superconducting_gap
plain-language theorem explainer
The superconducting gap function in Recognition Science is defined as E_coh times (1 - T/T_c) when T is below the critical temperature T_c and zero otherwise. Researchers deriving room-temperature superconductivity conditions from the φ-ladder would cite this when analyzing gap temperature dependence. The definition is a direct piecewise expression built from the coherence quantum E_coh = φ^{-5}.
Claim. The gap function is defined by $Δ(T, T_c) = E_{coh} (1 - T/T_c)$ for $T < T_c$ and $Δ(T, T_c) = 0$ otherwise, where $E_{coh} = φ^{-5}$ is the RS coherence quantum and $T_c > 0$.
background
The EN-002 module derives room-temperature superconductivity from the φ-ladder energy structure. Superconductivity requires Cooper pair formation with binding energy at least k_B T; in RS the pairing scale is quantized as E_n = E_coh · φ^n with E_coh = φ^{-5} ≈ 0.090 eV. This exceeds room-temperature thermal energy (≈ 0.026 eV), so φ-coherent materials can superconduct at ambient conditions. The module lists supporting results including rs_coherence_quantum_pos (E_coh > 0) and phi_ladder_tc_monotone (higher rung yields higher T_c).
proof idea
This is a direct definition via if-then-else on the comparison T < T_c. It inserts the sibling E_coh definition and the linear thermal suppression factor (1 - T/T_c).
why it matters
The definition supplies the explicit gap expression used by gap_max_at_zero, gap_zero_above_tc, superconducting_gap_positive and the central room_temperature_superconductivity_from_ledger theorem. It realizes the EN-002.7 slot in the hierarchy of conditions, linking the coherence quantum E_coh = φ^{-5} to the observable gap that vanishes above T_c. The construction sits inside the Recognition Science forcing chain by tying the T5 J-uniqueness and T6 φ fixed-point to an engineering-scale prediction for ambient superconductivity.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.