Exact Observation Distances and Historical Identifiability in a 24-State Fibonacci System

This preprint presents an exact mathematical and computational study of historical identifiability in a finite Fibonacci dynamical system. The underlying system is the 24-periodic orbit of the Fibonacci recurrence modulo nine. The principal objective is to determine which properties of a single phase-jump event can be reconstructed from complete or partial observations, including observations affected by a finite number of arbitrary substitutions. The manuscript establishes finite-versus-infinite observation-distance criteria, exact Hamming-distance formulas, zero-error ambiguity classifications, and substitution-error thresholds. It also distinguishes reconstruction of the current state from reconstruction of the historical event that produced it. The accompanying research package contains the manuscript in PDF and LaTeX formats, computational verification scripts, review documentation, bibliographic references, and reproduction instructions. Research scope: The analysis assumes a known initial phase and exactly one nonzero jump. Multiple jumps, unknown initial phases, stochastic observation models, and insertion/deletion errors are outside the scope of the present work. Publication status: Version 1.0, mathematical preprint, not peer reviewed. AI assistance: OpenAI ChatGPT (GPT-6) assisted with mathematical exposition, computational implementation, proof checking, literature research, and manuscript preparation. The human author retains responsibility for the scientific content, interpretations, and publication. Author approval: Granted by the author for preparation of this publication package.