Hyperbrane Relativity V41.4: Mathematical Companion — V35 Madelung Tier Integration

This Mathematical Companion paper documents the V35 Madelung-fluid foundation ofHyperbrane Relativity (HBR), as integrated into the V41.4 Main Paper. The V35 sprint(2026-05-09) established the quantum-fluid mathematical basis on which V41.4's geometricarticulations rest. Six Tier sections are presented: Tier 1 (Madelung foundation): The bulk wave function ψ_bulk = √ρ_bulk · exp(iS_bulk/ℏ)introduces the bulk-as-fluid-medium reading. The bulk fluid energy density ε_bulk =ρ_bulk · c² articulates Birth Energy (バースエネルギー) as the static distribution,with c² as the natural fluid energy density scale rather than a creation barrier(E=mc² trap-decoded reading). Tier 2 (Force unification): All classical forces decompose into tension repulsion(f_tension = -c²·∇ρ_bulk) and W-flow drag (f_drag = -η_flow·(u-v_W)), both sourcedfrom the same Madelung field. The HBR mass identity m_i ≡ m_g = ∫ρ_bulk·d³x dissolvesthe additive-shift problem of the V40 T2 EP attempt. Tier 3 (Density trichotomy): Gross-Pitaevskii nonlinearity yields three local fluid-density classes (vacuum / matter / flow), re-read as a temporal phase cycle in V41.3bicone scale transition reading. Tier 4 (Brane projection kernel + Eternal Generation): The OCS weight functionW_OCS(Δw) is derived as a Gaussian-Dirichlet kernel from Madelung coherence + biconecycle cutoff, reducing the V40 4-free-function structure to 2 V35-derived parameters.The Eternal Generation source Γ(x,t) = Γ_0·Θ[|ψ_bulk|²-ρ_crit]·|ψ_bulk|² eliminatesthe need for a Big Bang singularity; Birth Energy is already distributed at c² scalethroughout the bulk. Tier 5 (Aggregate vortex + SPARC tanh + Friedmann β H^4): Galaxy-scale collectiveMadelung fluid coherence yields the SPARC tanh rotation-curve form recovered from theV32 Paper A LOCKED ensemble fit (175/171 galaxies, BTFR α = 3.56 ± 0.11, χ²_ν = 1.29).The Friedmann β H^4 correction articulates Big Bang elimination at cosmological scale. Tier 6 (AHS Euclidean twistor, optional): The Atiyah-Hitchin-Singer twistor formalcorrespondence with bulk Euclidean signature (0,4) provides a programme outline forHBR-internal QFT (V42+ scope). The mathematical content is integrated into V41.4 Main Paper through ~110 cross-references (45 section + 65 equation citations), articulating the V35 sprint resultas the quantum-fluid foundation of the V41.4 geometric framework. V41.4 Companion is intended for journal submission to JCAP / CQG / Foundations ofPhysics or as a Zenodo mathematical record supporting the Main Paper.