rs_coherence_quantum_pos
The coherence quantum E_coh equals phi to the power of negative five and is strictly positive. Researchers deriving room-temperature superconductivity conditions from the Recognition Science phi-ladder would cite this as the base positivity fact for pairing energy scales. The proof is a term proof that unfolds the definition and applies positivity of integer powers of a positive base.
claim$E_{coh} := ϕ^{-5} > 0$, where $ϕ$ denotes the golden ratio.
background
In the module on Room-Temperature Superconductivity Structure, the coherence quantum is introduced as the fundamental pairing energy scale on the phi-ladder. E_coh is defined as phi raised to the power minus five in RS-native units, approximately 0.090 eV. This sets the scale for Cooper pair binding energies E_n = E_coh · phi^n. The local theoretical setting derives superconductivity conditions from the phi-ladder energy structure, requiring E_binding ≥ k_B T for Cooper pair formation. The module states that E_coh exceeds thermal energy at room temperature (0.090 eV > 0.026 eV), enabling coherent pairing at ambient conditions. Upstream results include the definition of K as phi to the power one half from Constants, and structural theorems from foundation modules ensuring algebraic consistency.
proof idea
The proof is a term-mode one-liner. It unfolds the definition of E_coh to phi ^ (-5 : Z), then applies the lemma zpow_pos instantiated at phi_pos to conclude the power is positive.
why it matters in Recognition Science
This theorem is EN-002.1, the base positivity result in the room-temperature superconductivity hierarchy. It feeds directly into the certificate en002_certificate and the theorem superconducting_gap_positive, where it is applied via mul_pos to show the gap exceeds zero for T < T_c. In the Recognition framework it anchors the phi-ladder energy structure with E_coh = phi^{-5} > 0, supporting the claim that coherent pairing overcomes thermal fluctuations at room temperature. It relates to the T6 phi fixed point and the eight-tick octave in the forcing chain through the constants module.
scope and limits
- Does not establish existence of materials supporting phi-coherent ledger states.
- Does not derive numerical values of E_coh beyond the definition.
- Does not address pressure tuning of the phi-rung.
- Does not compute explicit critical temperatures T_c.
formal statement (Lean)
54theorem rs_coherence_quantum_pos : E_coh > 0 := by
proof body
Term-mode proof.
55 unfold E_coh
56 apply zpow_pos phi_pos
57
58/-- Room temperature thermal energy in units where E_coh is natural.
59 k_B · T_room ≈ 0.026 eV at T = 300 K.
60 In RS units: k_B · T_room / E_coh ≈ 0.026 / 0.090 ≈ 0.289 < 1. -/