Domain τau

Complete Analysis of the Tau Block Pattern

Preprint — March 2026
MSC 2020: 11A63, 11B85, 11J81, 11Y11, 37E10

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Abstract

We study the discrete structures that emerge when integer arithmetic — specifically the floor function and modular reduction — acts on the transcendental constant τ = 2π. The function mod(⌊n/τ⌋, 9) generates a hierarchy of nested cycles: a 710-number cycle containing exactly 113 blocks with lengths drawn from {6, 7} in a self-similar pattern {7, 6, 6, 7, 6, 6, 6} interrupted by two symmetric phase breaks; a 6390-number complete value cycle; and super-cycles at 25560 and 102240 numbers preserving this architecture at scale.

We identify a family of prime-generating functions of the form ⌊c · 10ⁿ / τ⌋ for c ∈ {1/τ, τ, 71, 73,996,200} that produce verified primes at specific exponents. The repeating decimal tails of ⌊710 · 10^y / τ⌋ encode the period of 1/71 and stand in clean rational ratios (4/11, 33/14, 7/6) across scales y = 6, 261, 3932.

In the complex-exponential extension sin(n^(iτ^k)), all orbits converge to sin(1), and the critical crossing points are identified as e^(π/2), e^(π) (Gelfond's constant), and e^(3π/2) — integer multiples of e^(τ/4) — revealing that the dynamics are governed by quarter-turn exponentials of τ in the complex plane.

The average block length of the 710-cycle is 710/113 ≈ τ, a self-referential signature: the transcendental constant reappears as the mean of the discrete structure it generates. We frame these phenomena not as consequences of Diophantine approximation but as properties of the interaction between discrete operators and transcendental constants — what is here termed Symbolic Necessity: the structure that arises when a formal system acts on a quantity it operationally requires but can never internally resolve.

Keywords: tau, transcendental constants, floor function, modular arithmetic, prime generation, Sturmian sequences, symbolic necessity, discrete–transcendental interaction

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Preamble: Discreteness Over Transcendence

This document records a body of research into what happens when discrete arithmetic — the floor function, modular reduction, integer projection — acts directly on a transcendental constant it can never capture. The central object of study is the function mod(floor(n/τ), 9), and the structures that fall out of it: a 710-number cycle containing exactly 113 blocks, a self-similar phase pattern {7, 6, 6, 7, 6, 6, 6} with two symmetric breaks, nested super-cycles at 6390, 25560, and 102240 numbers, and a family of prime-generating functions anchored to these same scales.

The thesis here is not Diophantine approximation. It would be easy to observe that 710/τ ≈ 113.00000096 and attribute the near-integer to the classical convergent 355/113 ≈ π, which is famously accurate for its denominator size. But that framing treats the discrete structure as a side effect of approximation quality — it says the rationals chase the irrationals well, and the rest is noise. This research asks a different question. It starts with floor and mod as operators — hard quantisation and finite-group projection applied to a transcendental — and asks what structure the interaction itself produces. The near-integer isn't the explanation; it is the phenomenon to be explained.

By Lindemann–Weierstrass, τ is transcendental: no algebraic equation over the rationals has τ as a root. Integer arithmetic cannot construct it. But integer arithmetic can act on it, and when it does, the result is not noise. It is 113 blocks. It is a base pattern with phase breaks at positions 56 and 109. It is prime numbers at scales 710, 6390, 73,996,200. The coherence is in the dance between the discrete and the transcendental — not in either partner alone.

This is one expression of what the author terms Symbolic Necessity (⊙): τ as something the discrete system operationally requires but can never fully resolve from within. The block pattern, the cycle hierarchy, and the prime-generating behaviour are what that irresolution looks like from the integer side.

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Structural Insights and Identifications

Several constants and relationships embedded in the numerical data deserve explicit identification, as they connect the system to known mathematical structures.

The Settle Point Is sin(1)

Throughout the n-dependent sequence algebra (Section XII), all points in the system sin(n^(iτ^k)) converge to the value 0.84147098. This is sin(1) = 0.84147098480789... — the sine of unity. The system's attractor is not an arbitrary constant; it is the image of the multiplicative identity under the most fundamental periodic function.

The Crossing Points Are Powers of e^(τ/4)

The three critical crossing points identified in the convergence analysis are not arbitrary:

Crossing	Value	Identity
First imaginary crossing	N_firstCross = 4.810477380965351	e^(π/2) = e^(τ/4)
First real crossing	N_SecondCross = 23.140692632779263	e^(π) = e^(τ/2), i.e. Gelfond's constant
Period completion ratio	Q ≈ 111.31777859	e^(3π/2) = e^(3τ/4)

The system's imaginary-line crossing, real-line crossing, and period completion are spaced at exact integer multiples of e^(τ/4). This is a structural claim: the dynamics of sin(n^(iτ^k)) are governed by the exponential of quarter-turns of τ in the complex plane. The period completion occurs at N_SecondCross × Q = e^(τ/2) × e^(3τ/4) = e^(5τ/4), itself a clean multiple of the base unit.

The Repeating Tail at y = 6 Contains 1/71

The 35-digit repeating tail at y = 6 — 16901408450704225352112676056338028 — encodes the decimal period of 1/71. The period of 1/71 is exactly 35 digits: 01408450704225352112676056338028169... This is structurally inevitable: since 710 = 10 × 71, the function floor(710 × 10^y / τ) probes the decimal structure of 71/τ, and the remainder cycles with the period of the reciprocal of 71. The prime 71 is not incidental to the pattern; it is woven into the repeating decimal fabric of the output.

The Average Block Length Self-Refers to τ

The average block length of the 710-cycle is 710/113 ≈ 6.28318... ≈ τ. The mod-9 structure of floor(n/τ) produces blocks whose average length converges back to τ itself. This self-reference — the transcendental constant reappearing as the mean of the discrete structure it generates — is a signature of the discrete-transcendental interaction at the heart of this work.

355/113 as Phenomenon, Not Explanation

The rational approximation 355/113 = 3.14159292... ≈ π is accurate to seven significant figures and is present throughout: 710 = 2 × 355, and 113 is the block count. But in this framework, 355/113 is not the cause of the pattern's coherence — it is one of its most visible symptoms. The question is not "why does 355/113 approximate π so well?" (a question about continued fractions and Diophantine theory). The question is: "what does the integer lattice see when it looks at τ through the floor function?" The answer is 113 blocks, phase-locked cycles, and primes — and 355/113 is the rational shadow that the lattice casts.

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I. Introduction to Tau

A. Definition of Tau

Tau (τ) is defined as the ratio of a circle's circumference to its radius, equal to approximately 6.28318. Mathematically, τ = 2π, where π (pi) is the ratio of a circle's circumference to its diameter.

B. Mathematical Foundations of Tau vs Pi

1. Primitive Measurement Proof

In geometry, the defining property of a circle is the set of points equidistant from a center point. This distance is the radius, not the diameter.

• A circle is defined as C = {(x,y) | √((x−h)² + (y−k)²) = r} where (h,k) is the center and r is the radius
• The circle constant relating this defining measure (radius) to circumference is τ = C/r
• The diameter-based constant π = C/d = C/2r is derived by introducing an additional factor
• Therefore, τ is the primitive constant arising directly from the circle's definition

2. Number Theory Perspective

In the multiplicative structure of these constants:

• If we consider the field extension ℚ(τ)/ℚ vs ℚ(π)/ℚ
• ℚ(τ) = ℚ(π) since they generate the same field
• However, when examining their roles in circle geometry, τ requires fewer additional factors to express fundamental relationships
• The minimal polynomial of τ over ℚ is the same as that of 2π, demonstrating that τ is not more algebraically complex than π

3. Symmetry and Group Theory

The circle group S¹ (the group of rotations of a circle) has period τ:

• The group operation on S¹ completes one cycle exactly when reaching τ radians
• The fundamental homomorphism from ℝ to S¹ given by θ → e^(iθ) has kernel τℤ
• τ is the primitive generator of the kernel, making it more fundamental from a group-theoretic standpoint

4. Information Theory Argument

• Using τ eliminates the factor 2 in numerous formulas
• By Kolmogorov complexity arguments, the simplest description of circle relationships requires fewer additional constants when using τ
• This reduction in descriptive complexity suggests τ is more primitive

5. Tau and Pi in the Multiplicative Group

• In the multiplicative group of positive reals, τ and π generate the same cyclic subgroup
• The reciprocal relationship τ⁻¹ = π⁻¹/2 shows that τ⁻¹ < π⁻¹
• When constructing the natural ordering, τ comes before π in the fundamental sequence of circle constants

Summary Table

Aspect	Mathematical Argument
Circle Definition	The radius (r) is the primary defining measure. Since τ = C/r, τ directly connects circumference to the circle's fundamental defining property.
Group Theory	In S¹, τ radians completes exactly one period. The kernel of R → S¹ is generated by τℤ, making τ the primitive generator.
Multiplicative Relations	τ⁻¹ = π⁻¹/2 shows τ⁻¹ < π⁻¹, giving τ precedence in the natural ordering.
Natural Parametrisation	Unit-speed parametrisation yields period τ: γ(t) = (r·cos(t), r·sin(t)) where 0 ≤ t < τ.
Fourier Analysis	Fourier series with τ as base period eliminates factor 2.
Differential Equations	In harmonic motion (y″ + ω²y = 0), period T = τ/ω is more direct than T = 2π/ω.
Information Theory	Using τ reduces Kolmogorov complexity by eliminating repeated factors of 2.
Subgroup Containment	⟨τ⟩ properly contains ⟨π⟩, as π = τ/2 means π ∈ ⟨τ⟩, but τ ∉ ⟨π⟩.
Radian Measure	A complete circle spans exactly τ radians: a 1:1 mapping between circle and constant.
Complex Analysis	e^(iτ) = 1 represents one complete rotation in the complex plane directly.

These arguments collectively demonstrate why τ can be considered the more primitive or fundamental circle constant from various mathematical perspectives.

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II. Base Function and Block Formation

A. Core Function

For any natural number n:

mod( floor(n · 1/τ), 9 )

Returns an integer 0–8, forming blocks of consecutive numbers that share the same value.

B. Block Properties

Each block has:

1. Block ID (sequential from 0)
2. Block Integer (value 0–8 from mod 9)
3. Block Length (either 6 or 7)
4. Block Start n (first number in block)
5. Block Stop n (last number in block)
6. Block Sum (Length × Integer)
7. Block Start n mod 9
8. Block Stop n mod 9

C. Block Formation Example

First 7 blocks showing all properties:

ID	Integer	Length	Start n	Stop n	Sum	Start mod 9	Stop mod 9
0	0	7	0	6	0	0	6
1	1	6	7	12	6	7	3
2	2	6	13	18	12	4	0
3	3	7	19	25	21	1	7
4	4	6	26	31	24	8	4
5	5	6	32	37	30	5	1
6	6	6	38	43	36	2	7

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III. The 710-Cycle Complete Analysis (113 Blocks)

A. Length Pattern Structure

Phase Pattern (Blocks 0–112, n = 0–709):

• Base pattern {7, 6, 6, 7, 6, 6, 6} repeats with two interruptions
• 113 blocks total
• 710 numbers in each sequence (n = 0 to n = 709)

First Phase (Blocks 0–59, n = 0–376):

• 60 blocks, 377 numbers
• Core Pattern (Blocks 0–55, n = 0–351):
  • Base pattern {7, 6, 6, 7, 6, 6, 6} — 7 blocks, 44 numbers per sequence
  • 56 blocks, 352 numbers, 8 complete sequences (counting from 0)
  • Last block start: n = 346–351
• First Break (Blocks 56–59, n = 352–376):
  • Extra {7, 6, 6, 6} group appears — 4 blocks, 25 numbers
  • Then returns to {7, 6, 6, 7, 6, 6, 6} pattern

Second Phase (Blocks 60–112, n = 377–709):

• 53 blocks, 333 numbers
• Core Pattern (Blocks 60–108, n = 377–684):
  • Base pattern {7, 6, 6, 7, 6, 6, 6} — 7 blocks, 44 numbers per sequence
  • 49 blocks, 308 numbers, 7 complete sequences (counting from 0)
  • Last block start: n = 679–684
• Second Break (Blocks 109–112, n = 685–709):
  • Extra {7, 6, 6, 6} group appears — 4 blocks, 25 numbers
  • Then returns to {7, 6, 6, 7, 6, 6, 6} pattern

B. Integer Value Progression

Block Integer sequence (first 44 numbers):

Block ID	Integer	Length
0	0	7
1	1	6
2	2	6
3	3	7
4	4	6
5	5	6
6	6	6

• Complete sequence progresses through all values 0–8
• Pattern progression maintains integrity through breaks
• Integer values cycle independently of length pattern
• Total sum of Block Integers = 4068

C. Length Distribution

For the complete 710-cycle:

• 81 blocks of length 6
• 32 blocks of length 7
• Total: 113 blocks
• Sum of lengths: 710
• Average block length: 710/113 ≈ 6.2832...

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IV. Important Sums and Their Relationships

A. Column Sums

1. Block Integer Column — Sum = 4068 = floor(25560/τ)
2. Block Length Column — Each 6 contributes 6n, each 7 contributes 7n; Total = 6390
3. Block Sum Column — Sum = 25560 = 4 × 6390
4. Mod 9 Columns — Both Start n mod 9 and Stop n mod 9 sum to 4068 (matching Block Integer sum)

B. Tau Ratio Connections

These sums connect directly to tau ratios:

710   / τ  ≈  113.00000959524569    (block count)
6390  / τ  ≈  1017.0000863572112   (total blocks in complete value cycle)
25560 / τ  ≈  4068.000345428845    (appears in multiple sums)
102240/ τ  ≈  16272.0013817

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V. Complete Value Cycle (6390 Numbers)

A. Block Structure

Total 1017 blocks:

• 729 blocks of length 6 (= 81 × 9)
• 288 blocks of length 7 (= 32 × 9)
• Maintains patterns from 710-cycle

B. Integer Value Properties

• Each value 0–8 appears exactly 710 times
• Complete return to starting values
• Block Integer sum = 4068

C. Length Pattern Properties

• Contains 9 complete 710-cycles
• Each subcycle maintains break points
• Pattern preserves {7, 6, 6, 7, 6, 6, 6} base structure

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VI. Prime Factor Relationships

A. Critical Numbers

Pattern Numbers:

710    = 2 × 5 × 71
6390   = 2 × 3² × 5 × 71
25560  = 2³ × 3² × 5 × 71
102240 = 2⁵ × 3² × 5 × 71

Tau Ratio Numbers:

113   = 113  (prime)
1017  = 3² × 113
4068  = 2² × 3² × 113
16272 = 2⁴ × 3² × 113

B. Key Relationships

Pattern Ratios:

710    = (16 + 3/22) × 44
6390   = 9 × 710 = (3195/22) × 44
25560  = 4 × 6390 = 36 × 710 = (6390/11) × 44
102240 = 4 × 25560 = 16 × 6390 = 144 × 710
73996200 = 723.74 × 102240 = 2895 × 25560 = 104220 × 710

Block Count Ratios:

1017 = 9 × 113
4068 = 4 × 1017 = 36 × 113

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VII. Pattern Preservation at Scale

A. Block Pattern Scaling

The pattern preserves its structure across cycles:

• Every 710 numbers: Length pattern index (n = 0 to 709; note 709 is prime)
• Every 6390 numbers: Complete value cycle (n = 0 to 6389; note 6389 is prime)
• Every 25560 numbers: Super cycle (n = 0 to 25559; 25559 is not prime)
• Every 102240 numbers: Full cycle (n = 0 to 102239; 102239 is not prime)
• After 102240: the CODA pattern cycle of 6390 numbers

B. Value Evolution

Values progress through four scales:

1. 710-cycle: 113 blocks — values progress but don't complete
2. 6390-cycle: 1017 blocks — complete value cycle
3. 25560-cycle: 4068 blocks — maintains all subcycle properties
4. 102240-cycle: 16272 blocks — includes a 6390-segment that differs slightly, then the overall pattern shifts

Shift Points in the 102240-Cycle

Starting from index n = 0 of the cycle, shifts occur at regular 710-intervals:

n	Index	Δ Index
104703	2463	—
105413	3173	710
106123	3883	710
106833	4593	710
107543	5303	710
108253	6013	710
108963	6723	710
109673	7433	710
110383	8143	710
111093	8853	710
111803	9563	710
112513	10273	710
113223	10983	710
113933	11693	710
114643	12403	710

Note: The number of blocks, their ratios, and all other structural properties are believed to remain the same throughout.

C. Large Scale Behaviour

73996200 / 710 = 104220
73996200 / τ   = 11776861.000016505...
floor(11776861.000016505...) = 11776861    ← PRIME

Cycle starts at massive scale:

Cycle 1 starts: n = 665,953,020
Cycle 2 starts: n = 1,331,906,040
Cycle 3 starts: n = 1,997,859,060

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VIII. Prime at Scale

A. Core Variables

a = 710.0
y = 6
b = 10^y
k = a × b
n = k / τ
x = floor(n)

B. Summary of All Prime Candidates

Found in floor(1/τ · 10ⁿ), floor(τ · 10ⁿ), and floor(71 · 10ⁿ/τ) for n = 1 to 9999. Note: the exponents n in the 71-family correspond to y+1 in the repeating-tail analysis because 710 × 10^y = 71 × 10^(y+1).

floor(1/τ × 10^90)
floor(1/τ × 10^71)
floor(1/τ × 10^155)
floor(τ × 10^344)
floor(τ × 10^382)
floor(τ × 10^521)
floor(τ × 10^3779)
floor(τ × 10^5754)
floor(71 × 10^7 / τ)
floor(71 × 10^262 / τ)
floor(71 × 10^3933 / τ)
floor(73996200 × 10^0 / τ)
floor(73996200 × 10^5 / τ)
floor(73996200 × 10^74 / τ)
floor(73996200 × 10^193 / τ)
floor(73996200 × 10^282 / τ)
floor(73996200 × 10^775 / τ)

C. Repeating Tail Patterns in floor(710 × 10^y / τ)

When y = 6

After the initial digits:

15915494225352112676056338028

The repeating tail emerges:

16901408450704225352112676056338028 (repeating)

When y = 261

After the initial digits:

15915494309189533576888376337251436203445964574045644874766734405889679763422653509
01138027662530859560728427267579580368929118461145786528779674107316998392292399669
37409077573077746396925307688717392896217397661693362390241723629011832380114222699
755715940461888732394366197183098591549295 77

The repeating tail emerges:

46478873239436619718309859154929577 (repeating)

When y = 3932

After the initial digits (full string preserved):

15915494309189533576888376337251436203445964574045644874766734405889679763422653509
01138027662530859560728427267579580368929118461145786528779674107316998392292399669
37409077573077746396925307688717392896217397661693362390241723629011832380114222699
75715940461890086902673956120489410936937844085528723099946443400248672347739459610
89832309678307490616698646280469944865218788157478656696424103899587413934860998386
80991999624428755851711788584311175187671605465475369880097394603647593337680593024
94496635305327156775503220324777816397166022946748119598165840606016803035998133911
98749883278665443527975507001624067756438884957131088012219937614768137776473789063
30680464579784817613124273140699607750245002977598570890569027967851315252100163177
46020924811606240561456203146484089248459191435211575407556200871526606802217159140
75747458272259774628539987515532939081398177240935825479707332871904069997590765770
78493470393589828087173425640366895116625457059433276312686500261227179711532112599
50438667945037625560836317116952597581282249416233343145106123536878563113636692167
14206974696012925057833605311960859450983955671870995474651043162381551758083944297
99709995052543875661294458833068460507852915151410404892988506388160776196993073410
38999578691890598093737772061875432227189301366255261238780387538881106814067654340
82827852693342679955607079038606035273899624512599574927629702359409558430116482964
11855777124057544494570217897697924094903272947702166496035653181535440038406898747
17691588763190966506964404776970687683656778104779795450353395758301881838687937766
12481495305996558021908359875103512712904323158049871968687775946566346221034204440
85549785037927386942935366193778292873593784347032302371458379235571186363419294601
83182291964165008783079331353497790997458649290267450609893689094588305033703053805
47312321580943197676032283131418980974982243833517435698984750103950068388003978672
35996080240027390108749548547879235682611399489032689974270834961149208289037767847
43035504568456083671479308456723327035485392556202086839324099562211753318394020970
79357077496549880868606636096866196703747454210283121925184622483499116114956655603
79696761399312829960776082779901007830360023382729879085402387615574454309260119100
54337998389046549212482951607072853005227210236017523313173179759311050328155109373
91363964530579260718008361795487672464598047397729244810920093712578691833289588628
39904358686666397567344514095036373271917431138806638307259230275973450605482127780
370653377830321709877349665684908003269885067417914646835082816168533143361607309951
498531198197337584442098416559541522506433943128644403838835615087977164501706470675
187745605916087168578579392262347563317111329986559415968907198506887442300575191977
056900382183925622033874235362568083541565172971088117217959368325648851874997487085
531165983061013921445446016148845277025114110702485217397451038667364038728600996748
931735618120711740478899368886556923078485023057057144063638632023685201074100574859
228111572196800397824759530016695852212303464187736504354676464565659719011230847670
993097085912836466691917769387914333155665066981321641521008957117286238426070678451
760111345080069947684223569896248805157759809533970808547505975362656490343944542058
178864356830420003150955947434392525448506749142908647514423033213324569511634945677
539394240360905438335528292434220349484366151466322860247766666049531406573435755301
409082798809147866934349227376026349978299570181619643212331404757628974840828911740
974782637899181699939487497715198981872666294601830539583275209236350685388922846824
725997252830076685693758365972291982442974740616381831139583067443485169285973832373
926624024345019978099404021896134834273613676449913827154166063424829363741850612261
086132119986334628470994183994274295591562833399048038211750116121166720519125793035
529292411344031161341124953183859269584904438468078490973982808855297045153053991400
988698840883654788732394366

The repeating tail emerges:

19718309859154929577464788732394366 (repeating)

(This last repeating tail persists up to y = 9999.)

D. Ratios Between Repeating Tails

Labelling the three repeating tails as a (y=6), b (y=261), c (y=3932):

a/a = b/b = c/c = 1   (diagonal)
a/b ≈ 0.3636363636... = 4/11
b/c ≈ 2.3571428571... = 33/14
c/a ≈ 1.1666666667... = 7/6

These clean repeating decimals (ratios of small integers) suggest a fundamental structural relationship, not coincidence. The spacing between the y-values shows:

• First gap: 261 − 6 = 255
• Second gap: 3932 − 261 = 3671

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IX. Primes from Tau Ratios

The following floor values are all confirmed prime:

floor(71 × 10^7    / τ)     ← prime
floor(71 × 10^262  / τ)     ← prime
floor(71 × 10^3933 / τ)     ← prime

And from the 710/τ chain:

710 / τ ≈ 113.0000095952...
1         ← primish
11        ← prime
113       ← prime
113000009 ← prime

Also: 1177686100001 is prime.

The tau-ratio numbers are all built from prime factors of 113 and 71:

1017  = 3² × 113
4068  = 2² × 3² × 113
16272 = 2⁴ × 3² × 113

710   = 2 × 5 × 71
6390  = 2 × 3² × 5 × 71
25560 = 2³ × 3² × 5 × 71
102240= 2⁵ × 3² × 5 × 71

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X. Comprehensive Overview of the Tau-Wave System

1. Core Mathematical System

The system involves wave functions defined with τ = 2π that exhibit connections to prime numbers:

• Wave function a: W_a = r_a · sin(x · r_a⁻¹) where r_a = 113 / (710 · τ⁻¹) ≈ 1.00018π
• Wave function b: W_b = r_b · sin(x · r_b⁻¹) where r_b = τ⁻¹
• Wave function c: W_c = r_a · sin(x · r_a⁻¹ + 6390·h)
• Wave function d: W_d = r_b · sin(x · r_b⁻¹ + 6390·h)

The parameter h controls phase shifting, with h = 0 creating perfect alignment between waves a/c and b/d.

2. Wave Synchronisation Properties

• Perfect synchronisation occurs at intervals of 710 units on the real line
• The waves drift apart but return to perfect synchronisation after approximately 11,580 cycles (h ≈ 11,580)
• The value 6390 × h ≈ 73,996,200 represents a complete cycle where phase relationships return to their initial state

3. Prime-Generating Properties

The function floor(71 × 10ⁿ / τ) generates primes for specific exponents:

• floor(71 × 10⁷ / τ) is prime
• floor(71 × 10²⁶² / τ) is prime
• floor(71 × 10³⁹³³ / τ) is prime

Additionally, floor(73,996,200 / τ) = 11,776,861 is prime, with 73,996,200 = 104,220 × 710, connecting back to the wave synchronisation cycle.

4. Mathematical Structure

The system reveals key relationships:

710   / τ ≈ 113.00000959...   (→ 113 blocks in a 710-cycle)
6390  / τ ≈ 1017.00008635...  (complete value cycle)
25560 / τ ≈ 4068.00034542...  (appears in multiple sums)

Prime factorisations share fundamental factors:

• Pattern numbers: 710 = 2×5×71, 6390 = 2×3²×5×71, 25560 = 2³×3²×5×71
• Tau-ratio numbers: 113 (prime), 1017 = 3²×113, 4068 = 2²×3²×113

5. Connection to Complex Analysis

The insight that sin(x) = sin(e^(iτ)) provides a geometric interpretation:

• Wave functions can be viewed as projections of complex rotations onto the real line
• Synchronisation points represent where rotations align despite different angular velocities
• This creates a geometric realisation of prime number sieves

6. Proposed Connection to the Riemann Hypothesis

The Riemann Hypothesis concerns prime distribution through the zeros of the Riemann zeta function. The tau-wave system may provide a geometric realisation of what RH describes analytically:

• Tau-wave synchronisation points may correspond to resonances in the spectral decomposition, similar to how zeta zeros act as frequencies controlling prime distribution
• The interference pattern (W_a − W_b) − (W_c − W_d) might act as a continuous generalisation of the Sieve of Eratosthenes
• The prime-generating function floor(71 × 10ⁿ / τ) could reveal structure related to the zeta zeros

7. Parameterised Transformation

A transformation function interpolates between the tau-wave and the Riemann zeta function:

F(s, t) = ((1-t) · tau_wave_b(n)  +  t · n^(-s)) / n^(t · s_real)

Where s is the complex input (typically on the critical line s = 1/2 + iy) and t is the parameter (0 = pure tau-wave, 1 = pure zeta).

This transformation visualises how the discrete prime-generating structure of the tau-wave transforms into the continuous spectral structure of the Riemann zeta function.

8. Desmos Formulation

The system can be simply expressed as follows (adapted from Desmos):

r_a = 1·τ⁰ · {n_normalize=0: 1, n_normalize>0: m}
m   = 113 / (710 · τ⁻¹)
r_b = τ⁻¹
W_a = r_a · sin(x · r_a⁻¹ + 6390·0)
W_b = r_b · sin(x · r_b⁻¹ + 6390·0)
W_c = r_a · sin(x · r_a⁻¹ + 6390·h)
W_d = r_b · sin(x · r_b⁻¹ + 6390·h)
Output = (W_a − W_b) − (W_c − W_d)

The system returns to its initial state when h ≈ 11,579.99999998377, so 6390 × h ≈ 73,996,200.

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XI. Complex Exponential Sequence Algebra

Notation and Formalism

This system works with sequences and complex exponentials, extending Euler's identity into higher dimensions through sequence operations.

Basic Notation:

• n = [−0...Z] — Defines a sequence of integers from 0 to Z (where Z = 6390)
• τ — Represents tau (2π)
• n[2] — The 2nd element in sequence n (equals 2)
• n[0...(k+1)] — Slice containing elements 0 through k+1 (inclusive)

Variables:

• k — Variable parameter for exponential powers
• K — Variable parameter used in defining v
• v = τ·K — A scaled version of K by factor τ

Core Sequences:

r   = n · e^(iτ^k)
b   = (n[01...(k+1)] / k) · e^(iτ^k)
b_b = −(n[01...(k+1)] / k) · e^(iτ^k)
c   = e^(iτ^n)
c_b = −e^(iτ^n)

Trigonometric Extensions:

a   = sin(n · e^(iτ))
a_c = sin(n · e^(iτ^(−v)))
a_e = sin(n · e^(iτ^(−k)))
a_b = a⁻¹

Key Properties

1. Zero-Difference Property: For all valid k (k ≥ 1 and k+1 ≤ Z): b[k+1] − c[k+1] = 0 and b_b[k+1] − c_b[k+1] = 0

2. Complex Distribution: As k increases, e^(iτ^k) creates points that rapidly circle the unit circle:

   • k=1: One rotation (e^(i2π) = 1)
   • k=2: ~6.28 rotations
   • k=3: ~39.5 rotations
   • k=4: ~248 rotations

3. Sequence Behaviour: Operations between sequences are performed element-wise. When sequences of different lengths are combined, the result is truncated to the shorter length.

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XII. n-Dependent Exponential Sequence Algebra

System Definition

sin(n · e^(iτ^k))
n = [−N...N]
N = number of points plotted
k = n(Zτ)

As Z increases, all points arrange themselves from two spirals outward to all being on the line between −1 and 1. The points fill in the line, and there is one that starts at some low N or Z and orbits before eventually settling:

61.46517051810098 < settle point < 61.46517051810099

Behaviour of sin(n^(iτ^k))

All points settle at the same value:

0.84147098  (may continue with more digits)

At around 180–600 N with Z near max, points are approximately evenly distributed, about 0.01724173 apart. Increasing N further fills in the gaps between existing points.

Convergence and Crossing Points

Focusing on Z = 0 and increasing N:

• First imaginary crossing: N_firstCross = 4.810477380965351, landing at N_firstCrossi = 0.04322737i
• First real crossing: N_SecondCross = 23.140692632779263
• Period completion: N = N_SecondCross · Q, where Q ≈ 111.31777859 (111.3177776 < Q < 111.317779094)

At the scale N = N_SecondCross · Q, one can view enough information in the system while also having crossed some other important state (yet to be determined).

Further analysis shows that X_Fixed ≈ 64.52642412788921 is potentially where things settle, but all sin(n^(iτ^k)) points do eventually settle to 0.84147098.

Behaviour as Z Increases

Most of the "action" in sin(n·e^(iτ^k)) happens by Z = 0.015 — most points have settled, though most of sin(n^(iτ^k)) have not.

Key observations:

• At Z = 0, the e-form spirals out to infinity in two lines
• As Z increases, all points follow curved paths and sort into place on the real line between 0 and 1
• At Z = 0.000284, the first point of sin(n^(iτ^k)) crosses the real line again, and the points from sin(n·e^(iτ^k)) are within radius 1

Point Accumulation Pattern

For sin(n·e^(iτ^k)) at N = 0, Z = 0:

• At N = 0.25: 2 points
• At N = 0.75: 3 points
• At N = 1.25: 4 points
• At N = 1.75: 5 points

Each new point falls exactly on the real line in order (point 1 at 1N, point 2 at 2N, etc.). The points start flying around but fall into a circle of radius 0.84147098. Once the circle is "full," they move to a larger shape best described as a cell mid-division — reaching this state when N = N_SecondCross · Q.

Notable Z Values

• Z = 0.00021108579925487037 — An interesting transition point; increasing by τ⁻²/360 causes the system to collapse to a new configuration
• Coalescence of points requires N > ~3440–3480 at very small Z, or around Z ≈ 0.000172 at normal scales (radius 2 from origin)
• With sin(n^(iτ^k)) at Z = 0.000097 and N = 4000, the larger (green) shape can be targeted

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End of document.