Internal structure of the S₆ vacuum: from Pauli matrices to the Standard Model gauge algebra

We analyse the internal structure of the 2-Sylow subgroup P = D8 × Z2 of S6 beyond the level of 3, and identify increasingly rich algebraic structures up to the complexied Lie algebra of the Standard Model gauge group. The analysis is organised in six groups of results. (i) Pauli matrices and Cliord obstruction: the two irreducible 2-dimensional representations of P realise σx and σz on the temporal and spatial generators (Theorem 2. 1) ; no Cliord algebra Cl (1, 3) closes on V3, 2, 1, with the analytic bound Tr (ρ (g), ρ (h) 2) ≥ 28 > 0 for every pair of involutions (Theorem 2. 5). (ii) Structure of involution pairs: the values Tr (A2) and Tr (D2) are completely determined by the order of gh (Theorem 2. 6) ; the bound Tr (A2) ≥ 28 is saturated by exactly 1200 pairs, partitioned as 480 pairs with ord (gh) = 3 and 720 with ord (gh) = 6 (Proposition 2. 7). The 1200 saturating pairs are in canonical correspondence with the 160 subgroups S3 and the 120 subgroups D12 of S6 (Theorem 2. 15), and the corresponding subgroup classes are Out (S6) -symmetric with explicit palindromic exchange (Theorems 2. 17, 2. 18, 2. 19). The 930 active dierences in the anti-commutation graph are completely characterised: they are exactly the dierences with ord (gh) ∈ 2, 3, with 45 exceptions in the commuting case corresponding to a specic class of Klein four-groups (Theorems 2. 13, 2. 14). The intra-S3 structure is analytically decoupled (Theorem 2. 16). (iii) Fermionic families and parity: the eight linear characters partition as 1+3+4 with bijective kernel correspondence (Theorem 3. 1), the three generational charges satisfy q1q2q3 = +1 (Theorem 3. 2), and the alignment Vab ⊕ Vnab = V3, 2, 1|A6 holds (Theorem 4. 1). (iv) Block structure and Fock space: the 15 transpositions classify in a 3 + 4 + 8 pattern with o- diagonal ranks 0, 4, 6, derived analytically from V3, 2, 1|S4×S2 (Theorem 5. 2). (v) Standard Model gauge algebra: the dihedral subgroup D10 = Sym (pentagon) ⊂ S6 has centralizer Z (D10) ∼= M (2, C) 2 ⊕ M (3, C) 2 of dimension 26 in End (V16) (Theorem 6. 4). V3, 2, 1 is the unique irreducible representation of S6 producing M (3, C) in its D10-centralizer (Theorem 6. 6), and D10 has a unique Out (S6) -invariant conjugacy class in S6 (Corollary 6. 8). Within Z (D10) we explicitly construct sl (3, C) ⊕sl (2, C) ⊕u (1) of dimension 12, the complexied Lie algebra of the Standard Model gauge group SU (3) × SU (2) × U (1) (Theorem 6. 9). (vi) Symmetry breaking: the chain D10 ⊂ F20 ⊂ S5 ⊂ A6 ⊂ S6 corresponds to a sequence of progressive symmetry breakings of the gauge algebra (Corollary 6. 10). All results are obtained without any continuous parameter or deformation, and veried by independent computation in GAP 4.