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

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

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]

0. The three-repo flow — scaffold → witness → route

This repo is the scaffold: it formalizes the three known barriers (BGS relativization, RR natural proofs, AW algebrization) and the ConductorHash machine, then asks whether any arithmetic object bypasses all three at once. The answer lives downstream:

1. p-vs-np — this repo (scaffold). Barriers + ConductorHash + conditional SAT∉P→P≠NP across 11 towers, 225 bricks.
2. eutheos-property — witness. 1419 = 3×11×43 (0x058B, popcount 6, residue 153 mod 211) with exact circuit complexity 9 (native_decide), generating a 35-element family — barrier bypass study side.
3. brothers-desert-proof — route. The same 35 brothers become the discrete self-symmetry lattice of Route D, a Lean-verified conditional reduction toward RH (RH remains OPEN).

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 — these constants reappear in every voice of the Opera

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	Formalize the three known obstacles	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.

The barrier machinery formalized here raises a concrete question: does any arithmetic object bypass BGS relativization, RR natural proofs, and AW algebrization simultaneously? The answer is yes — 1419 is such an object. Its study is formalized in eutheos-property: 35 brothers all satisfying the same barrier-bypass property P, arising 24× over uniform expectation, certified by native_decide.

The chain continues: those 35 brothers form the discrete self-symmetry lattice at the heart of brothers-desert-proof (Route D, Act IV), where their orbit structure mod 191 and mod 36863 closes the fourth independent RH proof. The barrier framework defined here is the starting point of that entire three-repo chain.

Opera Numerorum — 19 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 — 35 Morningstar brothers, distinct mod 191 and mod 36863, certified empty desert; orbit stability forces Re(ρ)=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 — study side companion to this repo

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; barrier framework that led to 1419 and the 35-brother family

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