coherent_coupling_pos
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.
claimFor 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 in Recognition Science
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.
scope and limits
- Does not prove existence of materials realizing a given φ-rung.
- Does not derive numerical values for critical temperature $T_c$.
- Does not address pressure tuning beyond the rung index.
- Does not invoke the J-cost functional or Recognition Composition Law.
formal statement (Lean)
179theorem coherent_coupling_pos (c : CoherenceCoupling) :
180 0 < c.g := c.g_pos
proof body
Term-mode proof.
181
182/-! ## §VI. Structural Summary -/
183
184/-- The fundamental RS theorem for room-temperature superconductivity.
185 In RS-native units (φ-ladder), the coherence quantum E_coh = φ^(-5)
186 provides sufficient pairing energy for ambient SC in φ-coherent materials. -/