zero_param_forces_unit_coefficients
Zero-parameter minimality in discrete ledger composition forces binary recurrence coefficients to be exactly unity. Researchers deriving the T5 to T6 bridge in Recognition Science cite this to obtain the Fibonacci relation from arithmetic alone. The proof is a direct term application of the upstream minimality lemma on positive integers.
claimLet $a, b$ be natural numbers with $a ≥ 1$, $b ≥ 1$, and max$(a, b) = 1$. Then $a = 1$ and $b = 1$.
background
The Hierarchy Dynamics module closes the gap between T5 (J-uniqueness via the cost functional) and T6 (phi as self-similar fixed point) by deriving the Fibonacci recurrence from zero-parameter ledger axioms. Zero-parameter posture means no free real parameters appear in inter-level ratios, which forces uniform scaling and minimal integer coefficients on a uniform scale ladder. The upstream theorem additive_composition_is_minimal states that among recurrence coefficients (a, b) with a ≥ 1 and b ≥ 1, the pair (1, 1) uniquely minimizes max(a, b); this is pure arithmetic.
proof idea
The proof is a one-line term wrapper that applies additive_composition_is_minimal directly to the hypotheses ha, hb, and hmin.
why it matters in Recognition Science
This supplies the minimality step required by bridge_T5_T6, which concludes that the uniform scale ratio equals phi, and by minimal_recurrence_forces_golden_equation. It realizes the module derivation chain: zero parameters force uniform ratio, locality forces order-two recurrence, discreteness forces integer coefficients, and minimality forces (1,1). The result supports the forcing chain landmarks T5 (J-uniqueness) and T6 (phi forced by self-similarity) without extra axioms.
scope and limits
- Does not prove existence of the uniform scale ladder.
- Does not compute the numerical value of the ratio.
- Does not extend to recurrences of order higher than two.
- Does not incorporate physical constants or continuous parameters.
Lean usage
theorem apply_zero_param (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. -/