X-Space Theory: (π−√3)/2 as a Geometric Constant of the A₂ Lattice — Blaschke–Lebesgue Identity, Reuleaux Duality, and Symmetry Breaking in Hexagonal Superconductors

This preprint investigates the numerical constant (π−√3)/2 ≈ 0.7048, establishing its identity in five independent geometric domains: Reuleaux triangle area, Blaschke–Lebesgue minimum-area coefficient, hexagonal close-packing deficit, QFT path-integral measure on the 120° lattice metric, and the Fourier spectrum of the Reuleaux triangle. A single mechanism underlies all five appearances: circular geometry (π/2) minus hexagonal projection loss (√3/2 = cos 30°). The 120° branching angle is proved geometrically necessary via the Maxwell rigidity theorem and the Steiner minimum tree theorem. A physical prediction (P1) is derived: in hexagonal-lattice superconductors, rotational symmetry of the superconducting order breaks from C₆ to C₂ at the Lifshitz transition pressure. Qualitative agreement with CsV₃Sb₅ high-pressure measurements is documented. All claims are graded by rigour. Gaps and negative results are explicitly itemised. Keywords: Reuleaux triangle, Blaschke–Lebesgue, A₂ lattice, hexagonal superconductor, symmetry breaking, Lifshitz transition, CsV₃Sb₅, geometric constant, Steiner tree, constant width