Part 27: Falsifiability and Cross-Consistency of Dual–H4 Geometry: Layered Constraints from Cosmic Rays, Gamma Emission, Bulk Gravitational Modes, and Direct Detection

Overview Origin Geometry (OG) proposes that effective spacetime may admit a dual-sector architecture, H₄ ∪ φH₄, in which the visible (H₄) sector and the phase-shifted (φH₄) sector coexist within a shared geometric substrate while remaining strongly misaligned at the level of boundary-supported modes. In this framework, dark-sector phenomenology arises not primarily from a separately postulated population of unknown particles, but from pinned, electromagnetically suppressed, gravitationally active geometric structure in the (φH₄) sector. Parts 21–26 developed this sequence through dark-sector geometry, topological pinning, bulk-mediated relaxation, effective dark collapse, antimatter-like sequestration, WKB-like leakage, and positron-biased mass filtering 1–6. Falsifiability and Cross-Consistency The present Part introduces no new physical mechanism. Its purpose is to formulate the falsifiability and cross-consistency structure of the dual-(H₄) dark-sector branch. A theory that connects dark matter geometry, antimatter-like sequestration, positron leakage, gamma emission, antiproton suppression, direct-detection null results, and bulk gravitational-mode relaxation cannot treat these signals independently. The same topological phase barrier that suppresses visible-sector coupling must also allow rare leakage in extreme-curvature environments. The same WKB mass filtering that favors positron-like leakage must also suppress heavy antimatter. The same dark-sector energy reservoir that remains cosmologically stable must not leak catastrophically into the visible sector. The same bulk-mediated relaxation picture that stabilizes compressed dark-sector structures must not be replaced by dominant electromagnetic discharge. Layered Falsifiability Framework This Part therefore develops a layered falsifiability framework. It distinguishes between: Fatal tests of the dual-sector architecture Fatal tests of the leakage branch Constraints on secondary phenomenological signatures This distinction is essential. The absence of a low-energy antihelium signal, for example, does not falsify the entire dual-(H₄) architecture. A robust detection of large low-energy dark-matter–nucleon scattering incompatible with topological suppression would be much more damaging. Likewise, a failure of positron–black-hole correlation would challenge the leakage mechanism of Parts 25–26, but would not by itself erase the dark-sector interpretation of Parts 21–24. Conclusion The central claim of this Part is that OG becomes scientifically stronger, not weaker, when its failure modes are made explicit. The dual-sector dark-sector branch survives only if it satisfies a coupled constraint network 15–18, 20–22: Topological Suppression + Dark-Sector Stability + WKB-Filtered Leakage + Bulk-Dominant Relaxation + Suppressed Direct Detection If these conditions cannot be maintained simultaneously, the framework must be modified or abandoned. Part 27 therefore converts the dark-sector branch of Origin Geometry from a conceptual proposal into a structured, testable, and potentially falsifiable theoretical program.