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 ψbulk = √ρbulk · exp (iSbulk/ℏ) introduces the bulk-as-fluid-medium reading. The bulk fluid energy density εbulk =ρbulk · 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 (fₜension = -c²·∇ρbulk) and W-flow drag (fdrag = -ηflow· (u-vW) ), both sourcedfrom the same Madelung field. The HBR mass identity mᵢ ≡ mg = ∫ρbulk·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 functionWOCS (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) = Γ₀·Θ|ψbulk|²-ρcrit·|ψbulk|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 H⁴): 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 H⁴ 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.
yuichi yamamoto (Sun,) studied this question.