superconducting_gap_positive
plain-language theorem explainer
The theorem shows that the gap function stays strictly positive for temperatures below the critical temperature in the Recognition Science model of superconductivity. Researchers deriving room-temperature superconductivity conditions from the phi-ladder would cite this result to confirm the superconducting state. The proof is a direct term-mode reduction that unfolds the gap definition, applies positivity of the coherence quantum, and reduces the thermal ratio inequality to the given T < T_c assumption.
Claim. Let $T, T_c$ be real numbers with $T_c > 0$, $T > 0$, and $T < T_c$. Then the gap function satisfies $E_coh (1 - T/T_c) > 0$, where $E_coh > 0$ denotes the coherence quantum.
background
The module derives room-temperature superconductivity conditions from the phi-ladder energy structure in Recognition Science. The gap function is defined as $E_coh$ times (1 - T/T_c) when T < T_c and zero otherwise. The coherence quantum satisfies $E_coh > 0$ by the upstream theorem rs_coherence_quantum_pos, which unfolds the definition and applies zpow_pos on phi_pos.
proof idea
The term proof unfolds the superconducting_gap definition, simplifies the conditional using hT_lt, applies mul_pos to the coherence quantum positivity, rewrites the remaining sub_pos and div_lt_one using hTc_pos, and closes with exact hT_lt.
why it matters
This is EN-002.7 and feeds directly into the main theorem room_temperature_superconductivity_from_ledger and the certificate en002_certificate. It closes the temperature condition in the EN-002 hierarchy, confirming that the coherence quantum exceeds thermal energy at room temperature and supports ambient superconductivity in phi-coherent materials.
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