Conditional Queries with Finite Historical States and Periodic Replacement

Abstract

We study a deterministic conditional-query problem in which a sampled index is encoded before a later query, the current index may change, and every surviving input-dependent distinction belongs to a finite historical alphabet. On a logarithmic probability grid with \(P\) points, displacement bound \(A\), spacing \(h\), and at most \(K\) historical states, we obtain the exact worst Euclidean query error \(\tanh(hD_*/4)/\sqrt2\). The optimal expanded-fiber diameter is \(D_*=P-1\) when \(2A\ge P-1\), and \(D_*=2A+\lceil(P-2A)/K\rceil-1\) otherwise. Consecutive encoder blocks and query-dependent midpoint responses attain the bound. Rational geometric parameters give a bit-machine implementation with separately counted workspace and output-precision error. A two-slot periodic construction retains bounded-age codes under correct fresh acquisition, bounded acknowledged service and initialized-only scoring. After a finite attack on stored symbols or forward payload fields, clean overwrite restores current-task accuracy while acquisition, executable encoder/decoder and control remain correct. A separate one-hop replicated crash-erasure model admits an exact period-endpoint law under all-copy lifetime, full-drain and no-side-channel contracts. Quadratic objectives transfer the query law to optimizer-estimation error. A normalized family expresses the conditional error in terms of retained mass. The results have distinct information, timing and fault assumptions; they do not establish a general network resource optimum or Byzantine source/control recovery.