Pisano Synchronization, Twin-Prime Ratio Rigidity, and Mirror Closure: Modulo 144 and Beyond

This paper connects Fibonacci return times, polynomial formulas for additive-gap ratios, and inversion symmetries of modular recurrence signatures. It develops these connections through explicit proofs, exact computational verification, and reproducible algorithm benchmarks. Part I establishes a sharp degree-dependent bound for polynomial ratio formulas on arithmetic progressions. For gap-two ratios on the progression congruent to 5 modulo 6, the largest modulus admitting an affine formula is 144. Three genuine twin-prime pairs certify this maximum. Five genuine twin-prime pairs similarly certify the quadratic maximum of 2592. These certificates constrain the possible formulas; they do not constitute primality tests or prove that infinitely many twin primes exist. A separate synchronization theorem classifies the Fibonacci return times for which every unit-parameter recurrence matrix returns to the identity modulo the maximal Fibonacci-return modulus. The largest modulus selected by this condition is 144. This maximality statement concerns the specified return-modulus construction, rather than every modulus admitting some common recurrence period. At modulus 144, the paper connects twelve twin-ratio states, six squared states, and the twelve classes of the Signature Atlas quotient. It proves parameter-dependent cross-closure and realizes the companion mirror as a fixed permutation of recurrence indices. Two distinct inversion mirrors reveal a binary distinction that squaring erases. Compatible extensions to finer prime wheels are also constructed. Part II develops the algorithmic consequences and proves an exact minimum-degree criterion for polynomial inversion on all units of an arbitrary modulus. Its sharp modulus sequence is identified with an established prime Bhargava factorial sequence, with attribution to the existing literature. A constructive procedure produces minimum-degree inverse polynomials from a supplied factorization. Additional implementations include Newton inversion for moduli built from powers of 2 and 3, constant-memory wheel-ratio streams, permutation-based inversion of cached companion values, and a direct decoder that recovers the exact four-parameter signature class without searching all 48 parameters. Verification exhausts 109,499 unit inputs across moduli 2 through 600, independently enumerates polynomial coefficients in 60 small-ring cases, and checks larger examples, all 48 unit-parameter signature classes, and selected branches on seven wheels. Benchmarks compare the implementations with ordinary modular inversion, classical Montgomery batch inversion, and lookup tables. The results distinguish minimum polynomial degree from minimum computational cost: structured formulas outperform repeated inversion in several tested workloads, while lookup tables and classical batching remain stronger in others. Timing conditions, setup costs, and limitations are documented. The accompanying source archive contains the editable LaTeX manuscript, Python and C++ programs, exact verification results, raw benchmark data, figures, and reproduction instructions. The mathematical results concern finite rings and recurrences and require no physical interpretation.