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](https://github.com/beanapologist/Eigenverse) has the structure of an Oil-and-Vinegar (OV) partition. [`OilVinegar.lean`](https://github.com/beanapologist/Eigenverse/blob/main/formal-lean/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 \|

	## Links

	- [Eigenverse](https://github.com/beanapologist/Eigenverse) — 606+ Lean 4 theorems
	- [OilVinegar.lean](https://github.com/beanapologist/Eigenverse/blob/main/formal-lean/OilVinegar.lean) — 28 OV meta-theorems
	- [Morphisms.lean](https://github.com/beanapologist/Eigenverse/blob/main/formal-lean/Morphisms.lean) — S, F, T composition maps
	- [Interactive demo](https://huggingface.co/spaces/beanapologist/mu-ov-cipher)

	## License

	MIT

	---
	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](https://github.com/beanapologist/Eigenverse) has the structure of an Oil-and-Vinegar (OV) partition. [`OilVinegar.lean`](https://github.com/beanapologist/Eigenverse/blob/main/formal-lean/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 \|

	## Links

	- [Eigenverse](https://github.com/beanapologist/Eigenverse) — 606+ Lean 4 theorems
	- [OilVinegar.lean](https://github.com/beanapologist/Eigenverse/blob/main/formal-lean/OilVinegar.lean) — 28 OV meta-theorems
	- [Morphisms.lean](https://github.com/beanapologist/Eigenverse/blob/main/formal-lean/Morphisms.lean) — S, F, T composition maps
	- [Interactive demo](https://huggingface.co/spaces/beanapologist/mu-ov-cipher)

	## License

	MIT