An Exact Eight-Probe Reconstruction Certificate for Shifted Quadratic Threshold Tests Modulo 81

We establish that exactly eight distinct fixed shifted-quadratic binary threshold tests are necessary and sufficient to identify every residue modulo 81. For each shift (t\in\mathbb{Z}/81\mathbb{Z}), the binary observation is defined by [ b_t(x)=\mathbf{1}{(x+t)^2\bmod81<27}. ] An explicit eight-shift reconstruction schedule is [ T={0,2,7,14,17,21,31,73}, ] which produces 81 distinct binary signatures and therefore identifies every possible residue. Optimality is established through the information-theoretic lower bound and two exhaustive finite search algorithms. The first algorithm uses pair-separation constraints and branch-and-bound, while the second uses canonical subset enumeration and signature partitions. Translation symmetry permits normalization to schedules containing shift zero. Both algorithms independently exclude every seven-shift solution. The accompanying reproducibility package includes Python source code using only the standard library, computational verification records, and documentation of the mathematical assumptions and search procedures. The result concerns a specific finite family of nonadaptive binary tests. It does not establish a universal optimality theorem for arbitrary prime powers or error-correcting reconstruction codes. The manuscript includes a discussion of related work on separating systems, minimum test cover, and combinatorial reconstruction. The preprint has not undergone external peer review. Computational verification. The exact eight-probe optimum for the specified finite model is supported by an explicit reconstruction witness and two completed exhaustive computational searches. The accompanying Python implementation and verification records are provided to facilitate independent reproduction. AI assistance statement. OpenAI ChatGPT was used substantially in mathematical exploration, algorithm design, programming, computational certificate development, literature searching, mathematical review, and manuscript drafting. The named author is responsible for reviewing the mathematical arguments, checking the released files, verifying bibliographic references and originality claims, and deciding whether to submit and publish the work. Author approval. GRANTED — author-directed designation for the release candidate. Final responsibility for the publication and approval of the uploaded version remains with the named author. Peer-review status. This manuscript is a preprint and has not undergone independent external peer review. Licensing. The manuscript is distributed under Creative Commons Attribution 4.0 International (CC BY 4.0). The accompanying Python source code is distributed under the MIT License, as specified in the included license file.