Exact Causal Reconstruction under Hidden Interventions: Gauge Obstructions, Finite Memory, and Robust Binary Probes

This preprint investigates exact causal reconstruction in a finite dynamical model over (\mathbb{F}_3^2), involving three candidate linear evolution laws, unknown initial states, a single hidden additive intervention, and partial observations. The study establishes exact finite-state identification results for fixed and adaptive observation strategies, including restricted-memory policies. An explicit algebraic gauge construction demonstrates that different hidden interventions can produce identical observable histories under every common adaptive policy, yielding a causal non-identifiability result valid for arbitrary dynamical horizons. The introduction of nondisturbing binary probes before the intervention restores identifiability. Under one adversarial marked erasure, the exact minimum worst-case number of binary probes is three; under one adversarial substitution, it is five. These lower bounds remain valid independently of the length of subsequent dynamical observation. Explicit error-correcting codes and exhaustive adaptive decision-tree certificates establish attainability. The contribution is model-specific and combines established concepts from finite-state system identification, adaptive distinguishing sequences, causal identifiability, and error-correcting codes. The accompanying archive contains the mathematical manuscript, LaTeX source, ten standalone Python verification programs, reproducibility information, and an internal referee-style audit. This manuscript is an AI-assisted, non-peer-reviewed mathematical preprint. The results have undergone internal computational and proof-oriented checks but have not been independently evaluated by an external human referee. No definitive priority claim is made.