529,281=94092=ALL PPTs.
• Squared #s that are NOT, remain NOT when x2 or √x, as above.

As to answers to the open questions called above:

• what exactly is the relationship between PPTs and PRIMES?

• a loosely threaded connection is quite apparent;
• why do they BOTH land on ARS?

• they both must be ODD #s, not ➗3, whose (x2-1)/24 is true;
• why do some ARS have both, neither, or one or the other?

• some clouds, some clarity, at least for the PPTs;
• can the PRIMES be used to predict the PPTs?

• yes, in the sense that if the PPT candidate is ➗Prime Factor (>5), it will not be a PPT;
• can the PPTs be used to predict the PRIMES?

• currently, NO, yet the threaded connections are so great that the pattern will eventually emerge!

The difference in the squared values of any two PTs/PRIME #s (>3) is ALWAYS a multiple of 24!

On the Prime Diagonal, the ODD #s follow the same pattern as on the Axis (see No.7)

BIM÷24: SubMatrix Sidebar: What is the role of 24 in the underlying structure?

SEE: Tables: 33a, 33b and 33c. towards the end of Appendix B for some very NEW INFO on the BIM÷24.

The underlying geometry of the BIM÷24 PRE-SELECTS the Axis Rows into TWO Groups: ARs and NON-ARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NON-ARs. While both Groups follow (PD2 - PD2)÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.

The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.

The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The Pythagorean-Inverse Square Connection, and the PRIMES.

Open in separate browser tab/window to see all.

BIM➗PPTs and PRIMES

BIM➗PPTs and PRIMES: (Latest: as this work was being prepared, a NEW relationship was found. SEE: Tables 31-32 in Appendix B for a great deal more info and proofs!)

A dovetailing of PPTs and PRIMES on the BIM

The discovery of the Active Row Sets (ARS) — the direct result of the BIM ➗24 — in which it has been found that ALL PPTs and ALL PRIMES are exclusively found on, was in and of itself, a slow an arduous journey.

Once found, it has added a great deal of visual graphic clarity! In simple terms, it simply marks out the obvious. Both the PPTs and the PRIMES can not be on Axis Row #s that are EVEN, nor ➗3. This leaves ONLY Rows that are ODD #s and not ➗3.

The BIM ➗24 marked those Active Row Sets indirectly, by being on either side — i.e., +/- 1 — of the Axis Row # intercepted by the ➗24. Directly, the ARS was shown to be picked out by Sub-matrix 1 and 2 values of the 1st cell column of those Rows.

So we have the PPTs and the PRIMES occupying the same footprint rows, the ARS Rows. Both as a necessary, but not sufficient for primality requirement, i.e., some ARS Rows do NOT have a PPT or PRIME, or both. ALL PPTs and ALL PRIMES are ALWAYS found on an ARS Row, NEVER on a non-ARS Row. Some ARS Rows may have none, either a PPT or a PRIME, or both.

Nevertheless, on this vast matrix array of ISL whole integer numbers, that the PPTs and PRIMES exclusively occupy the same ➗24-based footprint points to an underlying connection!

The 1st connection was found and written about in the three white papers of 2005-6 on PRIMES:

The 2nd connection, as referenced below, has been the latest discovery that Euler’s 6n+1 and 6n-1 pick out, as a necessary — but not sufficient for primality — condition ALL the PRIMES.

When you look at the BIM24, you can readily see how this theorem simply picks out the very same ARS Rows! (For ARS 5 and above.)

The BIM24 becomes a DIRECT GRAPHIC VISUALIZATION of EULER’s PRIMES = 6n+1 and 6n-1, where n=1,2,3,..

The 3rd connection is that for those ARS Rows that do NOT contain PRIMES — e.i., 25, 35, 49, … and have been shown to negate the possibility of the # being prime because it is itself prime factorable — divisible by another set of primes — is ALSO DIRECTLY VISUALIZABLE ON THE BIM24 AS THE INTERSECTING PRIME COLUMNS!!!

The 4th connection reveals that the BIM Prime Diagonal (PD) — the simple squares of the Axis #s — defines:

• The ISL itself, as every BIM Inner Grid cell value is simply the difference between its horizontal and vertical PD values;
• The Pythagorean Triples, as every PD cell value points to a PT when one drops down from it to its ARS intercept;
• The PRIMES, as the difference in the squares of any two PRIMES (≧5) — every PD value of a PRIME Axis # — is evenly ➗24.

The 5th connection is that for those ARS Rows— that may or may not contain PPTs and/or PRIMES — their 1,5,7,11,13,17,19,23,25,.. ODD intersecting Columns NOT ➗3, ARE ALL➗24, and, this is ALSO DIRECTLY VISUALIZABLE ON THE BIM24 AS THE INTERSECTING COLUMNS (usually depicted graphically in YELLOW-ORANGE boxes/cells on the BIM as part of the diamond with centers pattern) !!!

￼ ￼ If P = X, evenly ➗24, then P = a PRIME Candidate where, X=evenly➗24, P=Prime, P22=larger Prime Squared, P21=smaller PRIME Squared, n=1,2,3,…

TPISC, The Pythagorean-Inverse Square Connection, has evolved into a connection with the PRIMES (TPISC-P).

How Visualization of the PRIMES on the BIM ÷24 Simplifies Much of the Mysteries of the PRIME Conjectures and What This Means!

Much is made of the PRIME Conjectures: https://primes.utm.edu/notes/conjectures/ & site:primes.utm.edu Prime Gap conjecture

1. Goldbach (Strong) Conjecture (every even # is made of two primes)
2. Goldbach Tertiary (Weak) Conjecture (every odd # is made of three primes)
3. Twin Prime Conjecture (there are an infinite # of primes separated by 2)
4. Twin Primes Conjecture (there are an infinite # of primes separated by a fixed # gap)
5. Prime Gap (size between consecutive primes)
6. Prime Triple Conjecture (there are an infinite # of 3 consecutive primes with ∆ of 6, first and last)
7. Prime Quadruple Conjecture (there are an infinite # of 4 consecutive primes with ∆ of 8, first and last)
8. Prime k-tuplet Conjecture (there are an infinite # of prime k-tuplets for each* k*)
9. Dickson’s Conjecture (there are a lot of primes, Twin, Sophie Germain, k-tuplet,…)
10. Pythagorean Primes Conjecture(Pythagorean triples with prime # hypotenuse)

A look at the BIM÷24 grid will quickly give one a visualization of the limits — and expanses — of these conjectures!

1. (1 above.) This was examined and a proof-solution was offered in 2010 with the Periodic Table of Primes (PTOP), hidden within the BIM.

2. (2 above.) Work remains.

3. (3 above.) ALL Primes fall on Active Rows (ARs) within an Active Row Set (ARS) of three Axis #s: two ODDs with an EVEN in-between. The difference (∆) between the ODDs = 2. Thus ANY and ALL Twin Primes — separated by 2 — are seen right here, and only here, directly, on the ARS.

4. (4-8 above.) Because the ARSs follow a strict Number Pattern Sequence (NPS) in that the EVEN #s are ÷12 — being the endpoint of BIM÷24 — and the bookcased ODD #s are NEVER ÷3, there is a built-in ∆, a natural gap, between the ARS and ODD #s ÷ 3. This means the ARs — that provide a necessary, but not sufficient requirement for ANY PRIME — must necessarily have gaps that are:

taken from the lower ARS #:

• (Twins of) 2, 6, 8, 12, 14, 18, 20,… i.e., up 2, 4, 2, 4, 2, 4, 2,….

taken from the higher ARS #:

• (Twins of) 4, 6, 10, 12, 16, 18, 22,… i.e., up 4, 2, 4, 2, 4, 2, 4,….
5. (9 above.) Dickson’s Conjecture leads to many of the individual conjectures above. One, the Sophie Germain Prime Conjecture states that if a PRIME #, p, has another PRIME # generated at 2p + 1, it is a Sophie Germain PRIME and that there are an infinite number of these. On the BIM, it is easy to see that any Sophie PRIME is simply taking the lower AR # and ADDING a multiple of 6 — 6x — to get to the next PRIME. Try it: (≧5)5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, ... OEIS: A005384. (https://en.wikipedia.org/wiki/Sophie_Germain_prime)

*the lower AR + 2 = higher AR — neither is ÷ by 3, while the lower AR - 2 = ODD# that is ÷ by 3 and NOT an AR.

This works for p≧5:

5+6=11, 11+(2)6=23, 23+6=29, 29+(2)6=41, 53+(5)6=83, 83+6=89, 89+(4)6=113, 113+(3)6=131, 131+(7)6=173, 173+(1)6=179, 179+(2)6=191, 191+(7)6=233, 233+(1)6=239, 239+(2)6=251, 251+(5)6=281, 281+(2)6=293, 293+(11)6=359, 359+(10)6=419, 419+(2)6=431, 431+(2)6=443, 443+(8)6=491, 491+(3)6=509, 509+(14)6=593, 593+(8)6=641, 641+(2)6=653, 653+(1)6=659, 659+(4)6=683, 683+(6)6=719, 719+(4)6=743, 743+(3)6=761, 761+(8)6=809, 809+(17)6=811, 911+(7)6=953, …

It appears that this conjecture really is simply restating Euler’s 6n+1 and 6n-1 — where n=1,2,3,… method for determining ALL PRIMES.

6. (10 above.) Ironically, the Pythagorean Triples were first to be found on the ARS, and only after reviewing earlier work — the Butterfly Primes — that the connection of the PRIMES to the BIM÷24 was made. Both the Primitive Pythagorean Triples (PPTs) and the PRIMES have the same — again, necessary, but not sufficient requirement — that they occupy EXCLUSIVELY the ARs potentially of any ARS. When the PPTs overlap the PRIMES, we get the Pythagorean Primes. An interesting note is that unlike the PRIMES that can occupy BOTH ARs of a given ARS, the PPTs will NEVER do this, occupying either one or the other, but NEVER BOTH, ARs within a ARS.

7. Fermat's (Fermat-Euler) 4n + 1 = Sum of Two Squares Theorem, where 4n + 3 ≠ Sum of Two Squares. The 4n + 1 = Sum of Two Squares = Pythagorean Primes (PTs where the hypotenuse, c = Prime #).

8. Remember, not only does the BIM÷24 reveal the ARs within the ARS, the∆ BETWEEN THE SQUARES OF ANY PAIR OF AR Axis #s (5) IS A MULTIPLE OF 24. The latter becomes a necessary, but not sufficient test of both primality, and, PPT validity.

9. Note that this same ∆ occurs with pairs of two non-AR ODD #s, i.e. 9 and 15. Both are ÷3. 152 - 92 = 144, and 144/24 = 6. A hybrid of an AR and a non-AR ODD pair set will NOT show a ∆ of 24x.

10. This points to underlying NPS within the BIM (see the pattern below the links):

For the ODD #s2:

∆÷24 vs NOT ∆÷24

∆÷24 in ARS vs ∆÷24 in non-ARS, NEVER a mix of the two

For the EVEN #s2:

∆÷24 vs NOT ∆÷24

∆÷24 includes those in progressive NPS series ∆2, ∆4, ∆6, ∆8, ∆10,… (see Table in Appendix ).

11. The PPTs and PRIMES are STRICTLY following this NPS within the BIM!

12. Perhaps the biggest finding here is: The PRIMES — Inverse Square Law Connection! Just like the PPTs! TPISC stands for both The Pythagorean-Inverse Square Connection, AND, The PRIMES-Inverse Square Connection. TP-P-ISC, TP/P-ISC, TP/PISC, TPPISC, or TP-IS-PC,...

The connection between that Universal Law, the ISL, — the underlying law of ALL of spacetime —and the PRIMES — the fundamental "quarks" of the number quantity system, AND, the Pythagorean Triples — the fundamental right-triangle/rectangle form of that same geometry — is without out a doubt the most intriguing, beguiling, and misunderstood relationship we are only just NOW getting a real glimpse at. The future looks very promising!

https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture

https://en.wikipedia.org/wiki/Pythagorean_triple

https://en.wikipedia.org/wiki/Twin_prime

https://en.wikipedia.org/wiki/Prime_gap

https://en.wikipedia.org/wiki/Prime_triplet

https://en.wikipedia.org/wiki/Prime_k-tuple

https://en.wikipedia.org/wiki/Dickson%27s_conjecture

https://en.wikipedia.org/wiki/Sophie_Germain_prime

https://en.wikipedia.org/w/index.php?search=Pythagorean+Primes&title=Special%3ASearch&go=Go

https://en.wikipedia.org/w/index.php?search=Pythagorean+Primes+Conjecture&title=Special%3ASearch&go=Go

https://en.wikipedia.org/wiki/Prime_k-tuple

https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares

https://en.wikipedia.org/wiki/Pythagorean_prime

https://en.wikipedia.org/wiki/Fermat%27s_little_theorem

https://en.wikipedia.org/wiki/Euler%27s_theorem

https://en.wikipedia.org/wiki/Euclid–Euler_theorem

http://mathworld.wolfram.com/Eulers6nPlus1Theorem.html

What is the underlying NPS of the BIM ÷24?

Earlier we found:

For the ODD #s2

• ∆÷24 vs NOT ∆÷24
• ∆÷24 in ARS vs ∆÷24 in non-ARS, NEVER a mix of the two

For the EVEN #s2

• ∆÷24 vs NOT ∆÷24
• ∆÷24 includes those in progressive NPS series ∆2, ∆4, ∆6, ∆8, ∆10,… (see Table in Appendix ).

The PPTs and PRIMES are STRICTLY following this NPS within the BIM!

But what about just the BIM ÷24. How does this pattern out?

Let’s dispense with the EVENs first.

1. If one takes the EVEN #s on either side of the ARS + the EVEN # in the MIDDLE of the ARS, we have a new set. Let’s call it the EVENs Set (ES).

2. Well, it turns out there are two versions of the ESs and they alternate down the PD. Let’s call them ES1 and ES2 and they go: ES1-ES2-ES1-ES2,…

3. If we plot the ESs along the PD it will look like this (remember these are the squares of the EVEN Axis #s):

16 — 36 — 64 100 —144 — 196

256 —324— 400 484 — 576 — 676

784 —900—1024 1156—1296—1444

1600—1764—1936 2116—2304—2500

The ES1 are on the Left, the ES2 are on the Right.

The ÷24 NPS of the ES1 andES2 play out as:

• Both group sets bookend the ARS with an ODD #÷3 (NON-ARS ODD) between.

• The OUTER ends of ES1 are ∆/24 and the OUTER ends of ES2 are ∆/24.

• The MIDDLE of ES1 are ∆/24 and the MIDDLE of ES2 are ∆/24.

• There is NO MIXING between sets and NO MIXING OUTER with MIDDLE #s.

• The MIDDLE ES1 is ÷ by the 1st MIDDLE of the whole ES1 set.
• The MIDDLE ES2 is ÷ by the 1st MIDDLE of the whole ES2 set.
• ES1 ÷ 16 ÷ 8 ÷ 4 ÷ 2 OUTER; MIDDLE ÷ 36, NEVER ÷ 24. (NOTE: these are the PD#s and not the ∆ in PD#s.)

• ES2 ————÷ 4 ÷ 2 OUTER; MIDDLE ÷ 36, ALWAYS ÷ 24. (NOTE: these are the PD#s and not the ∆ in PD#s.)

• A look at the BIM ÷24 shows that ALL MIDDLE PDs of their ESs are ÷12 and graphically:Table 31d. Snapshot of the PDF showing the EVENS role in the BIM ÷24 Distribution.

• ES1 fall along the IN-BETWEEN YELLOW DIAMOND PATTERN formed from the BIM ÷24.

• NEVER ÷ 24.
• ES2 fall at the Axial POINTS of the YELLOW DIAMOND PATTERN.

• ALWAYS ÷ 24.

Table31d PDF sequence. The EVENS. Open the PDFs in a separate tab/window to see all pages.

Table 33c. The EVENS

So now that we have established that the EVENS, too, reveal a ÷24 expression within the BIM, let's move on the ODDS.

We know that the ODDs ÷3 are the separators of the EVEN's ES1 and ES2 . They are also a separate player in the whole ODDs set in that though their PDs are ÷24, they are NOT part of the ARS. And as NON-ARs they do NOT MIX with the AR ODDs.

We will continue to label the two groups as ARs and NON-ARs.

Earlier we found:

For the ODD #s2

• ∆÷24 vs NOT ∆÷24
• ∆÷24 in ARS vs ∆÷24 in non-ARS, NEVER a mix of the two.

More precisely:

1. ALL ARs2 are ∆/24, and, NEVER ÷3, and after subtracting (-1) are ALWAYS ÷24, and, ALL now ÷3.

2. ALL NON-ARs2 are ∆/24, and, ALWAYS ÷3, and after subtracting (-1) NEVER ÷24, but ALL ÷9 ÷3.

3. There is NO MIXING between groups.

4. Additionally, the ODD AR group may be further divided into:

ODD Set 1 (OS1): (Like the EVENS, ES1) for IN-BETWEEN YELLOW DIAMOND Rows 6,18,30,42,...

ODD Set 2 (OS2): (Like the EVENS, ES2) for the POINTS of the YELLOW DIAMOND Rows 12,24,36,...

However, here, BOTH groups ARE MIXABLE, i.e. ∆/24 across OS1 and OS2.

5. What really becomes interesting about the ODDs on the BIM ÷24 is that, all by themselves without regard to the PPTs or PRIMES, they form a deep, underlying NPS ALL BASED ON THEIR ∆s/24!

6. In a nutshell, if you take each ODD #, square it (that's the PD#) and now sequentially substract the ODD PDs below it, and then ÷24, the results will generate a tightly connected NPS across the ODDS on the BIM.

The ARs2 1-25-49-121-169-289-361-529-625,… (12-52-72-112-132-172-192-232-252,) subtraction=ALL ARs.

The NON-ARs2 9-81-225-441-729,… (32-92-152-212-272,…) subtraction=ALL NON-ARs.

Here is a quick look at the AR and NON-AR Distribution. Open the PDFs below in a separate tab/window.

Here is a quick look at the AR and NON-AR Distribution. Open the PDFs below in a separate tab/window.

Table 33a. ODDs AR and NON-AR Distribution PDF. Open the PDFs in a separate tab/window to see all pages.

Table 33b. ODDs sequenced PDF. Open the PDFs in a separate tab/window to see all pages.

Table 33c ODDs_more detail PDF. Open the PDFs in a separate tab/window to see all pages.

Table 33d-2-1

Table 33d-2-2

Table 33d-2

~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~

NEWLY ADDED (after TPISC IV published):

Back to Part III of the BIM-Goldbach_Conjecture.

BACK: ---> TPISC IV: Details: BIM + PTs + PRIMES on a separate White Paper

BACK: ---> TPISC_IV: Details:_PRIMES_vs_NO-PRIMES on a separate White Paper

BACK: ---> PRIME GAPS on a separate White Paper

BACK: ---> PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper

BACK: ---> Make the PTOP with Fractals on a separate White Paper

BACK: ---> TPISC IV: Details White Paper

Artwork referencing TPISC.

PRIMES vs NO-PRIMES ebook in Apple Books.

~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~ ~~

PTs: 40+ FEDM derived profiling focus point parameters:

1. While the actual Area (A) and Perimeter (P) of the PT in question can be shown directly on the Matrix as grid cell areas — as can the PT proof — it is the 4x the A, and, the difference (∆) in the length of the sides, squared, that present the true Pythagorean - Inverse Square Connection.

4A + (b-a)2 = c2 = a2 + b2

​ where, (b-a)2 = (t-s)2 = ƒ2, giving 4A + ƒ2 = c2 = a2 + b2

2. Finding the A and P on the Matrix necessarily introduced some additional key focus point values — giving rise to the FEDM.

3. The 40+ key focus points are found consistently in each and every r-set Template — defining the PTs.

4. Every PT Row contains at least 8 key values directly;

5. And another 8 values indirectly by counting STEPS;

• r/4 — r/2 — ratpsb
6. Giving another 8 values, by calculation/STEPS or a mixture of both;

• APUeƒƒ2Fr2

• 8A@Ƀd/pghiJ — √Jklmnp2qvVW√W¥.
8. If you plot these out along their r-set STEPS (r-steps) spacings, the ∆ between values and their PD for any given Column follows 1r2 — 4r2 — 9r2 — 16r2 — 25r2 —…;

c2-o=1r2 169-153=16=1r2, c2-d=4r2 169-105=64=4r2, c2-g=9r2 169-25=144=9r2

9. Naturally, the PD sequence, up from the Row values, follows the same 1—4—9—16—25—...

10. The "downward" Diagonal, perpendicular from the PD, back to the Row Axis gives the ɃV¥ and c-ƒ points, and, 8A is always r-steps down the grid from the 4A location on that PT Row.

PTs: 10 ways to approach the BIM :

1. The PT Axis Row: ALL PTs have cXfob24Ada2c2

• c is on the Left SIDE Axis;
• Xƒ is r/2 steps in from Axis;
• o is r-steps in from Axis;
• b2 is a-steps in from Axis;
• 4A = c-ƒ steps coming back from the PD;
• d is 2r-steps in from the Axis;
• a2 is b-steps in from the Axis;
• c2 is on the PD at the intersect of the PT Row.
2. The Main Diagonals (Template) for ALL PTs:

​ PD — O (origin)—>c2 as oƒ2r2a2b2c2;

​ ⊥PD — c2—>2c (Axis) follows 4c8c12c16c sequence.

3. The Secondary Diagonals:

• pUd this important diagonal defines the Common Diagonal of the Golden Diamond ToPPTs;
• ded/p (on Axis).
4. The Tertiary Diagonals:

• √Wo;
• o√J (on Axis).
5. The Horizontal Axis (TOP) of Columns:

• 0—ALL of the above—from c —>c2;
• r/4—r/2—r—2r—3r—4r—.. and b-a=t-s.
6. The Vertical Axis (Left SIDE) of Rows:

• 0—ALL of the above—from c —>2c;
• sp√Wc√Jd/ptn—2cr—2r—3r—4r—....
7. The 4A8Ac2ƒ2 Rectangle Connection:

8. The Proof a2 + b2 = c2 = 4A + ƒ2:

9. The Complementary Pair (Square Pairs) Sets of ANY PT:

10. ALL PTs have matching 4A values on both the PD Row itself and r-steps down that PD Column:

10 Basic, fundamental rules of the symmetrical BBS-ISL Matrix

1. Basic BBS-ISL Rule 1: All numbers (#s) related by the 1-4-9-...PD sequence

2. Basic BBS-ISL Rule 2: Every # in the PD sequence is the square of an Axial #.

3. Basic BBS-ISL Rule 3: The Odd-Number Summation sequence forms the PD sequence.

4. Basic BBS-ISL Rule 4: Every EVEN Inner Grid (IG) # is divisible by 4 & all are present.

5. Basic BBS-ISL Rule 5: Every IG# is:

A: The difference (∆) between its two PD-sequence #s. (Note: A=B=C=D, and, E.)

• Ex: PD 25 - PD9 = 16

B: The sum (∑) of the ∆s of each of its PD#s between its two PD-sequence #s (as above).

• Ex: (PD 25 - PD16) + (PD16 - PD9) = 16

C: The ∆ between the squares of the two Axial #s forming that IG# (as above).

• Ex: 5^2 - 3^2 = 16

D: The product of the Addition & Subtraction of the two Axial #s forming that IG# (as above).

• Ex: (5 + 3) x (5 - 3) = 16

E: Also, the product of its 2 Axial #s intersected by that IG#'s 90° diagonals.

• Ex: 2 x 8 = 16
6. Basic BBS-ISL Rule 6: Every ODD IG# is NOT PRIME & all are present.

• Corollary: NO EVEN NOT divisible by 4 #s are present on the IG.
7. Basic BBS-ISL Rule 7: The ODD-Number sequence, and the 1-4-9-...PD sequence, forms the sequential ∆ between ALL IG#s.

8. Basic BBS-ISL Rule 8: The ∆ between #s within the Parallel Diagonals is a constant 2 x its Axial #.

9. Basic BBS-ISL Rule 9: The ∆ between #s in the Perpendicular Diagonals follow:

A: From EVEN PD#s, √PD x 4 starts the sequence & follows x1-x2-x3-x4....

B: From ODD PD#s, √PD x 4 starts the sequence & follows x1-x2-x3-x4....

C: From ODD Perpendicular Diagonals between the EVEN-ODD diagonals (above), the sequence starts with the same value as the Axis number ending the diagonal, the sequence following x1-x3-x5-x7..

10. Basic BBS-ISL Rule 10: Every #, especially the #s in the ONEs Column, informs both smaller and larger Sub-set symmetries (much larger grids required to demonstrate).

BBS-ISL Matrix Inner Grid Golden Rules (IGGR)

5 Basic, fundamental rules of the symmetrical BBS-ISL Matrix Inner Grid

1. IGGR 1: The IG is formed of two equal & symmetrical 90°-right, isosceles triangles that are bilaterally symmetrical about the PD — and, infinitely expandable.

2. IGGR 2: The 90°-right-triangle — inherent to ALL squares and rectangles by definition — both forms the alternating EVEN-ODD square grid cells within the Matrix, and, is responsible for all major patterns and sequences, thereupon.

3. IGGR 3: Subtraction-Addition: Every IG# is simply the ∆ between its two PD#s (subtraction), and, the sum (∑) of any IG# + its PD# above = the PD# on the end of that Row (or, Column).

4. IGGR 4: Multiplication-Division: Every IG# is simply the product of the two AXIAL #s intersected by the two diagonals — of that said IG# — pointing back to the Axis at a 90° angle (multiplication), and, the dividend of the Axial divisor and quotient (division).

5. IGGR 5: The actual # of grid-cell steps — i.e., the actual # of STEPS from a given IG# to another by a strictly horizontal, vertical, or 45° diagonal path — forms a simple, yet often fundamental descriptor to the pattern-sequence templates that inform the more advanced patterns, e.i., Exponentials and especially the Pythagorean Triples (PTs). STEPS are particularly important in the geometric visualizations within the BBS-ISL Matrix (as alluded to in IGGR 2, above).

Pythagorean Triples and BBS-ISL Fundamentals (TPISC: The Pythagorean-Inverse Square Connection)

3 Basic, fundamental rules of the symmetrical BBS-ISL Matrix Inner Grid that encompass the PTs.

1. TPISC-BBS-ISL Rule 1: Every IG EVEN Squared # is part of a Square Paired- Set (SPS) that:
• A: Has reciprocal SPS members on the PD vertically above.
• B: Both SPS members reside on the SAME Row.
• C: They represent the a2 and b2 values of a PT, whose c2 value is on the PD intersection
1. TPISC-BBS-ISL Rule 2: Every PT is found on the BBS-ISL Matrix and can be located by this intersection of EVERY PD (9>) and a Row with SPSs.

2. TPISC-BBS-ISL Rule 3: Every PT — including its sides, perimeter, area and proof — can also be found and fully profiled (and, predicted) as r-set, s-,t-set members of the Dickson Method (DM), Expanded Dickson Method (EDM), and the Fully Expanded Dickson Method (FEDM), shown herein. The role of the r-value for any given PPT (or nPT) can not be overstated: the r-value is as equally important in defining any PT as the a-, b- and c-sides.

Witness:

r = a + b - c

and,

(a + b + c + r)/2 = c + r = a + b.

And as

a = r + c - b = (c + r -ƒ)/2

solving for ƒ gives:

ƒ= b-a

and,

ƒ2= (b-a)2

giving the The Proof

a2 + b2 = c2 = 4A + ƒ2.

With

p = c - 2r

we can subsitute in

c = a + b - r

to give:

p = a + b - 3r

giving us all the basics to profile any PT, knowing that:

4A = rc + r2 = 2ab

Matrix Flow:

1. ALL WINs;
2. Axis (vertical, horizontal) +IG + PD = BIM;
3. BIM - (Axes & PD) = Inner Grid (IG);
4. IG - 1st Parallel Diagonal (P∥D)= Strict Inner Grid (SIG);
5. Axes = ALL WINs (X) sequentially on both vertical (Left SIDE) and horizontal (TOP) Axis;
6. PD = Axis2 = X2 = a2b2c2 of ALL Pythagorean Triples (PTs);
7. Axis Rows & Columns define the PD, IG, ALL PTs and ALL Exponentials Xn;
8. Parallel Diagonals (to the PD) define ALL Exponentials Xn (Exp Xn);
9. ALL PTs and ALL Exp Xn are on the BIM;
10. ALL Squared #s, by definition, are on the PD;
11. ALL Squared #s are ALSO on the IG;
12. ALL Squared Sides (a2, b2, c2) of ALL PTs (across Rows) & ALL Exp X>2 (down Parallel Diagonals) are also on the IG;
13. Therefore, ANY Squared # on the IG is PART OF A PT & some, but NOT ALL, are also Exp X>2;
14. ALL Exp Xn are found on the IG, specifically along their Parallel Diagonals (P∥Ds) ;
15. ALL Exp Xn values, like ALL IG #s, are simply the ∆ between their two PD Area values.

Exponentials Summary (see TPISC V: Exponentials):

1. For any given # X, located on the Axis, its respective x2 is, of course, located on the PD, while X3, X4, X5,.. (Xn), are ALL found on that X Diagonal Parallel ( P∥D) to the PD;

2. The distance — # of steps diagonally — between successive Exponentials X1,2,3... for a given X, follows a Number Sequence Pattern (NPS) equal to is Xn sequence value:

3. The Sum () of the Axis Column & Row # values x that Diagonal Axis # X, equals the Xn value:

• ∑(AxisCol + AxisRow) x X = Xn as does the AxisCol x AxisRow product;
• ∑(6+10) x 4 = 64 = 43, where X=4, and, AxCol x AxRow = 4 x 16 =64 = 43.
4. The Sums (s) of the ∆s between the PD #s of a given X35=X3 and is simply an expression of the IGGR:

• 23 = 1-4-9 with ∆s of 3 & 5, where 3 + 5 =8 = 23.
5. The Area (# of grid cells) of a given Xx = Area Xx~uo~ - Area Xx~LD~, and flollows the same NPS progression sequence as X3, X4, X5,… in Area and in # of PD steps.

• 1st P∥D = # ∆ 2, ODD #s (Prime & Not-Prime [NP]);
• 2nd P∥D = # ∆ 4, EVEN #s ➗ 4 = ALL Exp 2n;
• 3rd P∥D = # ∆ 6, ODD #s NP= ALL Exp 3n;
• 4th P∥D = # ∆ 8, EVEN #s ➗ 4 = ALL Exp 4n;
• 5th P∥D = # ∆ 10, ODD #s NP= ALL Exp 5n;
• 6th P∥D = # ∆ 12, EVEN #s ➗ 4 = ALL Exp 6n;
• 7th P∥D = # ∆ 14, ODD #s NP= ALL Exp 7n;
• 8th P∥D = # ∆ 16, EVEN #s ➗ 4 = ALL Exp 8n.

Summation:

Here’s the thing. Amongst a myriad of other connections, there exist an intimate connection between three number systems:

1. The ISL as laid out in the BIM;
2. PTs — and most especially PPTs — as laid out on the BIM;
3. The PRIME numbers — PRIMES — as laid out on the BIM.

The BIM is the FIXED GRID numerical array of the ISL.

Amongst its vast array of inter-connecting Number Pattern Sequences (NPS) — i.e., number systems — two such systems stick out and do so in such an overtly visual — as well as mathematical — way that their connection to each other is more than implied.

You see, both the PPTs and PRIMES strictly align themselves on the SAME paths within the BIM.

Now, their footprints upon these paths are not identical, yet their paths chosen are. If you divide the number array — i.e., the Inner Grid numbers — of the BIM by 24, a Sub-Matrix 1 grid is formed.

Upon that Sub-Matrix 1, pathways are formed on every ODD Axis number NOT ➗3.** EVERY PPT and PRIME lies on these paths!**

Yes, while in the details we show how:

1. The 1st Col. on the BIM — that which is the square of the Axis number (i.e., the Prime Diagonal number) - 1, when then ➗ by 24, equals a Whole Integer Number (WIN);
2. This defines the path — the “Active Row” upon the BIM;
3. While every Active Row path may or may not contain a PPT and/or PRIME, every PPT and/or PRIME is ALWAYS located on one of these paths;
4. The difference (∆) between the squares of any two PRIMES (>5) is also ➗24;
5. The serial — and exponential — products of ANY and ALL PPTs remain PPTs, whilst those NOT remain NOT.

Furthermore, by distilling the BIM to Sub-Matrix 2 — i.e., every number across a Row is progressively divided by a growing sequence series starting with the Col. 1 as the Axis number - 1 — every such serial-exponential PPT is clearly predicted by its neighbors within the sequence. One more example of the extremely intimate relationship between The Pythagorean - Inverse Square Connection (TPISC).

One simply can not ignore how the PRIMES, the PTs and ultimately the ISL define the architecture of SpaceTime!

In the original MathspeedST, an artificial division was made, separating the content into:

1. TAOST, The Architecture Of SpaceTime,
2. TCAOP, The Complete Absence Of Primes.

Now we have come full circle.

ALL PRIMES and ALL PPTs follow — although individually with their own respective footprints — the SAME, HIGHLY PATTERNED NPS path on the BIM.

This is no coincidence. The ➗24 Active Row Pattern that defines this path on the BIM does so in a highly ordered pattern. The energy density that expresses itself as curved ST does so precisely by the numerical architect of the built-in ISL.

We now have some very strong evidence that the numbers that define the PPTs and the numbers that define the PRIMES are ubiquitously linked throughout the entire number pattern array that defines the ISL. This is revealed on the BIM!

Surely their interplay provides some very significant contributions to the overall Architecture of SpaceTime!

feel the beat: + - + - - - + - + - - - + - + - - - + - + - - - ...

VIMEO "BIM_PRIMES_24"1! )

UPDATE: Finally, the NEW relationship between the BIM÷24 and the PRIMES has been found! See the bottom of APPENDIX B for the tables that document this work! From the APPENDIX B:

Back to the Table 31 sequences. Table 31a... series, along with the previous Table33 series, uncovers how the PRIMES were found within the BIM both by algebra and geometry. This has been reduced to a focus white paper: "PRIMES vs NO-PRIMES that is a condensed summary of this work. One may benefit from looking over this paper prior to the study of these more elaborate tables. A summary from the paper is presented here:

SUMMARY*

How do we go from the simple grid of the BIM (BBS-ISL Matrix) to identifying the PRIMES?

1. The BIM is a symmetrical grid — divided equally down its diagonal center with the Prime Diagonal (PD) — that illuminates the Number Pattern Sequence (NPS) of the Inverse Square Law (ISL) via simple, natural Whole Integer Numbers (WIN).

2. The BIM Axis numbers are 1,2,3,.. with 0 at the origin.

3. The Inner Grid (IG) contains EVEN and ODD WIN, but except for the 1st diagonal next to the PD — a diagonal that contains ALL the ODD WIN — there are NO PRIMES (NO-PRIMES, NP) on the SIG (Strict Inner Grid).

4. The PD WIN are simple the square of the Axis WIN.

5. ALL the IG WIN result from subtracting the horizontal from the vertical intersection of the PD.

6. Dropping down a given PD Squared WIN (>4) until it intersects with another squared WIN on a Row below will ALWAYS reveal that Row to be a Primitive Pythagorean Triple (PPT) Row, whose hypotenuse, c, lies on the intersecting PD. ALL PPTs may be identified this way.

7. Dividing the BIM cell values by 24 — BIM÷24 — forms a criss-crossing DIAMOND NPS that divides the overall BIM into two distinct and alternating Row (and Column) bands or sets:

• ODD WIN that are ÷3 and referred to as NON-ARs;
• ODD WIN that are NOT ÷3 and referred to as ARs, or Active Rows;
• The ARs ALWAYS come in pairs — with an EVEN WIN between — as the UPPER and LOWER AR of the ARS (Active Row Set);
8. ALL PPTs and ALL PRIMES ALWAYS are found exclusively on the ARs — no exceptions.

9. By applying:

(1)                  * $6yx ± y = NP$

let y = odd number (ODD) 3, 5, 7,… and x = 1, 2, 3,... one generates a NP table containing ALL the NP;

*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool;

10. Eliminating the NP — and the NP contain a NPS — from ALL the ODD WIN, reveals the PRIMES (P).

11. A necessary, but not sufficient confirmation — but not proof — of primality is found by finding the even division of 24 into the difference of the square of ANY two PRIMES as:

(2)                  (P2)2 - (P1)2 = $n24$

let $n = 1, 2, 3,...$

Be aware that this also holds true for ALL the AR NP. The P and NP are NOT ÷3, and are both part of the ARS and therefore any combination of the two squared differences will be ÷24:

(3)                  (NP2)2 - (NP1)2 = $n24$

(4)                  (NP2)2 - (P1)2 = $n24$

(5)                  (P2)2 - (NP1)2 = $n24$

The ÷3 NON-AR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:

(6)                  (NON-AR-NP2)2 - (NON-AR-NP1)2 = n24

(7)                  (NON-AR-NP2)2 - (NP1)2$n24$

(8)                  (NP2)2 - (NON-AR-NP1)2$n24$

(9)                  (P2)2 - (NON-AR-NP1)2$n24$

(10)                (NON-AR-NP2)2 - (P1)2$n24$

The division into AR and NON-AR sets has a NPS that ultimately define the elusive pattern of the P.

Furthermore, $6yx ± y = NP$ may be re-arranged to:

(11)                  $(NP ± y)/6yx = 0$

(12)                  $(NP ± y)/6x = y$

asking whether any given ODD (>3) is a P or NP, it is exclusively a NP if, and only if, y reduces to the same value after applying x. As y is effectively an ODD of either a PRIME or composite of PRIMES factor*, one only needs to satisfy a single instance to validate NON-Primality.

12. One can also obtain ALL the P by eliminating the BIM SIGO and O2 from the 1st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O2 = ODDs2, and the 1st Diagonal = the 1st Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O2 — the O2 being the PD values — such that we now have:

(13)   SIGO(A2) = Strict Inner Grid ODDS & ODD AXIS2 giving a distinct visualization advantage;

(14)                  NP = SIGO(A2)

(15)                  1st Diagonal - SIGO(A2) = P.

This second method — the algebraic geometry method — as presented here.

Introduction

Two methods — one pure algebraic and the other a more visual algebraic geometry presented here — have been found that capture ALL the NO-PRIMES (NP). While they process slightly different, they dovetail nicely into a very visual Number Pattern Sequence (NPS) here on the BIM. They both give identical NP results.

So what is the significance of capturing ALL the NP?

The NP are the highly NPS that define the elusive pattern of the P. P + NP = ALL ODD WINs (≥3).

In any group of WIN, if you know the NP, you also know the P. Here is the highly visualizable geometric method for capturing ALL NP. In fact, it is just as simply stated in 12. of the SUMMARY.

Algebraic Geometry Method

One can obtain ALL the P by eliminating the BIM SIGO and O2 from the 1st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O2 = ODDs2, and the 1st Diagonal = the 1st Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O2 — the O2 being the PD values — such that we now have:

• One can also obtain ALL the P by eliminating the BIM SIGO and O2 from the 1st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O2 = ODDs2, and the 1st Diagonal = the 1st Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O2 — the O2 being the PD values — such that we now have:

(13)   SIGO(A2) = Strict Inner Grid ODDS & ODD AXIS2

(14)                  NP = SIGO(A2)

(15)                  1st Diagonal - SIGO(A2) = P.

• The PDFs will follow these animated gifs. Videos and other supporting graphics thereafter.

Animated Gifs:

PRIMES vs NO-PRIMES-2: algebraic method.

PRIMES vs NO-PRIMES-1: algebraic-geometry method.

PRIMES vs NO-PRIMES-3: algebraic and algebraic geometry method.

PRIMES vs NO-PRIMES-4: algebraic method in detail.

PDFs +

PRIMES vs NO-PRIMES: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The NPS of the NP define the elusive pattern of the P.

PRIMES vs NO-PRIMES: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES PDF: snapshot-4 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

Table31a4: PRIMES vs NO-PRIMES PDF: Open in a separate tab/window to see all 11 pages. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--are shown individually and collectively as sets. The full Upper and Lower tables conclude.

Table31a6_2: PRIMES vs NO-PRIMES PDF: If you look at the ODD Axis ÷3 NO-PRIMES (NP) that lie in the paths between the L-shaped Double-wide x-base sets (x=1,2,3,)…), one finds a distinct Number Pattern Sequence (NPS) between successive NP values. Those shown in BLUE are NOT included in the criss-crossing L-shaped Double-wide paths (seen in the snapshots above), while those in GRAY are. The NPS seen here, based ultimately on the 1,3,5,7,… ODD number summation series that defines the whole BIM distribution (including the 1st Diagonal, the PD, and the successive differences in sequential Inner Grid cell values) reiterates that of both the L-shaped Double-wide paths as well as the individual x paths. Both give a NPS of the NP that reveal the elusive pattern of the PRIMES. There remains little doubt that the PRIMES , as well as the Primitive Pythagorean Triples (PPTs), are intimately related to the INVERSE SQUARE LAW (ISL)!

Videos:

...more supporting graphics and tables:

Table31a4-5_+RunDiff_EQUATIONsDEMO(divide3Filter)+10x10+. A simple table to demo the NP process.

Table31a4_+RunDiff_EQUATION-PATTERNS-LOWER+UPPER+Annotated. NP Table with layout notes.

Table31a4_+RunDiff_EQUATIONs(SqrdAxis-seqPD)+50x500+.pdf. From the BIM to the NP/P Tables.

BIM35-Table31a4-5_NO-PRIMES_Factores-withARS-YELLOW-numbAnnotated. The NP pattern on the BIM.

Table33a,c,d_ODDsqrddivide24_sheets. The NP pattern – an amazing NPS – defines the P.

BIMrows1-1000+Primes_sheets+Sub-Matrix2. A BIM reference showing ÷24 ARS, PPTs, NP an P.

For the full documentation of all the graphics and pdfs — including history, development, expanded tables, and more — see the link references immediately below. Appendix B has all the documentation.

REFERENCES

While there are no specific references for this work other than referring back to my own original work, there are many references involved in the study and research on the PRIMES in general. These have been well documented in the TPISC_IV: Details_BIM+PPT+PRIMES focus white paper. This paper also contains the background graphics and tables leading up to this current work. The focus page is part of the much larger TPISC_IV: Details ebook project that is freely available as an HTML webpage. The quick reference outline can be found here.

VIII. Conclusion

From SQUARES to RECTANGLES — CIRCLES to OVALS — ISOSCELES to non-ISOSCELES TRIANGLES: That's the story of TPISC: The Pythagorean - Inverse Square Connection.

The Areas found on the PD become the very SAME Areas on the IG in the BIM. These IG Square Areas are the Squared Sides of the PTs, and thus are part of the "PROOF" of the Pythagorean Theorem, and, they are each present on the Matrix as both a grid cell value and literally as actual square grid cell units giving us a beautiful visual proof of the algebraic geometry they express!

In forming the proofs, each PT — because of its presence 4x — becomes engaged in an even larger set of correlations to itself, individually, as well as to other PTs. The inter-connections are beyond extensive, they are so deeply ingrained into the PTs as a whole — as with the Tree of Pythagorean Triples (ToPPT) — that structurally they become a fractal-driven sub-structure framework within the larger confines of the BIM.

The simple beauty of the 4A and 8A Areas both proving the Theorem and informing another PT generation, belies what also is going on here: as we have pounded home repeatedly, strict symmetry is given focus nodes for the introduction of some slight asymmetry — provided by PTs whose Template-driven fractal expression gives both harmony and balance to the evolving SpaceTime structure that the BIM is forming — and this very process, so intimately informed by the BIM, that pure ST is given the possibilities of unique expression. Expression of ST energy in the form of wave-particle ST units whose categorical identities are dependent on just such variations in the ST configuration patterns — mass, spin, charge,… — and yet are always related back to the very ISL driven ST from which they arose.

The implication for these PT-driven ST expressions become an intriguing challenge to the interpretation of the Double-Slit Experiment and the very notion of Quantum Entanglement. The DSEQEC (Double-Slit Experiment Quantum Entanglement Conjecture) suggests that these two phenomena are essentially just two-sides of the same coin. The wave-particle ST unit supposition — that a ST unit may simultaneously present itself as the sum total of all its possible quantum states and yet definitely collapse its "wave-" portion to be expressed as a "particle-" ST unit is — the conjecture maintains — a built in property of the very BIM and ToPPT forming any and all ST units pulse-propagating from their singularities to an expression within our Universe.

The quantum universe begs the question: what is the quantum?

Perhaps we can say the basic pulse-propagation of ST from its singularity — a singularity connected to the larger pool of all singularities by the Conservation of Energy — out into full spacetime extension, and back, is the fundamental quantum. The ST so formed has articulation points, nodes of structural formation made by the embedded Pythagorean Triples. These nodes are also quantized as they both are dependent upon the unfolding Inverse Square Law-based ST, and they have distinct vector parameters inherently build into their asymmetric forms. They have both magnitude and direction — and as vectors their net expression can be the result of the total added or subtract interfering ST units. The fact that every primitive PT comes in both its parent PPT form as well as its child nPPT form provides a built in fractal-like structure, that, when combined with the fact that each and every PT — PPT and nPPT — has 4 iterations of expression within the unfolding ST unit pulse-propagation, and we we now have a fractal-based, holographic quantum universe(s)! The Creation and Conservation of SpaceTime (CaCoST) model brings it all together in an attempt to answer some of the most outstanding unanswered questions of physics today.

Final Thoughts

Even mathematically, Nature can be expressed in more than one way. While the view may change, Nature seems to prefer efficiency.

In order to satisfy the Conservation Laws — from the initial Conservation of SpaceTime to its outter flanks: Conservation of Momentum, Angular Momentum, Charge and, of course, Energy — that means an accounting system that is primarily "pay-as-you-go." Rather than balancing the entries across the expanse of the Universe at some distant point in time, it's much more expedient to balance the parts as you go.

When ST — and that includs all ST unit entities — is formed, its balance sheet requries both a matter and antimatter accounting. The Equal And Opposite (EAO) quantum state parameters — mass, charge and spin — that are parts of its energy expression, are accounted for right from the get go. A pairing — a virtual pairing is one way of looking at it — that is part and parcel of the very creation of ST.

The BIM itself informs that ST, and, if, as proposed, the 4 iterations of each PT within the BIM represent the quantum state numbers of any ST unit — and their information is ubiquitously known for every ST unit p-p — we now have great insights into the Creation and Conservation of SpaceTime (CaCoST).

Like the Big Bang itself, every little bang, p-p ST unit is but a fractal mimicking the process and accounting for the quantum as:

• Singularity
• PEH
• Inflation-expansion
• Matter - Antimatter
• Dark Matter - Dark Energy
• DSE
• QE
• (EP = EPR)
• CaCoST

This is the CaCoST model!

Lastly, the hugely important finding that ALL PTs and ALL PRIMES ALWAYS fall on ONLY Active Rows — either side of BIM ÷24 Rows, and, the very same Rows algebraically described by Euler's Prime=6n =/-1 Theorem — is not to be dismissed as it points to yet another very deep relationship between The Pythagorean - Inverse Square Connection (TPISC), the "Details" of which we have laid out in this work!

The difference in the Squares of EVERY Active Row number — including ALL PTs and ALL PRIMES — is ALWAYS an even multiple of 24! This is built into the Prime Diagonal (PD) of the BIM itself!

CaCoST-DSEQEC with Hands representing "spin."

Every p-p unfolds a BIM grid where ALL info is known throughout its expansion. Letting the Pythagorean Triples (PTs) represent "spin" and their vector "direction" orientation (Left, L or Right, R), one finds 4 iterations of spin (2-L and 2-R) automatically built into the ISL matrix. Those 4 iterations are graphically represented as the opposing Arrow Circles. We have two simple possibilities: 1. The Equal And Opposite (EAO) "spin" values are simply those in the AV Antimatter; 2. Every ST unit actually reveals as a virtual-pair with both EAO "spin" values in BOTH the OV Matter AND the AV Antimatter. In #1, the single photon interferes (and entangles) with its EAO Antimatter partner. In #2, interference (and entanglement) occur in BOTH Matter and Antimatter domains. In Entanglement, we have 2 ST units — e.i. photons — created & sharing the same Quantum State numbers/parameters, i.e., "spin." Regardless of distance of separation, they ALWAYS maintain EAO "spin" values. Within the BIM, each PT has 4 iterations of its triangle simultaneously present. One can designate the short side "a" to be the "spin" vector direction. That means, there are 4 "spin" vector direction information-directions ALWAYS known. In either scenario #1 or #2, measuring the "spin" direction on one will always yield the EAO "spin" direction vector on the other photon. Scenario #1 — being simpler — gives an easier visualization of the process. Scenario #2 — where both the Matter & Antimatter photon are virtually present as pairs of EAO entities — gives a more complex, but perhaps closer to "reality" picture as here the "KNOWN" information of "spin" direction allows the photon to be in either-or/both "spin" directions at once, and coinciding with known results, the manner, method and actual act of measurement "selects" — at that moment — for a particular "spin" vector. If you orientate your test to select the "spin" state, it will automatically be mirrored to the information — and thus state — of the EAO entangled photon. There is no information transported, it is ALWAYS known! DSEQEC Every ST unit forms from the instruction set of the Inverse Square Law (BIM). The PTs provide the basis of the Quantum State Numbers/Parameters. Each PT has its 4-iterations of its triangle ubiquitously KNOWN for all ST. The formation of any & all ST units generates EAO Matter-Antimatter expressions. Both the Matter & Antimatter expressions are guided by the ISL of the BIM. Thus, any given ST unit —e.i., photon —is Entangled from the start, acting like a virtual-pair. The "spin" information —encoded in the BIM —is EAO in the Entangled pairs. The Double-Slit Experiment (DSE) exemplifies how a single photon can interfere with itself. Quantum Entanglement (QE) exemplifies how 2 entangled photons —born of and sharing the same quantum state — reveal how ALL ST units are formed as 2 Entangled, EAO Matter-Antimatter virtual-pairs, the INFORMATION of their QUANTUM STATE ALWAYS built right into their ST formation via the EAO iterations of the PTs within the expanding BIM. There is NO HIDDEN or TRANSFERRED INFORMATION. ALL INFORMATION IS ALWAYS KNOWN.

Appendix A: BIM PPTs: Details of their Portrait Profiles

(Table number reference is for Appendix A.)

Note: Often both quick view png/jpg and full view pdf versions presented.

Appendix B: BIM ➗24 PPTs and PRIMES

(Table number reference is for Appendix B and TPISC IV.)

TABLES 1-9 (r-sets, Z➗24,Axis squared, differences, groundwork tables)

Table VI b fully expanded 1-1010 PTs and Primes with ALL ACTIVE Rows color-coded in Violet

(Once renamed, ALL Tables — as pdf/pngs—will be listed here with links to and fro main article)

TABLES 10-19 (PRIMEs-gaps and random differences)

TABLES 20-23 (PRIME-PPT or not with r-values)

TABLES 24-28 (Sub-Matrix 2 )

TABLES 29-30 (Sub-Matrix 2 Sidebar: Exponentials of the PPTs)

Sub-Matrix 2 Sidebar: Exponentials of the PPTs

• Table 29 Exponentials of the first 10 PPTs c-values to be used in Tables 30a-g.

• Tables 30a-g The Sub-Matrix 2, when ➗4, and the difference (∆) between this and the next exponential PPT treated this way, is subsequently ➗ by its Sub-Matrix 2 variable, the PREVIOUS exponential within the series is revealed. Restated as an example: When one subtracts 1 from the exponential values of c (the c-value of the PPT) you get the Sub-Matrix 2 value. Divide that by 4 and take the difference (∆) between it and the next. Divide that by 3 to give the PREVIOUS PPT c-value in the series.

The Sub-Matrix 2 variable divisor = 3 = Sub-Matrix 2 value/4 = 12/4. These variables run: 1,3,4,6,7,9,10,...

Tables: 31a, a+, a++, ++MP, a+++ and 31b

Tables: 32a, 32b and 32c

Tables: 33a, 33b and 33c.

The underlying geometry of the BIM÷24 PRE-SELECTS the Axis Rows into TWO Groups: ARs and NON-ARs. The PPTs and PRIMES are EXCLUSIVELY — as a sufficient, but not necessary condition — found on the ARs and NEVER on the NON-ARs. While both Groups follow (PD2 - PD2)÷24, they do so ONLY within their own respective Groups. They do NOT crossover. This Grouping divide occurs naturally within the BIM as shown in these images below.

The ISL as presented in the BIM is deeply, intimately structured around the number 24 — and its factors: 4,6, 3,8 2,12, and 1,24.

The interplay between these small sets of Numbers generates an incredible amount of richness and complexity with seemingly simplistic BIM itself. This has led to TPISP: The Pythagorean-Inverse Square Connection, and the PRIMES.

(NOTE: OPEN in separate browser tab/window to see all.)

Supporting Graphics forTables: 33a, 33b, 33c and 33d. Open in separate browser tab/window to see all.

Back to the Table 31 sequences. Table 31a... series, along with the previous Table33 series, uncovers how the PRIMES were found within the BIM both by algebra and geometry. This has been reduced to a focus white paper: "PRIMES vs NO-PRIMES that is a condensed summary of this work. One may benefit from looking over this paper prior to the study of these more elaborate tables. A summary from the paper is presented here:

SUMMARY*

How do we go from the simple grid of the BIM (BBS-ISL Matrix) to identifying the PRIMES?

1. The BIM is a symmetrical grid — divided equally down its diagonal center with the Prime Diagonal (PD) — that illuminates the Number Pattern Sequence (NPS) of the Inverse Square Law (ISL) via simple, natural Whole Integer Numbers (WIN).

2. The BIM Axis numbers are 1,2,3,.. with 0 at the origin.

3. The Inner Grid (IG) contains EVEN and ODD WIN, but except for the 1st diagonal next to the PD — a diagonal that contains ALL the ODD WIN — there are NO PRIMES (NO-PRIMES, NP) on the SIG (Strict Inner Grid).

4. The PD WIN are simple the square of the Axis WIN.

5. ALL the IG WIN result from subtracting the horizontal from the vertical intersection of the PD.

6. Dropping down a given PD Squared WIN (>4) until it intersects with another squared WIN on a Row below will ALWAYS reveal that Row to be a Primitive Pythagorean Triple (PPT) Row, whose hypotenuse, c, lies on the intersecting PD. ALL PPTs may be identified this way.

7. Dividing the BIM cell values by 24 — BIM÷24 — forms a criss-crossing DIAMOND NPS that divides the overall BIM into two distinct and alternating Row (and Column) bands or sets:

• ODD WIN that are ÷3 and referred to as NON-ARs;
• ODD WIN that are NOT ÷3 and referred to as ARs, or Active Rows;
• The ARs ALWAYS come in pairs — with an EVEN WIN between — as the UPPER and LOWER AR of the ARS (Active Row Set);
8. ALL PPTs and ALL PRIMES ALWAYS are found exclusively on the ARs — no exceptions.

9. By applying:

(1)                  * $6yx ± y = NP$

let y = odd number (ODD) 3, 5, 7,… and x = 1, 2, 3,... one generates a NP table containing ALL the NP;

*True if ÷3 ODDs are first eliminated, otherwise ADD exponentials of 3 to the NP pool;

10. Eliminating the NP — and the NP contain a NPS — from ALL the ODD WIN, reveals the PRIMES (P).

11. A necessary, but not sufficient confirmation — but not proof — of primality is found by finding the even division of 24 into the difference of the square of ANY two PRIMES as:

(2)                  (P2)2 - (P1)2 = $n24$

let $n = 1, 2, 3,...$

Be aware that this also holds true for ALL the AR NP. The P and NP are NOT ÷3, and are both part of the ARS and therefore any combination of the two squared differences will be ÷24:

(3)                  (NP2)2 - (NP1)2 = $n24$

(4)                  (NP2)2 - (P1)2 = $n24$

(5)                  (P2)2 - (NP1)2 = $n24$

The ÷3 NON-AR set is separately ÷24, but can NOT be mixed with members of the AR set (ARS) as:

(6)                  (NON-AR-NP2)2 - (NON-AR-NP1)2 = n24

(7)                  (NON-AR-NP2)2 - (NP1)2$n24$

(8)                  (NP2)2 - (NON-AR-NP1)2$n24$

(9)                  (P2)2 - (NON-AR-NP1)2$n24$

(10)                (NON-AR-NP2)2 - (P1)2$n24$

The division into AR and NON-AR sets has a NPS that ultimately define the elusive pattern of the P.

Furthermore, $6yx ± y = NP$ may be re-arranged to:

(11)                  $(NP ± y)/6yx = 0$

(12)                  $(NP ± y)/6x = y$

asking whether any given ODD (>3) is a P or NP, it is exclusively a NP if, and only if, y reduces to the same value after applying x. As y is effectively an ODD of either a PRIME or composite of PRIMES factor*, one only needs to satisfy a single instance to validate NON-Primality.

12. One can also obtain ALL the P by eliminating the BIM SIGO and O2 from the 1st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O2 = ODDs2, and the 1st Diagonal = the 1st Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O2 — the O2 being the PD values — such that we now have:

(13)   SIGO(A2) = Strict Inner Grid ODDS & ODD AXIS2 giving a distinct visualization advantage;

(14)                  NP = SIGO(A2)

(15)                  1st Diagonal - SIGO(A2) = P.

This second method — the algebraic geometry method — as presented here.

Introduction

Two methods — one pure algebraic and the other a more visual algebraic geometry presented here — have been found that capture ALL the NO-PRIMES (NP). While they process slightly different, they dovetail nicely into a very visual Number Pattern Sequence (NPS) here on the BIM. They both give identical NP results.

So what is the significance of capturing ALL the NP?

The NP are the highly NPS that define the elusive pattern of the P. P + NP = ALL ODD WINs (≥3).

In any group of WIN, if you know the NP, you also know the P. Here is the highly visualizable geometric method for capturing ALL NP. In fact, it is just as simply stated in 12. of the SUMMARY.

Algebraic Geometry Method

One can obtain ALL the P by eliminating the BIM SIGO and O2 from the 1st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O2 = ODDs2, and the 1st Diagonal = the 1st Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O2 — the O2 being the PD values — such that we now have:

• One can also obtain ALL the P by eliminating the BIM SIGO and O2 from the 1st Diagonal WIN, where SIGO = Strict Inner Grid ODDs, O2 = ODDs2, and the 1st Diagonal = the 1st Diagonal Parallel to the PD.

This itself is further simplified by switching out the ODD AXIS values with the O2 — the O2 being the PD values — such that we now have:

(13)   SIGO(A2) = Strict Inner Grid ODDS & ODD AXIS2

(14)                  NP = SIGO(A2)

(15)                  1st Diagonal - SIGO(A2) = P.

• The PDFs will follow these animated gifs. Videos and other supporting graphics thereafter.

Animated Gifs:

PRIMES vs NO-PRIMES-2: algebraic method.

PRIMES vs NO-PRIMES-1: algebraic-geometry method.

PRIMES vs NO-PRIMES-3: algebraic and algebraic geometry method.

PRIMES vs NO-PRIMES-4: algebraic method in detail.

PDFs +

PRIMES vs NO-PRIMES: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-1 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The NPS of the NP define the elusive pattern of the P.

PRIMES vs NO-PRIMES: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-2 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths. The larger PDF is below.

PRIMES vs NO-PRIMES PDF: snapshot-3 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

PRIMES vs NO-PRIMES PDF: snapshot-4 showing dovetailing visualization of the algebraic method of calculating the NP with that of the geometric method on the BIM. The larger PDF is below. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--as double-wide, L-shaped paths beginning just below the 1st Diagonal. These double rows alternate with divisible by 3--or divisible by 9 Axis squared--paths.

Table31a4: PRIMES vs NO-PRIMES PDF: Open in a separate tab/window to see all 11 pages. Here the x=1,2,3,... base value sets -- Lower -y an Upper +y--are shown individually and collectively as sets. The full Upper and Lower tables conclude.

Table31a6_2: PRIMES vs NO-PRIMES PDF: If you look at the ODD Axis ÷3 NO-PRIMES (NP) that lie in the paths between the L-shaped Double-wide x-base sets (x=1,2,3,)…), one finds a distinct Number Pattern Sequence (NPS) between successive NP values. Those shown in BLUE are NOT included in the criss-crossing L-shaped Double-wide paths (seen in the snapshots above), while those in GRAY are. The NPS seen here, based ultimately on the 1,3,5,7,… ODD number summation series that defines the whole BIM distribution (including the 1st Diagonal, the PD, and the successive differences in sequential Inner Grid cell values) reiterates that of both the L-shaped Double-wide paths as well as the individual x paths. Both give a NPS of the NP that reveal the elusive pattern of the PRIMES. There remains little doubt that the PRIMES , as well as the Primitive Pythagorean Triples (PPTs), are intimately related to the INVERSE SQUARE LAW (ISL)!

Videos:

...more supporting graphics and tables:

Table31a4-5_+RunDiff_EQUATIONsDEMO(divide3Filter)+10x10+. A simple table to demo the NP process.

Table31a4_+RunDiff_EQUATION-PATTERNS-LOWER+UPPER+Annotated. NP Table with layout notes.

Table31a4_+RunDiff_EQUATIONs(SqrdAxis-seqPD)+50x500+.pdf. From the BIM to the NP/P Tables.

BIM35-Table31a4-5_NO-PRIMES_Factores-withARS-YELLOW-numbAnnotated. The NP pattern on the BIM.

Table33a,c,d_ODDsqrd÷24_sheets. The NP pattern – an amazing NPS – defines the P.

BIMrows1-1000+Primes_sheets+Sub-Matrix2. A BIM reference showing ÷24 ARS, PPTs, NP an P.

For the full documentation of all the graphics and pdfs — including history, development, expanded tables, and more — see the link references immediately below. Appendix B has all the documentation.

REFERENCES

While there are no specific references for this work other than referring back to my own original work, there are many references involved in the study and research on the PRIMES in general. These have been well documented in the TPISC_IV: Details_BIM+PPT+PRIMES focus white paper. This paper also contains the background graphics and tables leading up to this current work. The focus page is part of the much larger TPISC_IV: Details ebook project that is freely available as an HTML webpage. The quick reference outline can be found here.

xxxxxxxx

Various supporting graphics

xxxxxxxxxx​

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BIM + PT + Decagon (double Pentagons)

NEW! Connections between:

1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples)

\2. PT + Pentagon & Decagon (double pentagons of DNA)

1. BIM + PT + DNA
2. BIM + PT + DNA + Zika Virus (cryo-em imagery)

It is suggested in the previous work that the pentagonal geometry of the virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA double-helical axis, looking for a simple match opening!

In this work, a closer look at how the base 3-4-5 PT relates to the pentagon, double-pentagon (decagon) and the 3 concentric decagonal geometry found in the axial view of the DNA double-helix molecule will be visually examined.

BIM + PT + Decagon (double Pentagons of the DNA double-helix)

See a full slideshow of the BIM + PT + Decagon : here

See the video: here on Vimeo at https://vimeo.com/263223746

See white papers: here

See more on TPISC: The Pythagorean - Inverse Square Connection: here

Comment: Here we continue with TPISC: The Pythagorean - Inverse Square Connection. This time looking for a connection between the 3-4-5 Pythagorean Triple and the concentric, decagons (double pentagons) that form the geometric structure of the DNA double-helix molecule as view down its axis composited from 1 full, 360 degree spiral.

While emphatically NOT AN EXACT MATCH, the 3-4-5 PT is ridiculously close to being an exact match both from the the angles of a given pentagon to that of the decagon and most especially connecting key angles and vertices between the concentric decagons.

In 2001, I wrote up “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA” about the pure double-pentagon (decagon) geometry of the double-helix molecule. It is the basis for the concentric decagonal geometry — acting a template.

The question remains: is the "ridiculously close to being an exact match" good enough — sufficient enough — to define an actual connection between the BIM + PT, and, the decagon? Does this slight wiggle room actually allow for the connection to be made within the organic presentation and interaction of Nature. "Soft geometry" versus the crisp, hard-edged, no room for anything less than perfect fits "hard geometry" of pure mathematics? Are the strict mathematics that informs all of spacetime operative in a slightly looser, more generous manner in the actual physical manifestations of the organic world of Nature? Are not the Fibonacci sequence numbers of the Golden Mean "approximations" to the actual value of the irrational number, phi (Φ)? (for more on 5, Φ and pentagons see "The Golden Number.")

here

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Appendix E: BIM + PT + DNA + Zika, Epstein-Barr and other Icosahedral-Structured Human Viruses

BIM + PT + DNA + Zika Virus

NEW! Connections between:

1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples)
2. PT + Pentagon & Decagon (double pentagons of DNA)
3. BIM + PT + DNA
4. BIM + PT + DNA + Zika Virus (cryo-em imagery)

It is suggested that the pentagonal geometry of the virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA double-helical axis, looking for a simple match opening!

See the video: here on Vimeo at https://vimeo.com/262123322

See white papers: here

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/arthry5.htm

See more on TPISC: The Pythagorean - Inverse Square Connection: here

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/MSST-TPISC_resources/MSST-TPISC_resources.html

Comment: In pushing for a further connection between pure, elemental natural whole integer numbers and the simple geometry that they inform, I was looking for a straightforward connection between the BIM, the PTs, and the pentagons as the BIM heavily favors the 5 resonance (base 10).

Waiting in the dentist’s office, I saw the magnificent imagery of the Zika virus as generated from the Nobel Winning cryo-electron micrograph imaging technique right there in the pages of the latest National Geographic.

The pentagonal structure of the Zika Virus leaped out and rang up all sorts of bells.

In 2001, I wrote up “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA” about the pure double-pentagon (decagon) geometry of the double-helix molecule. Surely, there is a connection between these two geometries: both based on the pentagon!

We know viruses insert themselves into the chromosome and redirect it to make copies of itself (and more).

That the 3-4-5 fundamental PT — parent to all subsequent PTs (both primitive and non-primitive) — nicely (though NOT PERFECTLY) matches the angles and slopes of the pentagon and decagon, gives one wonder if the simple geometries of the Inverse Square Law (BIM) and those ever so simple Pythagorean Triples might be in the perfect position to inform the formation of the pentagonal geometry that itself informs so much of the sublime beauty and structure we see manifested throughout the cosmos!

Would not the highly geometric viral population be of the same or similar geometry of the host population it exploited to its own ends?

References:

https://www.nationalgeographic.com/magazine/2018/03/explore-wellness-nobel-prize-cryo-electron-microscopy/

http://www.cbc.ca/news/technology/nobel-prize-in-chemistry-shared-by-3-for-cryo-electron-microscopy-1.4325570

https://www.wired.com/story/cryo-electron-microscopy-wins-the-nobel-prize-in-chemistry

https://www.nobelprize.org/nobel_prizes/chemistry/laureates/2017/press.html?source=techstories.org

http://www.cell.com/trends/biochemical-sciences/abstract/S0968-0004(17)(17)30048-8

https://cdn.rcsb.org/pdb101/learn/resources/zika/zika-paper-model.pdf

https://www.nih.gov/news-events/nih-research-matters/zika-virus-structure-revealed

https://chemistryandcomputers.wordpress.com/2016/06/19/pymol-and-very-large-pdb-files-the-zika-cryo-em-structure-as-a-case-study/

https://www.sciencedaily.com/releases/2016/04/160419144741.htm

http://science.sciencemag.org/content/early/2016/03/30/science.aaf5316/tab-figures-data

https://www.chemistryworld.com/news/explainer-what-is-cryo-electron-microscopy/3008091.article

http://www.virology.ws/2016/04/05/structure-of-zika-virus/

https://en.wikipedia.org/wiki/Zika_virus

https://reliawire.com/structure-immature-zika/

https://www.ncbi.nlm.nih.gov/Structure/pdb/5IRE

Cryo-em Structure of Zika Virus

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BIM + PT + DNA + Epstein-Barr Virus

NEW! Connections between:

1. BIM (BBS-ISL Matrix) + PTs (Pythagorean Triples)

\2. PT + Pentagon & Decagon (double pentagons of DNA)

1. BIM + PT + DNA
2. BIM + PT + DNA + Zika Virus + Epstein-Barr Virus + many more(cryo-em imagery)

Number 4. should read: BIM + PT + DNA + the Icosahedral structure of the majority of human viruses

It is suggested that the pentagonal geometry of the virus allows it to insert itself within the decagonal geometry of our DNA, perhaps spiraling in along the DNA double-helical axis, looking for a simple match opening!

BIM + PT + DNA + Epstein-Barr virus

See a full slideshow of the BIM + PT + DNA + Epstein-Barr : here

See the video: here on Vimeo at https://vimeo.com/266783704

See white papers: here

See more on TPISC: The Pythagorean - Inverse Square Connection: here

Comment: In pushing for a further connection between pure, elemental natural whole integer numbers and the simple geometry that they inform, I was looking for a straightforward connection between the BIM, the PTs, and the pentagons as the BIM heavily favors the 5 resonance (base 10).

Previously, I wrote on the connection between the BIM + PT + DNA and specifically the Zika Virus.

A recent paper on the Epstein-Barr Virus lead to this brief follow up that ultimately relates to the majority of viruses that infect humans, from the common cold to HIV.

The pentagonal structure — at the core of the icosahedral structure that the majority of human viruses reveal — is at the heart of the connection between geometry and biology.

In 2001, I wrote up “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA” about the pure double-pentagon (decagon) geometry of the double-helix molecule. Surely, there is a connection between these two geometries: both based on the pentagon!

We know viruses insert themselves into the chromosome and redirect it to make copies of itself (and more).

That the 3-4-5 fundamental PT — parent to all subsequent PTs (both primitive and non-primitive) — nicely (though NOT PERFECTLY) matches the angles and slopes of the pentagon and decagon, gives one wonder if the simple geometries of the Inverse Square Law (BIM) and those ever so simple Pythagorean Triples might be in the perfect position to inform the formation of the pentagonal geometry that itself informs so much of the sublime beauty and structure we see manifested throughout the cosmos!

Would not the highly geometric viral population be of the same or similar geometry of the host population it exploited to its own ends?

more...Zika connections)

more...(3-4-5 PT + Decagon connections)

References:

https://pdbj.org/emnavi/emnavi_detail.php?lgc=1&id=2092

https://pdbj.org/emnavi/emnavi_movie.php?id=2092

https://pdbj.org/emnavi/quick.php?id=2092

https://pdbj.org/emnavi/doc.php?id=developer

https://pdbj.org/emnavi/doc.php?tag=

http://epstein-barr-virus.purzuit.com

https://3dprint.nih.gov/discover/3dpx/001280/x3d

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4128116/

https://www.sciencedaily.com/news/plants_animals/viruses/

http://virology.net/big_virology/BVretro.html

http://virology.net/big_virology/Special/Nermut/Retro.html

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3075868/

https://www.liebertpub.com/doi/abs/10.1089/aid.1993.9.929

What is the basic structure of a virus?

http://www.discoverbiotech.com/wiki/-/wiki/Main/Eukaryotic+Viruses;jsessionid=84878465DEF388737ED2FFD6B247FC14

“Structure of eukaryotic viruses The viral genome is surrounded by a protein shell known as capsid. Capsid encloses the genetic material of the virus and consists of protein subunits known as capsomeres. The nucleic acid genome plus the protective protein coat is called the nucleocapsid which may have icosahedral, helical or complex symmetry. The majority of viruses have capsids with either helical or icosahedral structure. The icosahedral shape, which has 20 equilateral triangular faces, approximates a sphere. All faces of an icosahedron are identical. In the icosahedral structure, the individual polypeptide molecules form a geometrical structure that surrounds the nucleic acid.”

Icosahedral Viruses include: Adenovirus Papovaviruses Herpes viruses Picornaviruses Reoviruses Retroviruse

https://en.wikipedia.org/wiki/Capsid

https://en.wikibooks.org/wiki/Structural_Biochemistry/Proteins/Cryo-Electron_Microscopy

https://en.wikibooks.org/wiki/Structural_Biochemistry/Carbohydrates/Virus

https://www.britannica.com/science/virus/The-protein-capsid#ref256445

The-3D-image-stop-cold-scientists-virtual-look-inside-virus.html

Phi (ϕ), Fibonacci, Pentagon Connections to the PTs

While the LINKS section below the Comments references the historically established connections, the Comments section will address some specific findings — some of them new!

The phi (ϕ), Fibonacci and pentagon connections — along with the Kepler Triangle — have some interesting properties that deserve a closer look. Even more so in that they relate back to the DNA double-helix molecule and the geometry of many viruses that transform it.

For reference (2-3 decimals here):

(✓ϕ)2 = 1.27

ϕ = 𝞪2/𝛾 = 𝞪𝛾 = 1.618

ϕ2 = 𝞪3 = 2.618

𝞪 = ϕ/𝛾 = 1.378 (*fine-structure constant-like approximation, see NOTE)

𝞪2 = 𝛾 ϕ = 1.90 ( as 1.8988)

𝞪3 = ϕ2 = 2.618

𝞪 ϕ = 2.23

𝞪2ϕ = 3.07

𝛾 = 3✓ϕ = ✓𝞪 = ϕ/𝞪 = 𝞪2/ϕ = 1.174

NOTE:

ϕ = 1.61803 3√ϕ = 1.174 = 𝛾 𝛾3 = 1.1743 = ϕ 𝞪 = 𝛾2 = 1.3782 𝛾= ϕ/𝞪 = 1.61803/1.3782 = 1.1739 These are the original phi (ϕ), 𝞪 and 𝛾 relationships found in the DNA Master Chart. PLEASE NOTE: In all references to the early DNA triangle work, 𝞪 should be correctly designated 𝞪-1. Back in 2001, the 𝞪 was used as 𝞪 = 1.3782, when in fact it is the reciprocal of 𝞪, as 𝞪 = 1/137, thus 𝞪-1 is, from this current work on, correctly used as 𝞪 = 0.72…….. The differences in 𝞪-like numbers are shown below this Kepler Triangle-DNA Triangle section in the Tables sections. The correction starts there. This does not preclude the relationships described here, it is only one of proper symbol use. Here 𝞪 = 1.3782, down below in the Tables section, 𝞪 = 0.72…….

What happens when you square the Kepler Triangle? (square all sides) ;

you get

If we set the Kepler Triangle = to the DNA Triangle with hypotenuse=phi (ϕ), it works as we solve for 1.

𝞪2 + (1/𝛾)2 = ϕ2 = 12 + (✓ϕ)2

𝞪2 + (1/𝛾)2 = 12 + (✓ϕ)2

𝞪2 + (1/𝛾)2 - (✓ϕ)2 = 12 = 1

1.90 + 0.73 - 1.618 = 1

NOTE: The DNA triangles are taken directly from earlier work, NOT scaled here to match the Kepler Triangle.

Another interesting relationship:

as, ϕ2 = 𝞪3 = 2.618

And:

as, ϕ2 = 𝞪3 = 2.618

And:

are equal to each other.

*My use of the close approximation to the numbers of alpha, not the actual value, was based on number theory work I was doing back then (2001 and before) on the axial view of the DNA double-helix molecule. I used the 1.3784... number because it kept popping up with my work on phi and the decagons. Something is most definitely there, but my use of it as the fine structure constant per se would probably require a more rigorous proof to be fully accepted as alpha in the physics community. Its approximation to alpha is maintained by the 𝞪 symbol. Key relationships:

• ϕ = 𝞪2/𝛾 = 𝞪𝛾 = 1.618
• ϕ2 = 𝞪3 = 2.618
• 𝞪 = ϕ/𝛾 = 1.378
• 𝞪2 = 𝛾 ϕ = 1.90 ( as 1.8988)
• 𝞪3 = ϕ2 = 2.618
• 𝞪 ϕ = 2.23
• 𝞪2ϕ = 3.07
• 𝛾 = 3✓ϕ = ✓𝞪 = ϕ/𝞪 = 𝞪2/ϕ = 1.174

As a pure number — 1/1.378 = 0.7255.. — is a fascination that its not to be be ignored! SEE DOWN BELOW!!!

From wikipedia https://en.m.wikipedia.org/wiki/Fine-structure_constant see Feynman quote:

"There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!

— Richard Feynman, Richard P. Feynman (1985). QED: The Strange Theory of Light and Matter. Princeton University Press. p. 129. ISBN 0-691-08388-6."

The above connection of the concentric double-pentagons (decagons) of the axial view of the DNA double-helix molecule to the Kepler Triangle provides yet another connection between phi (ϕ), right triangles and the Pythagorean Theorem, and by inference, the right triangles of the Pythagorean Triples, as they too, have a phi (ϕ) connection via their strong connection with the Fibonacci numbers. The original work, “GoDNA: the Geometry of DNA (axial view),” and, “SCoDNA: Structure & Chemistry of DNA,” generated 3 concentric decagons that were related to each other by phi (ϕ), alpha (𝞪) and the ratio of the two, gamma (𝛾 = ϕ/𝞪). The relationship generated the DNA Master Chart in which every segment of any one decagon relates mathematically to all other segments in the other two. From this a DNA Triangles image was formed:

For more, see the original paper…here.

*The connection between phi (ϕ), the Fibonacci numbers and the BIM (BBS-ISL Matrix) has been extensively covered in the original white paper and ebook "Brooks Base Square and The Inverse Square Law" (2010-11): the specific reference (Rules 161-168) to phi (ϕ), the Fibonacci numbers and the IBM (BBS-ISL Matrix) connection: here

See more of the white papers: here

See more on TPISC: The Pythagorean - Inverse Square Connection: here

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If A = B and B = C, then A = C.

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APPENDIX D TABLES I - IV: A deeper look:

Table I. The Fibonacci Numbers are sequentially divided by phi (ϕ) in each Column. Each resolves down to a 𝛾 value in BLUE and a 𝞪 like value in ORANGE. The reciprocal of the 𝞪-like value, 𝞪-1, is shown on the last ROW.

Table II. While the ratios of random, non-Fibonacci numbers added sequentially resolve to phi (ϕ), successive divisions by phi (ϕ) — as shown in each Column — do NOT re-generate themselves, nor do they resolve to 𝞪- or 𝛾-like number values like the Fibonacci does.

Table III. Below, corresponding to three slightly different iterations of √α-1 = 𝛾 as seen in Columns 1,2 — Columns 4,5 — Columns 6,7

Column 1 Reciprocal of 𝞪** -1= 1.3819 as
𝞪** = 0.7236292487 taken from working down Fibonacci Number 89 (Table I)— dividing each Fibonacci number sequentially by ϕ.

Working from the 0.7236292487 on up — multiplying sequentially by ϕ — only works up to 10946 before becoming too large. Col 1,2

Column 4 Reciprocal of the published fine-structure constant = 𝞪**−1** = 137.035999139(31) as α = 0.0072973525664(17).

Working from the 0.72973525664 on up — multiplying sequentially by ϕ — only works up to 55 before becoming too large. Col 4,5 Reference: https://en.wikipedia.org/wiki/Fine-structure_constant

Column 6 Reciprocal of the natural fitting 0.7236067977 = 1/1.3819660113

Working from the 0.7236067977 on up — multiplying sequentially by ϕ — works perfectly at least up to 20,365,011,074. Col 6,7

1.3819660113 × 1.17082 = 1.6180334454 1.170822 = 1.3708194724 1.170823 = 1.6049828547

Here we see that this nearly “perfect” fit 𝞪 = 0.7236067977 = 1/1.38196601 completely re-generates the Fibonacci Numbers Sequence.

As, 𝞪** -1= 1/𝞪 and (1/𝞪)ϕ = ϕ, so 1.3819660113 × 1.17082 = 1.6180334454 and 𝛾2 = 𝞪 -1 , so ** 1.170822 = 1.3708

Less perfect, than the original DNA Triangle values (see above), is 1.170823 = 1.6049828547 — NOT quite ϕ.

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Clearly, these numbers are close — very close — to defining a definitive connection between the Phi (ϕ), the Fibonacci Number Sequence, the pentagon, the Kepler Triangle, the Pythagorean Triples, the DNA double-helix molecule, and perhaps even the fine-structure constant.

Tweaking the values slightly one way satisfies some sets, but not all. Tweaking the values a little the other way satisfies those sets outside, but sacrifices some of the formerly included sets. It may also be simple a matter of scale, i.e. molecular versus atomic.

On the level of physics, and their constants, one would prefer a more definitive connection.

On the level of biology — even though physics informs the chemistry that informs the biology — we can appreciate how the template of the perfect match may enjoy the fruits of diversity by exactly that type of subtle variation from an exact fit!

Table IV. ϕ = 1.61803 3√ϕ = 1.174 = 𝛾 𝛾3 = 1.1743 = ϕ 𝞪 = 𝛾2 = 1.3782 𝛾 = ϕ/𝞪 = 1.61803/1.3782 = 1.1739 The Columns in ORANGE are from the Original DNA Triangle data (𝞪-1 designated as simple 𝞪)

SUMMARY

We have three (3) sets of ϕ—𝞪—𝛾 relationships:

1. From the original DNA Triangle (as shown in ORANGE in Table IV, Cols. 1-6 ), 𝛾 = 1.1739 or 1.174
2. From the Fibonacci Number Sequence (Table I, Col. 89; Table III, Col. 1,6; and Table IV, Cols. 7,8 ), 𝛾 = 1.17082 or 1.17086
3. From the fine-structure constant numbers (Table III, Col. 4), 𝛾 = 1.1807

The latter, somewhat mimicking the actual fine-structure constant, strays the furthest from the others.

No. 1 and 2 are quite close, yet not exactly interchangeable as one plays out the ratios.

The DNA Triangle set is derived from the measurements and calculations of the concentric, double-pentagonal geometry of the axial view (i.e. one complete 360° rotation) of the double-helix spiral. Its tight correlations make for some direct DNA Triangle correlations and a self-consistent DNA Master Chart relating the various geometric segments of the concentric decagons to each other.

The Fibonacci set is taken strictly from calculations: dividing the known Fibonacci Numbers by ϕ. The difference comes from where you start in the sequence and what direction do you go. We know the ratio more perfectly registers the ϕ value, the further along the sequence one goes.

The common factors here are ϕ, the pentagon, and the Fibonacci numbers…and, now perhaps, some common binding factor that has been referred to as gamma, 𝛾.

The Fibonnaci numbers can generate the Pythagorean Triples, and, because of their relationship to ϕ, they are related to pentagons — especially in the form of concentric, double-pentagons.

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Review of the Pythagorean Triples from Wikipedia: here

While there are many excellent reference links available online, the GoldenNumber.net is a great starting point for organization, clarity and thoroughness:

https://www.goldennumber.net

https://www.goldennumber.net/site-map/

https://www.goldennumber.net/geometry/

https://www.goldennumber.net/triangles/

https://www.goldennumber.net/five-phi/

https://www.goldennumber.net/math/

https://www.goldennumber.net/fibonacci-series/

https://www.goldennumber.net/phi-pi-great-pyramid-egypt/

Another great site covering Numbers (and much more) for education:

http://schoolbag.info/mathematics/numbers/index.html Numbers: Index

http://schoolbag.info/mathematics/numbers/76.html Pythagorean Triples

http://schoolbag.info/mathematics/numbers/77.html Fibonacci +Pythagorean Triples

http://schoolbag.info/mathematics/numbers/82.html Pythagorean Curiosities

http://schoolbag.info/mathematics/numbers/107.html Fibonacci list

http://schoolbag.info/mathematics/numbers/109.html Mers.Primes list

http://schoolbag.info/mathematics/numbers/110.html Perfect Numbers

Golden Ratio, phi (ϕ)

https://en.wikipedia.org/wiki/Golden_ratio

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi.html

Platonic Solids

http://www.3quarks.com/en/PlatonicSolids/ (images courtesy 3quarks.com)

http://www.treenshop.com/Treenshop/ArticlesPages/FiguresOfInterest_Article/Introduction.htm

https://en.wikipedia.org/wiki/Geometry

https://en.wikipedia.org/wiki/Platonic_solid

https://en.wikipedia.org/wiki/Regular_dodecahedron

https://en.wikipedia.org/wiki/Regular_icosahedron

https://en.wikipedia.org/wiki/Icosahedral_symmetry

http://mathworld.wolfram.com/PlatonicSolid.html

Phi (ϕ) , Fibonacci Numbers and Pentagons

https://en.wikipedia.org/wiki/Fibonacci_number

https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers

http://www4.ncsu.edu/~njrose/pdfFiles/GoldenMean.pdf

http://www.matematicasvisuales.com/english/html/geometry/goldenratio/pentagondiagonal.html

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section3

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phi2DGeomTrig.html

https://www.mathsisfun.com/numbers/fibonacci-sequence.html

https://math.temple.edu/~reich/Fib/fibo.html

Images

https://www.pinterest.com/pin/10414642860397615/

https://www.pinterest.com/goodjolt/fibonacci/

https://www.pinterest.com/goodjolt/fibonacci/

Pythagorean Triples and Fibonacci Numbers

http://nextlevelmaths.com/resources/wow/pythag_fibonacci/

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section3

https://summathfun.wordpress.com/2010/12/02/fibonacci-method-of-finding-pythagorean-triples/

Kepler's Triangle and phi (ϕ) and the Pythagorean Theorem

https://en.wikipedia.org/wiki/Kepler_triangle

https://www.goldennumber.net/triangles/

https://summathfun.wordpress.com/2010/12/02/fibonacci-method-of-finding-pythagorean-triples/

http://www.studentguide.org/the-ultimate-resource-on-the-fibonacci-sequence/

https://www.mathsisfun.com/numberpatterns.html

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Appendix F: BIM misc...

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BIM (BBS-ISL Matrix): How to Make

10 Easy steps to making the symmetrical BIM

After you set up your table with 10 x 10 Columns and Rows:

• (Example using the Numbers app.)

• line Axis # = x PD # = x2 x2 -1 x2 -4 x2 -9 x2 -16 x2 -25 36 A B C E F G H I J
1. Column Headers. After -25, simply copy and paste in the PD sequence 36-49-64-… that you can copy from your own list: 12, 22, 32, 42,... as found in Column 3 (C) below.
2. Column 2 (B) is just like Column 1 (A): type in 0,1,2,3… and drag down.
3. POWER('Axis # = x',2) is the formula for squaring the Axis #s giving the PD#=x2 in Column 3 (C).
4. $'PD # = x2'−1 is the formula for ALL the remaining Columns ADJUSTED by changing the “1” with the PD cell value found in the Header immediately above. The “$” symbol means “absolute” and all subsequent calcs will use the fixed value from C.
5. Once selected, you can drag-copy the formula across the Table and then make the ADJUSTMENTS quickly.
6. Autofill the Table by selecting the TOP Row (under the HEADER) and pull down the orange-dot symbol to the bottom of the Table.
7. Because the BIM is bilaterally symmetrical about the PD, and, because the same PD #s forming the horizontal ROWS are also forming the vertical Columns, the BIM upper triangle — above the PD — will show a negative (-) sign in front of the cell values and this sign is to be ignored.
8. Below the PD, all the cell values will be (+) numbers, without any sign.
9. The PD itself — because it reflects subtracting the same PD # from itself — will be “0” and this may be color coded to reflect that and/or the ACTUAL PD values, as found on the HEADER above, can be inserted.
10. To expand or to fill in a particular Axis # — or range of #s — simply add the appropriate Columns and Rows and duplicate the above process for those values.

Color coding the Table, Columns, Rows and Cells can be done in the Numbers App Cell section. Text in the Text section, etc.

Why the 10th Column and Row? Fill that in to check your work here.

Larger examples of How to Make the BIM with a spreadsheet app (Numbers) found here in Table 31 series found in Appendix B of TPISC IV: Details.

Of course for smaller BIM grids, one can simply enter the cell values by hand as all the Inner Grid (IG) values are simply the difference between their Prime Diagonal (PD) Column and Row intercepts, e.i. 16 = 25 - 9 PD values.

UPDATE: Once you get the basics down, there is a cleaner, more direct method that is also easily expandable.

.

Open below image in a new tab to see all 5 example sheets.

NOTE: for extending beyond allowed (Numbers, Excel is much higher) max 255 Columns:

• 1. Duplicate sheet made with the NEW method, rename new sheet;
• 2. On Row 2, replace with the NEW squared #s that you are extending;
• 3. On Row 3 (first non-header row), place the first formula and drag across:
• $'PD # = x2'−D$2
• 4. Drag Row 3 down to autofill the cells below. Each Col should have SAME formula, but will give individual values appropriate for its location. Double check. SAVE.
• 5. Color code the PD (Prime Diagonal “0”s), Row 3, and Col. C (blue) to separate out the Axis and PD from the Inner Grid cell values. Replace the PD #s with the Row 2 values, center all cells, and adjust format as needed.

A central hub for all math works can be found on the MathspeedST Media Center page.

BIM 50x50 (open image in separate tab/window)

BIM 75x75 (open image in separate tab/window)

BIM 125x125 (open image in separate tab/window)

BIM 250x250 (open image in separate tab/window)

Appendix G: BIM the PTOP and the Goldbach Conjecture

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Reference:

"A New Kind of Prime

The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any even gap that you might want."

Quote is from Quanta Mag 9/26/19 article: Big Question About Primes Proved in Small Number Systems

by Kevin Hartnett

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• Twin Primes
• Primes separated by other EVEN numbers
• Euler's 6n+1 and 6n-1
• Fermat-Euler's 4n + 1= Sum of Two Squares Theorem (Pythagorean Primes)
• Dickson's Conjectures: Sophie Germain Primes
• Goldbach Conjecture (Euler's "strong" form)
• Primes - BIM - Pythagorean Triples

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A new and easy visualization!

PRIME GAPS

Game Board

There is a new game board in town. It has actually been around for awhile, but few know of it. It's basically a matrix grid of natural numbers that defines the Inverse Square Law (ISL). It's called the BIM, short for the BBS-ISL Matrix. Every grid cell is uniquely occupied by a given number that is simply the difference between the horizontal and vertical intercept values of the main "Prime Diagonal" (not Prime number diagonal) that mirror-divides the whole matrix.

If you show all the matrix values that are evenly divisible by 24, a criss-crossing pattern of diamond-diagonal lines will appear and this ends up giving us a unique visual on the distribuition of ALL PRIMES!

Basically, all ODDs (3 and greater for this presentation) fall into a repetitive pattern of:

ODD ÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3—Not÷3—Not÷3—÷3...

The matrix ÷24 described above, nicely picks out these sets of —Not÷3—Not÷3 that we call Active Row Sets (ARS). Each ODD member of the ARS is referred to as an Active Row (AR).

The ÷3 ODD Rows that lie between are called Non-Active Rows, or NA.

It is a necessary, but not sufficient condition, that ALL PRIMES are strictly located on ARs. NO exceptions, except for some ARs have NO PRIMES.

Another striking visualization is that ALL Primitive Pythagorean Triples (PPTs) also lie only on the ARs, following the same necessary, but not sufficient, condition. NO exceptions here as well, except for some ARs also have NO PPTs. In addition, only one of the ARs within an ARS can have a PPT.

So any AR within an ARS can have 0/1 PRIMES and/or 0/1 PPTs in any combination, only with the one caveat: that only 1 PPT/ARS is allowed.

Now, not to belabor this BIM game board, let's talk PRIME GAPS!

Prime Gaps

Let's refer to the lower number value for the ARS as "Lower" and the higher as "Upper." All are ODDs. As will become apparent, you can always tell if your ODD AR is "Lower/Upper" simply by adding 2 and ÷3: if the result evenly ÷3 it is "Upper" and if not, "Lower."

The Even Gap between primes strictly follows this pattern: (all are necessary but not sufficient conditions for primality)

For any given set of twin primes, or any single prime, that is the smaller "Lower" of an AR set:

• Even Gap must be 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50,…
• Example: P=5: 7, 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55)
• P + 2, P + 2+4, P + 2+6, P + 2+10, P + 2+12, P + 2+16, P + 2+18, P + 2+22, P + 2+24,…
• P+21, P+2+22, P+23, P+22+23, P+2+22+23, P+2+24, P+22+24, P+23+24, P+2+2324
• The Even Gap difference pattern follows: 2-4-2-4-2-4-….

For any given set of twin primes, or any single prime, that is the larger "Upper" of an AR set:

• Even Gap must be 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54.…
• Example: P=7: 11, 13, 17, 19, 23, (25), 29, 31, (35), 37, 41, 43, 47, (49), 53, (55), 59, 61
• P + 4, P + 2+4, P + 2+8, P + 2+10, P + 2+14, P + 2+16, P + 2+20, P + 2+22, P + 2+26, P + 2+28,
• P+22, P+2+22, P+2+23, P+22+23, P+24, P+2+24, P+2+22+24, P+23+24, P+22+23+24, P+2+22+23+24,
• The Even Gap difference pattern follows: 4-2-4-2-4-2-…. and is identical in the middle, not the start!

One can readily see that all this is simply the natural result of two ARs alternating with a NA row:

AR—AR—NA—AR—AR—NA—AR—AR—NA—AR—AR—NA—…

“Lower” AR number

AR—5

AR—7

NA—9

AR—11

AR—13

NA—15

AR—17

AR—19

NA—21

AR—23

AR—25

NA—27…

It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.

To any “Lower” Active Axis Row ODD:

• Add 2, 4, or 6,…
• If sum is ÷3, it is a NA
• If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.

“Upper” AR number

AR—7

NA—9

AR—11

AR—13

NA—15

AR—17

AR—19

NA—21

AR—23

AR—25

NA—27…

To any “Upper” Active Axis Row ODD:

• Add 2, 4, or 6,…
• If sum is ÷3, it is a NA
• If sum is NOT÷3, it is an AR number and a PRIME (or PPT) candidate.

Again, It is 6 steps, i.e. +6, from any NA to the next NA, from any “Lower” AR to the next “Lower” AR, from any “Upper” AR to the next “Upper” AR. ALL of the various EVEN GAPS may be shown to be a direct consequence of the natural number sequence and easily visualized on the BIM.

The ARS pattern on the BIM clearly shows the above patterns and may be extrapolated to infinity:

PRIME Conjectures

A number of PRIME conjectures have been shown to be easily visualized on the BIM:

Reference: TPISC IV: Details: BIM + PTs + PRIMES

The factors of 24 — 1,24–2,12–3,8–4,6 — when increased or decreased by 1, ultimately pick out ALL ARs. Euler's 6n +/-1 is the most direct, Fermat's 4n + 1 gets the Sums of Two Squares = Pythagorean Primes(while 4n + 3 gets the rest).

Fermat's Little Theorem (as opposed to the more familiar "Fermat's Last Theorem") tests for primality.

But now there is a dead simple way to test for primality:

The difference in the squares between ANY 2 PRIMES (≧5) ALWAYS = n24.

For example, take any random ODD # - 25 —> it must be ÷24 n times to be PRIME. (n=1,2,3,...)

• 741

7412-52/24 = n= 22877.3 NOT PRIME

• 189

1892-52/24 = n = 1487.3 NOT PRIME

• 289

2892-52/24 = n = 3479 PRIME

PRIMES vs NO-PRIMES

I would be remiss if I did not mention that throughout this long journey that began with the Inverse Square Law and the Primtive Pythagorean Triples that the PRIMES kinda of just fell out. They just kept popping up. Not the least, but certainly not without effort, the BIM actually directly visualizes ALL of the NO-PRIMES. In doing so, one is left with information that is simply the inverse of the PRIMES. Subtract the NO-PRIMES information from the list of ODDS (disregarding 2) and what remains are ALL the PRIMES! A simple algebraic expression falls out from that:

NP = 6yx +/- y

letting x=1,2,3,… and y=ODDs 3,5,7,… with the only caveat is that if you don't first eliminate all the ÷3 ODDS, you must include exponentials of 3 (3x) in the NP tally.

PRIMES vs NO-PRIMES

Goldbach Conjecture (Euler's "strong" form)

In 2009-10, a solution to Euler's "strong" form of the Goldbach Conjecture "that every even positive integer greater than or equal to 4 can be written as a sum of two primes" was presented as the BBS-ISL Matrix Rule 169 and 170. This work generated a Periodic Table of Primes (PTOP) in which Prime Pair Sets (PPsets) that sequentially formed the EVEN numbers were laid out. It is highly patterned table. A recently annotated version is included with the original below. (See above link for details.)

It turned out this PTOP was actually embedded — albeit hidden — within the BIM itself as shown in Rule 170 and here, too, a recently annotated version is included below. (See above link for details.)

Rule 169: Periodic Table of Primes.

Rule 169: annotated

Rule 170: Periodic Table of Primes (PTOP): embedded within Brooks (Base) Square.

Rule 170: annotated

Table35: PTOP 200+

MathspeedST: TPISC Media Center

MathspeedST: eBook (free)

Apple Books

NEWLY ADDED (after TPISC IV published):

Back to Part III of the BIM-Goldbach_Conjecture.

BACK: ---> Simple Path BIM to PRIMES on a separate White Paper

BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper

BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper

BACK: ---> PeriodicTableOfPrimes(PTOP)_GoldbachConjecture on a separate White Paper

BACK: ---> Make the PTOP with Fractals on a separate White Paper

With that review of the ongoing work, the full presentation of this work starts below. It will cover everything above plus the NEW work leading to the findings and proof of the Goldbach Conjecture. This will be the basis for the ebook: PTOP: Periodic Table of PRIMES & the Goldbach Conjecture

PTOP: Periodic Table Of PRIMES & the Proof of the Goldbach Conjecture

PTOP Goldbach Conjecture from Reginald Brooks on Vimeo.

Goldbach Conjecture

INTRODUCTION

While the (strong) Goldbach Conjecture has been verified up to 4x1018, it remains unproven.

A number of attempts have demonstrated substantial, provocative and often beautiful patterns and graphics, none have proven the conjecture.

Proof of the conjecture must not rely solely on the notion that extension of a pattern to infinity will automatically remain valid.

No, instead, a proof must, in its very nature, reveal something new about the distribution and behavior of PRIMES that it is absolutely inevitable that such pattern extension will automatically remain valid. The proof is in the pudding!

Proof offered herein is just such a proof. It offers very new insights, graphical tables and algebraic geometry visualizations into the distribution and behavior of PRIMES.

In doing so, the Proof of the Euler Strong form of the Goldbach Conjecture becomes a natural outcome of revealing the stealthy hidden Number Pattern Sequence (NPS) of the PRIMES.

STATEMENT: Layout & Essentials

Proof of the Goldbach Conjecture (strong form, ≥6)

1. Natural (n), Whole Integer Numbers (WIN) — 0,1,2,3,…infinity — form horizontal and vertical Axis of a simple matrix grid.

2. The squares of such WINs — n2=12=1, 22=4, 32=9,…infinity — forms the central Diagonal of said grid — dividing it into two bilaterally symmetric triangular halves.

3. Every Inner Grid (IG) cell within is simply the difference (∆) between its horizontal and vertical Diagonal intercept values. They extend to infinity. The Diagonal WINs form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex.

4. Every IG cell within is also the product of two Axis WINs (Either horizontal or vertical, not both), that form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex.

5. The complete matrix grid extends to infinity and is referred to as the BIM (BBS-ISL Matrix).The BIM forms — and informs — a ubiquitous map (algebraic geometry) to:

​ The Inverse Square Law (ISL);

• The Pythagorean Triples (PT);
• The PRIMES (stealthily hidden, but revealed by NPS.
6. The 1st Diagonal that runs parallel to either side of the main Prime Diagonal (PD, not of PRIME numbers, but primary), is composed of the ODD WINs: 1,3,5,…infinity.

7. If we add +1 to each value, that 1st Diagonal now becomes a sequence of ALL the EVEN WINs (≥4): 4, 6, 8,…infinity. NOTE: this is why the PTOP is hidden, in the normal, base BIM these remain ODDs.

8. Select ANY EVEN WIN and plot a line straight back to its Axis WIN — that Axis WIN = EVEN/2 = core Axis #.

9. Upon that same Axis, PRIME Pair sets (PPsets) — whose sum (∑) equals the EVEN WIN (on the 1st Diagonal) — will be found that form the base of 90° R-angled isosceles triangle(s) whose apex lie(s) on that straight line between the EVEN and its 1/2 Axis WINs. PPsets with identical PRIMES = 1/2 Axis value.

10. The proof that every EVEN WIN has ≥1 PPsets can be seen in the Periodic Table Of PRIMES (PTOP) that stealthily informs the BIM of how each and every EVEN WIN is geometrically related to one or more PPsets.

11. These PPsets are NOT randomly contributing their ∑s to equal the EVEN WINs, rather they come about as the consequence of a strict NPS: the sequential — combining, linking, concatenation — addition of the PRIMES Sequence (PS) — 3,5,7,11,13,17,19,…—to a base PS — 3,5,7,11,13,17,19,….

12. The NPS of this addition forms the PTOP: for each vertical PS — the 1st PRIME (P1) remains constant (3), the 2nd PRIME (P2) sequentially advances one (1) PS WIN — is matched diagonally with the 2nd PS, but now the 1st PRIME sequentially advances, while the 2nd PRIME remains constant within a given PPset.

13. This matching addition of the 2nd PS at the bifurcation point of the common 2nd PRIMES, forms the zig-zag diagonal PPset Trails that are the hallmark of the PTOP.

14. For every subsequent vertical PPset match, the Trail increases by one (1) PPset.

15. The rate of such PPset Trail growth far exceeds the PRIME Gap rate.

16. The zig-zag diagonal PPset Trails combine horizontally on the PTOP to give the ∑# of PPsets whose ∑s = The EVEN WIN.

17. More than simply proving the Goldbach Conjecture, the PTOP hidden within the BIM reveals a new NPS connection of the PRIMES: PRIMES + PRIMES = 90° R-angle isosceles triangles.

18. The entire BIM, including the ISL—Pythagorean Triples—and, PRIMES, is based on 90° R-triangles!

19. Similar to how the grid cell values of the Axis, PD and IG of the Pythagorean Triples reveal additional, intimate connections within the BIM, so too do the PPsets: the 1st PRIME values of each set points to the # of STEPS from the PD that intersects the given EVEN WIN (Axis2), on a straight line path back to its Axis, at the apex where its other PPset member intersects — this is no mere coincidence — and that apex is, of course, the 90° R-angle isosceles triangle that results. For example: EVEN = 24, Axis = 12, PD = 144, # of STEPS from PD towards Axis = 5 and 19 for the 5+19 PPset and 7 and 17 for the 7+17 PPset, and 11 and 13 for the 11+13 PPset that each forms the EVEN 24.

20. As the bifurcation concatenation of the PS — 3,5,7,11,13,17,19,..— with the same base PS — 3,5,7,11,13,17,19,…— of those EVEN WIN — that when “3” is subtracted, the remainder is a next-in-the-sequence PRIME — remains one of a similar split with the Pythagorean Triples: for every “Primitive” parent PT, there are multiple “Non-Primitive” child PTs and it is the PPTs (Primitive Pythagorean Triples) that ultimately form the inter-connectedness of ALL PTs back to the original PPT — the 3-4-5. With the PRIMES, one raises the question: why are these “EVENS” preselected to be the “parent” EVENS forming the “beginning” or “start” of every PPset Trail with all other EVENS hitching on to that Trail further down the sequence?

21. Another set of STEPS (S) from the core Axis value , directly on the Axis, identifies each symmetrical pair of a given PPset that forms that given EVEN. These STEPS may by Universally calculated from the EVENS, P1 and P2 values. Examples are given further down.

As every new discovery unlocks many more questions, it follows that the details of the PTOP and BIM visualizations should both satisfy the proof offered and, more importantly, provide provocative data that will advance the field for the next researcher!

There are five sets of data to consider:

1. BIM (BBS-ISL Matrix): grid visualizations that overview the entire work

• Fig. 1-PTOP: Periodic Table Of PRIMES (100, original)
• Fig. 2-PTOP: Periodic Table Of PRIMES (100, annotated)
• Fig. 3-PTOP: Periodic Table Of PRIMES (100, upgraded)
• Fig. 4-PTOP: Periodic Table Of PRIMES (200, annotated)
• Fig. 5-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (original)
• Fig. 6-BIM: Symmetrical STEPS of the PPsets for EVEN 24 (annotated)
• Fig. 7-Table 46: Symmetrical STEPS of the PPsets for EVEN 128 annotated snapshot
• Fig. 8-BIM: Symmetrical STEPS of the PPsets for EVEN 126 snapshot
• Fig. 9-BIM: Symmetrical STEPS of the PPsets for EVEN 126
• Fig. 10-BIM: Symmetrical STEPS of the PPsets for EVEN 128 snapshot
• Fig. 11-BIM: Symmetrical STEPS of the PPsets for EVEN 128
• Fig. 12- animated gif of video (below)
• Video: PTOP rule 169-170 Annotated
2. PTOP: the actual Table

• Table 34: Original PTOP from 2009 (EVENS 6-100)
• Table 35: Upgraded PTOP (EVENS 6-200)
• Table 36: Upgraded PTOP (EVENS 6-400)
• Table 37: Upgraded PTOP (work in progress, EVENS 6-1000)
3. PTOP: Analysis

• Table 38: Distribution and NPS of the PPset Trails (EVENS 6-404)
• Table 39: Distribution and NPS of the PPset Trails (EVENS 6-914)
• Table 40: Distribution and NPS of the PPset Trails (EVENS 6-2360 and up and up)
• Table 41: Bifurcation Addition and PPset ∑s
4. Reference

• Table 42: PRIME Gaps
• Table 43: PRIME Partitions = PPsets per EVENS (4-2000)
• Table 44: Summaries of Table 43 (EVENS 4-2000)
• Table 45: Equations for PTOP Tables 38-41
5. PRIME PPset Trails

• Table 46: PRIME PPset Trails (3-10007 and up)
• Table 47: PRIME PPset Trails simplified and extended Table 46 (3-568201)
• Table 48: Working example: EVEN 8872, core Axis 4436 with 93 PPsets.
• Table 49: PRIME PPset Trails & EVENS divisible by 6,12 or 24 (3-10007 and up).

The DATA has been grouped in its own section down below. It is highly recommended that you, the reader, preview first — review, thereafter.

~~~~

FINDINGS & PROOF

The definitive proof depends on demonstrating that the PPset Trails grow and extend to cover the “next” EVENS at a rate that exceeds the Prime Gap rate. PRIMES Sequence, PS is the key.

The PS is well established. How many primes are there?

As every beginning PPset — of 3,P2 — forms from successively increasing the 2nd PRIME by the next number in the PS (3,5,7,11,13,17,…), the Trail formed by bifurcating off from that point increases the total number sum (∑) of PPsets by one, i.e. Trail, Trail+1, (Trail+1)+1, ((Trail+1)+1)+1,…

A necessary, and sufficient, condition is that the Trail lengths — i.e. the total number of PPsets, in their overlapping aggregate, always exceed both the number of PRIMES and their Gaps for any and all numbers.

Specifically from Table 46 and Fig.__, locate a P2 PRIME with a large Gap, e.i. P2= 23, Gap=6.

line#———PRIMES, P2—∑#set/Trail—PRIME Gap—∆Trail-Gap— EVEN:∑#sets—EC——Eending——EVEN, E

82386226: 3114626

We see that 23 has 8 PPsets in its Trail, it has a Gap of 6 to the next PRIME (for a ∆=2). It’s EVEN=26 and there are 3 PPsets (going horizontally across the PTOP) that will make 26. The 11 is the number of EVENS Covered (EC) as the PPsets zig-zag diagonally down the PTOP. If we add double this EC - 1 to the EVEN 26, we get the EVEN Ending of 46 as: 2(11-1) + 26 = 46. So this Trail alone inclusively covers the EVENS 26-46, although their will be some holes. To fill the holes, we look at the Trails that started up above this 23 Trail.

For the Specific set of equations exclusively for the 3,P2 sets, where ∆ = P2 - 3 (see Table 45: Equations):

2EC = P2 -1

Ending EVEN Covered = Ee = 2(EC - 1) + EVEN

51354116: 262616
61762420: 283420
71974322: 393822

We can readily see that the 17 and 19 Trails, when their EVENS are added to their 2EC-1, respectively, will equal or exceed the EVEN of the NEXT PRIME up by the GAP=6.

Take Trail 17: EVEN = 20 with EC = 8. 2(8-1) + 20 = 34.

Take Trail 19: EVEN = 22 with EC = 9. 2(9-1) + 22 = 38.

And these Trails 17 and 19 will overlap the Trail 23 EC span, filling in any and all holes between it and the next Trail 29.

92992732: 2145832

For completeness, we can see that Trail 13, while reaching Trail 23 at it’s start, does not overlap any further.

Take Trail 13: EVEN = 16 with EC = 6. 2(6-1) + 16 = 26. We also can see that 13 + 13 = 26.

~~~

The PPset Trails for EVEN 26 gives 3 PPsets via the overlapping Trails 17, 19 and 23, AND, their overlapping Trails extend past the next 3 EVENS (28, 30 and 32), ensuring that they are “covered” with PPsets in the Gap jump to the next PRIME (29). All very neat and clean.

• The actual PPsets ∑s equal the EVEN 26:

• 3 + 23 = 26
• 7 + 19 = 26
• 13 + 13 = 26
• which comes about more simply as the core Axis value ± steps away:

• 13 ± 10 = 3 and 23
• 13 ± 6 = 7 and 19
• 13 ± 0 = 13 and 13
• where the core Axis value = 1/2 the EVEN, or 26/2 = 13 and we locate it on Column 2, PRIMES (line 5) for reference. It is not actually the PRIME, as we will see. Here it happens to land directly on the PRIME 13 as it is one of the PPsets. Other times it may well be somewhere between two PRIMES in this column. We will clarify this further on, but for now it is a reference to numbers on the Axis of the BIM that, being core Axis values, are key in showing that the PPsets will ALWAYS have their Pair members symmetrically located on either side of this reference marker.

But what about a larger Gap? How about Trail 113, EVEN = 116 and a Gap of 14 to the next Trail 127? (See image from Table 36.)

Trail 113 is overlapped by Trails 73, 79, 97, 103, and 109 that account for the 6 PPsets that will form the EVEN = 116 (the 3,113 PPset is included in the tally and you can follow it in Table 43). Again, from Table 46:

1312-16: 1166
252208: 12108
3734-110: 231410
41142214: 252214
51354116: 262616
61762420: 283420
71974322: 393822
82386226: 3114626
92992732: 2145832
1031106434: 4156234
1137114740: 3187440
12411221044: 3208244
1343134946: 4218646
1447146850: 4239450
1553156956: 32610656
16591621462: 32911862
17611761164: 53012264
18671841470: 53313470
19711921774: 53514274
20732061476: 53614676
21792141782: 53915882
22832261686: 54116686
23892381592: 44417892
249724420100: 648194100
2510125223104: 550202104
2610326422106: 651206106
2710727225110: 653214110
2810928424112: 754218112
29113291415116: 656226116
3012730426130: 763254130

What about the Gap = 14 going to the Trail 127, EVEN = 130? Overlapping Trails 71, 83, 89, 101,107 and 113 combine with the 127 to account for the 7 PPsets that will form EVEN = 130.

Of course to account for all the EVENS between Trail 113 and 127 — EVENS 116 to 130, we have Trails 59 to and including 127 in various combinations to account for that. We can easily see this on Table 46, lines 16-30, where each has an Ee (118–254) that meets or exceeds those EVENS (116–130). Altogether, there are some 80 PPsets that account for these 15 EVENS 116-130. Easily done.

What if you want to know specifically who covers, say EVEN 128? Looking at the full table, line 18, we see that Trail 67 and above have Ee > 128. How can we determine that the 19+109, 31+97 and 61+67 PPsets are the ones we are looking for?

Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick:

• on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 128 = 64

• see that PRIMES 61 & 67 on either side = 128 when added together and both are PRIMES

• knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 64 slot like the 61 & 67

• the next pair out would be 31 & 97, followed by 19 & 109

• this is easily calculated as the core Axis value ± steps away:

• 64 ± 3 = 61 and 67
• 64 ± 33 = 31 and 97
• 64 ± 45 = 19 and 109
• where the steps away are literally the number of steps along the Axis to either side of the core Axis value that goes to a PPset
• their ∑s equal the EVEN 128:

• 61 + 67 = 128

• 31 + 97 = 128

• 19 + 109 = 128.

• We can confirm this on the PTOP (Table 35), and, of course, directly on the BIM. Tables 43 and 44 can also confirm ALL EVENS up to 2000. One can also use an online calculator to get the results.

• it can be quite helpful to follow these examples directly on the BIM. The visualizations of how the symmetrical PPsets contributed from the separate, individual PRIME Trails line up their 90° R-angled isosceles triangles over the central core with each of their respective apexes inline and pointing towards the EVEN that they are forming. On the Axis, one can easily see the symmetrical steps from the core to each PPset.

• What if you want to know specifically who covers, say EVEN 126? Looking at the full table, line 18, we see that Trail 67 and above have Ee > 126. How can we determine that the

• 13+113

• 17+109

• 19+107

• 23+103

• 29+97

• 37+89

• 43+83

• 47+79

• 53+73

• 59+67

PPsets are the ones we are looking for?

• Without even knowing the values for the Column 6 (Table 46): EVEN, ∑# of PPsets, one can:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick:

• on Table 46, find where EVEN 128 would be located between line 17 and 18 as 1/2 of 126 = 63

• see that PRIMES 59 & 67 on either side = 126 when added together and both are PRIMES

• knowing from the BIM that ALL PPsets are symmetrical about the center Axis core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 63 slot like the 59 & 67

• the next pair out would be 53 & 73, followed by 47 & 79.

• this is easily calculated as the core Axis value ± steps away:

• 63 ± 4 = 59 and 67

• 63 ± 10 = 53 and 73

• 63 ± 16 = 47 and 79

• 63 ± 20 = 43 and 83

• 63 ± 26 = 37 and 89

• 63 ± 34 = 29 and 97

• 63 ± 40 = 23 and 103

• 63 ± 44 = 19 and 107

• 63 ± 46 = 17 and 109

• 63 ± 50 = 13 and 113

• their ∑s equal the EVEN 126:

• 59 + 67 = 126
• 53 + 73 = 126
• 47 + 79 = 126
• 43 + 83 = 126
• 37 + 89 = 126
• 29 + 97 = 126
• 23 + 103 = 126
• 19 + 107 = 126
• 17 + 109 = 126
• 13 + 113 = 126
• The key is seeing that the PPsets are symmetrical about the core Axis value of the EVEN and applying these sequential, linear # of steps out from the core in identifying the PPsets that form that EVEN! EVERY PPset that forms an EVEN is part of a symmetrical pair of PPs, that together forms the EVEN. This is a critical and paramount finding in the behavior of the PRIMES and in the PROOF of the strong form of the Goldbach Conjecture.
• The ONLY EVENS (>4) with less than 2 PPsets are the EVENS 6, 8 and 12. EVEN 6 has only 1 PPset of 3,3 and EVEN 8 has only 1 PPset of 3,5. The EVEN 8 is the beginning of the 3 + 5 Trail, while the EVEN 6 is BOTH the start and end of the 3 + 3 Trail that begins the PTOP. The EVEN 12 is the one isolated example where the middle of a PRIME Trail — here the 3 + 7 Trail — is the only PPset to form EVEN 12, as the 3 + 5 Trail is — being the early start of Trails — not yet sufficient enough in length to cover the EVEN 12 slot. All 3 EVENS — 6,8,12 — are the exceptions precisely because they are the very beginning of the PTOP where the PRIME Trails are first being established.
• So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically flanked on either side by ODD WINs, and, as we have shown here on the PTOP and the PTOP as hidden within the BIM, will have two (2) or more ODD PPs pairs within that symmetrical flanking that will, as PPsets, form said EVEN.
• The symmetry simply falls out of the inherent inter-connection between the number values of each cell within the BIM. Every 1st Parallel EVEN has a straight line path back to its Axis and along that path the cell values must reflect the product of the two Axis values located symmetrically from the core Axis value outward, such that at their 90° R-angle, isosceles intersection on that path, one finds that product.
• The real wonder — and part of this amazing new finding — is that when such Axis numbers are found that are PRIME, they ALWAYS have a symmetrical ODD # counterpart on the other side of their common, core Axis number: and at the minimum, two or more candidates will have their ODDs be PRIMES, thus completing two or more PPsets!
• To remember, the Axis core number may itself be either EVEN or ODD as it is simply 1/2 of the EVEN that the path points to from its 1st Parallel Diagonal location. Also, the above rule does NOT mean that ALL ODD #s found on the Axis are part of the PPsets that form the EVEN. Even though symmetrical, and forming the same products on the path, BOTH PPset MEMBERS FORMING THE PAIR MUST BE PRIMES, e.i. an Axis 9, 15, 21,.. are NOT PRIME and NOT part of a PPset forming the EVEN. The geometry defines the algebra = algebraic geometry.
• The growth rate of the staggered, zig-zagging diagonal Trails of the PS ensures that the number of overlapping Trails will always exceed the growth rate of the PRIME Gaps by a sizable margin.

Towards the bottom of Table 46, lines 1092 - 1107, there is a good run of PRIME Gaps in close proximity. Below is the table in plain form with the same column headings:

line #PRIMES, P**2≥3**∑# of 3+P**2PPsets/Trail**Prime Gap∆ **Trail-Gap**EVEN: ∑# of**PPsets**EC=# of EVENs coveredE**e = Ending EVEN covered**EVEN
10808681108081072944340173628684
108186891081410771004344173788692
10828693108261076894346173868696
10838699108381075964349173988702
108487071084610781324353174148710
10858713108561079924356174268716
1086871910861210741124359174388722
108787311087610811014365174628734
108887371088410841364368174748740
10898741108961083934370174828744
109087471090610841384373174948750
109187531091810831044376175068756
1092876110921810741114380175228764
10938779109341089914389175588782
109487831094201074954391175668786
109588031095410911254401176068806
1096880710961210841244403176148810
109788191097210951024409176388822
109888211098101088904410176428824
10998831109961093934415176628834
110088371100210981414418176748840
110188391101101091934419176788842
110288491102121090914424176988852
11038861110321101934430177228864
110488631104411001184431177268866
1105886711052010851214433177348870
110688871106611001484443177748890
110788931107301077944446177868896
11088923110861102954461178468926
110989291109411051254464178588932
11108933111081102954466178668936
1111894111111011011014470178828944
1112895111121211001044475179028954
11138963111361107964481179268966
11148969111421112844484179388972
1115897111152810871044485179428974
111689991116211141104499179989002
11179001111761111954500180029004
111890071118411141334503180149010
11199011111921117964505180229014
1120901311201611041194506180269016

Without even knowing the values for the Column 6 (Table 46) which have been inserted here: EVEN, ∑# of PPsets, one can take EVEN 8872:

• look at an expanded PTOP

• look at an expanded BIM

• calculate using this trick: (see *below for a simple Universal calculation method as shown in Table 45: Equations.)

• on Table 46, find where EVEN 8872 would be located as the core Axis value between lines 601 and 602 as 1/2 of 8872 = 4436

• see that PRIMES 4423 & 4441 on either side = 8864 when added together and both are PRIMES we are close

• knowing from the BIM that ALL PPsets are symmetrical about the center core that points to the 90° R-angled isosceles triangle that each set forms, look for two PRIME Trails that are symmetrical to the 4436 slot like the 93 PPsets that equal EVEN 8872:

• 5+8867
• 11+8861
• 23+8849
• 41+8831
• 53+8819
• 89+8783
• 131+8741
• 173+8699
• 179+8693
• 191+8681
• 263+8609
• 359+8513
• 443+8429
• 449+8423
• 503+8369
• 509+8363
• 599+8273
• 641+8231
• 653+8219
• 701+8171
• 761+8111
• 863+8009
• 953+7919
• 971+7901
• 1019+7853
• 1031+7841
• 1049+7823
• 1181+7691
• 1223+7649
• 1229+7643
• 1283+7589
• 1289+7583
• 1373+7499
• 1439+7433
• 1523+7349
• 1619+7253
• 1721+7151
• 1871+7001
• 1889+6983
• 1901+6971
• 1913+6959
• 1973+6899
• 2003+6869
• 2039+6833
• 2069+6803
• 2081+6791
• 2111+6761
• 2153+6719
• 2213+6659
• 2273+6599
• 2309+6563
• 2351+6521
• 2381+6491
• 2399+6473
• 2423+6449
• 2543+6329
• 2549+6323
• 2609+6263
• 2699+6173
• 2729+6143
• 2741+6131
• 2819+6053
• 2843+6029
• 2861+6011
• 2969+5903
• 3011+5861
• 3023+5849
• 3089+5783
• 3203+5669
• 3221+5651
• 3299+5573
• 3371+5501
• 3389+5483
• 3491+5381
• 3539+5333
• 3593+5279
• 3701+5171
• 3719+5153
• 3821+5051
• 3833+5039
• 3851+5021
• 3863+5009
• 3929+4943
• 4001+4871
• 4073+4799
• 4079+4793
• 4139+4733
• 4229+4643
• 4289+4583
• 4349+4523
• 4391+4481
• 4409+4463
• 4421+4451
• as before, these can be simplified as:

• 4436 ± 5 = 4431 and 4441. Together, 4431 + 4441 = 8872. See Table 48 for the complete list.
• 4436 ± 11
• 4436 ± 23
• 4436 ± 41
• 4436 ± 53
• 4436 ± 89
• 4436 ± 131
• 4436 ± 173
• 4436 ± 179
• 4436 ± 191
• 4436 ± 263
• 4436 ± 359
• 4436 ± 443
• 4436 ± 449
• 4436 ± 503
• 4436 ± 509
• 4436 ± 599
• 4436 ± 641
• 4436 ± 653
• 4436 ± 701
• 4436 ± 761
• 4436 ± 863
• 4436 ± 953
• 4436 ± 1019
• 4436 ± 1031
• 4436 ± 1049
• 4436 ± 1181
• 4436 ± 1223
• 4436 ± 1229
• 4436 ± 1283
• 4436 ± 1289
• 4436 ± 1373
• 4436 ± 1439
• 4436 ± 1523
• 4436 ± 1619
• 4436 ± 1721
• 4436 ± 1871
• 4436 ± 1889
• 4436 ± 1901
• 4436 ± 1913
• 4436 ± 1973
• 4436 ± 2003
• 4436 ± 2039
• 4436 ± 2069
• 4436 ± 2081
• 4436 ± 2111
• 4436 ± 2153
• 4436 ± 2213
• 4436 ± 2273
• 4436 ± 2309
• 4436 ± 2351
• 4436 ± 2381
• 4436 ± 2399
• 4436 ± 2423
• 4436 ± 2543
• 4436 ± 2549
• 4436 ± 2609
• 4436 ± 2699
• 4436 ± 2729
• 4436 ± 2741
• 4436 ± 2819
• 4436 ± 2843
• 4436 ± 2861
• 4436 ± 2969
• 4436 ± 3011
• 4436 ± 3023
• 4436 ± 3089
• 4436 ± 3203
• 4436 ± 3221
• 4436 ± 3299
• 4436 ± 3371
• 4436 ± 3389
• 4436 ± 3491
• 4436 ± 3539
• 4436 ± 3593
• 4436 ± 3701
• 4436 ± 3719
• 4436 ± 3821
• 4436 ± 3833
• 4436 ± 3851
• 4436 ± 3863
• 4436 ± 3929
• 4436 ± 4001
• 4436 ± 4073
• 4436 ± 4079
• 4436 ± 4139
• 4436 ± 4229
• 4436 ± 4289
• 4436 ± 4349
• 4436 ± 4391
• 4436 ± 4409
• 4436 ± 4421

We can further simplify by applying these Universal (vs Specific, see Table 45:) equations:

• By definition, EVEN = P1 + P2 = PPset
• let S = steps, E = EVEN = 2(core Axis value) = 2(Ax), as E/2 = Ax

• S = P2 - E/2 = P2 - Ax

• the PPset for a given EVEN:

• P2 = S + E/2 = S + Ax

• P1 = P2 - (2S)

• 2S = P2 - P1

• S = AX - P1

• E = P1 + P2

Example: 3,5 PPset for EVEN = 8: S = P2 - E/2 = P2 - Ax 1 = 5 -4

P2 = S + E/2 = S + Ax

5 = 1 + 4

P1 = P2 - (2S) 3 = 5 - (2*1)

And, as P1 = E - P2, or E = P1 + P2

P1 = P2 - (2S) = E - P2

2P2 - E = 2S 2(5) - 8 = 2(1) S = 1

As S = (2P2 - E)/2

S = (2P2 - E)/2 = P2 - E/2

2P2/2 - E/2 = P2 - E/2

P2 - E/2 = P2 - E/2 = S

P2 - Ax = P2 - Ax = S

The built-in symmetry of the PPsets around the core Axis value is easily calculated as these four examples show:

• EVEN 24 with core Axis value (Ax) = 24 / 2 = 12 with 3 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 11,13 13-12 = 1 13 - (2x1) = 11

• 7,17 17-12 = 5 17 - (2x5) = 7

• 5,19 19-12 = 7 19 - (2x7) = 5

• EVEN 26 with core Axis value = 26 / 2 = 13 with 3 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 13,13 13-13 = 0 13 - (2x0) = 13

• 7,19 19-13 = 6 19 - (2x6) = 7

• 3,23 23-13 = 10 23 - (2x10) = 3

• EVEN 22 with core Axis value = 22/2 = 11 with 3 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 11,11 11-11 = 0 11 - (2x0) = 11

• 5,17 17-11 = 6 17 - (2x6) = 5

• 3,19 19-11 = 8 19 - (2x8) = 3

• EVEN 100 with core Axis value = 100 / 2 = 50 with 6 PPsets:

• P1 = P2 - (2S)

• as S = P2 - E/2 = P2 - Ax

• 47,53 53-50 = 3 53 - (2x3) = 47

• 43,57 57-50 = 7 57 - (2x7) = 43

• 41,59 59-50 = 9 59 - (2x9) = 41

• 29,71 71-50 = 21 71 - (2x21) = 29

• 17,83 83-50 = 33 83 - (2x33) = 17

• 11,89 89-50 = 39 89 - (2x39) = 11

~~~~~~~~

As one moves successively along the EVENS, the Trails get longer and longer, adding one new member for each successive PS (See Tables 44-46.). This rate of increase far exceeds the size and incidence rate of the PRIME Gaps (See Tables 42 and 46.), ensuring that for every EVEN ≥6, there is at least one PPset of ODD PRIMES that will form it. Actually, as we have shown, there are always a minimum of 2 sets of PPsets that form the EVENS (≥14, including 10).

Table 44 shows how ∑s of the number of PPsets/ EVEN grows. It increases such that for every 60 successive EVENS on average, the ∑ increases by 3 (as calculated from 4-1080).

For example:

Columns generally have 60 entries ending in multiples of 120.

∑of PPsets per column Ave/Col

Col A 262 / 59 = 4.4 = 4

Col B 516 / 60 = 8.6 = 9

Col C 718 / 60 = 12 = 12

Col D 930 / 60 = 15.5 = 16

Col E 1076 / 60 = 17.9 = 18

Col F 1267 / 60 = 21.1 = 21

Col G 1302 / 60 = 21.7 = 22

Col H 1534 / 60 = 25.6 = 26

Col I 1687 / 60 = 28.1 = 28

SUMMARY and CONCLUSION

It is worth repeating here from above:

• So we are claiming here — and giving as proof — that ALL EVENS ≥ 14 (and including EVEN 10), have a MINIMUM OF 2 PPsets. It is within the geometry of the number distribution of simple, natural, Whole Integer Numbers (WIN) that ALL EVEN WINs shall have their EVEN/2, Axis core value be symmetrically