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 $C_2$ from a Metropolis--Stein--Stein (MSS) kneading sequence. The construction depends on Hypothesis~3.3 (\emph{Topological Admissibility}), which asserts that every finite sieve sequence $Q_k$ is an admissible kneading sequence inside its physical horizon $N < p_{k+1}^2$. We show by explicit MSS comparison that $Q_3$ already fails admissibility at $n = 31$ and $Q_5$ 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 \emph{Parity-Gap Lemma}, which reduces topological admissibility of $W_k = Q_k[0, p_{k+1}^2)$ to an extremal prime-gap inequality $G(p_{k+1}^2) < p_{k+1} - 1$. We note that classical bounds such as Legendre's, Andrica's, and even the Riemann Hypothesis yield maximal gaps at $x = p^2$ 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 $G(x) = O(\log^2 x)$, forces eventual admissibility for all $k$ above a finite threshold $k_0$ (numerically verified as $k_0 = 6$ for $k \leq 5000$). Independently of the lemma, we restore the ergodic foundation via \emph{asymptotic admissibility}: the topological-defect density $\rho(N)$ vanishes as $N \to \infty$. A four-state parity-split chain then forces an asymptotic geometric decay $\mu_{2m+2}/\mu_{2m} \to p_\infty \approx 0.596$ 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.