hydrideSCOptimizationCert
The hydride superconductor optimization certificate packages five structural properties: the Coulomb pseudopotential lies inside (0,1), electron-phonon coupling stays positive at every phi-rung, an optimal rung exists on any finite candidate interval, the Tc landscape collapses to a single integer search, and the phonon scale is imported from the phi-ladder resonance module. Materials modelers working on high-Tc hydrides would cite it when reducing Eliashberg parameter searches to discrete rung selection under the Recognition Science phi-ladder
claimThe hydride superconductor optimization certificate is the structure asserting that the Coulomb pseudopotential satisfies $0 < mu^* < 1$, the electron-phonon coupling satisfies $lambda_k > 0$ for every rung $k$, an optimal rung $k_{opt}$ exists on any finite search range that maximizes $T_c$, the optimization reduces to a single integer parameter, and the phonon rung is supplied by the phi-ladder phonon resonance construction.
background
This module deepens the phi-ladder phonon resonance treatment for hydrogen-dominant superconductors (H3S, LaH10, YH6). The bare phonon frequency is fixed by hydrogen mass and lattice spring constant while the coupling takes the form lambda_k = lambda_0 * phi^k, so Tc optimization on any finite rung range reduces to integer search over k. The master certificate HydrideSCOptimizationCert collects the five required clauses: mu_star_in_band, lambda_pos, T_c_optimization_exists, phi_ladder_collapses, and phonon_rung_imported.
proof idea
The definition is a direct structure constructor. It supplies mu_star_in_band from the pair of sibling lemmas mu_star_pos and mu_star_lt_one, lambda_pos from lambda_at_rung_pos, T_c_optimization_exists from T_c_optimization_finite_search, phi_ladder_collapses from phi_ladder_optimization_collapses, and phonon_rung_imported by reflexivity.
why it matters in Recognition Science
The definition supplies the structural backing for RS_PAT_010. It shows that the Tc optimization landscape collapses to a single integer parameter once the phi-ladder form of lambda is adopted, consistent with the self-similar fixed point and eight-tick octave of the Recognition Science forcing chain. No downstream uses are recorded; the declaration closes the module by packaging the key lemmas into one object.
scope and limits
- Does not compute explicit numerical values for optimal Tc or rung index.
- Does not derive the phi-ladder form of lambda_k from microscopic interactions.
- Does not address non-hydrogen-dominant superconductors.
- Does not prove convergence of the Eliashberg equations themselves.
formal statement (Lean)
160def hydrideSCOptimizationCert : HydrideSCOptimizationCert where
161 mu_star_in_band := ⟨mu_star_pos, mu_star_lt_one⟩
proof body
Definition body.
162 lambda_pos := @lambda_at_rung_pos
163 T_c_optimization_exists := T_c_optimization_finite_search
164 phi_ladder_collapses := phi_ladder_optimization_collapses
165 phonon_rung_imported := fun _ _ => rfl
166
167/-! ## §5. One-statement summary -/
168
169/-- **HYDRIDE SC OPTIMIZATION ONE-STATEMENT.** Three structural facts:
170
171(1) The phonon rung is `ω_0 · φ^k`, derived from
172 `Materials.PhiLadderPhononResonance` (not asserted).
173(2) The Coulomb pseudopotential `μ* = 0.1` is in the standard Eliashberg
174 band `(0, 1)`.
175(3) The hydride superconductor T_c optimization on any finite φ-rung
176 range reduces to a single-parameter integer search (the structural
177 content of RS_PAT_010). -/