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 = w_brane(t)} via a Dirichlet-type condition (Ẋ^W|_Σ = c); E = mc² then arises as a geometric identity from the self-contraction |U_inflow|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 G_μν + β H_μν = 8π T_μν, with weak-field expansion Φ = 1 + Φ_N/c² + β Φ_N²/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:

1. 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.
2. OCS-1: formal Trichotomy theorem (Part 4 §sec:ocs_orthogonality): three-regime classification proved as thm:ocs_trichotomy with formal definition def:ocs_sub_regimes and two corollaries on equivalence-principle compliance and matter-species universality.
3. OCS-2: Part 4 refactor for 3-regime structure: new section sec:ocs_orthogonality_fusion, 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.
4. OCS-3: CN12 entry into core_novelties.tex with cross-references to BBN epoch fusion (Part VIII) and compact-object core fusion (Part VII).
5. 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.
6. OCS-5: CN12 Falsifiability sub-section (Part 4): four explicit falsifiable scenarios that would distinguish the 3-regime structure from a binary OCS dichotomy.
7. 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).
8. 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 c_GW = c_EM 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.
9. K-B2: GW170817 c_GW = c_EM Dirichlet structural identity (new section sec:egr_gw_dispersion_proof in the Extended GR part): rigorous proof that the V31 Dirichlet-type boundary condition Ẋ^W|_Σ = c forces gravitational and electromagnetic signals to share the same gate-inflow speed at any common gate, eliminating the GW170817 fine-tuning concern.
10. K-B3: Bullet Cluster honest status (new sub-section subsec:bullet_cluster_status): 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.
11. 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.
12. K-C2: PPN β as HBR-internal Prediction: explicit boxed equation for δβ_HBR 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.
13. 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).
14. 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).
15. 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:ocs_orthogonality 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 χ²_ν = 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 |δΨ_B| ~ 0.056 rad BNS dephasing, A(r_ph) = 0.333 photon-radius shift, D_rms ~ 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

• HBR_Master.pdf: 327 pages (V31 → V32: +24p), exit 0, undefined references 0
• HBR_Master_JP.pdf: 308 pages (V31 → V32: +23p), exit 0, undefined references 3 (pre-existing only: sec:foundation_w_axis_jp, tab:field_constants, eq:4d_field_eq)
• 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 (HBR_Master.pdf) and Japanese (HBR_Master_JP.pdf) editions.