Exact Static and Dynamic Phase Reconstruction in a Periodic Fibonacci Product Field

In the static reconstruction problem, observations are restricted to the 63 lattice offsets within Manhattan radius three. We establish, by an explicit construction and an independently checked finite computational certificate, that exactly six point samples are necessary and sufficient to distinguish all 3,456 phases. The lower bound is obtained through an exact counterexample-guided hitting-set procedure and an independent exhaustive depth-first verification of the resulting necessary separation constraints. In the dynamic reconstruction problem, we consider a stipulated discrete wave recurrence on a weighted three-dimensional lattice, with edge coefficients determined by the Fibonacci product field. Exact integer arithmetic demonstrates that the response of a single fixed sensor at times 5, 6, and 7 is individually injective over the complete phase space. The resulting three-coordinate observation code has minimum Hamming distance three and therefore corrects one arbitrary coordinate substitution. The accompanying research package includes the mathematical manuscript, Python source programs, computational verification logs, proof and literature audits, reproducibility documentation, and SHA-256 checksums. The manuscript is distributed under Creative Commons Attribution 4.0 International (CC BY 4.0), while the accompanying Python programs are distributed under the MIT License. All results concern explicitly defined finite mathematical models. They do not constitute experimental validation of physical laws, proof of universal reconstruction properties, or an established claim of scientific novelty. Declaration of AI Assistance OpenAI ChatGPT was used as an AI-assisted research and writing tool during the development of this work. Its contributions included mathematical exploration, formulation and refinement of computational algorithms, preparation and revision of Python source code, assistance with mathematical exposition, LaTeX manuscript preparation, literature-oriented analysis, and assembly of the publication package. The author reviewed the research materials, examined the mathematical results, executed the accompanying programs, and explicitly approved the publication package. The mathematical claims are supported by the definitions, exact computational procedures, and verification materials supplied with the manuscript. AI-generated content is not treated as independent mathematical verification or as evidence of scientific priority. The author assumes responsibility for the submitted manuscript and its contents. Copyright and licensing Copyright © 2026 Ivo Vajc. The mathematical manuscript and accompanying textual research documentation are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0), except where otherwise indicated. The accompanying Python source programs are distributed under the MIT License, as specified in the included license file. The licenses apply to their respective components. The software license does not replace the manuscript license, and the manuscript license does not replace the software license. Author approval: GRANTED, 10 November 2026.