P27: The Planck Constant: Derivation from Condensation Geometry and Its Consequences Across Quantum Mechanics, Planck Units, and the Fine Structure Constant

The Big Flare-Up Theory (BFUT) Paper 16 derives the reduced Planck constant from the condensation geometry of the first stable substrate excitation as ħ = m_p · c · r_p / (π · R₀), where R₀ = 1.27348 is obtained as the energy minimum of the P16 free-energy functional with all coefficients derived from first principles. This paper systematically substitutes the BFUT expression for ħ into the major formulas of quantum mechanics. The Compton and de Broglie wavelengths, uncertainty principle, quantum tunnelling, harmonic oscillator, angular momentum quantisation, and Schrödinger kinetic operator are all shown to follow directly from substrate condensation geometry. The spin-statistics theorem is derived from the 720° versus 360° embedding topology of the condensation. All Planck units are obtained as derived quantities with 0.0003% agreement. A structural cross-check between the P16 derivation of ħ and the P19 derivation of the fine structure constant α confirms the value of R₀ to 0.0007% from two independent routes — one purely geometric and the other purely empirical. The vacuum energy discrepancy is resolved by recognising that zero-point energy is a property of organised condensations rather than of empty field modes, yielding ρ_vac = ρ_s · c².