Posner molecule Ca₉(PO₄)₆: 18-spin ³¹P qubit register in rhombohedral R3c lattice — E8 Intelligence Research

FINDING: Posner molecule Ca₉(PO₄)₆ — a nuclear-spin qubit lattice proposed by Fisher as the physical substrate for neural quantum processing | MATH: Ca₉(PO₄)₆ has 6 phosphate groups × 3 = 18 ³¹P nuclear spins (I=½) + 9 ⁴⁰Ca²⁺ (I=0, NMR-silent) → 18-spin register; phosphate tetrahedron PO₄³⁻ with P–O ≈ 1.54 Å, O–P–O ≈ 109.5° (tetrahedral); Ca₉(PO₄)₆ unit cell is rhombohedral (R3̄c), a=10.44 Å, c=37.3 Å, Z=6 | CONNECTION: Rhombohedral R3̄c = trigonal crystal system; 6-fold screw axis (6₃) → base-6 / hexagonal close-packing of phosphate tetrahedra around Ca²⁺; tetrahedral angle 109.47° = arccos(−1/3), the canonical sp³/root-system A₃ angle; 18 = 6×3 encodes the 6 phosphate sites × 3 oxygens, echoing hexagonal (6-fold) symmetry | DEPTH: 7 FINDING: Calcium-phosphate solubility product governs both biomineralization (bone/Posner) and pathological calcification in neurons | MATH: Ksp[Ca₅(PO₄)₃OH] ≈ 2.3×10⁻⁵⁹ (hydroxyapatite); Ksp[Ca₃(PO₄)₂] ≈ 2.0×10⁻²⁹; Posner cluster Ca₉(PO₄)₆ is the pre-nu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com