mf_temp_eq_six_kappa
plain-language theorem explainer
The mean-field critical temperature equals six times the GCIC stiffness constant. Researchers deriving thermodynamic relations in the GCIC phase model cite this identity when linking critical behavior to the underlying stiffness from the phase potential. The proof is a direct algebraic reduction obtained by unfolding the two definitions and applying ring normalization.
Claim. The mean-field critical temperature satisfies $T_c = 6k$, where $k$ is the GCIC stiffness constant arising from the uniform convexity of the phase potential.
background
The GCIC Phase Thermodynamics module formalizes constants from the GCIC Response paper 'Two Upgrades for the GCIC Paper' (Feb 2026), focusing on stiffness, barrier height, and phase structure. The stiffness parameter is obtained from the second derivative of the phase potential, which equals cosh(t) and is therefore at least 1. The mean-field critical temperature is constructed from the eight-tick phase definitions and J-cost minimization structures imported from PhiForcingDerived and SpectralEmergence.
proof idea
The proof is a one-line wrapper that unfolds the definitions of the mean-field critical temperature and the GCIC stiffness, then applies the ring tactic to verify the algebraic identity.
why it matters
This equality supplies a concrete numerical link inside the GCIC phase thermodynamics, feeding the broader Recognition Science forcing chain at the eight-tick octave (T7) and the emergence of three spatial dimensions (T8). It connects directly to upstream structures for J-cost convexity and 8-tick phases without introducing new hypotheses.
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