Weyl Symmetry Breaking in Posner Molecules Constrains Quantum Processing Viability — E8 Intelligence Research

Abstract

FINDING: The Posner molecule (Ca₉(PO₄)₆) exhibits a dynamical ensemble that is *not* symmetric under the full Weyl group A₃ root system, breaking the naive nuclear-spin singlet subspace symmetry — a direct mathematical constraint on its viability as a biological quantum processor. MATH: - Posner molecule: Ca₉(PO₄)₆ — 9 Ca²⁺ ions, 6 PO₄³⁻ tetrahedra. - Nuclear spin Hilbert space: 9 ³¹P nuclei (I=1/2) → dimension 2⁹ = 512. - Singlet subspace: total spin I_total = 0 → dimension given by Clebsch–Gordan decomposition: dim(singlet) = (1/2⁹) Σ_{k=0}^{9} C(9,k)² = 42 (for 9 spin-1/2). - Weyl group A₃: order |W(A₃)| = 4! = 24; root system has 12 roots, 6 positive roots; Cartan matrix: [[2, -1, 0], [-1, 2, -1], [0, -1, 2]] (type A₃). - Key result (arXiv:2108.08822): The dynamical ensemble — the set of accessible states under the molecule's internal Hamiltonian — does *not* decompose into irreducible representations of the full A₃ symmetry. Specifically, the permutation symmetr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com