Paper IV: The Planck Constant as a Cosmological Relic: ℏ(t) ∝H(t)3 in the Pre-Electroweak Epoch, Electroweak Freezing, and the Heisenberg Uncertainty Principle as a Property

The reduced Planck constant ℏscales as ℏ(t) ∝H(t)3 in the pre-electroweak epoch. This arises from the quantisation condition on the brane oscillation modes: the quantum of action is the minimum phase-space area of one brane oscillation cycle, which scales as ℓP mP c = ℏwhere ℓP = p ℏG/c3 is constant (since ℏ/c3 = const is protected by the G2 brane geometry) and mP = p ℏc/G and c both vary with H. The electroweak freezing theorem of Paper III shows that ℏdecouples from H at TEW ∼160 GeV and is constant throughout all astronomical epochs. The Heisenberg uncertainty principle ∆x ∆p ≥ℏ0/2 with frozen ℏ0 is the minimum phase-space area of the frozen brane fabric a property of the set G2-symmetric viscoelastic solid, not a cosmologically varying constraint. The combination ℏ(t)c(t) ∝H(t)4 is the pre-EW brane acoustic action. After freezing, ℏ0c0 is the xed acoustic action of the frozen brane. In the pre-EW epoch, the product ℓP = p ℏG/c3 is exactly constant throughout the Planck length is the one combination protected at all epochs by the G2 geometry. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.