The Minimal Primitive Renormalization Law of a Discrete Reciprocal Ledger Is Fibonacci, and Its Perron–Frobenius Eigenvalue Is φ

We prove that a discrete reciprocal ledger with zero adjustable parameters admits a unique minimal primitive renormalization law, and that law is the Fibonacci substitution L → LS, S → L. The proof proceeds in five steps: (i) zero parameters force a uniform inter-level scale ratio σ; (ii) locality of ledger posting forces an order-2 recurrence ℓk+2 = aℓk+1 + bℓk with a, b ∈ N+; (iii) the zero-parameter posture selects the unique pair (a, b) = (1, 1) minimising max(a, b); (iv) the resulting Fibonacci transfer matrix M = [[1, 1], [1, 0]] is primitive, and its Perron–Frobenius eigenvalue is φ = (1 + √5)/2; (v) φ is the smallest Perron–Frobenius eigenvalue among all 2 × 2 primitive non-negative integer matrices. Every theorem in the main chain is formalised in Lean 4 with zero sorry.