Generative Structural Theory v1.3: Zero-Strength Relations, Effective Critical Sets, and the Asymmetry of Threshold and Perturbation Stability

Generative Structural Theory v1.3 extends GST v1.2.1 by removing the strict-positivity assumption on off-diagonal relation strengths and admitting non-empty zero-strength layers. The central result is a separation between two critical sets. The effective critical set C_thr(R) = C(R) ∩ R_{>0} governs changes produced by varying the threshold, whereas the full critical image C(R) governs perturbations of the relation strength. When zero belongs to C(R), it produces no threshold event but remains perturbatively critical. This yields a precise asymmetry between threshold stability and perturbation stability. The paper proves an exact decomposition of the threshold range over the effective critical set, corrects the candidate-family count for zero-strength models, distinguishes the layer and loss readings of the edge-event flag, classifies perturbations supported on the zero layer, and gives an exact criterion for admissibility of the first threshold interval. The analysis also identifies which results inherited from GST v1.2.1 survive unchanged, which become model-dependent, and which return under explicit additional hypotheses. In particular, the component bound for admissible regions survives through a new combinatorial proof that does not assume admissibility at the first interval. Five finite zero-strength models provide counterexamples, witnesses, and concrete illustrations. All published model quantities are recomputed in exact rational arithmetic by a primary probe and independently checked using boolean matrices, Warshall transitive closure, order-ideal recursion, and bisection over critical values. The verification code also compares its results directly with the values printed in the paper source. This record contains: the complete 54-page preprint; the LaTeX source; six vector PDF figures and six fixed-resolution PNG counterparts; figure source files and shared styling definitions; primary and independent verification scripts; byte-identical expected computational output; a README, license, and internal SHA-256 manifest. The deposited paper contains no embedded raster figures and no Type 3 fonts. All figures are incorporated as vector PDF assets. The finite computations support transcription, arithmetic, and reproducibility checks; the mathematical claims are established by proofs in the paper. Version relationship: GST v1.3 is a new version of GST v1.2.1 (DOI: 10.5281/zenodo.22209196), which is the canonical corrected release of GST v1.2 (DOI: 10.5281/zenodo.22202708). GST v1.3 is an extension, not a correction of GST v1.2.1. External SHA-256 of the complete reproducibility deposit: ee93b1dea1048ba7b9d5124cfa7ff0fcd3b347edd02a41ef4a3a85e736f2ee34 GST_v1.3_deposit.zip Author: Koji Okino, SD Lab LLCORCID: 0009-0003-9273-9813License: Creative Commons Attribution 4.0 International (CC BY 4.0)