Generative Structural Theory v1.7: Zero-Boundary Events and the Extension to the Full Real Threshold Line

Abstract

Generative Structural Theory v1.7 extends the threshold domain from the non-negative half-line to the full real line while retaining non-negative relation strengths. For a finite sector set and a relation-strength function R: Π → ℝ≥0, the strict threshold cut is Eε = {(P,Q) ∈ Π : R(P,Q) > ε}. At every negative threshold, the cut is the full off-diagonal relation Π. Crossing zero removes exactly the zero-strength layer D₀: E₀₋ = Π, E₀ = Π \ D₀. Consequently, zero carries a threshold-boundary event if and only if D₀ is non-empty. The paper proves the complete zero-boundary event spectrum Σzero = {111, 110, 101, 100, 010} = Σev \ {000}. The states 011 and 001 are excluded by the inherited implication E5 ⇒ E2. The state 000 is excluded by a zero-boundary-specific result: if reachability and anchored transitivity both remain unchanged after leaving the full cut, then D₀ must be empty, so no zero-boundary event exists. The extension introduces no new cut-flag states: the realizable cut-state spectrum over the full real threshold line is identical to that over the non-negative threshold line. The paper also proves that the zero-boundary state depends only on the zero layer, not on the positive critical values. Two inequivalent notions of minimum witness complexity are introduced: an event-centred order minimizing |D₀| and a state-centred order minimizing |E₀|. Their coincidence is characterized exactly, and the absence of a single canonical minimum witness is made explicit. Finally, the post-boundary cut state is shown to determine the zero-boundary event state only partially. The resulting correspondence is surjective but not single-valued: the cut states 0010 and 0000 each admit the two event states {110, 010}, distinguished precisely by the reachability-change flag E2. The analytic results are accompanied by explicit realizability certificates, finite censuses through five sectors, independent recomputation paths, document and rendering guards, and mutation-tested verification programs. The frozen deposit contains the paper, source files, certificate data, verification code, manifest, archive, and SHA-256 sidecar. Verification archive SHA-256: e3933d0436a5d6fb2e2f17e04a69aa5425aa4b24fd7a3956204f5bceb0819057