Generative Structural Theory v1.4: Admissibility–Separation Compatibility and the Mutual-Pair Criterion for Region Inclusion

Generative Structural Theory v1.4 completes the region-inclusion problem left open in GST v1.3. It gives exact structural criteria for the relationship between the admissibility region 𝒜(R) and the Kolmogorov separation region 𝒯₀(R) in finite non-negative relation-strength models. For each threshold ε, the strict cut is defined by Eε = {(P,Q) : R(P,Q) > ε}. The central result is the mutual-pair criterion: under the admissibility condition N4(ε), failure of T₀ separation is equivalent to the existence of distinct sectors P and Q satisfying both (P,Q) ∈ Eε and (Q,P) ∈ Eε. Thus, among admissible cuts, a mutual pair is the exact obstruction to separation. The paper proves the following exact inclusion criteria: • 𝒜(R) ⊆ 𝒯₀(R) if and only if no admissible cut contains a mutual pair. • 𝒯₀(R) ⊆ 𝒜(R) if and only if every acyclic cut is transitive, equivalently, no acyclic cut contains an open two-path. These two criteria yield a complete four-way taxonomy: equality, either strict inclusion, or incomparability. Minimal finite witnesses are established, including the asymmetry that 𝒜(R) ⊊ 𝒯₀(R) requires at least three sectors, whereas 𝒯₀(R) ⊊ 𝒜(R) already occurs with two. Further results show that admissibility together with T₀ separation is equivalent to ordinary transitivity of the strict cut; no bounded initial collection of threshold intervals suffices for deciding the forward inclusion; the interval-indexed taxonomy depends only on the ordered level-set pattern and zero layer; and the four inclusion states have sharply different behavior under arbitrarily small perturbations. The deposit includes the complete preprint, source files, exact finite-model tables, verification programs, frozen build auxiliaries, a GST v1.3 reference package, and a SHA-256 manifest. All finite census claims and named witnesses are independently machine-checked. These computations support reproducibility and error detection; the general mathematical conclusions rest on proofs rather than finite enumeration. Public archive SHA-256: ec246734355a03643aaa6c2c5eea0d39a5fc0ca630cea5c896dbef2fac98b1f7 Author: Koji Okino, SD Lab LLCVersion: 1.4Resource type: Publication — PreprintLanguage: English