Generative Structural Theory v1.2.1: Critical Thresholds, Stability Margins, and the Exact Relational Filtration

Generative Structural Theory (GST) v1.2.1 is a corrected release of GST v1.2. Definition 6.3 now states that, when the off-layer margin is +∞, the norm hypothesis of Theorem 6.4(ii) is redundant for the conclusion rather than vacuous. No theorem, proof conclusion, computed value, figure content, phase region, or claim boundary is changed. GST v1.2.1 replaces the sampled threshold grids of GST v1.1 with an exact decomposition of the full threshold range. For a relation-strength functional R with finite, non-empty, strictly positive off-diagonal image, the critical set C(R) partitions [0, ∞) into finitely many left-closed, right-open constancy intervals. On every such interval, the threshold cut, candidate neighbourhoods, direct and transitive preorders, reachability closures, Alexandrov topology, admissibility condition N4, separation properties, and—when the sector set is finite—the loss statistic Lε remain constant. The paper establishes the exact interval-decomposition theorem, one-sided behaviour at critical thresholds, regular and off-layer stability margins, fixed-threshold perturbation invariance, layer-preserving isomorphism of the full filtration, and a classification of critical events. It also determines the exact admissibility and separation regions. The admissibility region is a finite union of half-open components and need not be monotone, whereas the T0 region is a principal upward-closed interval, and the T1 and T2 regions equal the final interval beginning at the maximum critical strength. For the five-sector model inherited from GST v1.1, the complete phase diagram contains ten constancy intervals, ten distinct candidate families, and six distinct topologies. The exact regions are A(R) = [0, 0.1) ∪ [0.8, ∞), T0(R) = [0.7, ∞), and T1(R) = T2(R) = [0.9, ∞). Additional three-sector and two-sector models provide witnesses for the component-count bound and the one-event separation case. The reproducibility deposit includes the canonical paper PDF and TeX source, six figures in vector and raster formats with TikZ sources, two independently implemented exact-rational verification scripts, fixed expected output, a SHA-256 manifest, and clean-room reproduction instructions. Computational checks verify the tabulated finite-model values but are not presented as proofs of the mathematical theorems. GST v1.2.1 builds directly on GST v1.1 (DOI: 10.5281/zenodo.22190635) and corrects GST v1.2 (DOI: 10.5281/zenodo.22202708). Zero-strength boundary models, systematic infinite-sector or infinite-critical-image extensions, general attainability of the admissibility-component bound, and non-identity homeomorphism classification across topology blocks remain outside the claims of this release.