Generative Structural Theory v1.105 — Depth-Four Probing on Selected Two-Node Extensions of J15a
Year: 2026
Type: preprint
Abstract: This version establishes that depth-four probing does not decide every undecided stop without a feasible set, even when D = ∅ (no additional core deletion). It presents J17a, a seventeen-node input whose forcing procedure stops undecided with no feasible set and whose depth-four probing result remains “undecided”. Consequently, if a common probing depth d as asked in (Q33) of v1.99 exists, then d ≥ 5. The investigation starts from 93 stops identified in v1.104. Each is detected as impossible at depth four, although some candidate pairs remain outside the excluded set I₄. Their detection structure is examined, and a specified family G is formed by adding one further node to their sixteen-node inputs. The family contains 639,778 labelled pairs and 3,603,330 admissible choices of leaf and side with D = ∅. The exhaustive check of G finds 1,438,361 undecided stops, of which 339,455 have no feasible set. Depths one, two, three and four first detect impossibility at 91,734, 95,731, 64,746 and 87,043 of these stops, respectively. The remaining 201 are not detected at depth four. All 201 were rechecked and confirmed using the earlier implementation from v1.93 and v1.99. For the representative input J17a (candidate 50), Proposition 1 proves both the absence of a feasible set and the failure of depth-four probing to decide impossibility. The infeasibility certificate uses eight cuts and 27 obstructions covering all 576 choices of one candidate from each cut. A directed acyclic witness graph with 103 nodes establishes the depth-four result. The supporting finite lists are supplied in GST_v1.105_proof_supplement.txt. The counts for G and the confirmation of the other 200 candidates are computational findings. The paper does not claim that depth five suffices, that J17a is a smallest counterexample, or that (Q33) is resolved. The distribution ZIP contains the paper in PDF and plain text, the proof supplement, its README, and a SHA-256 manifest. The paper and manifest are also available as separate downloads.
Author: Okino, Koji
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225562570