Generative Structural Theory v1.110 — Complete Chains for Eleven Fixed Inputs with Certified Local Obstructions
Year: 2026
Type: preprint
Abstract: This version provides explicit complete chains for nine further fixed eighteen-node inputs with certified local obstructions. Together with the two chains established in v1.109, complete chains are now available for eleven of the twelve fixed inputs M1–M12. Versions v1.107 and v1.108 established that each of these twelve inputs has a certified local obstruction at leaf 3, side 1, with D empty: an undecided stop of the forcing procedure with no feasible set, first detected at depth six. Version v1.109 nevertheless exhibited complete chains for M1 and M12 through alternative choices. The present version applies the same limited construction to the remaining ten inputs, with the first-step order, witness rules and resource limits fixed before execution. For M2, M3 and M5–M11, explicit chains of 13–15 relations are given. Each chain consists of minimal strongly connected relations from the original relation h to the bidirected tree relation P_S, remains inside h ∪ P_S, and deletes at most two ordered pairs per transition. Each new chain passes two checkers, including a separate implementation that reads the original input records, together with first-step comparisons and controls. These checks do not constitute third-party verification. The eleven chains are compared by their operations and states. One operation occurs in all eleven chains, and six further operations occur in nine or ten. These are observations about the displayed chains; no general necessary or sufficient condition is inferred. For M4, no complete chain has been obtained by the specified trials. All five fixed first steps ended without a chain within the permitted operations. At one specified reduced state and choice, with D empty, all 560 three-pair subsets of a sixteen-pair residual set failed the witness test. Combined with the preceding check for added sets of at most two pairs, this excludes witnesses with an added set C of at most three pairs at that state and choice only. These trials do not establish M4 as a counterexample to (Q1). The supplement gives the ten inputs, all nine new chains and their transitions, check results, comparison tables and the M4 records. The accompanying ZIP includes a verification appendix with input data, chain files, first-step witnesses, checking scripts and reproduction commands. The checking scripts require only Python 3 and its standard library. The general question (Q1) remains open. No claim is made that these chains are shortest or that a deletion budget of two is necessary.
Author: Okino, Koji
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225830509
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