Transient Chaos and Topological Bounds in Prime Dynamics: Revisiting the One-Dimensional Sieve Mapping

A recent line of work models the prime distribution as the symbolic dynamics of a one-dimensional unimodal map at the band-merging point, recovering the twin-prime constant C₂ from a Metropolis--Stein--Stein (MSS) kneading sequence. The construction depends on Hypothesis~3. 3 (Topological Admissibility), which asserts that every finite sieve sequence Qₖ is an admissible kneading sequence inside its physical horizon N < p₊+₁². We show by explicit MSS comparison that Q₃ already fails admissibility at n = 31 and Q₅ fails at the prime gap 113--127, and that no bound on prime gaps of any strength can patch the proof: the breakdown is driven by parity inversion, not by gap size. We then prove a sharper structural result, the Parity-Gap Lemma, which reduces topological admissibility of Wₖ = Qₖ[0, p₊+₁²) to an extremal prime-gap inequality G (p₊+₁²) < p₊+₁ - 1. We note that classical bounds such as Legendre's, Andrica's, and even the Riemann Hypothesis yield maximal gaps at x = p² that asymptotically exceed the topological shield p - 1, and are therefore mathematically insufficient to guarantee admissibility. Only a strongly sub-root bound, such as Cram\'er's probabilistic conjecture G (x) = O (² x), forces eventual admissibility for all k above a finite threshold k₀ (numerically verified as k₀ = 6 for k 5000). Independently of the lemma, we restore the ergodic foundation via asymptotic admissibility: the topological-defect density (N) vanishes as N. A four-state parity-split chain then forces an asymptotic geometric decay ₂₌+₂/₂₌ p_ 0. 596 of the even-gap measure, ruling out internal mod-3 resonance and identifying the 1D model as an abelian / mod-2 holographic projection of the prime universe. We close with three open problems pointing to Langlands-style arithmetic shadowing, non-autonomous kneading metrics, and sub-quadratic admissibility verifiers.