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 pᵥoid = tanh² (1/√2) + rₜet²·rₒct/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·rₒct. The BPS kink evaluated at x = rₒct gives tanh (rₒct/ξ) = tanh (1/√2), with rₒct cancelling identically. The floor pfloor = 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·ΨA·ΨB between two independently saturated condensates (O (α₀), not O (α₀²) ). The BPS amplitude ΨA = tanh (1/√2) cancels exactly in the product: 2· (rₜet·rₒct/π) ·ΨA·rₜet/ (2π·ΨA) = rₜet²·rₒct/π² The NLO correction is purely geometric. Open — Appendix G: Why does ΨB = rₜet/ (2π·ΨA) 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 (rₜet). 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: rₒct = (√2−1) /2 = 0. 207106781. . . rₜet = (√6−2) /4 = 0. 112372435. . . α₀ = rₜet·rₒct/π = 0. 007408055. . . ξBCT = √2·rₒct = 0. 292893218. . . Part of the BCT Superfluid Lattice Model programme. 229+ Letters, 278+ Predictions. Zero free parameters.