Materials Hydride Scoptimization T C Optimization Finite Search

The finite search theorem

Superconducting hydrides, compounds of hydrogen and a metal such as sulfur or lanthanum, can carry electrical current without resistance at high temperatures. The standard Eliashberg-McMillan formula estimates that temperature, T_c, from a few physical inputs: the characteristic vibration frequency of the hydrogen atoms, the strength of the electron-phonon coupling, and a Coulomb repulsion term. In the Recognition Science framework, these inputs are not treated as fully independent. The framework models the coupling strength as following a ladder of discrete steps, each step a power of the golden ratio φ, so the coupling at step k is λ₀·φ^k, where λ₀ is a per-material starting value. The framework also models the phonon frequency as scaling with the same ladder.

The declaration T_c_optimization_finite_search proves a limited but exact claim: on any finite list of candidate rungs, from step 0 up to step n-1, there exists a rung whose predicted T_c is at least as high as every other rung on that list. The proof is a direct application of the fact that a finite set of real numbers always has a maximum. This is a THEOREM in the framework's machine-checked library of formal theorems. It is the structural backing for a patent claim about single-parameter optimization: it certifies that, within the model, searching over the integer rung index is enough to find a best candidate, rather than needing a continuous search over multiple parameters.

What the theorem does not claim is where the honesty lies. It does not prove that any real hydride material's measured T_c will match the framework's prediction. The model's assumptions, that coupling follows a φ-ladder and that the phonon scale is fixed by hydrogen mass and lattice constant, are definitional choices, not proved facts about nature. The theorem only says that if you accept those definitions, then a finite search over rungs has a well-defined winner. It also does not establish that the winning rung is unique, nor that the optimal T_c is positive. Positivity is a separate theorem requiring the coupling to exceed the Coulomb repulsion, the standard Eliashberg condition. The finite search theorem itself is silent on whether any candidate is physically realizable.

The practical upshot for a reader is modest but precise. The framework's contribution here is not a new prediction about a specific material. It is a formal guarantee about the structure of a search problem: once the model is accepted, the optimization over a finite range is a solved problem, reducible to checking a finite list. The falsifier for the broader model is a clean published hydride whose measured T_c lies more than 5% off the φ-ladder pattern, meaning its actual coupling ratio is not a φ-rational number. That falsifier targets the model, not the finite search theorem, which would remain true as a piece of mathematics regardless of any measurement.

/-- **THEOREM.** On any finite candidate range, an optimal rung
exists. This is the single-parameter optimization claim of RS_PAT_010. -/
theorem T_c_optimization_finite_search
    (omega_0 lam : ℝ) (n : ℕ) (h : 0 < n) :
    ∃ k_opt ∈ Finset.range n,
      ∀ k ∈ Finset.range n, T_c_phi_rung omega_0 lam k ≤ T_c_phi_rung omega_0 lam k_opt := by
  have hne : (Finset.range n).Nonempty := ⟨0, by simp [Finset.mem_range]; exact h⟩
  exact Finset.exists_max_image (Finset.range n) (T_c_phi_rung omega_0 lam) hne

/-- **THEOREM.** On any finite candidate range, an optimal rung
exists. This is the single-parameter optimization claim of RS_PAT_010. -/
theorem T_c_optimization_finite_search
    (omega_0 lam : ℝ) (n : ℕ) (h : 0 < n) :
    ∃ k_opt ∈ Finset.range n,
      ∀ k ∈ Finset.range n, T_c_phi_rung omega_0 lam k ≤ T_c_phi_rung omega_0 lam k_opt := by
  have hne : (Finset.range n).Nonempty := ⟨0, by simp [Finset.mem_range]; exact h⟩
  exact Finset.exists_max_image (Finset.range n) (T_c_phi_rung omega_0 lam) hne

/-- **THEOREM.** On any finite candidate range, an optimal rung
exists. This is the single-parameter optimization claim of RS_PAT_010. -/
theorem T_c_optimization_finite_search
    (omega_0 lam : ℝ) (n : ℕ) (h : 0 < n) :
    ∃ k_opt ∈ Finset.range n,
      ∀ k ∈ Finset.range n, T_c_phi_rung omega_0 lam k ≤ T_c_phi_rung omega_0 lam k_opt := by
  have hne : (Finset.range n).Nonempty := ⟨0, by simp [Finset.mem_range]; exact h⟩
  exact Finset.exists_max_image (Finset.range n) (T_c_phi_rung omega_0 lam) hne

What this page does not claim

The theorem does not claim that the optimal rung is unique. The theorem does not claim that the optimal T_c is positive or physically realizable. The theorem does not claim that the φ-ladder model itself is empirically correct for any specific hydride.

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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