{"id":"doi:10.5281/zenodo.20447245","kind":"article","title":"Transient Chaos and Topological Bounds in Prime Dynamics: Revisiting the One-Dimensional Sieve Mapping","authors":["liang wang"],"author_count":1,"year":2026,"published":"2026-05-29","venue":"Zenodo (CERN European Organization for Nuclear Research)","cited_by":0,"doi":"10.5281/zenodo.20447245","arxiv_id":null,"nct_id":null,"source_url":"https://doi.org/10.5281/zenodo.20447245","archive_url":"https://space-kitty.com/papers/doi:10.5281/zenodo.20447245","updated":null,"pmid":null,"pmcid":null,"openalex_id":"W7162758733","pdf_url":null,"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_license":"CC-BY (publisher license, via OpenAlex)","license":"cc-by","is_open_access":true,"retracted":false,"peer_reviewed":null,"categories":["Analytic Number Theory Research"],"details":null,"provenance":"openalex","attribution":"Abstract of \"Transient Chaos and Topological Bounds in Prime Dynamics: Revisiting the One-Dimensional Sieve Mapping\" by liang wang, https://doi.org/10.5281/zenodo.20447245, used under its Creative Commons Attribution license (CC-BY); markup removed, otherwise unmodified.","topics":[{"slug":"landmark-math-results","name":"Landmark Results in Pure Mathematics","domain_slug":"math-cs"}],"cited_in_dossiers":[]}