Exact Binary Reconstruction and Symmetry-Compressed Obstruction Certificates for Fibonacci Carry Dynamics

We investigate the exact reconstruction of an unknown element of GL(2, F₃) from fixed, noiseless, binary observations associated with a finite Fibonacci recurrence. The specified zero-base observation model has 48 hidden hypotheses and 216 indexed probes, which reduce to 32 distinct nonconstant binary response functions. Although the construction is embedded in a 24-position Fibonacci framework, the effective recurrence modulo 3 has period eight. An explicit algebraic representation expresses the observations through the inverse action of the hidden matrix, eliminating the need to construct the canonical time lift. We prove, with exhaustive finite computational certificates, that no set of seven binary probes distinguishes all 48 hypotheses, while an explicit set of eight probes does. Thus the exact minimum fixed, nonadaptive, noiseless binary reconstruction budget is eight. We further classify the 10,368 seven-probe subsets leaving exactly one unresolved hypothesis pair into 14 symmetry orbits. The corresponding obstruction hypergraph contains 1,104 inclusion-maximal hyperedges, partitioned into five orbits under a verified permutation group of order 1,152. These five orbit representatives provide a symmetry-compressed description of the complete finite obstruction certificate. The work connects Fibonacci recurrence dynamics, finite group actions, separating systems, minimum test cover, and computational combinatorics. Reproducible standard-library Python implementations support the exact finite results. The conclusions are restricted to the specified zero-base model with fixed, noiseless, nonadaptive binary observations. No extension to arbitrary base states, adaptive reconstruction, noisy measurements, or physical dynamics is claimed. This manuscript is an independent research preprint and has not undergone external peer review. Research status: Independent mathematical research preprint, version 1.1. Not peer reviewed. Mathematical scope: Exact reconstruction of hidden elements of GL(2, F₃) in a specified zero-base Fibonacci carry observation model. The 24-position construction has an effective period of eight modulo 3. Computational verification: The exact minimum of eight fixed, nonadaptive binary probes is supported by exhaustive finite enumeration and an explicit separating witness. An independently implemented algebraic model reproduces the principal seven-probe obstruction results. The five-orbit hypergraph classification is supported by the original computational audit suite and has not yet undergone a separate independent reimplementation. Reproducibility: The accompanying archive includes the mathematical manuscript, LaTeX source, Python verification programs, review notes, and supporting documentation. The programs use the Python standard library and require no external packages. Licensing: The mathematical manuscript and its LaTeX source are released under Creative Commons Attribution 4.0 International (CC BY 4.0). The accompanying Python source programs are released separately under the MIT License. The applicable license notices are included in the archive. AI assistance disclosure: OpenAI ChatGPT was used extensively during mathematical exploration, algorithm development, computational verification, literature research, and manuscript preparation. The named author retains responsibility for the correctness of the scientific claims, the verification of references, and the publication decision. Corresponding author: Ivo Vajc, Independent Researcher. Contact: ivovajc@web.de ORCID: https://orcid.org/0009-0005-5145-6491