An All-Level Three-Round Minimax Theorem for Active Identification of Unknown Fibonacci-Type Dynamics

We investigate the exact adaptive identification of an unknown dynamical law selected from three reversible Fibonacci-type maps over the finite rings (\mathbb{Z}/3^k\mathbb{Z}), for every integer (k \geq 1). The initial state is unknown, and no reset to a known state is permitted. In each experimental round, the observer selects one of five additive unit interventions, including the zero intervention, applies the unknown dynamical law once, and measures one of the two state coordinates. The objective is to identify the governing law rather than reconstruct the initial state. We establish that the exact worst-case adaptive minimax identification depth is three for every (k \geq 1). The lower bound is obtained through an integer-affine obstruction certificate covering all 100 ordered pairs of experimental actions. The certificate consists of 75 cases with invertible integer minors and 25 structural cases exhibiting a law-independent observation identity. The upper bound is provided by an explicit three-round adaptive identification policy over (\mathbb{F}_3), which extends to every higher power of three through compatibility with modular reduction. The accompanying reproducibility package includes the LaTeX manuscript, a standard-library Python verification program, finite certificate outputs, and technical review notes. The contribution is a precisely scoped exact minimax theorem for the specified family of Fibonacci-type dynamics. It does not claim originality for the general concepts of active system identification, adaptive distinguishing experiments, or finite-ring dynamics. This manuscript is a preprint and has not undergone independent external human peer review. Preprint status: Version 2.0, dated 26 October 2026. This manuscript has undergone an AI-assisted technical and mathematical audit, but has not been independently peer-reviewed by a human referee. Scope of the theorem: The result applies to the three explicitly defined Fibonacci-type maps F, G, and K over (\mathbb{Z}/3^k\mathbb{Z}), the ten permitted intervention-and-observation actions, noiseless deterministic measurements, and identification of the governing law with an unknown initial state and no reset. Reproducibility: The accompanying Python 3 program uses only the standard library and can be executed offline, including in Pydroid 3. It checks the finite affine certificates, the adaptive identification policy, and selected modular instances. Originality statement: The claimed contribution is the exact all-level three-round minimax theorem and its constructive arithmetic certificate for this specific controlled dynamical model. No claim is made that active system identification or adaptive distinguishing sequences are new mathematical concepts.