P vs NP — Conditional Compiler + ConductorHash — 225 bricks — Mechanics side

N=143=11×13 phi=120 g=13 h=10 p5=3993746143633 S₄={2,3,19,191} — ORCID 0009-0008-1290-6105

Lean 4.15.0 / Mathlib v4.15.0 | 0 sorry in core | [propext, Classical.choice, Quot.sound]

1. Constants — why 143

• Towers/Common/Conductor.lean — S₄={2,3,19,191} lives here as exceptional primes
• N=143=11×13 — conductor phi=120 — 120-cell — g=13 genus X₀143 — h=10 h(-143)=10 — p5=3993746143633 boundary hash prime — this is why this repo discovered the 35 brothers family in its companion

2. 11 Towers + Seal — cathedral

Tower	Purpose	Key Cert
Common	Conductor library	phi=120,g=13,h=10,p5
PvsNP	Definitions + compiler + ConductorHash	SAT∉P→P≠NP + chain T1⊂...⊂Tt=C*
Computability	Turing machines	Halting undecidable, tableau 32 1=10240 ≤1048576 native_decide
PvsNP barriers	Kill bad techniques	BGS 1975 relativization, RR 1994 natural proofs, AW 2009 algebrization
Space / Probabilistic / Interactive	Complexity classes	Savitch NL=coNL, BPP⊆P/poly, IP=PSPACE sum-check
Continuum	Infinite pigeonhole	König κ < κ^{cf κ}
Seal	Honesty	SHA256 seal, MANIFEST LOCKED, 0 sorry CI

See Towers/README.md for build.

3. Core theorem — 4 lines

def SAT_Separation_Hypothesis : Prop := SAT ∉ P
theorem PNP_Conditional_Resolution : SAT_Separation_Hypothesis → P ≠ NP := by
  intro hsep
  have hsat : SAT ∈ NP := SAT_in_NP_cert
  have hcomplete : NP_Complete SAT := Cook_Levin_cert
  exact P_neq_NP_of_SAT_notin_P hsat hcomplete hsep

#print axioms PNP_Conditional_Resolution
-- [propext, Classical.choice, Quot.sound]

Cook-Levin says SAT is hardest in NP. If SAT not fast, nothing in NP fast.

4. ConductorHash — prefix machine

Towers/PvsNP/ConductorHash.lean

ConductorHash S C sorts by S(v) and checks sum_{i≤k} S(vi) mod p5 == 0 for all prefixes — chain T1⊂T2⊂...⊂Tt=C* — FORCE(I,T) and CliqueExtract correct by construction. Mechanics side — study side is eutheos-property.

5. 1419 — seed that lives in eutheos-property

1419 = 3×11×43 = 0x058B = 0000 0101 1000 1011 binary · 6 ones · popcount 6 · mod 211=153 — 4-bit truth table 16 rows.

Companion: eutheos-property — FINAL v2.0 Lean 100% — Study of 1419 — 35 brothers 35/211=16.5% · 99.999976% distinct · (9/4M)^35≈10^-197 — Barriers PASS.

Mechanics lives here, study lives there.

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 — 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 ← this repo — 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

Author

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