Generative Structural Theory v1.106 — Depth-Five Probing at J17a and the Remaining Stops of G
Year: 2026
Type: preprint
Abstract: This version establishes that the least detecting depth at the undecided stop of J17a is exactly five, and that five is the common least detecting depth for the undecided stops without a feasible set in the specified family G of v1.105, with D = ∅ (no additional core deletion). Version v1.105 proved that J17a, a seventeen-node input, has an undecided stop with no feasible set that remains undetected by depth-four probing. The present version supplies a depth-five AND–OR certificate with 39 nodes and 86 candidate entries. A soundness lemma, proved by induction on the certificate depth, shows that acceptance of the certificate implies an “impossible” decision by the probing rule. Together with the depth-four result and depth monotonicity, this establishes that the least detecting depth at this stop is exactly five. The investigation also covers all 201 stops of G that remained undetected at depth four. Each has an accepted depth-five certificate containing 18–48 nodes. The final comparison with the fixed target list confirms complete coverage, with no missing, duplicate or extra targets. Deliberately corrupted certificates were rejected by the prescribed controls. Of the 339,455 undecided stops of G with no feasible set, the other 339,254 were detected at depths one through four by finite check 49 of v1.105. Depth monotonicity extends those results to depth five. Thus depth five detects every such stop in G, while J17a shows that depth four does not suffice. The evidence is distinguished explicitly: the 339,254 cases rely on the earlier finite-check counts and monotonicity, whereas the remaining 201 are supported by individual certificates. All 201 certificates passed the checker and a rerun of the same checker in another environment. This is not verification by a separately implemented checker. The distribution includes the paper, supplement, all 201 certificates, the target list, the checker, a batch verification script, expected results, instructions and SHA-256 manifests. The conclusions are restricted to the specified family G. They do not establish sufficiency of depth five for arbitrary inputs or all seventeen-node inputs, resolve the general common-depth question (Q33), or assert that J17a is minimal in vertex count or counterexample size.
Author: Okino, Koji
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225661416