The c = -2 Foundation of Constraint Geometry Deriving the Cosmic Budget from Logarithmic Conformal Field Theory

Abstract

Abstract We establish that the Constraint Geometry Framework (CGF) admits a precise mathematical realization as a c = -2 Logarithmic Conformal Field Theory (LCFT) embedded in three-dimensional sphere packing geometry. This identification is not phenomenological fitting but follows from structural necessity: CGF describes constraint accumulation (negative central charge), coupled primary-partner dynamics (Jordan blocks), and threshold-dominated physics (percolation universality class). The vacuum lattice is derived from CGF’s n = 14 confinement threshold, which uniquely selects the Kelvin cell (truncated octahedron with 14 faces)-the Wigner-Seitz cell of the BCC lattice. The correspondence yields a complete operator dictionary: every entry in the c = -2 Kac table maps to CGF physics, from the Jordan vacuum pair (Δ = 0) through chiral shear (Δ = 1/8) to matter operators (Δ = 15/8). The fusion algebra validates cosmic history-inflation, radiation, and matter formation respect the constraint grammar, with the critical result that matter requires chiral precursors (No Spin → No Shear → No Matter). From LCFT structure, we derive the chiral asymmetry δ = 1/96 as the boundary operator weight screened by the kissing number, and the Hubble tension ΔH/H = 1/12 as the Jordan coupling amplified by BCC coordination. Matter regions crystallize as C15 Laves phase, yielding the dark matter to baryon ratio Ω_DM/Ω_b = 5.31. A competing modular derivation (Weber functions at τ = i) suggests ε = 2^(1/8) = 1.0905, yielding Ω_DM/Ω_b = 5.34. Both predictions lie within current uncertainty (observed: ~5.4); we present them honestly for observational adjudication. The entire cosmic budget-dark energy (0.1%), dark matter ratio (0.2–1.7%), Hubble tension (0.04%), baryon fraction (0.4%)-emerges from LCFT operator spectrum and crystallographic integers with zero free parameters. The Jordan block dynamics admits two interpretations: Inherited Torsion (relaxation toward equilibrium) and External Pull (linear growth toward yield). DESI DR2 dark energy measurements [13] show w₀ > -1 and w_a < 0, favoring the relaxation model; a preliminary tension slope analysis provides additional suggestive evidence. The framework is generative: the Ramanujan connection reveals that hyperconvergent 1/π series encode the Jordan block structure of the vacuum.

Keywords

Affiliated Institutions

• Laboratoire de Physique Nucléaire et de Hautes Énergies FR
• CEA Paris-Saclay FR
• Institut de Recherche sur les Lois Fondamentales de l'Univers FR
• Commissariat à l'Énergie Atomique et aux Énergies Alternatives FR
• Institut d'Astrophysique de Paris FR

Related Publications