T_c_rung
plain-language theorem explainer
T_c_rung defines the critical temperature on the nth rung of the phi-ladder as phi^n in normalized units where the coherence quantum sets the base scale to 1. Materials scientists and engineers applying Recognition Science to superconductivity would cite this when checking whether a coherent material meets the ambient temperature condition. The definition is a direct power expression that downstream results unfold to establish monotonicity and existence of suitable rungs.
Claim. The critical temperature on the nth rung of the phi-ladder is $T_c(n) = phi^n$ in units normalized so that the coherence energy scale $E_{coh}/k_B$ equals 1.
background
In the Room-Temperature Superconductivity Structure module the phi-ladder supplies quantized binding energies for Cooper pairs. The coherence quantum is defined as $E_{coh} := phi^{-5}$ in RS-native units, approximately 0.090 eV. The module document states the full formula $T_c(n) = E_{coh} · phi^n / k_B$, yet T_c_rung absorbs the base scale into the normalized definition phi^n.
proof idea
Direct definition T_c_rung n := phi ^ n. Downstream proofs such as phi_ladder_tc_monotone and cooper_pair_binding_exceeds_thermal simply unfold the definition and apply the known inequality 1 < phi.
why it matters
T_c_rung supplies the temperature scale for the EN-002 hierarchy of ambient superconductivity conditions. It is invoked by ambient_superconductivity_possible to exhibit a rung satisfying T_c(n) >= T_room, by phi_ladder_tc_monotone to prove higher rungs give higher T_c, and by phi_ladder_unbounded to reach arbitrary temperatures. This fills the temperature condition in the RS derivation from the phi-ladder, consistent with phi as the self-similar fixed point.
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