Generative Structural Theory v1.112 — Complete Chains for All Twelve Fixed Inputs under Deletion Budget Two
Year: 2026
Type: preprint
Abstract: This version provides an explicit complete chain for the remaining fixed eighteen-node input M4 under deletion budget two. Together with the chains established in v1.109 and v1.110, complete chains are now available for all twelve fixed inputs M1–M12 of v1.108. The main question (Q1) asks whether, for a minimal strongly connected relation h and a spanning tree S, there is a chain from h to P_S, the relation containing both orientations of every edge of S, such that every intermediate relation is inclusion-minimal strongly connected, lies within h ∪ P_S, and each transition deletes at most two pairs. For M4, the supplied chain consists of 13 relations and 12 transitions. Its successive deletion counts are 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2. The chain is verified by the drafting-side checkers and by a separately implemented direct checker. These are computational checks of an explicit finite object. The result is consistent with the local nonexistence established in v1.111. After the first step #4, the chain uses another choice: leaf 12, side 1, with the additional deletion D = {(0, 7)}. The previously established local obstruction remains valid. The supplement records the current classification of 72 choices at that state: 22 not applicable, 22 with no feasible set detected by the forcing procedure, one with no feasible set established in v1.111, 14 with a confirmed step, and 13 unresolved. Five newly confirmed steps were checked by the drafting side only; the separate direct verification concerns the M4 chain and does not individually verify those five steps. The general question (Q1) remains open. Neither the necessity of an additional deletion for every complete chain of M4 nor an exhaustive classification of the unresolved choices is claimed. The accompanying ZIP contains the manuscript and supplement in text form, the M4 chain and its input data, three verification programs, reproduction instructions, and SHA-256 manifests. The manuscript and supplement are also supplied as separate PDFs. AI-assisted tools were used in the research, verification, and preparation of the manuscript. The author is responsible for the claims and the decision to publish.
Author: Okino, Koji
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225807085
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