Exact and Asymptotic Optimal Threshold Reconstruction: A Geometric-Tail Sequential Decision Model

We investigate an exact sequential threshold-reconstruction problem with geometrically distributed hidden states and absolute-threshold query costs. The model arises as a scalar optimization subproblem motivated by finite-ring reconstruction, but its mathematical results are formulated independently of the original matrix classification problem. For a geometric survival factor of 27, we study the optimal expected cost of adaptive threshold queries under a restarted-cost convention. The infinite-horizon value function satisfies the Bellman equation [ V(a)=\min_{j\geq1}\left(a+j+27^{-j}V(a+j)\right). ] We establish convergence of finite-horizon value functions to the infinite-horizon problem, together with an explicit quantitative error bound. Our main analytical result proves that every optimal first jump j^*(a), for a\geq2, satisfies [ \log_{27}(25a-3)-1 \leq j^*(a) \leq \lceil\log_{27}a\rceil+3. ] Consequently, [ j^*(a)=\log_{27}a+O(1). ] The result provides bounded-additive localization of optimal threshold increments without requiring an explicit solution of the Bellman equation or prior knowledge of the exact switching boundaries. We additionally develop an exact integer Bellman formulation and a computational certification procedure for selected infinite-horizon optimal first decisions. Reproducible integer witnesses certify unique optimal jumps at six selected states, including the transitions at 26, 753, and 20434. The accompanying archive contains the manuscript, LaTeX source, exact integer witnesses, and an independent verification script implemented in Python using only the standard library. The results apply to the specified scalar geometric-tail threshold model. They do not establish global optimality for the original finite-ring matrix reconstruction problem. This manuscript is released as a research preprint. It has not undergone independent peer review, and priority relative to the complete existing literature has not been established.