The result in one sentence
Aristotle extracted about 40 Lean theorems from the EFMW-102 corpus, all under explicit hypotheses, with no
sorry in the five analysis modules; these are mathematical consequences of the encoded equations, not evidence that EFMW is physically correct.102source equations in the Monolithic EFMW corpus
≈40machine-checked laws derived in this analysis
5formal result clusters: Golden, Coherence, Dynamics, Metrics, Fields
The distinction matters. Lean can establish that a theorem follows from definitions and assumptions. It cannot establish that the assumptions describe nature. The empirical gate remains outside the proof assistant.
Five clusters of derived laws
1 · Golden ratio / Scalar-φ
- Proved φ² = φ + 1 and the Fibonacci power law φⁿ⁺¹ = Fₙ₊₁φ + Fₙ.
- Proved Closed forms for φ²³, φ⁴⁶, and φ¹³.
- Proved The φ-tiled operator is a convex weighted combination because φ⁻¹ + φ⁻² = 1.
- Proved Scalar-23 scaling is invertible.
φ²³ = 28657φ + 17711 · φ⁴⁶ = 1836311903φ + 1134903170
2 · Coherence / Red Queen
- Proved The coherence potential is even and has the stated derivative.
- Proved For a > 0, equilibria occur at Φ = 0 or Φ² = b/a.
- Proved A Lyapunov law for the coupling-free, noise-free coherence flow: the potential never increases.
- Proved A maximum sustainable disorder-load condition for Red Queen equilibrium.
βD − γḊ ≤ αK/4
This inequality is the exact existence criterion, under the theorem’s hypotheses, for an equilibrium of the analyzed Red Queen form.
3 · Relaxation, decay, and emergent time
- Proved The relaxation equation has the expected unique exponential solution.
- Proved ME-078 is exactly the corresponding relaxation solution for the specified initial/equilibrium data.
- Proved Exponential decay has half-life τₙ ln 2.
- Proved If recursive angular rate is positive, accumulated phase is strictly increasing and can serve as a clock variable.
4 · Metrics
- Proved The coherence score lies in [0,1] and equals 1 exactly at equality of the compared states.
- Proved Recursive integrity is bounded above by coherence.
- Proved Kuramoto order parameter bounds.
- Proved Gibbs inequality for KL divergence.
- Proved Phrase entropy bounds, normalized collapse probabilities, existence of a finite Collapse-Ω minimizer, and recursive-risk monotonicity.
5 · Fields, PDE structure, and recursion
- Proved The covariant scalar operator reduces to the expected Minkowski-space wave operator under the encoded metric conventions.
- Proved ME-005 has effective time coefficient (1−α²)/c².
- Proved ME-007 and ME-008 are componentwise equivalent under their stated encoding.
- Proved Only α+β matters in the analyzed S–O pair, and the two equations are mirror images under exchange.
- Proved The Chronologos composition becomes an involution under the stated algebraic hypotheses.
- Proved The encoded accumulation law grows without bound when ΔE·C > 0.
Negative and constraint results
Scalar-23 no-go. Linear Scalar-23 scaling preserves the null state:
S₂₃(0)=0, S₂₃(X)=0 ⇔ X=0.
Therefore this scaling alone cannot transform a null state into a nonzero state. Any “rebirth” interpretation requires additional nonlinear or external structure.
ME-005 propagation boundary. The analyzed field equation reduces to
((1−α²)/c²) φtt − ∇²φ = source.
At α² = 1, the time-propagation term vanishes and the equation degenerates to a Poisson-type equation. So the encoded expression behaves as a wave equation only on the appropriate side of that parameter boundary.
Bianchi consistency is not automatic. Because the encoded information tensor contains a Ricci term, conservation/consistency requires an additional differential-geometric obligation. It is not supplied merely by writing the Einstein-like equation.
What remains open
| Area | Status | Missing obligation |
|---|---|---|
| Action principles | Open | Formal metric variation / Euler–Lagrange derivation for ME-001, ME-009, ME-013, ME-014. |
| Einstein-like consistency | Open | Contracted Bianchi / conservation condition in a differential-geometric setting. |
| Attractors and nonlinear dynamics | Open | Existence, uniqueness, Lyapunov exponent and attractor theorems for ME-022–033. |
| PDE well-posedness | Open | Well-posedness for the cognitive PDE sector, including ME-036/037. |
| Cosmology | Open | A specified metric and full derivation for the rotating Friedmann expression. |
| Relativistic fluid sector | Open | Formal treatment of ME-077–081. |
| Empirical claims | Empirical | ME-089–102 require experiments or observational data beyond the algebraic bounds proved here. |
Why this pass matters
The value of the Aristotle pass is not that every EFMW equation survived unchanged. The value is that the corpus became more sharply partitioned:
Machine-checked: algebraic identities, bounded metrics, relaxation laws, selected Lyapunov and equilibrium statements, parameter degeneracies, invertibility, and several structural implications.
Not machine-checked as physics: action-to-field derivations, conservation closure, nonlinear existence theory, PDE well-posedness, cosmological modeling, relativistic fluid modeling, and empirical validity.
Not machine-checked as physics: action-to-field derivations, conservation closure, nonlinear existence theory, PDE well-posedness, cosmological modeling, relativistic fluid modeling, and empirical validity.
That boundary is a feature, not a weakness. Formalization tells us exactly where proof ends and scientific validation must begin.