coherence_at_phi_temp
plain-language theorem explainer
At the recognition temperature fixed by the golden-ratio self-similar point, the coherence measure of the system reaches its maximum value of unity. Researchers extending zero-temperature J-minimization to finite recognition temperature in the Gibbs ensemble would cite this critical-point result. The proof is a direct term-mode reduction that unfolds the coherence definition and simplifies via positivity of the temperature parameter.
Claim. At the recognition temperature $T_φ$ set by the golden-ratio fixed point, the coherence of the system equals one: $C(T_φ) = 1$.
background
Recognition Thermodynamics extends the strict J-cost minimization at zero temperature to finite recognition temperature TR. The J-cost is the functional J(x) = (x + x^{-1})/2 - 1 that serves as the energy; the Gibbs measure is the probability distribution p(x) ∝ exp(-J(x)/TR); recognition entropy quantifies degeneracy among near-minima; and recognition free energy is the quantity minimized by the dynamics. The phi-temperature is the specific TR value at which the system reaches a critical point of perfect coherence. The module connects these objects to the foundation through the phi-ladder discretization and the eight-tick cycle as the fundamental time unit τ₀. Upstream results supply the atomic tick as the discrete temporal unit and the inductive classification of crystal structures (BCC, FCC, HCP) used in spatial embeddings.
proof idea
The proof is a one-line term-mode wrapper. It unfolds the definitions of recognition coherence and the phi-temperature system, then applies simp using the non-equality of the phi-temperature to zero.
why it matters
This result marks the critical point where coherence equals unity inside the recognition thermodynamics framework, completing the finite-temperature extension of the T=0 cost minimum. It directly instantiates the phi fixed point from the forcing chain (T5–T6) and the eight-tick octave (T7) as the natural temperature scale. No downstream theorems are recorded yet, leaving open its use in explicit entropy or free-energy calculations.
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