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One Successful Seed Forces Universality: Selection Jump and Witness-Shape Laws in Exact Witness Architectures

Year: 2026
Type: article
Abstract: The central theorem of this paper is a selection jump. For a decidable source-independent stagewise verifier $\Cert(n,c)$, let\[S_{\Cert}=\{e:\forall n\,\exists t\,(\KleeneT(e,n,t)\wedge \Cert(n,\KleeneU(t)))\}.\]If $S_{\Cert}\neq\varnothing$, then $S_{\Cert}$ is $\Pi^0_2$-complete. Thus a single successful seed already forces maximal stagewise complexity. We prove a companion one-shot calibration theorem: for every decidable $D(c)$, the class\[O_D=\{e:\exists t\,(\KleeneT(e,0,t)\wedge D(\KleeneU(t)))\}\]is either empty or $\Sigma^0_1$-complete. Together these results yield a witness-shape universality principle for exact local verification: one-shot local exactness yields $\Sigma^0_1$-universality, while stagewise local exactness yields $\Pi^0_2$-universality. We place this principle inside a broader barrier hierarchy. At the lower level, finite stagewise prefixes are uniformly insufficient, and undecidable co-c.e.\ witness architectures admit neither decidable one-shot positive certifiers nor exact-domain witness compilers. At the higher level, same-theory adequacy along a universal $\Pi_1$ embedding yields full $\Pi_1$ reflection, while arithmetic exact terminality on a truth-faithful image exists exactly when the corresponding fragment truth set is arithmetical, yielding Tarski and diagonal barriers. For fixed propositions with exact two-sided decidable witness packages, these results assemble into a two-sided bridge trichotomy: isolated extensional bridges are vacuous; the effective bridge layer splits into a positive stagewise fibre that is empty or $\Pi^0_2$-complete and a negative one-shot fibre that is empty or $\Sigma^0_1$-complete; and assertion-enriched resolver layers are truth-universal.
DOI: 10.5281/zenodo.19561300
Language: English
Open Access status: green
OpenAlex ID: W7154036730
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