zero_param_forces_unit_coefficients
Zero-parameter minimality in discrete ledger composition forces recurrence coefficients to be exactly (1,1). Researchers closing the T5 to T6 gap in the Recognition Science chain cite this when deriving the golden ratio from integer recurrences on uniform scale ladders. The argument reduces directly to an upstream arithmetic lemma identifying the unique minimizer of max(a,b) among positive integers.
claimLet $a, b$ be positive integers satisfying $1 ≤ a$, $1 ≤ b$, and $max(a,b) = 1$. Then $a = 1$ and $b = 1$.
background
This theorem belongs to the Hierarchy Dynamics module, which resolves the gap between T5 (J-uniqueness via the cost functional) and T6 (self-similar fixed point φ) by deriving the Fibonacci recurrence from zero-parameter ledger axioms. The module traces multilevel composition to a uniform scale ladder, then uses locality and discreteness to obtain a finite-order recurrence with positive integer coefficients.
proof idea
The proof is a one-line wrapper that applies additive_composition_is_minimal from HierarchyForcing to the hypotheses ha, hb, and hmin.
why it matters in Recognition Science
The result supplies the minimality step in the module's derivation chain and feeds directly into minimal_recurrence_forces_golden_equation and bridge_T5_T6. The latter derives that a minimal local binary recurrence on a uniform scale ladder forces the ratio σ to equal φ. It realizes step 5 of the chain: zero-parameter posture selects coefficients (1,1), yielding σ² = σ + 1. This connects T5 J-uniqueness to the self-similar structure required for T6 and the eight-tick octave.
scope and limits
- Does not establish existence of a uniform scale ladder.
- Does not derive the recurrence relation from locality axioms.
- Does not treat non-integer or negative coefficients.
- Does not connect coefficients to physical constants such as α.
Lean usage
example (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b) (hmin : max a b = 1) : a = 1 ∧ b = 1 := zero_param_forces_unit_coefficients a b ha hb hminformal statement (Lean)
86theorem zero_param_forces_unit_coefficients
87 (a b : ℕ) (ha : 1 ≤ a) (hb : 1 ≤ b)
88 (hmin : max a b = 1) :
89 a = 1 ∧ b = 1 :=
proof body
Term-mode proof.
90 additive_composition_is_minimal a b ha hb hmin
91
92/-! ## Step 6a: Unit Coefficients → Fibonacci Relation -/
93
94/-- Integer recurrence with unit coefficients reduces to the
95Fibonacci relation L₂ = L₁ + L₀. -/