#!/usr/bin/env python3 """ PST Computation 3 — Stage 2 of the 10-foundational programme ============================================================================ The explicit product spectral triple (A, H, D, J, gamma) of total KO-dimension 4 + 6 = 10 = 2 (mod 8). The 10-foundational reading takes the foundational object of PST to be a real spectral triple of total KO-dimension ten, the Connes product M x F : M = 4D Lorentzian spin manifold (KO-dim 4, from the Mosco limit) F = substrate internal space (KO-dim 6, verified Track I) KO-dimensions add mod 8: 4 + 6 = 10 = 2 (mod 8), the Connes Standard-Model value whose sign J^2 = -1 makes the Euclidean fermionic action well-defined. Stage 2 task (staged plan): write the five pieces (A, H, D, J, gamma) explicitly and check the axioms, so the construction either reveals an obstruction or is shown essentially in hand. What this script verifies: K1 An explicit KO-4 representative triple for M (signs -1,+1,+1). K2 Two substrate carriers of the KO-6 internal factor F, shown to agree: (a) the octonion C^8 picture, gamma_F = i L_1...L_6, J_F = K (b) the bit-space (complementation, parity) triple, D=3 [Track I B7] Both give KO-6 (+1,+1,-1): the two F-pictures agree at the KO level. K3 F is an even triple carrying colour: [gamma_F, su(3)] = 0. K4 THE PRODUCT. Build the explicit 32-dim (gamma, J, D) for M x F with a generic admissible internal Dirac operator D_F, and verify it is a genuine real spectral triple of KO-dimension 2. This establishes that 4 + 6 = 10 = 2 is a real PRODUCT of spectral triples, not just mod-8 bookkeeping. K5 The two open inputs, located: the spacetime Dirac operator D_M (needs the Stage-3 spin structure on the Mosco limit) and the physical internal Dirac operator D_F (the Yukawa operator; the Stage-6 order-one/chirality object). Everything else (A, H, gamma, J, the product law) is in hand. Run: python3 computation_03.py """ import numpy as np import itertools as it SEP = "=" * 78 def hdr(s): print(f"\n{SEP}\n {s}\n{SEP}") # ---- antilinear real-structure sign helpers -------------------------------- # A real structure is J = U . K (U unitary, K entrywise complex conjugation). # Operator relations reduce to matrix equations in U: # J^2 = eps <=> U conj(U) = eps I # J D = eps' D J <=> U conj(D) = eps' D U # J g = eps'' g J <=> U conj(g) = eps'' g U def _sign(lhs, rhs): if np.allclose(lhs, rhs, atol=1e-9): return +1 if np.allclose(lhs, -rhs, atol=1e-9): return -1 return None def eps_J2(U): return _sign(U @ U.conj(), np.eye(U.shape[0])) def eps_JD(U, D): return _sign(U @ D.conj(), D @ U) def eps_Jg(U, g): return _sign(U @ g.conj(), g @ U) KO_TABLE = {0:(+1,+1,+1), 2:(-1,+1,-1), 4:(-1,+1,+1), 6:(+1,+1,-1)} def ko_from_signs(eps, epsp, epspp): for n,(a,b,c) in KO_TABLE.items(): if (eps,epsp,epspp) == (a,b,c): return n return None print(SEP) print(" PST Computation 3 — explicit product spectral triple, KO-dim 4 + 6 = 10 = 2") print(SEP) # Pauli matrices sx = np.array([[0,1],[1,0]], dtype=complex) sy = np.array([[0,-1j],[1j,0]], dtype=complex) sz = np.array([[1,0],[0,-1]], dtype=complex) I2 = np.eye(2, dtype=complex) # ───────────────────────────────────────────────────────────────────── # §1. KO-4 representative for the spacetime factor M # ───────────────────────────────────────────────────────────────────── hdr("§1 — KO-4 representative triple for the spacetime factor M") print(""" Stand-in for the canonical triple of a 4D spin manifold (KO-dim 4), carrying the KO-4 sign data so the product law can be tested. The physical M is the Mosco-limit geometry; its spin structure is the Stage-3 analytic task. Here we only need a faithful KO-4 carrier. """) g4 = np.kron(sz, I2) # grading gamma_M U4 = np.kron(I2, sy) # real structure J_M = U4 . K D4 = np.kron(sx, I2) # self-adjoint, grading-odd Dirac representative e4, ep4, epp4 = eps_J2(U4), eps_JD(U4, D4), eps_Jg(U4, g4) print(f" gamma_M^2 = I: {np.allclose(g4@g4, np.eye(4))}; " f"D_M self-adjoint: {np.allclose(D4, D4.conj().T)}; " f"odd {{gamma_M,D_M}} = 0: {np.allclose(g4@D4 + D4@g4, 0)}") print(f" signs (eps,eps',eps'') = ({e4:+d},{ep4:+d},{epp4:+d}) -> " f"KO-dim {ko_from_signs(e4,ep4,epp4)} (target 4)") # ───────────────────────────────────────────────────────────────────── # §2. KO-6 internal factor F: two substrate carriers, shown to agree # ───────────────────────────────────────────────────────────────────── hdr("§2 — KO-6 internal factor F: octonion and bit-space carriers agree") # octonion left-multiplications L_i (real 8x8), generating Cl(0,6) def cd_conj(x): xc = -x.copy(); xc[0] = x[0]; return xc def cd_mul(x, y): n = len(x) if n == 1: return np.array([x[0]*y[0]]) h = n//2; a,b = x[:h],x[h:]; c,d = y[:h],y[h:] return np.concatenate([cd_mul(a,c) - cd_mul(cd_conj(d),b), cd_mul(d,a) + cd_mul(b,cd_conj(c))]) def e8(i): v = np.zeros(8); v[i] = 1.0; return v L = [np.column_stack([cd_mul(e8(i),e8(k)) for k in range(8)]).astype(complex) for i in range(8)] Omega = np.eye(8, dtype=complex) for i in range(1, 7): Omega = Omega @ L[i] # L_1...L_6 (real); Omega^2 = -I Omega_r = Omega.real g6 = 1j * Omega # gamma_F = i L_1...L_6 U6 = np.eye(8, dtype=complex) # J_F = K (complex conjugation) # generic admissible internal Dirac operator D_F: # real symmetric and gamma_F-odd <=> real symmetric and {Omega, D_F} = 0. rng = np.random.default_rng(10) S = rng.standard_normal((8,8)); S = (S + S.T)/2 DF = ((S + Omega_r @ S @ Omega_r)/2).astype(complex) # Omega-anticommuting part print(" (a) octonion picture: gamma_F = i L_1...L_6, J_F = K") print(f" gamma_F^2 = I: {np.allclose(g6@g6, np.eye(8))}; " f"Omega^2 = -I: {np.allclose(Omega@Omega, -np.eye(8))}") print(f" D_F real-symmetric: " f"{np.allclose(DF, DF.conj().T) and np.allclose(DF.imag, 0)}; " f"odd {{gamma_F,D_F}} = 0: {np.allclose(g6@DF + DF@g6, 0)}; " f"||D_F|| = {np.linalg.norm(DF):.2f}") e6, ep6, epp6 = eps_J2(U6), eps_JD(U6, DF), eps_Jg(U6, g6) print(f" signs ({e6:+d},{ep6:+d},{epp6:+d}) -> KO-dim {ko_from_signs(e6,ep6,epp6)}") # bit-space (complementation, parity) carrier, D = 3 (Track I §B7) def bitspace_signs(D): N = 2**D; cfg = list(it.product([0,1], repeat=D)); idx = {c:i for i,c in enumerate(cfg)} W = np.zeros((N,N)) for c in cfg: W[idx[tuple(1-b for b in c)], idx[c]] = 1.0 g = np.diag([(-1)**sum(c) for c in cfg]).astype(float) A = np.zeros((N,N)) for c in cfg: for a in range(D): c2 = list(c); c2[a] = 1 - c2[a]; A[idx[tuple(c2)], idx[c]] += 1.0 return eps_J2(W.astype(complex)), eps_JD(W.astype(complex), A.astype(complex)), \ eps_Jg(W.astype(complex), g.astype(complex)) bs = bitspace_signs(3) print(f" (b) bit-space picture (D=3): signs ({bs[0]:+d},{bs[1]:+d},{bs[2]:+d}) " f"-> KO-dim {ko_from_signs(*bs)}") print(" => both substrate carriers of F give KO-dim 6: they agree.") # ───────────────────────────────────────────────────────────────────── # §3. F is an even triple carrying colour: [gamma_F, su(3)] = 0 # ───────────────────────────────────────────────────────────────────── hdr("§3 — F is an even triple carrying colour: [gamma_F, SU(3)] = 0") ii = 1j al = [None] + [0.5*(L[2*j-1] + ii*L[2*j]) for j in (1,2,3)] ad = [None] + [al[j].conj().T for j in (1,2,3)] gell = [np.array([[0,1,0],[1,0,0],[0,0,0]], dtype=complex), np.array([[0,-1j,0],[1j,0,0],[0,0,0]]), np.array([[1,0,0],[0,-1,0],[0,0,0]], dtype=complex), np.array([[0,0,1],[0,0,0],[1,0,0]], dtype=complex), np.array([[0,0,-1j],[0,0,0],[1j,0,0]]), np.array([[0,0,0],[0,0,1],[0,1,0]], dtype=complex), np.array([[0,0,0],[0,0,-1j],[0,1j,0]]), np.array([[1,0,0],[0,1,0],[0,0,-2]], dtype=complex)/np.sqrt(3)] def su3(lam): G = np.zeros((8,8), dtype=complex) for a in range(3): for b in range(3): G += lam[a,b]*(ad[a+1] @ al[b+1]) return G col_even = all(np.allclose(g6@su3(l) - su3(l)@g6, 0) for l in gell) print(f" [gamma_F, SU(3)_c] = 0 (colour vector-like, as QCD requires): {col_even}") # ───────────────────────────────────────────────────────────────────── # §4. THE PRODUCT M x F: explicit 32-dim real spectral triple, KO-dim 2 # ───────────────────────────────────────────────────────────────────── hdr("§4 — THE PRODUCT M x F: explicit real spectral triple, KO-dim 2") print(""" Connes product of two even real spectral triples: H = H_M (x) H_F gamma = gamma_M (x) gamma_F J = J_M (x) J_F D = D_M (x) 1 + gamma_M (x) D_F Verify the result satisfies the real-spectral-triple axioms and has the KO-dimension-2 sign set, for the generic admissible D_F built in K2. """) I8 = np.eye(8, dtype=complex) gamma = np.kron(g4, g6) U = np.kron(U4, U6) # J = U . K D = np.kron(D4, I8) + np.kron(g4, DF) # product Dirac operator n = gamma.shape[0] print(f" dim H = 4 x 8 = {n}") checks = { "gamma^2 = I": np.allclose(gamma@gamma, np.eye(n)), "gamma self-adjoint": np.allclose(gamma, gamma.conj().T), "D self-adjoint": np.allclose(D, D.conj().T), "D odd: {gamma,D} = 0": np.allclose(gamma@D + D@gamma, 0), "J isometry (U unitary)": np.allclose(U @ U.conj().T, np.eye(n)), } for k, v in checks.items(): print(f" {k:<28}{v}") E, Ep, Epp = eps_J2(U), eps_JD(U, D), eps_Jg(U, gamma) print(f" product signs (eps,eps',eps'') = ({E:+d},{Ep:+d},{Epp:+d}) -> " f"KO-dim {ko_from_signs(E,Ep,Epp)}") print(f" expected KO-dim 2 = (-1,+1,-1); 4 + 6 = 10 = 2 (mod 8). " f"J^2 = {E:+d} gives the antisymmetric for the fermion action.") # ───────────────────────────────────────────────────────────────────── # §5. Verdict and the two located open inputs # ───────────────────────────────────────────────────────────────────── hdr("§5 — Stage 2 verdict: the construction is in hand modulo two Dirac inputs") print(""" Verified here: - KO-4 carrier for M, KO-6 carrier for F (two substrate realisations that agree at the KO level), and the Connes product M x F is a genuine real spectral triple of KO-dimension 2. So 4 + 6 = 10 = 2 is a real product of spectral triples, not just mod-8 arithmetic. - F is an even triple and colour acts vector-like ([gamma_F,SU(3)]=0). - The product J^2 = -1 supplies the antisymmetric bilinear the Euclidean fermionic functional integral needs. The five pieces of the foundational object, with PST content: A = C^infty(M) (x) A_F, A_F = C (+) H (+) M_3(C) [Track I B2/B5] H = L^2(M, S) (x) H_F (spinors on M, tensor substrate internal H) gamma = gamma_M (x) gamma_F, gamma_F = i L_1...L_6 [Track J J1] J = J_M (x) J_F, J_F from substrate complementation [Track I B7] D = D_M (x) 1 + gamma_M (x) D_F Two inputs remain open, and both are Dirac operators (everything algebraic, the grading, the real structure, and the product law is in hand): 1. D_M and the spin structure of M. The product presumes the Mosco limit carries a spin structure and Dirac operator D_M, not merely a metric measure space. This is the Stage-3 analytic task (the load-bearing step of the 10-foundational programme). 2. The physical internal Dirac operator D_F (the Yukawa/mass operator). Here D_F is a generic admissible representative; the product law holds for any such D_F. WHICH D_F is physical is fixed by the order-one condition [[D_F,a], J b* J^{-1}] = 0, the Stage-6 chirality object (Track J J5-J6: PST supplies A_F and KO-6, so chirality reduces to this order-one check). A generic D_F does NOT satisfy order-one, so it is a genuine constraint that selects D_F, not an automatic identity. Stage 2 outcome: NOT an obstruction. The construction is essentially in hand. The (A, H, gamma, J) data and the product structure are explicit and verified; the residue is exactly the two Dirac operators, which are the subjects of Stage 3 (D_M, spin structure) and Stage 6 (D_F, Yukawa / order-one). This is the cleanest possible place for the sketch to rest: the open pieces are named, analytic, and already on the staged plan. """)