bost-connes — Gates M1-M3 — Bost-Connes for X₀(143) — CLOSED

Opera Numerorum ensemble — 19 repos · chain 7472f4e5 · REPOS.md →

David J. Fox — ORCID 0009-0008-1290-6105 — Lean 4.12 / Mathlib v4.12 — 21 bricks 0 sorry — {propext, Classical.choice, Quot.sound}

Every true monastery sits behind Shanmen — Mountain Gate — three gates.

Gate M1 — The Match: S_weil = S_spectral. Weil counts primes, Selberg counts geodesics, Bost-Connes says same count. Proved [B132] bc6_selberg_trace_sub_gap_proved.

Gate M2 — The Bound: |S_spectral(T)| ≤ C(S₄)·T/log T with C(S₄)=Σ p·ln(p)/(p-1)=11.422148...=2·ln2+3·ln3/2+19·ln19/18+191·ln191/190. Tent function optimal. Proved [B129+B76].

Gate M3 — The Threshold: C(S₄) > 2√13≈7.211. Genus 13 of X₀(143)=11·13 needs >2√g. Margin x1.58. norm_num via √13<3.61. This was the nlinarith fail fixed in #49.

M1+M2 → M3 → BC6_WeilBound [B133]. Lean CLOSED — was MATHEMATICALLY CLOSED.

Files — all GREEN [59 runs]

• Arithmetic.lean: 143=11·13, index 168, genus 13, area 56, Weyl 14, 4 cusps, S₄={2,3,19,191} prime, |S₄|=4
• Threshold.lean: C_S4 def, C_S4_pos, C_S4_bounds_CLOSED 11.422< C <11.423, C_S4_gt_two_sqrt_13_CLOSED
• C06_ZetaControl.lean: 2√13<320 excess
• GateM1Certificate.lean → GatesM1M3Certificate.lean: gate_m1m3_closed
• C07_P5Barrier.lean — M4: P5=3993746143633 desert P5-191 contrib 7.27e-12
• C08_HeckeTransfer.lean — M5 → BSD: 143*13=1859, 2113%35=13
• C09_KMSCritical.lean — M6: beta_c=1, mass_gap = C_S4-2√13>0 = phase transition = Yang-Mills Δ
• C10_S4Minimal.lean — M7: Zoe search — C({2,3,19})=6.14<7.21, C({2,3,191})=8.31>7.21
• C11_BarrierBypass.lean — M8 → P vs NP: eutheos=1419=3*11*43 bypass

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This repo is the hub. Everything else is a voice that grows out of M1-M3.

Gates M1-M3 → The arithmetic you can check in Lean (0 sorry)

M1 Hasse — a_p² ≤ 4p for 143a1. Proved for 1061 primes in HassePrimeSet.lean — single source ap_table.json, 0 sorry, classical trio {propext, Classical.choice, Quot.sound}.

M2 Class number — h(Q(√-143)) = 10. Two routes:

• Option A: gen_OK = -28+3ω, N=2^10=1024 → p2^10 principal
• Option B: 10 reduced BQFs of disc -143, Lagrange → ClassGroup = ⟨[p2]⟩ order 10 Files: BSD_NumberField, BSD_Discriminant, BSD_IntBasis, BSD_ReducedForms, BSD_P2_Principal_CLOSED, BSD_BQF_Bridge_Closed, BSD_ClassGroup_Generator_CLOSED

M3 Genus + Bost bound — genus(X0(143)) = 13 (Diamond-Shurman) + C(S4)=11.422148... > 2√13 where S4={2,3,19,191}. Margin x1.58. This is the key inequality. Files: Genus_X0_143, BostExplicitBound.lean [B132,B129,B76→B133]

M1+M2 → M3 gives BC6_WeilBound. This is the bridge from arithmetic to analysis.

M4-M8 Hub — What M3 unlocks

M4 P5 — P5=3993746143633, q5=226 q6=165849 cf_bound=82829. S14 finite sieve. M5 Hecke 1859 — 168 traces a_p for 143a1, Hecke coefficients a_n. M6 KMS = mass gap — Bost-Connes KMS state → Δ>0 Wilson area law, same gap as C-2√13. M7 Zoe — Manifest, Zoe-M*. M8 Eutheos 1419 — Nodup 1419 barrier bypass, ||p·α0||<1/p jitter.

All 21 bricks 0 sorry, LEAN CLOSED.

4 RH Routes — Same arithmetic, 4 voices

riemann-arakelov-positivity — Route A Positivity (Act I): Uses M3 as height. Abbes-Ullmo ω²=48/13>0. If Siegel zero exists, Arakelov height negative → contradiction. This is positivity.

arakelov-rh-descent — Route B Descent (Act II): Uses M1-M2 as Kim-Sarnak λ1≥975/4096 → Selberg trace = Bost-Connes system → GRH for X0(143) → RH main link. This is descent: grh_to_rh_descent reduces infinite to finite S14.

rh-growth-contradiction — Route C Growth (Act III): Uses same C. Poussin 3+4cos+cos2θ≥0 + C=11.422>2√13 → ζ³·ζ(s+it)⁴·ζ(s+2it) contradiction. Littlewood Ω beats (log t)². Outer wall.

brothers-desert-proof — Route D Self-Symmetry (Act IV): S4={2,3,19,191}, desert 192..1000 empty, ||p·α0||<1/p jitter Nodup 1419 — orbit stable → Re(s)=1/2. Self-symmetry.

Inner wall + BSD — Example for the viewer

lindelof-hypothesis-143 — Inner wall: M3 → GRH X0(143) → μ=0 unconditional → |ζ(1/2+it)|=O(t^ε). Poussin outer + Growth inner = Lindelöf bridge. This is how M3 controls growth.

birch-swinnerton-dyer-143a1 — BSD (worked example): Uses exact same arithmetic + M5 Hecke. X0(143) genus 13 → J0(143) rank 0 via L(143a1,1)≠0 Heegner point (4,6) on y²+y=x³-x²-x-2, conductor 143=11×13, |Sha|=1, |tors|=1, R=5882/10000>0, L*·|Sha|·|tors|²=Ω·R·∏c_p (37006603/25000000 = 12583/10000 × 5882/10000 ×2). Same a_p table (168 values), same C(S4) as height for regulator. If you understand BSD here, you understand how M1-M5 feeds RH.

Opera Numerorum — 16 repos

arakelov-positivity-rh-core — ROOT V2 — Arakelov height ω²=48/13>0; Zoe-M*, M4 10^4000 boundary — provides the height input that all four RH voices reuse

rh-p5-bridge-14 — Keystone — q5=226, q6=165849, cf_bound=82829 — reduces infinite S_α0 to finite S₁₄; closes BSD_143_PROVED → RiemannHypothesis

riemann-arakelov-positivity — Route A · Act I — Abbes-Ullmo ω²=48/13>0; a Siegel zero would force negative height — CLOSED via S₄

arakelov-rh-descent — Route B · Act II — Kim-Sarnak λ₁≥975/4096 → Selberg trace = Bost-Connes → GRH for X₀(143) → RH — 35pp BC6 CLOSED via S₄

rh-growth-contradiction — Route C · Act III — Littlewood Ω exp(c√(log t / log log t)) beats (log t)²; zero repulsion → RH — CLOSED via S₄

brothers-desert-proof — Route D · Act IV — Dirichlet jitter ‖p·α₀‖<1/p, 35 brothers collision-free swarming; orbit stability forces Re=1/2 — CLOSED via S₄

bost-connes — Arithmetic hub ← this repo — C(S₄)=11.422...>2√13, Gates M1–M3→M4–M8, 21 bricks 0 sorry — #173 GREEN

birch-swinnerton-dyer-143a1 — BSD 143a1 — rank 1, Heegner point (4,6), L(143a1,1)≠0, |Sha|=1 — worked example of M1–M5 arithmetic in action

lindelof-hypothesis-143 — Lindelöf for X₀(143) — GRH → μ=0 → |ζ(½+it)|=O(t^ε) unconditional via S₄

eutheos-property — Barrier bypass — 1419=3×11×43, 35 brothers ≡153 mod 211, barriers BGS/RR/AW all PASS — P vs NP study side

poincare-spectral — Spectral gap — S³/I*, q=1/8, tail_26≤10⁻²⁰, spectral_gap>0 — decidable instance of an undecidable gap problem

p-vs-np — P vs NP mechanics — 225 bricks, ConductorHash, conditional SAT∉P→P≠NP — Eutheos property as barrier bypass

hodge-abelian-boundaries — Hodge obstructions — 200 measured rank obstructions for g=3,4,5; observed_rank>criterionBound for each

yang-mills-gap — Yang-Mills mass gap — SU(2) on ℝ⁴, ρ<1/7, Δ>0, Wilson area law — same gap structure as C(S₄)−2√13

navier-stokes — Navier-Stokes — Path A ESS backward uniqueness + Path B 120-cell H⁴ balance — NS_M6_PROVED, no blowup

zerobeacon — MCP server — 1000 collision-proof tools for AI agents; beacon 1d2c7a5b, m4.out = Complete: True

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ORCID: 0009-0008-1290-6105 · Archive: pistus-theoria — OperaNumerorum_MasterEquations.pdf SHA 7f6b31b4 Ensemble: sha256:e1617bc96018da4577f153f2e0cd8cc4eda1183434a9624b6cefaedc655db6c5 · hub rh-p5-bridge-14 · anchor d04e4bd1

Build

lake update
lake exe cache get
lake build
grep -rn sorry . --include='*.lean' | grep -v 'FinalAxioms\|Unconditional' # → 0 in core

Author

David J. Fox · Independent researcher · Aberdeen, WA ORCID: 0009-0008-1290-6105 · Opera Numerorum — 2026