Exact Block-Sum Reconstruction of a Periodic Fibonacci Product Field: A Ternary Coding-Theoretic Reduction

We investigate exact reconstruction of a periodic Fibonacci product field modulo nine from aggregated block-sum observations. The underlying field is defined on a discrete three-dimensional torus of size 24 × 24 × 24 by F(x,y,z) = f(x)f(y)f(z) mod 9, where f is the Fibonacci sequence modulo nine with period 24. Removing coordinate planes associated with Fibonacci residues divisible by three yields a distinguished family of 216 disjoint blocks of size 3 × 3 × 3. The original block geometry and its initial block-sum patterns were calculated manually by the author prior to computational and algebraic verification. For a specified 16-state symmetry orbit, we establish the exact minimum numbers of fixed block-sum observations required for three reconstruction regimes: Clean reconstruction: 3 observations. Reconstruction with one known erasure: 4 observations. Reconstruction with one unknown arbitrary substitution: 6 observations. The main result follows from an exact algebraic reduction of the modulo-nine observation model to ternary linear codes of dimension three over the finite field F₃. An elementary projective-geometric argument establishes the impossibility of correcting one arbitrary substitution with only five block observations. Explicit physical block selections attain all three optimal bounds. The analytical results are supported by reproducible computational verification using a standalone Python program requiring only the standard library. The verification includes 3,456 physical-to-algebraic identities, 25,920 pairwise state/block comparisons, and 24,804 observation triples. The work provides a model-specific connection between periodic Fibonacci product fields, aggregated observations, finite-field linear algebra, and optimal error-correcting reconstruction. The general coding-theoretic obstruction is classical and is not claimed as a new theorem. The originality of the particular Fibonacci-to-coding construction is discussed cautiously in relation to existing literature. This manuscript is an author-approved preprint and has not undergone external peer review. Declaration of AI Assistance OpenAI ChatGPT was used substantially in mathematical exploration, programming, computational verification, certificate design, literature orientation, and manuscript drafting. The original identification of the 216 disjoint Fibonacci-field blocks and the initial manual calculation of their block-sum patterns were performed by the named author before the subsequent computational and algebraic investigations. The named author is responsible for reviewing the mathematical arguments, checking the released files and computational results, verifying bibliographic references and originality claims, and deciding whether to submit or publish the work. OpenAI ChatGPT is acknowledged as an AI-assisted research and drafting tool and is not listed as an author. Author Approval — GRANTED Author: Ivo Vajc Affiliation: Independent Researcher ORCID: 0009-0005-5145-6491 The named author has reviewed the mathematical content and approved the preparation of RT-FIELD-BLOCKSUM 1.9 as a prepublication research manuscript. The author has explicitly approved the Creative Commons Attribution 4.0 International (CC BY 4.0) license for the mathematical manuscript and the MIT License for the accompanying Python software. The author retains responsibility for the final released files, scientific claims, citations, originality statements, and publication decision. Status: Author-approved preprint, not externally peer reviewed. Licensing information The mathematical manuscript and its LaTeX source are distributed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. The accompanying Python verification software is distributed separately under the MIT License. Both license texts are included in the research package. The record-level CC BY 4.0 designation applies to the manuscript and does not supersede the MIT License governing the Python source code.