FCC Lattice Geometry and Particle Mode Decomposition from the SU(2) Pauli-Clifford Algebra
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| Natura: | Recurso digital |
| Lingua: | inglese |
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Zenodo
2026
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| _version_ | 1866901182870454272 |
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| author | Hunter, Bruce |
| author_facet | Hunter, Bruce |
| contents | <p>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, O_h symmetry confirmed); (ii) the speed ratio c_L/c_T = √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.</p> <p>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) ⊗ G_flavor. 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 ℓ_Pl², 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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19212216 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | FCC Lattice Geometry and Particle Mode Decomposition from the SU(2) Pauli-Clifford Algebra Hunter, Bruce Clifford algebra, SU(2), FCC lattice, Skyrmion, quantum gravity, Pauli algebra, bivector, cuboctahedron, Bravais lattice, particle spectrum, Higgs boson, gauge boson, graviton, fermion generations, Coleman-Mandula theorem, Wigner-Seitz cell, rhombic dodecahedron, holographic bound, Field of Resonance, pregeometry, Planck scale, phonon, Brillouin zone, spinor, idempotent, close-packing, Manton, Frescura-Hiley, Binz-de Gosson-Hiley, Bohm-Davies-Hiley, quantum potential, speed ratio, Cauchy relation, elastic tensor <p>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, O_h symmetry confirmed); (ii) the speed ratio c_L/c_T = √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.</p> <p>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) ⊗ G_flavor. 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 ℓ_Pl², 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.</p> |
| title | FCC Lattice Geometry and Particle Mode Decomposition from the SU(2) Pauli-Clifford Algebra |
| topic | Clifford algebra, SU(2), FCC lattice, Skyrmion, quantum gravity, Pauli algebra, bivector, cuboctahedron, Bravais lattice, particle spectrum, Higgs boson, gauge boson, graviton, fermion generations, Coleman-Mandula theorem, Wigner-Seitz cell, rhombic dodecahedron, holographic bound, Field of Resonance, pregeometry, Planck scale, phonon, Brillouin zone, spinor, idempotent, close-packing, Manton, Frescura-Hiley, Binz-de Gosson-Hiley, Bohm-Davies-Hiley, quantum potential, speed ratio, Cauchy relation, elastic tensor |
| url | https://doi.org/10.5281/zenodo.19212216 |