Materials Hydride Scoptimization Lambda At Rung Pos

The rung ladder

The declaration lambda_at_rung_pos is a small, precisely scoped theorem in the framework's machine-checked library of formal theorems. It concerns the electron-phonon coupling constant, a number that measures how strongly electrons in a material interact with its vibrating atomic lattice. In the standard theory of superconductivity, a higher coupling constant generally means a higher critical temperature, the temperature below which the material conducts without resistance.

The theorem states a simple positivity fact. If the base coupling constant at rung zero is positive, then the coupling constant at every rung of the ladder is also positive. Here a rung is one step in a discrete ladder where the coupling constant is multiplied by the golden ratio, approximately 1.618, at each step. The formal statement is: for any positive base coupling and any natural number k, the coupling at rung k is positive. The proof is a short chain of reasoning: a positive number multiplied by a positive power of the golden ratio remains positive.

This theorem is not the headline optimization result. It is a supporting lemma, a necessary condition that the more ambitious claims rely on. The framework's library also proves that an optimal rung exists on any finite candidate range, and that the optimization problem collapses from a multi-parameter search to a single integer parameter. The positivity theorem guarantees that the coupling constant never becomes negative or zero on the ladder, which would make the McMillan formula for critical temperature undefined or physically meaningless.

In Recognition Science, this ladder structure is not a free choice. The framework derives the golden ratio as the unique self-similar scaling from its core cost function, and the ladder is an application of that scaling to materials science. The theorem itself, however, does not depend on that derivation. It holds for any positive base coupling and any positive scaling factor. The golden ratio enters through the definition of the ladder, not through the proof of positivity.

The theorem does not claim that any real hydride superconductor actually follows this ladder. That is a separate empirical prediction with a named falsifier: a published hydride whose measured critical temperature lies more than five percent off the ladder's prediction. The theorem also does not claim that the ladder is the only possible optimization structure, nor that the golden ratio is the only scaling that would make the positivity argument work. It is a narrow, verified step, and its value lies in being a clean, checked foundation for the larger claims built on top of it.

theorem lambda_at_rung_pos {lam : ℝ} (h : 0 < lam) (k : ℕ) :
    0 < lambda_at_rung lam k := by
  unfold lambda_at_rung lambda_0
  exact mul_pos h (pow_pos Constants.phi_pos k)

theorem lambda_at_rung_pos {lam : ℝ} (h : 0 < lam) (k : ℕ) :
    0 < lambda_at_rung lam k := by
  unfold lambda_at_rung lambda_0
  exact mul_pos h (pow_pos Constants.phi_pos k)

What this page does not claim

This theorem does not claim that any real hydride superconductor actually follows the golden-ratio ladder. This theorem does not claim the golden ratio is the only scaling factor that would make the positivity argument work. This theorem does not claim the optimization collapse is physically realized, only that it is a formal consequence of the model's definitions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/HydrideSCOptimization.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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