SDT Mathematical Extensions Higher-Order Structures and Cross-Theoretical Interfaces for GD

This document develops the higher-order mathematical structures of the determination dependency graph G_D introduced in Structural Determination Theory (SDT) v6.0, together with the cross-theoretical interfaces that build on G_D's base structure. Working at the level of model theory, category theory, reproducing kernel Hilbert space theory, sheaf cohomology, and Alexandroff topology, it provides the mathematical machinery for which SDT's foundational paper and the companion graph-theoretic and formal-system documents leave room. The principal contributions are: (i) the chain-pair relational geometry of G_D, comprising the antichain spatial structure, predecessor overlap, joint constraint capacity, structural proximity graph, intrinsic light-cone saturation on lightlike pairs, the five-fold tolerance-relation classification of chain pairs, inter-chain information flow as reference-frame invariant, observer-relative simultaneity via common successor, the structural absence of Lorentz-type frame transformations at the meta-level, and the conditional emergence of Lorentz, Galilean, Carrollian, and Conformal synchronization regimes under entity-theoretic axioms (S1, S2, CL1); (ii) confluence predictability for ≺-incomparable chains, with two independent mathematical paths (H2-compatibility reduction; pushout/colimit construction) and Bell correlations as a pre-confluence transient (Section 11.11); (iii) the Jaccard metric on G_D nodes via predecessor cones with Aronszajn–Schölkopf RKHS embedding; (iv) structural entropy from A's monotonic contraction; (v) the token-parameterized time projection interfacing G_D's discrete chain structure with continuum time variables (QM, GR, thermal); (vi) the sheaf-theoretic perspective on the locally single-output subgraph and the multi-product extension to full G_D, with the SDT-KS no-obstruction theorem as the principal sheaf-theoretic result; (vii) the SDT–quantum-mechanics correspondence program, including the four-type classification's quantum-event interpretation, the read-out principle, and the structural alignment with Constructor Theory; (viii) the local-maximum principle for sub-graph identification supporting cross-application structural-pattern recognition. The document is the third in the SDT v6.0 mathematical series; the foundational paper (SDT v6.0), the graph-theoretic formalization (GD v6.0), and the formal system (FS v6.0) are the canonical sources for SDT's main theorems, G_D's graph-theoretic structure, and SDT's proof-theoretic properties respectively. Boundary statements identify what each construction does and does not provide; entity-theoretic content (specific physics, gauge group, particle spectrum, dimension) is consistently distinguished from the meta-structural machinery developed here.