gap_zero_above_tc
plain-language theorem explainer
The theorem asserts that the superconducting gap vanishes for all temperatures at or above the critical temperature T_c. Modelers of room-temperature superconductivity in Recognition Science cite this when confirming the normal-state transition in the phi-ladder energy hierarchy. The proof is a one-line wrapper that unfolds the gap definition and simplifies using the ordering assumption.
Claim. Let $T, T_c$ be real numbers with $T_c > 0$ and $T_c ≤ T$. Then the superconducting gap function satisfies $Δ(T, T_c) = 0$, where $Δ(T, T_c) = E_{coh} (1 - T/T_c)$ if $T < T_c$ and $0$ otherwise.
background
The module derives room-temperature superconductivity conditions from the Recognition Science phi-ladder. The coherence quantum is $E_{coh} = φ^{-5}$ eV, and pairing energies are quantized as $E_n = E_{coh} · φ^n$. The superconducting gap is defined to be positive below $T_c$ (where binding exceeds thermal energy) and zero above it, implementing the temperature condition in the EN-002 hierarchy. This theorem sits downstream of the gap definition and upstream of the overall certificate that coherent pairing can overcome thermal fluctuations at ambient temperature.
proof idea
The term proof unfolds the definition of superconducting_gap, exposing the if-then-else that returns zero whenever the test $T < T_c$ fails. It then applies simp with the lemma not_lt.mpr applied to the hypothesis $T_c ≤ T$, which directly discharges the conditional branch.
why it matters
This result supplies the upper half of the gap characterization required by the EN-002 certificate, which certifies that room-temperature superconductivity is derivable in the Recognition Science framework. It closes the temperature-threshold part of the hierarchy that begins with the coherence quantum exceeding room-temperature thermal energy and continues through the monotone phi-ladder dependence of $T_c$. The theorem therefore anchors the claim that materials supporting phi-coherent ledger states can remain superconducting at 300 K when placed on a sufficiently high rung.
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