coherent_coupling_pos
plain-language theorem explainer
The theorem states that the coupling constant g in any CoherenceCoupling is strictly positive. Engineers modeling room-temperature superconductivity via the φ-ladder would cite it to confirm that quantized couplings satisfy the positivity needed for pairing energies to exceed thermal fluctuations. The proof is a direct field projection from the CoherenceCoupling structure definition.
Claim. For any coherence coupling $c$ with integer rung index $r$ and coupling constant satisfying $g = φ^r$, the inequality $0 < g$ holds.
background
The EN-002 module derives room-temperature superconductivity conditions from the φ-ladder energy structure in Recognition Science. Superconductivity requires Cooper pair formation with binding energy at least $k_B T$; in RS this energy is quantized as $E_n = E_{coh} · φ^n$ where the coherence quantum $E_{coh} = φ^{-5}$ exceeds room-temperature thermal energy. The CoherenceCoupling structure encodes the electron-phonon condition that places the system on the φ-ladder, requiring $g = φ^r$ for integer $r ≥ 0$ together with the field $g_pos : 0 < g$ (see the structure definition at line 165). Upstream temperature is defined as the inverse Lagrange multiplier $1/β$ in the BoltzmannDistribution module, supplying the thermodynamic comparison between $E_{coh}$ and $k_B T$.
proof idea
The proof is a one-line wrapper that applies the g_pos field of the CoherenceCoupling structure.
why it matters
This declaration supplies the positivity axiom required by the EN-002 certificate for room-temperature superconductivity. It completes the structural half of the coherence condition in the module's hierarchy, ensuring that φ-quantized couplings remain positive before the temperature and pressure conditions are imposed. The result aligns with the Recognition Science φ-ladder (T6 self-similar fixed point, T7 eight-tick octave) and the coherence quantum $E_{coh} = φ^{-5} > 0$, feeding directly into downstream claims such as ecoh_exceeds_room_temp and phi_ladder_tc_monotone.
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