module module moderate

`IndisputableMonolith.Physics.ThermalFixedPoint`

This module derives the thermal fixed point from the characteristic polynomial of the Fibonacci recurrence a(n+2) = a(n+1) + a(n). It establishes that the positive root equals phi and equals the leading thermal eigenvalue. Physicists modeling discrete thermal systems or critical exponents in Recognition Science would cite these results. The module structure consists of algebraic definitions for the polynomial followed by root uniqueness and eigenvalue equality theorems.

claimThe characteristic polynomial of the recurrence $a_{n+2} = a_{n+1} + a_n$ is $x^2 - x - 1 = 0$, whose unique positive root is the golden ratio $phi = (1 + sqrt(5))/2$. The module further shows that the leading thermal eigenvalue equals this root and is the unique positive solution satisfying the self-similar fixed-point condition.

background

The module operates in the Physics domain of Recognition Science and imports the time quantum tau_0 = 1 tick from Constants, the proof that phi arises from self-similarity in a discrete ledger with J-cost from PhiForcing, and the spectral properties of the 3-dimensional hypercube Q_3 from CubeSpectrum. It introduces the Fibonacci characteristic polynomial whose positive root governs asymptotic growth rates on the phi-ladder. The local setting connects discrete recurrence relations to thermal eigenvalues that appear in critical exponent corrections.

proof idea

The module begins with the definition of the characteristic polynomial for the linear recurrence. It applies algebraic reduction to identify the positive root as phi. Subsequent results prove uniqueness of this positive root and its equality to the thermal eigenvalue via direct substitution into the fixed-point equation together with positivity arguments. The overall argument is a sequence of supporting lemmas rather than a single monolithic proof.

why it matters in Recognition Science

This module supplies the growth-rate foundation for thermal fixed points and feeds into derivations of mass formulas and critical exponents on the phi-ladder. It realizes the T6 step of the forcing chain in which phi is forced as the self-similar fixed point. The results also support the eight-tick octave structure and D = 3 spatial dimensions by providing the leading eigenvalue for thermal modes in the Recognition framework.

scope and limits

Does not treat recurrences of order higher than two.
Does not incorporate Berry creation thresholds or Z_cf corrections.
Does not compute full thermal spectra beyond the leading eigenvalue.
Does not address continuous limits or renormalization group flows.

depends on (3)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import
IndisputableMonolith.Foundation.PhiForcing module_import
IndisputableMonolith.Physics.CubeSpectrum module_import

declarations in this module (23)

def fibonacci_char_poly L48
theorem fibonacci_char_poly_root L51
theorem fibonacci_char_poly_unique_pos_root L55
theorem fibonacci_recurrence L64
theorem phi_ladder_growth L71
def thermal_eigenvalue L91
theorem thermal_eigenvalue_eq_phi L93
theorem thermal_eigenvalue_uniqueness L96
theorem thermal_eigenvalue_golden L101
theorem thermal_eigenvalue_pos L104
theorem thermal_eigenvalue_relevant L107
def nu_leading L114
theorem nu_leading_eq L116
theorem nu_leading_pos L118
theorem nu_leading_lt_one L122
def anomalous_nu_correction L139
theorem anomalous_nu_correction_zero L141
theorem anomalous_nu_correction_small L147
def nu_corrected L159
theorem nu_corrected_at_zero L162
theorem spectral_gap_multiplicity_eq_degree L169
structure ThermalFixedPointCert L175
def thermalFixedPointCert L182