phi_ladder_growth
plain-language theorem explainer
Consecutive rungs on the φ-ladder satisfy the exact scaling ratio φ. Researchers deriving the thermal eigenvalue in the recognition lattice RG flow would cite this result. The proof reduces the ratio algebraically after confirming φ is nonzero via a direct invocation of phi_ne_zero followed by field simplification and ring normalization.
Claim. $φ^{n+1} / φ^n = φ$ for every natural number $n$.
background
The module establishes the thermal fixed-point operator on the recognition lattice (ℤ³ with unit cell Q₃). The φ-ladder is the unique geometric scaling sequence forced by T6 self-similarity, satisfying φ² = φ + 1. Consecutive rungs obey the Fibonacci recurrence whose characteristic polynomial λ² − λ − 1 has φ as its unique positive root, so the thermal growth rate per ladder step equals φ and the leading correlation-length exponent is ν₀ = 1/φ.
proof idea
One-line wrapper that first obtains phi^n ≠ 0 from phi_ne_zero, then applies field_simp to clear the division and ring to normalize the powers.
why it matters
This supplies the direct scaling identity needed to reach the forced thermal eigenvalue y_t = φ in the derivation chain from PhiForcing (T6) through the Fibonacci cascade to ν₀ = 1/φ. It anchors the renormalization-group fixed point in the recognition lattice and feeds the thermal_eigenvalue family of results.
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