thermal_eigenvalue_eq_phi
plain-language theorem explainer
The thermal eigenvalue of the recognition-lattice renormalization group fixed point equals the golden ratio φ. Researchers deriving critical exponents on self-similar lattices cite this result to fix the leading correlation-length exponent at ν₀ = 1/φ. The proof reduces to a single reflexivity step because thermal_eigenvalue is defined to be phi.
Claim. The thermal eigenvalue $y_t$ of the recognition lattice equals the golden ratio $φ$, where $φ$ satisfies $φ^2 = φ + 1$.
background
The module derives the thermal fixed point on the recognition lattice ℤ³ with unit cell Q₃. The φ-ladder is the unique geometric scaling sequence forced by self-similarity (T6). Consecutive rungs obey the Fibonacci recurrence whose characteristic polynomial λ² − λ − 1 has unique positive root φ (PhiForcing). Upstream, the definition states: The thermal eigenvalue of the recognition-lattice RG fixed point. Why this value is forced: 1. The φ-ladder is the unique geometric scaling sequence in the recognition lattice (PhiForcing: r² = r + 1 ↔ r = φ). 2. Consecutive rungs satisfy the Fibonacci recurrence (fibonacci_recurrence), whose characteristic polynomial is λ² − λ − 1.
proof idea
The proof is a one-line wrapper that applies reflexivity to the definition of thermal_eigenvalue, which is set equal to phi.
why it matters
This equality closes the derivation chain from PhiForcing (T6) through the Fibonacci cascade to the thermal eigenvalue y_t = φ, which determines ν₀ = 1/φ. It sits inside the Thermal Fixed-Point Operator section and supports the eight-tick octave and D = 3 spatial dimensions in the broader framework.
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