Hyperbrane Relativity: Fusion-Contact-Observation OCS Trichotomy and Reviewer-Hardening Second Wave — CN12 Trichotomy Refinement, Theorem 2 Exhausted-Regime Extension (Appendix G), and Devil's Advocate Eight Objections (Master Edition V32)

Hyperbrane Relativity (HBR) is a Foundation-First geometric extension of general relativity (GR) describing the observable universe as a dynamical 3-brane uniformly translating in the −W direction within a flat 4D Euclidean bulk. The empirical 3D light speed c is identified with the gate-inflow speed at the dimensional boundary Σ = w = wbrane (t) via a Dirichlet-type condition (ẊW|Σ = c) ; E = mc² then arises as a geometric identity from the self-contraction |Uinflow|² = c². Lorentz invariance is treated as an emergent brane symmetry, not a fundamental postulate. The framework yields a single-parameter modification β of GR through the field equation Gμν + β Hμν = 8π Tμν, with weak-field expansion Φ = 1 + ΦN/c² + β ΦN²/c⁴ and PPN metric g00 = −1 + 2U − 2β U². V32 promotes the Observation-Contact Separation (OCS) regime structure to a refined 1-scale 2-channel 3-regime theory (CN12 Trichotomy): Fusion (winding-number coincidence at brane overlap), Contact (repulsive touch), and Observation (zero-mode-only when threads are well-separated). V32 NEW CONTENT (over V31) V32 sprint completed in four phases (commits 06eec1b / 069b459 / 5b66f97 / 21755f5) plus Block G final verification, totalling 15 sub-blocks of OCS-Trichotomy refinement and reviewer-hardening second wave: NEW Core Novelty 12 (CN12) — OCS Trichotomy (Fusion-Contact-Observation): the V31 OCS dichotomy (contact / observation) is refined into a 3-regime structure indexed by the same single crossover scale Δw. The Fusion sub-regime corresponds to W-thread winding-number coincidence (full back-action and identity merging) ; Contact corresponds to repulsive touch (incoherent back-action without identity merging) ; Observation corresponds to thread separation L ≫ Δw (zero-mode-only coupling). HBR thereby becomes a 1-scale 2-channel 3-regime theory: one geometric scale (Δw), two coupling channels (force law and quantum measurement), three sub-regimes within the contact channel. OCS-1: formal Trichotomy theorem (Part 4 §sec: ocsₒrthogonality): three-regime classification proved as thm: ocsₜrichotomy with formal definition def: ocsₛubᵣegimes and two corollaries on equivalence-principle compliance and matter-species universality. OCS-2: Part 4 refactor for 3-regime structure: new section sec: ocsₒrthogonalityfusion, principle environment for CN12, and a boxed-result twin pairing CN12 with the V31 CN10 (OCS-Orthogonality) so the relationship between regime structure and matter-orthogonality is explicit. OCS-3: CN12 entry into coreₙovelties. tex with cross-references to BBN epoch fusion (Part VIII) and compact-object core fusion (Part VII). OCS-4: BBN epoch fusion + compact object core fusion embedded into Part VIII (early universe) and Part VII (compact objects), respectively, so the Fusion sub-regime appears at both ends of the cosmological history. OCS-5: CN12 Falsifiability sub-section (Part 4): four explicit falsifiable scenarios that would distinguish the 3-regime structure from a binary OCS dichotomy. NEW Appendix G — Theorem 2 Exhausted-Regime Extension (~506 lines EN / ~458 lines JP, labelled Appendix E in the TOC by appendix counter): partial extension of the V31 Theorem 2 (Φ > 0 everywhere, no horizons) to the exhausted regime where matter inflow has effectively ceased. Theorem G. 1 with explicit T* time-scale and four-step proof using the Hopf strong maximum principle on the matter-sustained → exhausted boundary, addressing Kurando-kun review B-4 (honest framing of matter-sustained vs exhausted regime). Devil's Advocate Part: 5 → 8 objections: three new pre-emptive objections with detailed replies — (Obj. 6) GW170817 multi-messenger speed equality requires fine-tuning, replied via the cGW = cEM Dirichlet structural identity; (Obj. 7) Bullet Cluster requires particulate dark matter, replied via the honest Paper C-mini commitment; (Obj. 8) Theorem 2 fails in the exhausted regime, replied via the new Appendix G partial extension. K-B2: GW170817 cGW = cEM Dirichlet structural identity (new section sec: egrgwdispersionₚroof in the Extended GR part): rigorous proof that the V31 Dirichlet-type boundary condition ẊW|Σ = c forces gravitational and electromagnetic signals to share the same gate-inflow speed at any common gate, eliminating the GW170817 fine-tuning concern. K-B3: Bullet Cluster honest status (new sub-section subsec: bulletclusterₛtatus): honest acknowledgment that Bullet Cluster lensing/X-ray offsets are not yet derived in HBR, with explicit Paper C-mini commitment for the dedicated derivation. K-C1: f (w) ansatz Devil's Advocate Obj. 2 reply expansion: Obj. 2 reply now lists explicit candidate f (w) families (exponential, power-law, hybrid) and the observational handles that distinguish them, rebutting the "single function smuggling all of cosmology" criticism. K-C2: PPN β as HBR-internal Prediction: explicit boxed equation for δβHBR as a geometrically constrained quantity (not a free parameter), with the Cassini bound |β − 1| < 10⁻⁴ entering as an HBR-internal prediction that happens to be satisfied, not as an external fit. K-C3: SPARC vs alternative DM benchmarks preview: side-by-side benchmarks against Einasto, SIDM, and DC14 halo profiles (Paper A submission deferred for the full comparison). K-C4: Vortex C/r² constraints: sub-millimetre and atomic constraints on the C/r² vortex correction give an upper bound Δw ≲ 10⁻¹⁵ m on the brane thickness (one-scale theory falsifiability sharpened). K-D: Devil's Advocate D-1〜D-5 cleanup: editorial polish across the V31 Devil's Advocate part for V32 publication readiness. LOCKED canonical values (preserved through V32) V30 §1. 5 7 LOCKED equations (eq: foundation_*) intact V31 §sec: ocsₒrthogonality structure intact (OCS-2 refactor adds 3-regime layer without modifying the V31 statements) V31 Appendix F (Theorem 2 matter-sustained Hopf proof) untouched; new Appendix G is a separate exhausted-regime extension Theorem 1, Theorem 2 statement bodies untouched SPARC χ²ν = 1. 36 across 171 galaxies, BTFR slope α = 3. 56 ± 0. 11 (Paper A LOCKED, 15000-step MCMC) Cassini PPN |β − 1| < 10⁻⁴ (Yuichi 2026 LOCKED) Paper B Phase B-0 LOCKED 12 quantities including |δΨB| ~ 0. 056 rad BNS dephasing, A (rph) = 0. 333 photon-radius shift, Drms ~ 0. 4–0. 5 inner-shadow dispersion, NICER QPO null at 10⁻⁸ across nine mass decades Phase B-2 Sprint 1 NICER pipeline files (calculations/nicer_*) preserved verbatim (0 lines diff) Build state HBRMaster. pdf: 327 pages (V31 → V32: +24p), exit 0, undefined references 0 HBRMasterJP. pdf: 308 pages (V31 → V32: +23p), exit 0, undefined references 3 (pre-existing only: sec: foundationwₐxisⱼp, tab: fieldconstants, eq: 4dfieldₑq) Language ruleset compliance: 0 forbidden-phrase hits in semantic positions ('drift' / 'slowly' appear only in legitimate physics descriptions of astrophysical phenomena and Le Sage-mechanism refutations). EN/JP mirror parity: Devil's Advocate part 8 objections / 8 objections, new Appendix G EN / JP both present. Concept-DOI chain V28 (10. 5281/zenodo. 19869381, 2026-04-29) → V29 (10. 5281/zenodo. 20021551, 2026-05-04, withdrawn 2026-05-05) → V29. 1 (10. 5281/zenodo. 20028176, 2026-05-05) → V29. 2 (10. 5281/zenodo. 20031326, 2026-05-05) → V30 (10. 5281/zenodo. 20045184, 2026-05-06) → V31 (10. 5281/zenodo. 20054875, 2026-05-07) → V32 (this version). V32. 1 micro-patches (this revision): SPARC 175/171 galaxy unification (Devil's Advocate Obj. 7 reply now consistently cites the 171-galaxy quality-cut canonical run from the 175-galaxy Lelli et al. 2016 published catalog), external-readable clarification of the internal "LOCKED" terminology (added to glossary, EN+JP), and explicit identity disclosure of the AI-assisted internal "Kurando-kun" review series (added to glossary EN+JP and acknowledgments EN). These three micro-patches address Priority B-7/B-8/B-9 of the V32 Kurando-kun review and are honest disclosures that do not modify any LOCKED canonical numerical values. Released under CC BY 4. 0. Includes both English (HBRMaster. pdf) and Japanese (HBRMasterJP. pdf) editions.