Structural, Analytic and Arithmetic Reduction of the Twin Primes Conjecture to a Finite Primitive-Divisor Multiplicity Problem

We present a finite-state structural–analytic framework for studying the Twin Prime Conjecture based on a deterministic recursion (“Mode–A”) coupled to a transport-based obstruction mechanism (“Route–C”). The argument isolates the tail behavior of the recursion into a dichotomy between a square-multiplicity phenomenon and the recurrence of obstruction events. The obstruction branch canonically produces candidate twin-prime configurations through strict registration of boundary primes, return-map transport, and composite-elimination tests.

The analysis reduces the remaining arithmetic content to a square-upgrade mechanism driven by a ∆-channel forcing argument together with a CRT-based growth lemma and a coset-confinement elimination theorem. The logical dependencies are made explicit via a detailed directed-acyclic dependency graph (DAG) so that each step of the reduction can be independently verified.

A key structural feature of the framework is that after tail normalization the recursion only retains denominators supported on the primes {2,3,5}. Consequently every prime p>5 behaves uniformly in the tail dynamics, and the global residue structure reduces to the multiplicative group

G₅ = (ℤ/30ℤ)× = {1,7,11,13,17,19,23,29} ≅ C₄ × C₂.

The evolution of the recursion therefore induces a deterministic multiplier walk inside this eight-element group. Under the hypothesis that twin primes occur only finitely often, the dynamics would force this walk into a proper coset confinement inside G₅. A separate rigidity result rules out such confinement, yielding a structural contradiction within the Mode–A / Route–C framework.

The computational implementations included here are not numerical experiments but audit tools that mirror the exact objects defined in the manuscript. The code verifies:

• strict registration predicates (boundary insertion vs. collision channel)
• canonical window construction Wₚ,t = [k₁,k₂)
• faithful return-map transport (Ψ,Ψ′,Ψ″) mod p²
• the obstruction gate OC±1 (operationally: Δ = 0 and J = 0)
• certified outputs via primality tests for N_{k₂} ± 1 when obstruction events occur.

The implementations can emit complete audit logs of the form

(t, p, k₁, k₂, E, J, Δ)

allowing independent inspection of each transport and obstruction computation without altering the mathematical definitions used in the reduction.