BCT Session Report: The Void Fraction Derivation — Bogomol'ny Floor and Cross-Term Resonance (8 April 2026)

This session report documents the derivation of both terms in the BCT void fraction p_void = tanh²(1/√2) + r_tet²·r_oct/π² = 0.370974707015939... from first principles. Results are Python-verified to 15 significant figures. Term 1 — Bogomol'ny floor (CLOSED): The BCT ground-state axial ratio c/a = √2 (Appendix D.0) forces the GP coherence length ξ = √2·r_oct. The BPS kink evaluated at x = r_oct gives tanh(r_oct/ξ) = tanh(1/√2), with r_oct cancelling identically. The floor p_floor = tanh²(1/√2) = 0.370709726... follows from geometry alone. Zero free parameters. Term 2 — NLO cross term (DERIVED structurally): The NLO correction is identified as the cross term δp = 2α₀·Ψ_A·Ψ_B between two independently saturated condensates (O(α₀), not O(α₀²)). The BPS amplitude Ψ_A = tanh(1/√2) cancels exactly in the product: 2·(r_tet·r_oct/π)·Ψ_A·r_tet/(2π·Ψ_A) = r_tet²·r_oct/π² The NLO correction is purely geometric. Open — Appendix G: Why does Ψ_B = r_tet/(2π·Ψ_A) specifically? The physical origin of this tet-void condensate value requires the BCT GP action written on the full void network with the correct continuum boundary conditions on ∂B(r_tet). This is the target of Appendix G. The discrete graph Laplacian approach was ruled out by Python (eigenvalues are O(α₀), not π). Canonical constants: r_oct = (√2−1)/2 = 0.207106781... r_tet = (√6−2)/4 = 0.112372435... α₀ = r_tet·r_oct/π = 0.007408055... ξ_BCT = √2·r_oct = 0.292893218... Part of the BCT Superfluid Lattice Model programme. 229+ Letters, 278+ Predictions. Zero free parameters.