Universal Six-Probe Reconstruction of Unknown Reversible Companion Dynamics over Prime-Power Rings

This preprint presents a universal six-round reconstruction theorem for unknown reversible second-order linear dynamical systems over prime-power residue rings. The system is governed by an unknown companion matrix with two coefficients, together with an unknown initial state. The experimenter applies a predetermined sequence of additive interventions before successive transitions and observes only the first coordinate of the evolving system. No system reset is required. The main result establishes that six observations are sufficient to reconstruct both unknown coefficients and the initial state for every prime and every positive prime-power exponent. The proof uses an explicit system of linear reconstruction equations and establishes the existence of an invertible two-by-two minor. This algebraic certificate guarantees unique reconstruction over all prime-power residue rings. The accompanying research package contains the mathematical manuscript, LaTeX source files, an independent Python verification program, documentation of the mathematical and computational audits, and licensing information. The result provides a universal upper bound of six observations. Global optimality is not claimed. This manuscript is released as a mathematical preprint and has not undergone external peer review. AI assistance disclosure: OpenAI ChatGPT was used substantially in mathematical exploration, programming, certificate design, literature discovery, and manuscript drafting. The named author is responsible for reviewing the mathematics, checking the released files, verifying citations and originality claims, and deciding whether to submit the work.