Entangled Phosphorus Spins in Posner Clusters: Minutes‑Long Coherence as a Neuronal Quantum Substrate — E8 Intelligence Research

FINDING: Posner‑cluster Ca₉(PO₄)₆ (hydroxy‑apatite precursor) can host entangled ³¹P nuclear spins that retain coherence > minutes, offering a plausible physical substrate for quantum information processing in neurons. | MATH: • Spin‑½ Hamiltonian for two phosphorus nuclei in a Posner cluster \(H = J\,\mathbf{S}_1\!\cdot\!\mathbf{S}_2\) with exchange coupling \(J\approx 10^{-3}\) J mol⁻¹ (≈ 10 Hz). • Decoherence time \(T_2\) estimated from dipolar relaxation: \(1/T_2 \approx \gamma^2 \hbar^2 \rho\,\langle r^{-6}\rangle\) → \(T_2\sim 10^2\) s. • Entanglement survival probability \(P(t)=\exp(-t/T_2)\). • Posner‑cluster symmetry group \(T_d\) (tetrahedral) yields degenerate nuclear‑spin eigenstates; the singlet subspace dimension = 2⁴/2 = 8, supporting a logical qubit. | CONNECTION: The tetrahedral \(T_d\) symmetry of the Posner cluster maps onto the root system of the \(A_3\) lattice (face‑centered cubic), whose reciprocal lattice is the body‑centered cubic (BCC). The BCC packing Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com