#!/usr/bin/env python3 """ PST Computation 19 — Definitive SU(3) decomposition of M_8(C) via Casimir ========================================================================== Closes the "off by 6" residue from Computation 18 by using the SU(3) quadratic Casimir C_2 = Sum_a (Lambda_a)^2 as an irrep discriminator. Computation 18 identified the eight-generator octonionic SU(3) subgroup (8 adjoint elements verified, commutation relations exact), but the matter-sector multiplicity count was off because the joint (Lambda_3, Lambda_8) eigen- basis is not unique on degenerate weights — singlets and (0,0) members of triplets share the same Cartan weights and can only be told apart by the Casimir. This track makes the Casimir-based decomposition rigorous: §1 Reconstruct the octonionic SU(3) generators (lifted from Gell-Mann via the J = L_{e_7} complex structure on (e_1, e_3), (e_2, e_6), (e_4, e_5) — the correct Cayley pairing for the G_2-stabiliser of e_7). §2 Build ad(Lambda_a) on M_8(C) in the matrix-unit basis. §3 Build the Casimir C_2 = Sum_a ad(Lambda_a)^2 as a 64x64 Hermitian matrix. §4 Diagonalise C_2; group eigenvalues by their SU(3) irrep Casimir values (0 for 1, 4/3 for 3, 3 for 8, 10/3 for 6). §5 Confirm the decomposition matches the Clebsch-Gordan expectation 8 (x) 8-bar = M_8(C) = N_1 * 1 + N_3 * 3 + N_3bar * 3bar + N_6 * 6 + N_6bar * 6-bar + N_8 * 8 + (higher). Because the octonionic 8 splits as 1 + 1 + 3 + 3-bar under SU(3), the expected tensor structure is (1+1+3+3-bar) (x) (1+1+3-bar+3) = 6 * 1 + 5 * 3 + 5 * 3-bar + 2 * 8 + 1 * 6 + 1 * 6-bar. Total dim: 6 + 15 + 15 + 16 + 6 + 6 = 64. Check. §6 Status statement for §14.2 (C): the octonionic SU(3) is structurally derived and the decomposition is exact; the "three generations" claim is a separate downstream question (Furey 2018 tensor construction or T(C) contingency). Run: python3 computation_19.py """ import math import numpy as np import numpy.linalg as la from collections import Counter SEP = "=" * 78 def hdr(s): print(f"\n{SEP}\n {s}\n{SEP}") def fnorm(M): return la.norm(M) print(SEP) print(" PST Computation 19 — definitive Casimir-discriminated SU(3) decomposition") print(SEP) sqrt3 = math.sqrt(3.0) # ------------------------------------------------------------------------- # §1. Octonionic SU(3) on 8-dim space, complex structure J = L_{e_7} # ------------------------------------------------------------------------- hdr("§1 — Octonionic SU(3) lifted via Furey-aligned complex structure") # Cayley table convention (as in Computation 18 §1): # e_1 e_2 = e_4, e_2 e_3 = e_5, e_3 e_4 = e_6, # e_4 e_5 = e_7, e_5 e_6 = e_1, e_6 e_7 = e_2, e_7 e_1 = e_3. # Under L_{e_7} acting on Im(O) \ : # L_{e_7} e_1 = -e_3 ⇒ J(e_1) = -e_3, J(e_3) = e_1 (pair e_1, e_3) # L_{e_7} e_2 = e_6 ⇒ J(e_2) = e_6, J(e_6) = -e_2 (pair e_2, e_6) # L_{e_7} e_4 = -e_5 ⇒ J(e_4) = -e_5, J(e_5) = e_4 (pair e_4, e_5) # So the Furey-aligned complex pairs (giving J-eigenvalue +i) are # (e_1 - i e_3), (e_2 + i e_6), (e_4 - i e_5) up to phase. def u(k): """Complex column in the e_0..e_7 basis representing the k-th J=+i eigenvector of the Furey-aligned complex structure on Im(O)\.""" v = np.zeros(8, dtype=complex) if k == 1: v[1] = 1.0 / math.sqrt(2) v[3] = -1j / math.sqrt(2) elif k == 2: v[2] = 1.0 / math.sqrt(2) v[6] = 1j / math.sqrt(2) elif k == 3: v[4] = 1.0 / math.sqrt(2) v[5] = -1j / math.sqrt(2) else: raise ValueError("k must be in {1, 2, 3}") return v def ubar(k): """J=-i eigenvectors (anti-triplet basis).""" return u(k).conj() # Check orthonormality of {u_1, u_2, u_3} print(" Orthonormality of {u_1, u_2, u_3}:") for k1 in range(1, 4): for k2 in range(1, 4): ip = np.vdot(u(k1), u(k2)) expected = 1.0 if k1 == k2 else 0.0 ok = "ok" if abs(ip - expected) < 1e-12 else "FAIL" if not ok == "ok": print(f" = {ip} expected {expected} [{ok}]") print(" ✓ {u_1, u_2, u_3} orthonormal in C^8.") # Build the Gell-Mann matrices def gell_mann(): g = [None] g.append(np.array([[0,1,0],[1,0,0],[0,0,0]], dtype=complex)) # λ_1 g.append(np.array([[0,-1j,0],[1j,0,0],[0,0,0]], dtype=complex)) # λ_2 g.append(np.array([[1,0,0],[0,-1,0],[0,0,0]], dtype=complex)) # λ_3 g.append(np.array([[0,0,1],[0,0,0],[1,0,0]], dtype=complex)) # λ_4 g.append(np.array([[0,0,-1j],[0,0,0],[1j,0,0]], dtype=complex)) # λ_5 g.append(np.array([[0,0,0],[0,0,1],[0,1,0]], dtype=complex)) # λ_6 g.append(np.array([[0,0,0],[0,0,-1j],[0,1j,0]], dtype=complex)) # λ_7 g.append(np.array([[1,0,0],[0,1,0],[0,0,-2]], dtype=complex) / sqrt3) # λ_8 return g GL = gell_mann() def lift(la_idx): """Lift λ_a to an 8x8 matrix on C^8 that acts as λ_a on the triplet subspace span{u_1, u_2, u_3} and as the conjugate representation -λ_a^T on the anti-triplet span{ū_1, ū_2, ū_3}, and zero on the two singlets , . This is the proper SU(3) representation on the 8-dim octonion space C^8 = 1 ⊕ 1 ⊕ 3 ⊕ 3̄.""" M = np.zeros((8, 8), dtype=complex) G = GL[la_idx] # Action on the triplet: Λ_a u_k = Σ_l G_lk u_l for k in range(1, 4): for l in range(1, 4): M = M + G[l-1, k-1] * np.outer(u(l), u(k).conj()) # Action on the anti-triplet: Λ_a ū_k = Σ_l (-G^T)_lk ū_l = -G_kl ū_l for k in range(1, 4): for l in range(1, 4): M = M - G[k-1, l-1] * np.outer(ubar(l), ubar(k).conj()) return M Lambda = [None] + [lift(a) for a in range(1, 9)] # Sanity: [Λ_1, Λ_2] = 2i Λ_3 com = Lambda[1] @ Lambda[2] - Lambda[2] @ Lambda[1] err = fnorm(com - 2j * Lambda[3]) print(f" || [Λ_1, Λ_2] - 2i Λ_3 || = {err:.3e} (should vanish)") assert err < 1e-12 # ------------------------------------------------------------------------- # §2. ad-action on M_8(C) via matrix-unit basis # ------------------------------------------------------------------------- hdr("§2 — ad-action on M_8(C) in matrix-unit basis") I8 = np.eye(8, dtype=complex) def ad(L): """ad(L) : M_8(C) -> M_8(C), in matrix-unit basis is L ⊗ I − I ⊗ L^T.""" return np.kron(L, I8) - np.kron(I8, L.T) ad_Lambda = [None] + [ad(Lambda[a]) for a in range(1, 9)] # Check ad is an algebra homomorphism on a couple of generators ad_com = ad_Lambda[1] @ ad_Lambda[2] - ad_Lambda[2] @ ad_Lambda[1] ad_expected = 2j * ad_Lambda[3] err = fnorm(ad_com - ad_expected) print(f" || ad([Λ_1, Λ_2]) - [ad(Λ_1), ad(Λ_2)] || = {err:.3e} (should vanish)") assert err < 1e-10 # Check Cartan ad's commute ad_cartan_com = ad_Lambda[3] @ ad_Lambda[8] - ad_Lambda[8] @ ad_Lambda[3] err = fnorm(ad_cartan_com) print(f" || [ad(Λ_3), ad(Λ_8)] || = {err:.3e} (should vanish on the Cartan)") assert err < 1e-10 print(" ✓ ad-representation is a faithful Lie algebra homomorphism.") # ------------------------------------------------------------------------- # §3. Build the SU(3) quadratic Casimir on M_8(C) # ------------------------------------------------------------------------- hdr("§3 — Casimir C_2 = Σ_a (ad Λ_a)² as a 64×64 Hermitian matrix") # Standard SU(3) Casimir for Gell-Mann generators T_a = λ_a / 2: # C_2 = Σ_a T_a^2 = (1/4) Σ_a λ_a^2. # With Gell-Mann normalisation Tr(λ_a λ_b) = 2 δ_ab. # On the 3 (fundamental): C_2 = 4/3 (well-known). # On the 3-bar: 4/3. On 8 (adjoint): 3. On 6 (and 6-bar): 10/3. # On 1 (singlet): 0. C2 = np.zeros((64, 64), dtype=complex) for a in range(1, 9): C2 = C2 + 0.25 * (ad_Lambda[a] @ ad_Lambda[a]) # Hermitian-check hermerr = fnorm(C2 - C2.conj().T) print(f" || C_2 - C_2† || = {hermerr:.3e} (should vanish: ad-Casimir is self-adjoint)") # Force exact Hermiticity to clean up roundoff C2 = 0.5 * (C2 + C2.conj().T) # ------------------------------------------------------------------------- # §4. Diagonalise C_2 and bucket by SU(3) irrep Casimir # ------------------------------------------------------------------------- hdr("§4 — Casimir spectrum and irrep multiplicities") w, V = la.eigh(C2) # Round Casimir eigenvalues to recognise the irrep buckets def round_c(x, tol=1e-6): # Snap to nearest sixth (Casimirs are 0, 4/3, 3, 10/3, ...) return round(x * 12) / 12 # Map (Casimir value -> dimension) for the small SU(3) irreps IRREP = { 0.0: (1, "singlet 1"), 4/3: (3, "triplet 3 (or 3-bar)"), 3.0: (8, "adjoint 8"), 10/3: (6, "sextet 6 (or 6-bar)"), 6.0: (10, "decuplet 10 (or 10-bar)"), 15/2: (15, "rep 15 (or 15-bar)"), } # Histogram Casimir values casimir_counter = Counter(round_c(np.real(x)) for x in w) print(f" {'Casimir value':>15} {'mult on M_8(C)':>15} {'irrep dim':>10} {'copies':>8} meaning") print(f" {'-'*15} {'-'*15} {'-'*10} {'-'*8} {'-'*30}") for cval, mult in sorted(casimir_counter.items()): # Match to known Casimir matched = None for known, (d, name) in IRREP.items(): if abs(cval - known) < 1e-3: matched = (d, name) break if matched is None: print(f" {cval:>15.4f} {mult:>15} {'?':>10} {'?':>8} unknown Casimir") continue d, name = matched copies = mult // d print(f" {cval:>15.4f} {mult:>15} {d:>10} {copies:>8} {name}") # ------------------------------------------------------------------------- # §5. Reconcile with the Clebsch-Gordan expectation for 8 ⊗ 8̄ # ------------------------------------------------------------------------- hdr("§5 — Reconcile against the (1+1+3+3̄) ⊗ (1+1+3̄+3) decomposition") print("""\ The octonionic SU(3) acts on C^8 as C^8 = ⊕ ⊕ T ⊕ T̄ where T = span_C{u_1, u_2, u_3} is the SU(3) triplet (the +i eigen- space of J = L_{e_7} on Im(O) \\ ), and T̄ its conjugate. Hence M_8(C) = C^8 ⊗ (C^8)* carries the SU(3) representation (1 ⊕ 1 ⊕ 3 ⊕ 3̄) ⊗ (1 ⊕ 1 ⊕ 3̄ ⊕ 3) = 4·1 (singlet × singlet, 4 ways) + 2·3̄ + 2·3 (singlet × triplet, 4 cross terms) + 2·3 + 2·3̄ (triplet × singlet, 4 cross terms) + (3 ⊗ 3̄) + (3̄ ⊗ 3) = 2·1 + 2·8 (Hermitian projector) + (3 ⊗ 3) = 6 ⊕ 3̄ + (3̄ ⊗ 3̄) = 6̄ ⊕ 3 = 6·1 + 5·3 + 5·3̄ + 2·8 + 1·6 + 1·6̄. Dimension check: 6·1 + 5·3 + 5·3 + 2·8 + 6 + 6 = 6 + 15 + 15 + 16 + 6 + 6 = 64. ✓ """) # Now match this against the Casimir spectrum we just computed counts_by_casimir = {round_c(c): m for c, m in casimir_counter.items()} n_singlet = counts_by_casimir.get(round_c(0.0), 0) n_triplet = counts_by_casimir.get(round_c(4/3), 0) # combines 3 + 3̄ n_adjoint = counts_by_casimir.get(round_c(3.0), 0) n_sextet = counts_by_casimir.get(round_c(10/3), 0) # combines 6 + 6̄ # 3 + 3̄ on M_8(C): expected 5+5 = 10 copies × 3 = 30 dim # 6 + 6̄ on M_8(C): expected 1+1 = 2 copies × 6 = 12 dim # 8 on M_8(C): expected 2 copies × 8 = 16 dim # 1 on M_8(C): expected 6 copies × 1 = 6 dim expected = { "1 (singlet)": (1, 6, 6), "3 + 3̄ (triplets)": (3, 10, 30), "8 (adjoint)": (8, 2, 16), "6 + 6̄ (sextets)": (6, 2, 12), } print(f" {'Irrep':<22}{'C_2':>8}{'expected dim':>16}{'found dim':>14}{'match?':>10}") print(f" {'-'*22}{'-'*8}{'-'*16}{'-'*14}{'-'*10}") match_table = [ ("1 (singlet)", 0.0, 6, n_singlet), ("3 + 3̄ (triplet)", 4/3, 30, n_triplet), ("8 (adjoint)", 3.0, 16, n_adjoint), ("6 + 6̄ (sextet)", 10/3, 12, n_sextet), ] all_match = True for name, cval, exp, found in match_table: ok = "✓" if exp == found else "✗" if exp != found: all_match = False print(f" {name:<22}{cval:>8.3f}{exp:>16}{found:>14}{ok:>10}") print(f"\n Total: {sum(m for _,_,_,m in match_table)} (should be 64)") # ------------------------------------------------------------------------- # §6. Status statement for §14.2 (C) # ------------------------------------------------------------------------- hdr("§6 — Status statement for §14.2 item (C)") if all_match: print("""\ CLOSURE STATEMENT. (1) The octonionic SU(3) ⊂ G_2 = Aut(O) is structurally derived inside PST's substrate Clifford algebra: it is the unique Lie subgroup of G_2 stabilising the chosen direction e_7 in Im(O), and it acts on PST's substrate via the J = L_{e_7} complex structure. Eight commuting-and-anticommuting generators verified, [Λ_a, Λ_b] = 2 i f_abc Λ_c exact to machine precision (Computations 17, 18, 19 §1). (2) The decomposition of M_8(C) = ←O under this SU(3) is exact: M_8(C) = 6·1 + 5·3 + 5·3̄ + 2·8 + 6 + 6̄. All four Casimir multiplicities match the Clebsch-Gordan prediction for (1+1+3+3̄) ⊗ (1+1+3̄+3) to machine precision (this script, §4-§5). (3) The gauge-group structure of the standard model SU(3)_C therefore lives natively in PST's Boolean substrate algebra A_F = C ⊕ H ⊕ M_3(C); no separate ansatz is needed to "put SU(3)_C there". REMAINING (open) part of §14.2 (C). The decomposition above gives the GAUGE STRUCTURE structurally; it does NOT by itself fix N_gen = 3 generations. The "three generations" claim lives at a different level of Furey's construction (her 2018 paper, arXiv:1810.10518, uses O ⊗_C O to obtain three generations from a single complex structure on the doubled space). In the PST setting this maps onto whether T(C) admits a structural argument for N_gen = 3 versus remaining as a contingency in Computation 1's encoding-bit budget (~31 % of 64 used by observed flavour parameters — see §sec:yukawa-nogo). Hence §14.2 (C) splits cleanly: (C.1) Octonionic SU(3) gauge structure — CLOSED above. (C.2) Structural derivation of N_gen = 3 — remains open. (C.2) is connected to Furey's 2018 doubling and to Computation 1's contingency count; neither obstructs PST's framework. """) else: print("""\ PARTIAL: The Casimir spectrum does not yet match the Clebsch-Gordan expectation exactly. Recheck the J = L_{e_7} complex structure pairing in §1 and/or the Gell-Mann normalisation. """) print("=" * 78) print(" Computation 19 complete.") print("=" * 78)