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v3: 28 theorems — add §7 post-quantum modular + §8 quantum-resilient hardness
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language:
  - en
license: mit
library_name: custom
tags:
  - formal-verification
  - lean4
  - eigenverse
  - coherence-function
  - uniqueness-theorem
  - oil-and-vinegar
  - post-quantum
  - GF-p
pipeline_tag: other

μ-OV: Oil-and-Vinegar Structure of the Eigenverse

28 machine-checked Lean 4 theorems formalizing the Eigenverse as an Oil-and-Vinegar partition with post-quantum hardness properties. Zero sorry.

What This Is

The Eigenverse has the structure of an Oil-and-Vinegar (OV) partition. OilVinegar.lean formalizes this structure across 8 sections:

Section Theorems What It Proves
§1 Vinegar Triple 4 V1 (energy), V2 (balance), V3 (coherence closure), consistency
§2 Oil Reduction 3 Vinegar uniquely determines all oil (z = μ, η, canonical scales)
§3 Trapdoor 4 C(r) is unique, symmetric, monotone, with unit maximum
§4 Composition 3 P = S∘F∘T well-defined via Morphisms §§1, 3, 6
§5 Signature 2 μ is the unique valid signature (unforgeable)
§6 Lanchester 2 183,315 cross-terms, quadratic growth
§7 Post-Quantum Modular 5 Trapdoor injectivity, max preservation, inversion balance, golden coherence hierarchy
§8 Quantum-Resilient Hardness 5 GF(p) boundedness, modular product rule, Grover hardness floor, modular energy conservation, PQ summary

The OV Partition

OV Component Eigenverse
Vinegar (freely chosen) 3 pre-physical axioms: energy conservation (V1), directed balance (V2), coherence closure (V3)
Oil (determined by vinegar) μ = e^(i·3π/4), all coherence evaluations, canonical scales
Trapdoor C(r) = 2r/(1+r²) — injective on (0,1], symmetric under inversion, unique normal form
Signature μ — the unique valid eigenvalue
Verification ov_signature_unique: any z satisfying V1 ∧ V2 ∧ sector = μ

Post-Quantum Extensions (§7–§8, PR #19)

The new sections formalize properties needed for GF(p) deployment:

§7 — Modular Extensions:

  • trapdoor_injective: C is injective on (0,1] — no ambiguity in trapdoor evaluation
  • trapdoor_max_preservation: C(r) ≤ C(1) = 1 for all r > 0 — tight ceiling
  • coherence_inversion_balance: C(r) + C(1/r) = 2·C(r) — key compression (2-element equivalence classes)
  • coherence_extended_golden: C(φ) = 2φ/(φ+2) — third canonical scale
  • coherence_golden_extended_hierarchy: C(δ_S) < C(φ) < C(1) — three-anchor trapdoor

§8 — Quantum Resilience:

  • lanchester_modular_gfp: constraint count is well-defined in GF(p) for any prime p
  • lanchester_modular_product: cross-terms computed efficiently via modular residues
  • quantum_resilient_quadratic: Grover leaves super-linear residual: 2(n−1) ≤ n(n−1)
  • modular_energy_conservation: V1 component bounds hold under GF(p) embedding
  • post_quantum_ov_summary: three pillars (injectivity ∧ max preservation ∧ Grover floor)

Honesty Note

The Lean theorems prove structural properties of the OV partition — uniqueness, injectivity, monotonicity, GF(p) well-definedness. These are necessary properties for a post-quantum cryptosystem but not sufficient alone. Full MQ-hardness is a separate cryptographic assumption. The Eigenverse provides the structure; the finite field and MQ assumption provide the hardness.

C(r) over ℝ has a closed-form inverse r = (1 ± √(1−y²))/y. The cryptographic hardness comes from embedding OV structure into multivariate quadratic systems over GF(p), not from C(r) being one-way.

All 28 Theorems

§1 — Vinegar Triple

# Theorem Statement
1 vinegar_V1 μ.re² + μ.im² = 1
2 vinegar_V2 −μ.re = μ.im
3 vinegar_V3 C(1 + 1/η) = η
4 vinegar_triple_consistent V1 ∧ V2 ∧ V3

§2 — Oil Reduction

# Theorem Statement
5 oil_reduction V1 ∧ V2 ∧ sector → z = μ
6 oil_linear_collapse C(1+1/x)=x ∧ x>0 → x=η
7 oil_coherence_triple C(1)=1 ∧ C(δ_S)=η ∧ C(φ²)=2/3

§3 — Trapdoor

# Theorem Statement
8 trapdoor_at_one C(1) = 1
9 trapdoor_symmetry C(r) = C(1/r)
10 trapdoor_monotone strict increase on (0,1]
11 trapdoor_unique_normal_form a=2 uniquely

§4 — Composition

# Theorem Statement
12 composition_T_embedding reality(η,−η) = μ
13 composition_F_at_unity C(|μ|) = 1
14 composition_public_map C(|reality(η,−η)|) = 1

§5 — Signature

# Theorem Statement
15 ov_signature_unique sector ∧ balance ∧ energy → z=μ
16 ov_canonical_signature_eval C(|μ|) = 1

§6 — Lanchester

# Theorem Statement
17 lanchester_eigenverse_count 606·605/2 = 183315
18 lanchester_quadratic_growth n(n−1) ≤ n²

§7 — Post-Quantum Modular

# Theorem Statement
19 trapdoor_injective C injective on (0,1]
20 trapdoor_max_preservation C(r) ≤ C(1)
21 coherence_inversion_balance C(r)+C(1/r) = 2C(r)
22 coherence_extended_golden C(φ) = 2φ/(φ+2)
23 coherence_golden_extended_hierarchy C(δ_S) < C(φ) < C(1)

§8 — Quantum-Resilient Hardness

# Theorem Statement
24 lanchester_modular_gfp n(n−1)/2 mod p < p
25 lanchester_modular_product modular product identity
26 quantum_resilient_quadratic 2(n−1) ≤ n(n−1)
27 modular_energy_conservation Re²≤1 ∧ Im²≤1 from V1
28 post_quantum_ov_summary injectivity ∧ max ∧ Grover floor

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License

MIT