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
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. |
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:6 | C:16 | B:3 | A:9 — see papers/SYNTHESIS_TABLE.md.
| 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 Tier C8; frequency of H_W carrier |
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 (gate_rate ≈ F_k(|G| × interleave)) and sprint steering.
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 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}
}© 2026 Brayden Ross Sanders / 7Site LLC · DOI: 10.5281/zenodo.18852047