Brayden Ross Sanders / 7Site LLC · C. A. Luther · Monica Gish
DOI: 10.5281/zenodo.18852047
Branch: clay | Tag: v1.0-luther
python ck_run.py # All core theorems verified in < 1 second
python ck_sinc_demo.py # Matplotlib plot: pre-echo field + Montgomery bridge→ CLAY_QUICKSTART.md — one-page guide with a numerical example per Clay problem
Five things this project proved that are true all the way down. Each has a plain statement, an exact formula, a proof file, and one thing it explicitly does not claim.
1. Before the sieve starts, arithmetic is free.
Every integer b has a coprime window {1, 2, ..., SPF(b)-1} where every element is coprime to b — not most, all. The sieve hasn't fired yet. At k = SPF(b) it fires exactly once.
Formula: gcd(k, b) = 1 for all k < SPF(b); gcd(SPF(b), b) = SPF(b) > 1.
Proof: papers/proof_d11_d1_corollaries.py (D11a) and papers/proof_d15_sieve_isomorphism.py (D15).
Does not claim: The window property tells you anything about the distribution of composites beyond SPF(b).
2. The operator ring has exactly two kinds of harmony, and we can count them.
CK's algebra uses 10 operators over Z/10Z in two tables: TSML (73 harmony cells) and BHML (28 harmony cells). The counts follow from four disjoint zone partitions. We know why every cell is or isn't harmony.
Formula: TSML: 100 − |V0| − |V1| − |ECHO| = 100 − 9 − 8 − 10 = 73. BHML: |R_A| + |R_B| + |R_7| + |R_89| = 2 + 11 + 2 + 13 = 28.
Proof: papers/proof_d10_tsml_73_cells.py (D10) and papers/proof_d16_bhml_28_cells.py (D16).
Does not claim: The specific counts 73 and 28 have numerological significance; they follow mechanically from the zone partition rules.
3. The coherence threshold T = 5/7 was never designed — it emerged.*
T* = 5/7 was calibrated from TSML geometry and burned into silicon (Zynq-7020 FPGA). Independently, the operator map Phi = P_odd ∘ BHML ∘ W_op has a unique fixed point at CREATE = 5, and TSML's dominant output is HARMONY = 7. These were never designed to relate. They do.
Formula: Phi(5) = P_odd(BHML[5][W_op[5]]) = P_odd(BHML[5][7]) = P_odd(6) = 5. T* = CREATE/HARMONY = 5/7.
Proof: papers/proof_d7_phi_fixed_point.py (D7).
Does not claim: T* = 5/7 is universal across all semiprimes; it is the value for b = 35 specifically, and the emergence was a discovery not a design.
4. The prime corridor has an exact spectral mean.
The sinc² function appears as the continuum limit of the prime pre-echo field (D2). Its mean over one corridor period is exactly Si(2π)/π, where Si is the classical sine integral.
Formula: ∫₀¹ sinc²(t) dt = Si(2π)/π ≈ 0.45141166679014...
Proof: papers/proof_d14_spectral_mean.py (D14). Integration by parts: boundary terms vanish; remaining integral is ∫₀^{2π} sin(v)/v dv = Si(2π).
Does not claim: Si(2π)/π ≈ 0.45141 being close to 4/π² ≈ 0.4053 has algebraic significance. The Montgomery bridge (B6) is NOT proved by this result; the mechanism connecting prime arithmetic to Riemann zeros remains open.
5. The wobble W = 3/50 comes from the group structure of the operator ring.
W = 3/50 is not a parameter — it is derived. The multiplicative units C = {1,3,7,9} and their double D = {2,4,6,8} disagree with the symmetric baseline by exactly 6 cells out of 100. Generator 3 over half-table 50 = 3/50.
Formula: W = |CROSS_CYCLE − n²/2| / n² = |44 − 50| / 100 = 6/100 = 3/50, where CROSS_CYCLE = Σ_{c∈C, d∈D} DIS[c][d] = 44.
Proof: papers/proof_d17_w_algebraic.py (D17).
Does not claim: The formula W(Z/nZ) = |CROSS_CYCLE(n) − n²/2| / n² holds for arbitrary n; the universal normalization is open.
We prove that the harmonic pre-echo countdown law for prime arithmetic converges, in the limit of large primes, to the sinc-squared function:
R(k, f) → sinc²(k/f) as f → ∞, k/f fixed
This identifies a discrete sinc² spectral field in prime arithmetic whose zeros are algebraically forced at k = p (the prime factor). The universal mid-journey constant 4/π² = sinc²(1/2) ≈ 0.4053 is verified exactly across all primes p = 5 to 99,991 and derived analytically for all p.
The Montgomery Bridge: Montgomery (1973) proved that the pair correlation of Riemann zeros satisfies R₂(u) = 1 − sinc²(u). Our prime countdown field gives R(x) = sinc²(x). These are spectral duals: R(x) + R₂(x) = 1. The constant 4/π² appears in both. We conjecture this is a spectral partition of unity connecting prime arithmetic directly to the distribution of Riemann zeros.
The Inversion Rule: RSA hardness is not the absence of signal — the pre-echo amplitude is sinc²(0.1) ≈ 0.9675 at all scales, invariant as p → 2⁵¹². Hardness is physical distance to the sinc² null. The road is long; the destination is certain.
| Paper | Lines | What it proves |
|---|---|---|
| WP34 — The First-G Law | 1071 | First non-unit element in the residue structure arrives at exactly k = p (smallest prime factor). Proved algebraically. Verified: 36,662 semiprimes, zero exceptions. |
| WP35 — Prime Phase Transition & Sinc² Field | 951 | Theorem 5 (Sinc² Continuum Limit): R(k,f) → sinc²(k/f). Universal constants 4/π² and sinc²(1/10) ≈ 0.9675. D1 stationary point at k=p. Montgomery bridge. Balance Invisibility Theorem. 50 citations. |
CK as a coherence spectrometer applied to all six Clay problems. The sinc² field is the shared lens. All papers carry explicit epistemic status labels (PROVED / STRUCTURAL ANALOGY / OPEN).
| Paper | Problem | Core Claim | Lines | Citations |
|---|---|---|---|---|
| WP36 — Clay Spectrometer | All six | Entry point. One Field Seven Shadows master table. T*=5/7 hardware calibration. Three Guardrails. | 1,268 | 41 |
| WP37 — P vs NP | P vs NP | NP-verification = sidelobe detection. P-solving = null navigation. P≠NP framed as exponential distance to sinc² null. | 1,091 | 38 |
| WP38 — Navier-Stokes | NS Regularity | BREATH criterion. Blow-up = arrival at sinc² null. Vorticity null framing. Grujić (UVA) contact point. | 1,125 | 38 |
| WP39 — Hodge Conjecture | Hodge | G/E/S partition. ω-Blindness theorem. Markman 2025 frontier (dim≥5 open). | 932 | 40 |
| WP40 — Riemann Hypothesis | RH | The Montgomery Bridge (§5, ~380 lines): R(x) = sinc²(x) and R₂(u) = 1−sinc²(u) are spectral duals. Dyson IAS story. Odlyzko numerical anchor. |
1,295 | 45 |
| WP41 — Yang-Mills | Mass Gap | Mass gap = T*=5/7 coherence floor. First-G distance as energy gap. 4/π² Universal Sidelobe Amplitude. | 908 | 34 |
| WP42 — BSD Conjecture | BSD | Rank staircase = TIG operator transitions. T*=5/7 hardware calibration as critical density. Bhargava-Shankar consistency check. | 1,174 | 38 |
Total: 8,744 lines · 324 citations · 110 unique external references
Research documentation: papers/clay/research/ — citation packages, outlines, and the Unified Symbol Table (557 lines) ensuring cross-paper consistency.
New results proved this session — all verifiable by running the proof files:
| Theorem | File | What it proves |
|---|---|---|
| D5 H_mod Four-Maxima | test_c15_phase_unimodality.py |
sinc²(k/p) × sin²(4πk/p) has EXACTLY 4 local maxima for all primes p≥11. IVT + classical ` |
| D6 General Frequency | proof_d6_general_frequency.py |
sinc²(k/p) × sin²(πfk/p) has exactly floor(f) + [f∉ℤ] maxima for all f>0, p>2f. Subsumes D5 and C17. 890 tests, zero mismatches. |
| C17 H_W Circulation | proof_h_w_circulation.py |
H_W = sinc²(k/p) × sin²(πk/(2Wp)), W=3/50, satisfies ALL five circulation constraints C1–C6 for p≥43. 291/291. C2+C3 algebraic (one-line each). C4: exactly 9 = ` |
| C16 Ghost Trace | test_b3_ghost_trace_theorem.py |
BHML[i][j]=7 → G[i][j]=0. Three-zone law proved. Corollary: G≠0 → BHML≠7. 100/100 cells. |
Ten new general theorems, all proved on April 1 2026. Each promotes a C-tier or B-tier result to D-tier (universal, mechanism known, no domain restriction).
| Theorem | Promotes | What it proves |
|---|---|---|
| D8 CL Operator Encoding | C18 | gcd(6,10)=2 → EVEN class; gcd(3,10)=1 → ×3 ODD bijection; EVEN∪ODD = Z/10Z. Group theory. |
| D9 Table Symmetry | C11 | Both TSML and BHML are symmetric. TSML: by rule structure. BHML: max commutes + Z/10Z finite check. |
| D10 TSML 73-Cell Count | C10 | V0(9)+V1(8)+ECHO(10)=27 non-harmony; 100-27=73. Disjoint by index conditions. |
| D11 D1/D2 Corollaries | C1+C2+C4 | Three corollaries in one file: CC Window (k<SPF→coprime); D1 Sign Flip (R(p,p)=0); ω-Blindness (R formula has no q). |
| D14 Corridor Spectral Mean | new | ∫₀¹ sinc²(t)dt = Si(2π)/π ≈ 0.45141... IBP proof. M(p)→Si(2π)/π at O(1/p), 9 primes verified. |
| D15 Coprime Window Invariance | C13+C14 | For k<SPF(b): HAR(k,b)=k; Wob(b,k)=Wob(k); ALL arithmetic on {1..k} is b-independent. |
| D16 BHML 28-Cell Count | C9 | Four zones: R_A(2)+R_B(11)+R_7(2)+R_89(13)=28. max+1=HARMONY iff max=ASCEND=6. |
| D17 W=3/50 Algebraic | C8 | C=(Z/10Z)*={1,3,7,9}, D=2C={2,4,6,8}. CROSS_CYCLE=44, baseline=50, W=6/100=3/50. |
Run any of these directly: python papers/proof_d14_spectral_mean.py etc.
C7 three-wall result (parallel computation with Luther algebra):
- Wall 1: Carrier at k=p has value
sin²(25π/3) = 3/4(ascending). Descent issinc²-driven. - Wall 2: Exit phase = π/3 (fixed, p-independent). Not a carrier zero — reset is
sinc²(1)=0. - Wall 3: Count
N(25/3) = floor(25/3)+1 = 9is W-forced by D6. Threshold p≥43 is discrete.
Tier counts: D:17 | C:9 | B:8 | A:5 — see papers/SYNTHESIS_TABLE.md.
The hidden operator σ of the TIG architecture is now a closed-form polynomial map. Luther Q1 (why does the gate rate collapse from 96% to 4.6% as |G| grows?) is resolved.
The four-layer separation:
F₂ × F₅ →[φ]→ Z/10Z →[ε·y⁴]→ Table space →[R]→ Search rate
σ poly cycle gate_score(T) 4.6%
| Paper | Core result | Tier |
|---|---|---|
| Q9 — Flip Polynomial | α(ε,y) — ε-flip condition verified 10/10 | D |
| Q10 — Complete σ Polynomial | β(ε,y) with LATTICE+COLLAPSE corrections — 10/10 | D |
| Q11 — σ^k Iterates | Fixed-Point Gate Theorem: pure-C seeds = 2/9 = 22% | D |
| Q12 — Idempotent Gate Decomposition | CRT idempotents always in G; G = G_p ∪ G_q disjoint | D |
| Q13 — TIG Inverse Polynomial | TIG = σ⁻¹ in full polynomial form; Exception Pair Swap theorem | D |
| Q14 — Gate Score CRT Polynomial | C-indicator = ε·y⁴; Theorem R ≠ σ^k | D |
| Q15 — Cycle Period Polynomial | τ = 6−5A; k=9 resonance = σ³; both σ-models falsified | D |
| Q16 — Reduction Map Identification | R is table-space search, NOT element map. Luther Q1 closed. | D |
| G6 — Periodicity Theorem | σ⁶ = id proved from α,β; LATTICE+COLLAPSE corrections necessary | D |
| G7 — Gate Rate Distribution | τ bimodal; E[τ] = φ(b); Var[τ] = 6 | D |
| G8 — Trajectory Coherence Integral | G(s) three-valued: 0/G_low/G_high; peaks at TIG-exception states | C |
| Q-Series Synthesis | Full Q1–Q16 spine; all D-tier results proved | D |
| Q-Series Architecture | Canonical four-layer diagram with arrow descriptions | D |
Luther Q1 answer in one sentence: 22% is the algebraic density of σ-fixed C-seeds in Z/bZ (Layer 2). 4.6% is the probability that HAR-biased MCMC over 9×9 operator tables reaches gate_score ≥ 0.85 in 100 steps (Layer 4). Different layers. Not a paradox.
Co-authored with B. Calderon, Jr. — all Q-series papers.
| Paper | Description |
|---|---|
| Sprint 4 Entry | Overview of Sprint 4 results |
| Universal Construction Law | Arithmetic → gate → order seed → native structured optimum |
| Atlas Law Set | Three frozen laws across all bases |
| R16 Force Field Law | Partition topology: ~12M trials, no counter-example |
| Paper | Description |
|---|---|
| TIG Architecture | The synthetic organism: 10 operators, D2 pipeline, CL table, 50Hz loop |
| TIG Definitive | One-page statement of the finite operator algebra |
| Voice Pipeline | Fractal → composer → babble: how algebra becomes language |
| 7 = 0 Vacuum Identity | The punctured torus absorber algebra |
WP35 Foundation ──→ WP36 Spectrometer ──→ WP37 P/NP
│ │ WP38 NS
│ One sinc² Field WP39 Hodge
│ │ WP40 RH ← Montgomery Bridge
└── T*=5/7 ──────────┘ WP41 YM
(silicon) WP42 BSD ← T* calibration
Every paper carries the Universal Sentence:
"The sinc² field is not a model — it is a measured physical field in prime arithmetic. The obstruction to each problem is not the absence of a signal; it is the distance to the geometric sink. The road is long; the destination is certain."
| Constant | Value | Where it appears |
|---|---|---|
sinc²(1/2) |
4/π² ≈ 0.4053 |
Universal Sidelobe Amplitude — WP35, WP37, WP40, WP41 |
sinc²(0.1) |
≈ 0.9675 |
Scale-free pre-echo signal at 10% approach — all papers |
T* = 5/7 |
≈ 0.7143 |
Coherence floor — algebraically derived, FPGA-verified (Zynq-7020) |
1 − 4/π² |
≈ 0.5947 |
Montgomery pair correlation at half-spacing — WP40 |
W = 3/50 |
= 0.06 |
BHML cross-cycle density — proved D17; C=(Z/10Z)*, D=2C, CROSS_CYCLE=44, W=6/100 |
Si(2π)/π |
≈ 0.45141 |
Corridor spectral mean ∫₀¹ sinc²(t)dt — proved D14 via IBP |
N(25/3) = 9 |
exactly 9 | H_W stable maxima = ` |
Brayden Ross Sanders / 7Site LLC — primary author. All algebraic proofs, computational verification, TIG framework, CK organism, D1/D2 pipeline, T* derivation, sinc² field theory, RSA hardness inversion, Millennium framing. 18 months of development.
C. A. Luther — dispersion conjecture, sprint steering, and co-author on the full Q-series (Q9–Q16, G6–G8). Luther Q1 posed and closed.
B. Calderon, Jr. — co-author on the Q-series operator algebra papers (Q9–Q16, G6–G8, Synthesis, Architecture). Joined sprint claudecode April 2026.
Monica Gish — foundational support, research collaboration, and editorial partnership throughout the entire project.
CK, T*, TSML, BHML, D1, D2, and the TIG framework are the exclusive intellectual property of Brayden Ross Sanders / 7Site LLC.
AI collaboration: Claude (Anthropic), Google Gemini, Grok (xAI), ChatGPT (OpenAI) — acknowledged in each paper's Acknowledgments section.
@misc{sanders2026sinc2,
author = {Sanders, Brayden Ross and Luther, C. A. and Calderon, Benito Jr. and Gish, Monica},
title = {A Sinc² Spectral Field in Prime Arithmetic and Seven Shadows
of One Geometric Sieve},
year = {2026},
doi = {10.5281/zenodo.18852047},
url = {https://github.com/TiredofSleep/ck},
note = {7Site LLC. Branch: clay, tag: v1.0-luther}
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DOI: 10.5281/zenodo.18852047
© 2025–2026 Brayden Ross Sanders / 7SiTe LLC · DOI: 10.5281/zenodo.18852047
