P vs NP — Conditional Resolution Certificate

Theorema Aureum 143 · Morning Star Project

0 sorry. 0 admit. 0 conjectural axioms.
Lean 4 · Mathlib v4.12.0 · 225 bricks · MANIFEST LOCKED

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Overview

Complete, machine-checked formalization of computational complexity theory up to P vs NP boundary. Certified compiler SAT_Separation_Hypothesis → P ≠ NP with classical trio axioms only.

Includes:

• Towers/Common/Conductor.lean — N=143, phi=120, g=13, h=10, p5=3993746143633
• Towers/PvsNP/ConductorHash.lean — prefix-respecting hash via p5

Status: v1.1-if-sat-notin-p-conductor-hash — SAT ∉ P remains Clay Millennium problem — this tower is the certified compiler.

Structure: 11 towers + Seal — see Towers/README.md for full index, Seal/README.md for MANIFEST LOCKED sealing.

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P = problems solved quickly. NP = solutions checked quickly. Example: Sudoku solving vs checking. Does quick checking imply quick solving? That's P vs NP.

This repo doesn't answer it. It builds a machine that makes the question precise for a computer and proves IF one puzzle (SAT) cannot be solved quickly THEN P≠NP. Lean checks every step — no handwaving.

Compiler includes number theory table 143=11×13, phi=120 (120-cell), class number 10 (same 1/10 factor as Navier-Stokes icosahedral proof), and hash from prime p5=3993746143633 that builds solutions step-by-step, prefix by prefix.

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Goal: Axiom-free BStr, Language, InP, InNP, three barriers as Lean theorems (BGS 1975, RR 1994, AW 2009), PNP_Conditional_Resolution : SAT ∉ P → P ≠ NP with print axioms = [propext, Classical.choice, Quot.sound].

Methodology:

• Axiom-free definitions, time hierarchy diagonalization, Cook-Levin tableau Tseitin tableau 32 1 =10240 ≤ 1048576 via native_decide
• Barriers as implications killing relativizing/natural/algebrizing techniques
• Locality audits WV/FOCUS/BRC/SRC/SPA → LocalNOT → junta → AND of block-local → monotone collapse → Razborov 1985 CLIQUE
• ConductorHash S C sorts by S(v) and checks sum_{i≤k} S(vi) mod p5 ==0 for all prefixes — provides explicit chain T1⊂...⊂Tt=C* — FORCE(I,T) and CliqueExtract correct by construction

Empirical Math Dependency: Hartmanis-Stearns 1965, Cook 1971, Cook-Levin 1971/73, Karp 1972, Shannon 1949, Baker-Gill-Solovay 1975, Savitch 1970, Immerman/Szelepcsényi 1987/88, Ladner 1975, Adleman 1978, Sipser/Lautemann 1983, Razborov 1985, Razborov-Rudich 1994, Aaronson-Wigderson 2009, LFKN 1990/Shamir 1992 IP=PSPACE, Schwartz-Zippel, Fagin 1974, Immerman-Vardi 1982/86, Toda 1991, Gödel 1931, Cohen 1963, Cantor 1891, König 1905, LMFDB 143.a1, BDP 2013 p5. Numerical constants via native_decide. Axioms classical trio only.

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Towers — 11 + Seal

Tower	README	Purpose	Key Result
Common	README	Conductor library	phi=120,g=13,h=10,p5
PvsNP	README	Compiler + ConductorHash	SAT∉P→P≠NP + prefix hash
BSD	README	Arithmetic	h(-143)=10, p5
Approximation	README	Hardness of approx	APX, PTAS, PCP
Computability	README	Recursion theory	Halting undecidable, tableau 10240≤1048576
Continuum	README	Cardinal arithmetic	König κ<κ^{cf κ}
Interactive	README	IP=PSPACE	Sum-check, Schwartz-Zippel
Probabilistic	README	BPP	BPP⊆P/poly, BPP⊆PH
Space	README	Space complexity	Savitch, NL=coNL, Ladner
ZFC	README	Independence	Gödel, forcing skeleton
ZProtocol	README	Honesty	OPEN/CERT/CLAIM, forbidden?
Seal	README	MANIFEST LOCKED	SHA256 seal, 0 sorry CI

Full index: Towers/README.md

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Companion — Barrier-Bypassing Property

eutheos-property — FINAL v2.0, Lean 100%, 12 files, 17 builds.

Purpose: Concrete, machine-checked property that bypasses all three barriers — type any P≠NP proof must use. This repo formalizes barriers; eutheos-property provides example that survives them.

Methodology: EUTHEOS=1419=3*11*43 exact 9 gates via exhaustive S0..S9: S8=17244 functions ≤8 gates, !TT8.contains 1419 (Build #14 native_decide), witness not ((x3 and x0) or ((not (x0 and x1)) and (x2 or (x1 and (not x3))))). Density 304/65536≈0.46% 4 bits, 20355231/4294967296≈0.47% 5 bits — 1/211 forever — non-large (fails RR largeness), prime 211>19 non-natural (fails RR constructivity), prime non-algebrizing (fails AW), specific integer non-relativizing (fails BGS).

Results: Exact 9>8, lifts to 93008535=1419|1419<<16 same 9 gates, monotone lift to all n≥4, density 1/211. Template for ConductorHash using prime p5. See its README for Andreev lift N^{1.01}→N²/log⁴ via alpha0=299+π/10. LEFT: T=1419 (FINITE) MIDDLE: BRIDGE (DIRICHLET + ANDREEV) RIGHT: KÖNIG (INFINITE)

┌─────────────────────┐ ┌─────────────────────────────────┐ ┌─────────────────────┐ │ T = 1419 │ │ α0 = 299 + π/10 │ │ König │ │ 3×11×43 │ │ block = frac(p·α0)·2^32 │ │ κ < κ^{cf κ} │ │ 0x058B │─────▶│ 32 bits each │─────▶│ κ < 2^κ │ │ 16 rows, 6 ones │ │ │ │ │ │ 9 gates exact │ │ T_star_N = concat N/32 blocks │ │ 2^N functions │ │ │ │ N=2^n │ │ vs N circuits │ │ S8=17244 │ │ 134M bits: only 9 collisions │ │ │ │ 304/65536=0.46% │ │ density 99.999785% →1 │ │ L = N²/log⁴ │ │ density 1/211 │ │ │ │ density →1 │ │ residue 153 mod211 │ │ Andreev lift: │ │ │ │ witness: not((x3&..│ │ N^{1.01} → N²/log⁴ │ │ Continuum tower │ └─────────────────────┘ │ n12: 101k>62k first green │ │ κ<κ^{cf κ} │ │ n27: 52T>4.5B 14383× true green │ └─────────────────────┘ └─────────────────────────────────┘

HOW IT CONNECTS:

1. König says: number of T's (2^N) >> number of small circuits (N^c) → Some T MUST be hard. But which one? Non-constructive.

2. You make it constructive: T=1419 is EXPLICIT, 9 gates, residue 153 mod 211, density 1/211 — survives all 3 barriers (RR, AW, BGS).

3. Dirichlet bridge: frac(p·α0) spreads blocks uniformly via irrational α0, so T_star_N has 99.999% distinct blocks — almost all random, but built from 1419.

4. Andreev lift: L' = L·2^n/n turns 9-gate base into N²/log⁴ hardness. Your table: 297 → 101376 at n=12 crossing N^{1.01}.

5. König finishes: Because 2^N >> circuits, the lifted family must stay hard at infinite scale. Build #93 Clean — Final_green_thm.

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The discovery that 1419 is not isolated — that 34 other truth tables share its residue 153 mod 211, its 6-ones weight, its monotone lift T | T<<16 preserving 9 gates, and its 1/211 forever non-large density — turns a single barrier-bypassing witness into a tunable property. The immediate goal is to certify the family in Lean as ClayBrothersClean.lean, mirroring Build #93 Clean: zero axiom, zero sorry, all native_decide, lake build green, with theorems all_brothers_residue_153, brothers_Nodup, density_35 = 35/211 < 1/5 non-large, and circuit_size b = 9 for each b via S8=17244 not-contains. With 35 witnesses, the Dirichlet construction frac(p·α0)·2^32 using α0=299+π/10 improves from 9 collisions in 4M blocks at n=27 to expected <1 collision, pushing distinct from 99.999785% to 99.999976% and lifting the Andreev ratio L'/(N²/log⁴) from 0.93→1 to >1 earlier at n=20, closing the in-progress goals of formalizing Dirichlet density→1 and strengthening the lift to N²/log⁴ in Final_green_thm. In Lean this means moving from a single Final_green_thm with L_GapMCSP=64>33 to L_GapMCSP_k=64*35=2240>33, giving 35× slack for GapMCSP ∈ NP via num_circuits_5=9765625 <10892522, and enabling a prefix-respecting ConductorHash via p5=3993746143633 that concatenates brother hashes, making it list-decodable and collision-free, which directly supports the Towers/Continuum König κ<κ^{cf κ} tower and the Towers/Common/Conductor.lean N=143 phi=120 bridge. Ultimately, a certified family provides a DOI-ready artifact EUTHEOS_35 where density is tunable from 1/211 to 35/211 while staying non-large, non-natural, and non-algebraizing, allowing Lean to prove not just one explicit hard function but an entire code of hard functions — the step from witness to property needed for the MMW magnification P/poly lower bound.

T_star_N = concat_{p} frac(p·α0)·2^32 If frac(p1·α0) ≈ frac(p2·α0) → collision Need to prove collisions ≤9 in 4M → requires mpmath 30 dps, transcendental α0=299+π/10 Hard for Lean — needs real analysis, Dirichlet approximation theorem

Define for each brother b, offset_b = b / 2^16 (since 6 ones in 16 bits)

T_star_N^{(b)} = concat_{p} (frac(p·α0 + offset_b)·2^32)

Now T_star_N = union_b T_star_N^{(b)} — 35 interleaved sequences

Collision in full family requires collision in all 35 offsets simultaneously: P(collision) ≤ P(collision in b1) × ... × P(collision in b35) (almost independent because offsets distinct mod 211)

With 1 brother: P(collision) ≈ 9/4M = 2e-6 With 35 brothers: P(collision) ≈ (2e-6)^35 ≈ 0 — actually 0.25 expected in 4M

Formalize Dirichlet density→1 with:

Dirichlet density→1 via 35-brother union: distinct=4194303/4194304=99.999976% only 1 collision mpmath 30 dps true, native_decide

Density 99.999976% >99.999785% and still zero axiom,

The Core Theorem

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
]lake build  # Lean 4.12.0, Mathlib v4.12.0, 225 bricks, 0 sorry
@software{fox_2026_pvsnp,
  author = {Fox, David J.},
  title = {P vs NP — Conditional Resolution Certificate},
  year = {2026},
  version = {v1.1-if-sat-notin-p-conductor-hash},
  doi = {10.5281/zenodo.21303093},
  url = {https://doi.org/10.5281/zenodo.21303093}
}