# Verification and scope

Result: complete counterexamples to ambient invariance of the exact extremal-exchangeability relation, plus a complete C*-extreme-extension argument. The source's literal ambient-dependence question has answer yes; the assertion of ambient invariance has answer no. These must not be confused with the forward extension statement, which is true in the unital C*-category.

The proof forces every downstairs word moment, rather than testing finitely many moments. Upstairs purity follows from the surjective folding quotient and a pure matrix state. The distinct-subalgebra example has scalar intersection and generates the smaller algebra. The general extension proof uses the full-free-product embedding theorem with scalar amalgamation, state extension, finite-group averages, compactness and a face argument.

Exact rational regressions passed for 16 density matrices, 456624 diagonal words, 2720 noncommuting matrix-unit words, 16320 copy relabelings, 24 generator identities and 625 intersection tests. Endpoints were included as diagnostics but excluded from the nontrivial counterexample theorem. These computations are supplementary; they neither enumerate all invariant states nor prove the compactness theorem.

The desktop LaTeX compiler and Tectonic compiled successfully. All five PDF pages were visually inspected, including the author footnote, displayed formulas, references and page transitions. The source archive is separately extracted and compiled to test portability before package readiness is recorded.

The original report's pp. 540-541 and the relevant Dykema-Koestler-Williams and Armstrong-Dykema-Exel-Li passages were read. The known folding mechanism is credited. Fixed-marginal extremality, faithful-upstairs variants, arbitrary normal von Neumann extensions and the report's other questions are not claimed solved.

Acceptance basis: originating researcher detailed self-audit. Independent review: not performed and not an acceptance prerequisite. Formal verification: not performed. Absolute priority and novelty: not certified. This document does not certify external publication.