thermalFixedPointCert
plain-language theorem explainer
The thermal fixed point certificate assembles the root property of the Fibonacci characteristic polynomial at phi, its uniqueness among positive reals, the recurrence for powers of phi, the identification of the thermal eigenvalue with phi, and the leading exponent nu as one over phi into a single structure. Physicists deriving correlation-length exponents from renormalization flows on the recognition lattice would cite this when closing the phi-ladder analysis. The definition is a direct record construction that assigns each field from an preë
Claim. Let $phi$ denote the golden ratio. The thermal fixed-point certificate asserts that $phi$ is a root of the Fibonacci characteristic polynomial $lambda^2 - lambda - 1 = 0$, that it is the unique positive root, that the powers satisfy the recurrence $phi^{n+2} = phi^{n+1} + phi^n$ for all natural $n$, that the thermal eigenvalue equals $phi$, and that the leading correlation-length exponent satisfies $nu_0 = 1/phi$.
background
In the Recognition Science framework the renormalization group at a critical point on the three-dimensional recognition lattice (Z^3 with unit cell Q3) proceeds along the phi-ladder, the unique self-similar scaling sequence forced by T6 self-similarity. The Fibonacci recurrence arises from J-cost-optimal partitioning and is inherited from eight-tick periodicity at D=3. The characteristic polynomial of this recurrence is lambda^2 - lambda - 1, whose unique positive root is phi by the PhiForcing theorem (see upstream fibonacci_char_poly_unique_pos_root).
proof idea
The definition constructs an instance of the ThermalFixedPointCert structure by assigning char_poly_root to the theorem fibonacci_char_poly_root, uniqueness to fibonacci_char_poly_unique_pos_root, cascade to fibonacci_recurrence, and the eigenvalue and nu fields to reflexivity. Each assignment draws on prior results that phi satisfies the quadratic and that the recurrence holds by algebraic manipulation using phi^2 = phi + 1.
why it matters
This certificate closes the derivation chain from PhiForcing (T6) through the phi-ladder to the thermal eigenvalue y_t = phi and the correlation-length exponent nu_0 = 1/phi. It supplies the concrete values needed for thermal perturbation analysis on the Q3 lattice, where spectral-gap multiplicity equals the graph degree D=3. No downstream uses are recorded yet, but it completes the summary certificate section of the module.
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