phi_ladder_growth
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 in Recognition Science
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.
scope and limits
- Does not prove existence or uniqueness of the φ-ladder itself.
- Does not compute numerical values for physical constants or exponents.
- Does not address convergence or stability of the ladder under perturbations.
formal statement (Lean)
71theorem phi_ladder_growth (n : ℕ) :
72 phi ^ (n + 1) / phi ^ n = phi := by
proof body
Term-mode proof.
73 have h : phi ^ n ≠ 0 := pow_ne_zero _ phi_ne_zero
74 field_simp
75 ring
76
77/-! ## 3. The Forced Thermal Eigenvalue -/
78
79/-- The thermal eigenvalue of the recognition-lattice RG fixed point.
80
81 **Why this value is forced:**
82 1. The φ-ladder is the unique geometric scaling sequence in the
83 recognition lattice (PhiForcing: r² = r + 1 ↔ r = φ).
84 2. Consecutive rungs satisfy the Fibonacci recurrence
85 (`fibonacci_recurrence`), whose characteristic polynomial
86 is λ² − λ − 1.
87 3. φ is the unique positive root of this polynomial
88 (`fibonacci_char_poly_unique_pos_root`).
89 4. The thermal growth rate per ladder step is therefore φ and
90 nothing else. -/