Phase-Plane Geometry and Quotient Reconstruction in a Periodic Fibonacci Product Field: Exact Finite Certificates and Minimal Distinguishing Antichains

A rank-one Fibonacci product field modulo nine is summed over 216 disjoint (3 \times 3 \times 3) blocks. The resulting block sums exhibit a six-phase structure organized into parallel diagonal planes, providing an exact geometric representation of a finite periodic field. Within a stipulated 16-state reflection-and-phase model, we investigate four reconstruction tasks: identification of oriented plane normals, identification of the phase parameter, complete hidden-state reconstruction, and identification of unoriented plane families. Exhaustive finite certificates establish sharp minimum numbers of fixed nonadaptive block probes. For oriented-normal and complete-state reconstruction, the optimal probe counts are (3, 4, 6), respectively, under error-free observations, one known erasure, and one arbitrary substitution. For phase-parameter and unoriented-plane-family reconstruction, the corresponding optimal counts are (2, 3, 4). An algebraic parity argument proves that orientation reconstruction and complete-state reconstruction have identical distinguishing-mask families in the specified model. A sufficient condition for this equality is also established for a broader class of even-modulus exponent models. The study further develops a reconstruction comparison based on minimal distinguishing antichains. Equality of these antichains characterizes equality of minimum weighted distinguishing distances for all fixed probe multisets. This combinatorial principle is connected to established antichain and monotone Boolean-function theory and is not claimed as a new general mathematical discovery. The results are supported by exhaustive finite verification, explicit optimal probe constructions, algebraic arguments, and an independent implementation that directly evaluates the physical Fibonacci block sums. The geometric structure underlying this investigation was originally identified by the author through manual calculations and drawings before its subsequent computational formalization. The work concerns exact finite mathematical reconstruction. It does not establish physical wave propagation, temporal dynamics, or experimental physical effects. This release is an author-approved mathematical preprint accompanied by reproducibility software, mathematical verification materials, and a literature and novelty audit. It has not undergone independent external peer review. Declaration of AI Assistance OpenAI ChatGPT was used substantially during the mathematical exploration, algebraic formulation, development of Python verification programs, design of computational certificates, literature analysis, and preparation of the manuscript and accompanying documentation. The original geometric observations concerning Fibonacci block sums and their spatial arrangement were developed by the named author through independent manual calculations and drawings prior to the computational investigation. The named author, Ivo Vajc, has reviewed the mathematical results, executed the verification programs, examined the proofs and accompanying materials, and explicitly approved the release. Responsibility for the mathematical content, verification of references, assessment of originality claims, and decision to publish rests with the named author. The use of AI assistance does not constitute independent external peer review. AUTHOR APPROVAL — GRANTED I, Ivo Vajc, the named author of the preprint Phase-Plane Geometry and Quotient Reconstruction in a Periodic Fibonacci Product Field: Exact Finite Certificates and Minimal Distinguishing Antichains, confirm that I have reviewed the manuscript, mathematical arguments, verification programs, and accompanying research materials. I authorize the public release of version 1.5 as a mathematical preprint on Zenodo. I approve the use of the Creative Commons Attribution 4.0 International (CC BY 4.0) license for the manuscript and the MIT License for the accompanying original Python source code. I acknowledge the disclosed contribution of OpenAI ChatGPT to mathematical exploration, computational verification, literature analysis, and manuscript preparation. Licensing Notice The manuscript and original textual research materials are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). The accompanying original Python source code is separately licensed under the MIT License, as specified in the included license files. Both licenses permit redistribution and reuse under their respective conditions. Author: Ivo VajcApproval status: GRANTEDDate: 2026-11-04