Tail generation for symmetric free-product states: a general deduction

For every unital C*-algebra A and every state on its countable universal unital free product invariant under finite permutations, the recursively defined Dykema-Kostler-Williams tail C*-algebra D generates the tail von Neumann algebra T in the original GNS representation: D double commutant equals T. More precisely, the canonical expectation of the represented free-product algebra is weak-operator dense in T, with inputs and outputs bounded by the exact norm of each tail target. No faithful state or representation, separability, trace, factoriality, quantum symmetry or normal extension of the expectation is assumed. This is a short general deduction answering DKW Question5.3.5 and the separately posed Question 2 in OWR 28/2015 and 40/2016; catalog 30003208 and 30002897 are aliases. DKW coefficient stabilization and bounded-vector extension, Liu bilateral future-expectation methods, and Kaplansky density are prior and explicitly credited. The detailed supporting bridge makes the close corollary relationship explicit. No new coefficient lemma, new Kaplansky mechanism, exclusivefirstness or exhaustive historical novelty is claimed. The distinct question D subset pi(free product) is not answered. Both AMS full-body routes and the Liu dissertation body were inaccessible; actual inspected arXiv versions and source-access limits are disclosed. The proof was already public in project PR1329 on 13 October 2026; this note is dated 22 October 2026 and is not backdated. The deposit proposal consists of a four-page paper and its complete editable-source and verification ZIP. Portable standard-library exact finite mechanisms and ten deliberately false controls supplement the written universal proof; they do not certify arbitrary C*-algebras or priority. AI tools were extensively involved in research planning, mathematical exploration, proof development and reconstruction, computation, source and literature investigation, writing, editing and adversarial verification. Alec Kriebel is the named human author. The work is unrefereed, without conventional independent human peer review, journal acceptance or formal proof-assistant certification; internal AI audits do not provide those certifications. Authored prose is CC BY 4.0 and authored code is MIT licensed.