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

Article · liang wang

- Archive id: doi:10.5281/zenodo.20447245
- Published: 2026-05-29
- Venue: Zenodo (CERN European Organization for Nuclear Research)
- DOI: https://doi.org/10.5281/zenodo.20447245
- Citations: 0
- Source URL: https://doi.org/10.5281/zenodo.20447245
- Provenance: openalex

## Abstract

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.

_Abstract reproduced under CC-BY (publisher license, via OpenAlex)._

Topics: [Landmark Results in Pure Mathematics](https://space-kitty.com/topics/landmark-math-results)

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