Generative Structural Theory v1.5 — Sharp Bounds and Realizability of Distinct Structural Components

Generative Structural Theory v1.5 determines the sharp upper bound and full realizability spectrum for the number of distinct structural components across threshold-constancy intervals in finite non-negative relation-strength models. For a model with m positive critical threshold values, the threshold range is divided into m+1 constancy intervals. The number of structural components is identified with the number of maximal true runs in the corresponding admissibility truth string. The paper proves the sharp, gapless realizability theorem: 𝔠(m) = {1, 2, ..., ⌈(m+1)/2⌉}. Thus, every component count permitted by the inherited upper bound is realized by a finite model. A general construction using independent obstruction-window gadgets is provided. More strongly, a binary string of length m+1 occurs as an admissibility truth string if and only if its final entry is true. For every m ≥ 1, the same complete spectrum is realizable by strictly positive models. The positivity transformation used for this result is stated conditionally: it preserves the full truth string when the initial truth value is already true. A three-sector counterexample shows why this condition cannot be omitted. The case m = 0 is treated separately. The paper also develops an exact decreasing-chain representation of finite models, derives equality conditions and capacity restrictions, and isolates the fixed-sector extremal problem. For a fixed sector count n, exhaustive finite classification gives: Q(2) = 1, Q(3) = 3, Q(4) = 6, Q(5) = 8. In particular, the previously suggested formula based only on the capacity condition n(n−1) > m is refuted by the case n = 5, m = 16: nine components cannot be attained despite the available edge capacity. This non-attainment result is certified by an exhaustive isomorphism-reduced search, while no general formula for Q(n) is asserted. Admissible threshold cuts are additionally identified with preorders after adjoining the diagonal, connecting the finite enumeration with the classical counting of finite topologies. The deposit contains the paper, complete LaTeX sources, explicit extremal models, finite-search certificates, verification and document-guard programs, reference sources from GST v1.3 and v1.4, build materials, a SHA-256 manifest, README, and license information. Reproducibility archive: GST_v1.5_deposit.zip SHA-256: ccb202ea8d2fe24a51251173b8e44698b3fe2b31c179f0f7640d4d2f546f91ae This work is formal and mathematical. It makes no physical, temporal, empirical, or cosmological claim.