Predictive Reconstruction versus State Identification in Finite Fibonacci Dynamics over Z/9Z

Predictive Reconstruction versus State Identification in Finite Fibonacci Dynamics over Z/9Z This preprint investigates task-dependent predictive reconstruction in a finite Fibonacci dynamical system over the ring Z/9Z. The state transition is defined by T(a,b) = (b,a+b) modulo 9, with a distinguished orbit of length 24. We study whether two affine scalar observations suffice to determine a prescribed prediction target without necessarily identifying the exact dynamical state. The model includes 16 coordinate interventions and 19 reconstruction targets, comprising exact phase identification, the complete intervention-response vector, the set of possible outcomes under an unknown intervention, and individual intervention outcomes. For the 3,240 unordered pairs of distinct sensor slopes, we classify 22 distinct joint kernels and all 418 combinations of prediction targets and kernels. The central criterion states that a prediction target is reconstructible precisely when the joint observation kernel contains no difference between states having distinct target values. The resulting exact counting formula is N(Y) = 1944 + 216 L(Y) + 24 P(Y), where L(Y) counts the admissible order-three kernel lines and P(Y) indicates whether the distinguished order-nine subgroup is admissible. The classification is supported by a finite computational certificate, including 61,560 direct sensor-pair checks, explicit collision witnesses for excluded kernels, algebraic multiplicity checks, and SHA-256 checksums. The results demonstrate, within the stipulated finite model, that predicting a specified intervention outcome can require less state discrimination than exact phase identification. The contribution is a concrete, reproducible finite classification rather than a claim of a new general theory of functional observability. The manuscript is author-approved as version 2.8, dated 11 November 2026. It has not undergone external peer review. AI assistance in manuscript preparation, code development, and audit planning is explicitly disclosed. The manuscript is distributed under CC BY 4.0, while the accompanying verification software is licensed separately under MIT. Reproducibility and mathematical verification The finite classification was verified using Python 3.13.13 on Pydroid 3. The verification software uses only the Python standard library and runs offline without writing files. The archived verification output reports PASS for 418 target-kernel decisions, 61,560 direct sensor-pair checks, and the independent modulo-three multiplicity audit. SHA-256 checksums are provided for integrity verification. The analytical collision criterion and kernel-counting arguments are distinguished from the action-specific classifications established through exhaustive finite computation. No external peer review, empirical physical validation, or comprehensive proof of mathematical novelty is claimed. AI assistance disclosure ChatGPT (OpenAI, GPT-6) assisted with mathematical exposition, drafting, Python code development, bibliography discovery, formatting, and audit planning. The author reviewed the complete publication package, tested the programs, and explicitly approved the final manuscript on 11 November 2026. Responsibility for the published content remains with the named human author. Offline, standard-library Python verification program accompanying the preprint "Predictive Reconstruction versus State Identification in Finite Fibonacci Dynamics over Z/9Z" (version 2.8). The program constructs the finite Fibonacci dynamical model modulo nine, enumerates 22 distinct joint sensor kernels, and verifies their admissibility for 19 prediction targets. It checks 418 target-kernel decisions and 61,560 direct sensor-pair cases, including explicit collision witnesses and independent modulo-three multiplicity controls. Designed for Pydroid 3 and standard Python environments. No third-party packages, network connection, or file writing are required. The software is distributed under the MIT License. The associated mathematical manuscript is distributed separately under CC BY 4.0.