What question did this study set out to answer?

The aim is to show that FCC lattice geometry arises from the Pauli-Clifford algebra and analyze its implications in particle physics.

March 27, 2026Open Access

FCC Lattice Geometry and Particle Mode Decomposition from the SU(2) Pauli-Clifford Algebra

Key Points

The aim is to show that FCC lattice geometry arises from the Pauli-Clifford algebra and analyze its implications in particle physics.
Demonstrated geometric relationships using numerical verification.
Applied the Cauchy elastic relation to derive speed ratios.
Mapped particle modes to Clifford algebra decompositions.
Related lattice orientations to idempotents of the algebra.
Examined implications of the Coleman-Mandula theorem at pregeometric levels.
Identified 12 FCC nearest-neighbour directions as orbits of bivector generators.
Derived speed ratio c_L/c_T as √2, confirming elastic relationships.
Mapped particle modes to identify fundamental particles like the Higgs boson.
Connected FCC orientations with algebraic structures of the Clifford group.
Established a unique packing condition related to FCC geometry.

Abstract

We demonstrate that the face-centred cubic (FCC) lattice geometry postulated as the ground state of the Field of Resonance (FoR) substrate field is not an assumption but a necessary consequence of the Pauli-Clifford algebra C₃, ₀ — the algebraic structure of the SU (2) preon field. Five results are established: (i) the 12 FCC nearest-neighbour directions emerge as the quarter-turn orbits of the three bivector generators of C₃, ₀, forming the cuboctahedron, verified numerically (24 vertex pairs at distance 1. 0, Oₕ symmetry confirmed) ; (ii) the speed ratio cL/cT = √2 is derived from the Cauchy elastic relation applied to the quarter-turn geometry (verified: C₁₁/C₄₄ = 2 exactly for central-force FCC) ; (iii) the Brillouin-zone particle mode decomposition A₁g + T₂u + Eg maps term-by-term onto the Clifford algebra grade decomposition ℒℛ = S + V + B, identifying the Higgs boson, weak gauge bosons, and graviton tensor sector respectively; (iv) the four FCC sublattice orientations correspond exactly to the four primitive idempotents of C₃, ₀, related by inner automorphisms of the Clifford group; and (v) three fermion generations follow from the three independent bivector planes of three-dimensional Euclidean space, each associated with one coordinate axis via the Binz-de Gosson-Hiley symplectic plane construction. A new subsection (Section 7. 1) addresses the Coleman-Mandula theorem, demonstrating that it does not apply at the pregeometric level L1 where the generation structure originates: the theorem governs the S-matrix at level L2, where the three generations appear as internal quantum numbers satisfying G = ISO (3, 1) ⊗ Gflavor. The quarter-turn angle π/4 is shown to be the unique close-packing condition (neighbour-to-neighbour distance equals origin-to-neighbour distance) rather than an assumption. An appendix provides the exact holographic bound for the FCC Wigner-Seitz cell (rhombic dodecahedron: A = 3√2 ≈ 4. 243 ℓPl², correcting the commonly cited truncated-octahedron misidentification). These results connect the algebraic quantum mechanics programme of Frescura-Hiley, Bohm-Davies-Hiley, and Binz-de Gosson-Hiley with the Skyrmion crystal results of Kugler-Shtrikman and Manton.

FCC Lattice Geometry and Particle Mode Decomposition from the SU(2) Pauli-Clifford Algebra

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