Fixed Composition Testers for Strong Positivity

This paper characterizes the fixed finite positive semidefinite composition partners whose tensor powers test strong positivity of every finite Hermitian matrix: a partner is universal exactly when it has a nonreal entry in the specified history basis. Constructive proofs give explicit copy bounds, finite-depth spectral and tolerance certificates, a finite-family obstruction, and a quantum-history realization. For a noisy two-history family, all-event lower bounds and explicit exposing events establish sharp weak-imaginary resource scaling. Further results include an exact pure-phase onset and conditional finite-composition CHSH certificates. The accompanying ZIP contains the LaTeX source, six exact standard-library Python checkers and their results, a package verifier and file hashes. Priority remains unestablished. No derivation of a physical probability law is claimed.Robert Barritz directed the research and its preparation for publication. OpenAI’s ChatGPT and Codex systems contributed substantially to mathematical formulation, proof development and examination, examples, exact computational verification, literature comparison, and manuscript preparation. Anthropic’s Claude contributed mathematical, computational, and literature review that informed revisions. The AI contribution extended substantially beyond language editing. AI-assisted review does not constitute human expert or journal peer review.