thermal_ratio_pos
The theorem establishes that the room-temperature thermal ratio, defined as k_B T_room divided by the RS coherence quantum E_coh, is strictly positive. Engineers checking ambient superconductivity conditions in the Recognition Science φ-ladder would cite it to confirm the basic inequality before comparing binding energies to thermal fluctuations. The proof is a one-line wrapper that unfolds the explicit constant definition and applies norm_num.
claim$0 < k_B T_{room} / E_{coh}$ where $E_{coh} = φ^{-5}$ eV is the Recognition Science coherence quantum and the ratio evaluates numerically to 0.289 at $T_{room} = 300$ K.
background
The Engineering.RoomTempSuperconductivityStructure module derives superconductivity conditions from the φ-ladder energy structure. Superconductivity requires Cooper-pair binding energy E_binding ≥ k_B T, with quantized levels E_n = E_coh · φ^n and E_coh = φ^{-5} eV ≈ 0.090 eV. The thermal ratio is introduced as k_B T_room / E_coh ≈ 0.026 / 0.090 ≈ 0.289, which is less than 1, so the coherence quantum exceeds room-temperature thermal energy.
proof idea
The proof is a one-line wrapper that unfolds the definition of thermal_ratio_room_temp to its explicit numerical value 0.289 and applies norm_num to obtain the strict inequality.
why it matters in Recognition Science
This fills the EN-002.3 slot in the room-temperature superconductivity hierarchy, confirming positivity of the normalized thermal energy so that the coherence condition can be checked against the φ-ladder. It supports the module's claim that coherent pairing can overcome thermal fluctuations at ambient temperature and pressure, anchoring the temperature condition within the T0-T8 forcing chain and the phi-ladder rung structure. No immediate downstream theorem consumes it yet.
scope and limits
- Does not compute the numerical value of the ratio beyond confirming positivity.
- Does not assign specific φ-rungs to materials or derive critical temperatures.
- Does not address pressure tuning or structural coherence conditions.
- Does not prove existence of Cooper pairs or superconducting states.
formal statement (Lean)
70theorem thermal_ratio_pos : 0 < thermal_ratio_room_temp := by
proof body
Term-mode proof.
71 unfold thermal_ratio_room_temp
72 norm_num
73
74/-! ## §II. Critical Temperature from φ-Ladder -/
75
76/-- Critical temperature for the n-th rung of the φ-ladder.
77 T_c(n) = E_coh · φ^n / k_B (in suitable units).
78 The RS prediction: each material sits on a particular rung n. -/