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| Автор: | |
|---|---|
| Формат: | Recurso digital |
| Мова: | Англійська |
| Опубліковано: |
Zenodo
2026
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| Предмети: | |
| Онлайн доступ: | https://doi.org/10.5281/zenodo.19212216 |
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Зміст:
- <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>