Generative Structural Theory v1.111 — Certified Local Infeasibility at a Residual State of M4
Year: 2026
Type: preprint
Abstract: This version proves local infeasibility at a fixed residual state of the eighteen-node input M4. The first step #4 reaches a saved state on seventeen nodes with nineteen pairs. At this state, the choice of leaf 2, side 1, and D = ∅ leaves the forcing procedure undecided. The present result establishes that no feasible set exists for this choice, regardless of the size of the added set, and hence no corresponding one-sided step exists. The main result has a direct proof that requires no exhaustive enumeration. On the sixteen-node set V′ obtained by removing leaf 2, the only pairs entering W = {0, 4, 7, 8} from outside in H ∪ {v} ∪ U are (10, 0) and (15, 0). For any B ⊆ U making H ∪ {v} ∪ B strongly connected, a path from 9 first enters W at 0. Combining its initial segment with the fixed route 8 → 6 → 1 → 9 gives a route from 8 to 0 avoiding (8, 0). Thus every strongly connecting candidate bypasses this pair of H and fails the feasibility condition. A separate exhaustive computation tests all 2^16 = 65,536 sets M ∪ C with C ⊆ R. Of these, 18,944 satisfy strong connectivity and 8,512 bypass no pair; none satisfies both conditions. Exactly five inclusion-minimal sets C satisfy strong connectivity, and each bypasses (8, 0). The enumeration establishes the completeness of this five-set list; the direct impossibility proof does not depend on that list. Two separate implementations developed by the same drafting side agree on every candidate, including the connectivity result, bypassed pairs, and final verdict. Positive and deliberately modified controls are included. The distributed verification code was also rerun in another environment, reproducing the results and the SHA-256 hashes of the candidate tables, states, and minimal-set records. This rerun uses the same supplied code and is distinct from verification by a new implementation or formal verification of the proof. The conclusion concerns only this fixed state and the specified choice with D = ∅. It does not establish the absence of a complete chain for M4 or provide a counterexample to the main question (Q1). Complete chains remain established for eleven of the twelve fixed inputs M1–M12; M4 as a whole and (Q1) remain unresolved. The release contains the main paper, a detailed supplement, and a verification appendix with input records, executable code, reproduction instructions, and checksums.
Author: Okino, Koji
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225784424
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