Conservation, Canonical Realization, Equilibrium, and Smooth Hybrid Geometry in Quantum-Operational Geometry

This paper develops a staged informational-to-geometric construction in which propagation geometry emerges from a sequence of informational principles and derived structures. Beginning with regular realization geometry inherited from earlier work, the framework introduces Structural Quantum Informational Identity (QII), Least Additional Structure (LAS), and the Unified Provenance–Coherence Principle (UPCP) to construct an admissible continuation bundle, canonical realization structure, and intrinsic orientation field.

The synthesis of these ingredients yields the Quantum Horizon, identified as an informational-equilibrium separatrix and primary balance manifold. A response-threshold propagation architecture is then developed through the Response-Threshold Propagation Principle (RTP), the Response-Capacity Saturation Principle (RCSP), and the Boundary Emergence Theorem (BET). These results culminate in the Horizon Response-Capacity Equilibrium Law (HRCEL), which characterizes propagation through the balance between normalized informational advance and transverse response burden.

Version 5 completes a major derivational revision of Paper 4. PC4 is derived from independently supplied structural restoration/severity data, Restoration-Monotone Survival, survival-set comparability, linear survival-order representation, and smooth local integration. Pointwise RTP is derived from the qualitative propagation structure together with the generalized speed-limit resource architecture, Continuation-Speed Correspondence, physical realizability, and the Finite Process Resource Principle. SIA is then derived under Regular Local Continuation from pointwise HRCEL uniqueness, gluing, and the HRCEL Smoothing Law. The revision also strengthens the associated uniqueness, regularity, rank, and claim-boundary arguments and separates optional Cartan-compatible assembly from the direct smoothing chain.

Together, these results yield a smooth positive response-capacity field and support the construction of a global smooth HRCEL geometry. The resulting framework includes a Lorentz-signature propagation form, smooth cone and null-boundary structures, a normalized boundary sphere bundle, a smooth threshold graph, and a canonical bridge representative satisfying Strong Smooth Regular Representability (SSR).