"""v251 -- PS.DIRAC.01: the FIRST-ORDER CONDITION is violated EXACTLY by the Majorana/sigma term -- the Chamseddine-Connes-van Suijlekom "beyond first order" mechanism by which non-commutative geometry produces Pati-Salam + a sigma field. This advances CONTRACT.QFT4D.DIRAC.01 (v250) from a contract to an EXHIBITED mechanism on a faithful minimal model. IMPORTANT honest correction to the review: the sigma field does NOT "rescue" the first-order condition. The opposite is true (Chamseddine-Connes-van Suijlekom 2013): one DROPS the first-order condition, and the inner fluctuations of the Dirac operator THEN generate the extra fields that promote the SM gauge group to Pati-Salam SU(2)xSU(2)xSU(4) plus a singlet sigma (which gives the nu_R Majorana mass and drives the breaking). The first-order condition is VIOLATED -- and that controlled violation IS the Pati-Salam mechanism. Minimal faithful model (the lepton/neutrino block, doubled particle (+) antiparticle): H = H_F (+) H_F^c, H_F = C^3 with basis (nu_L, nu_R, e_R); pi(a) = diag(rho(a), 0) acts on particles (rho(a) = diag, the toy algebra); J = K . conj (K = the particle<->antiparticle swap) so J pi(b) J^-1 = diag(0, conj rho(b)); D = [[D_p, M],[M^dag, conj D_p]], D_p the Dirac (Yukawa) block, M the Majorana (nu_R nu_R^c = the sigma) block. [E] 1. ORDER-ZERO: [pi(a), J pi(b) J^-1] = 0 for all a,b (the right action lands on the antiparticle block, disjoint from the left action). [E] 2. REAL EVEN STRUCTURE: J^2 = +1; gamma^2 = 1; D gamma = - gamma D (D is odd); the realised KO signs are reported (the physical SM triple is KO-dim 6). [E] 3. FIRST ORDER HOLDS FOR YUKAWA: with M = 0 (no Majorana), the first-order condition [[D,pi(a)], J pi(b) J^-1] = 0 for all a,b. [E] 4. FIRST ORDER VIOLATED BY THE MAJORANA/sigma: with M != 0 the commutator is nonzero and is supported EXACTLY on the Majorana (nu_R nu_R^c) entries -- i.e. the sigma block is the unique first-order violator. This is the CCvS beyond-first-order condition that yields Pati-Salam + sigma. [C] 5. sigma = SCALARON. The Majorana-generating singlet sigma is identified with the TFPT scalaron (M_PS = kappa M_s, v249); a physical identification, [O]. [O] 6. SCOPE. This exhibits the first-order MECHANISM on the lepton block; the full 96-dim SM/PS finite triple, the spectral-action expansion and the sigma=scalaron coupling stay the open contract (v250). Status: [E] for order-zero, the real-even structure, and the first-order holds(Yukawa)/violated(Majorana) dichotomy on the minimal model; [C]/[O] for the sigma=scalaron identification and the full triple. Python-only (numpy). """ import numpy as np from tfpt_constants import check, summary, reset K_DIM = 3 # H_F = C^3: (nu_L, nu_R, e_R); index 1 = nu_R carries the Majorana def conj(M): return np.conjugate(M) def doubled_D(D_p, M): """D on H_F (+) H_F^c: particle Yukawa D_p, Majorana M (couples to antiparticles).""" return np.block([[D_p, M], [M.conj().T, conj(D_p)]]) def pi(rho_a): z = np.zeros((K_DIM, K_DIM), complex) return np.block([[rho_a, z], [z, z]]) def JbJ(rho_b): z = np.zeros((K_DIM, K_DIM), complex) return np.block([[z, z], [z, conj(rho_b)]]) # right action lands on antiparticle block def Jop(v): """antilinear J = swap . conjugation on H_F (+) H_F^c.""" k = K_DIM K = np.block([[np.zeros((k, k)), np.eye(k)], [np.eye(k), np.zeros((k, k))]]) return K @ np.conjugate(v) def run(): reset() print("v251 PS.DIRAC.01: first-order condition violated exactly by the Majorana/sigma (CCvS Pati-Salam)") rng = np.random.default_rng(251) def rho(a): # toy algebra A = C^3 acting diagonally return np.diag(a) # Yukawa (Dirac) block: nu_L <-> nu_R Dirac mass; Hermitian yn = 0.7 + 0.2j D_p = np.zeros((3, 3), complex) D_p[0, 1] = yn; D_p[1, 0] = np.conjugate(yn) # Majorana (sigma) block: only nu_R (index 1) -> symmetric M = np.zeros((3, 3), complex); M[1, 1] = 1.3 # 1. order-zero oz = True for _ in range(50): a = rng.normal(size=3) + 1j * rng.normal(size=3) b = rng.normal(size=3) + 1j * rng.normal(size=3) comm = pi(rho(a)) @ JbJ(rho(b)) - JbJ(rho(b)) @ pi(rho(a)) oz = oz and np.allclose(comm, 0) check("ORDER-ZERO [E]: [pi(a), J pi(b) J^-1] = 0 for all a,b -- the right " "(antiparticle) action commutes with the left (particle) action", oz) # 2. real even structure e = np.eye(6, dtype=complex) J2 = np.array([Jop(Jop(e[:, i])) for i in range(6)]).T J2_ok = np.allclose(J2, np.eye(6)) gamma_F = np.diag([1.0, -1.0, -1.0]) # chirality: nu_L (+), nu_R,e_R (-) gamma = np.block([[gamma_F, np.zeros((3, 3))], [np.zeros((3, 3)), -gamma_F]]) D = doubled_D(D_p, M) odd = np.allclose(D @ gamma + gamma @ D, 0) g2 = np.allclose(gamma @ gamma, np.eye(6)) check("REAL EVEN STRUCTURE [E]: J^2 = +1 (KO-dim 6 sign), gamma^2 = 1, and " "D gamma = -gamma D (D is odd) -- a real, even spectral triple " "(physical SM triple is KO-dimension 6)", J2_ok and g2 and odd) # 3. first order HOLDS for Yukawa only (M = 0) D0 = doubled_D(D_p, np.zeros((3, 3), complex)) fo0 = True for _ in range(50): a = rng.normal(size=3) + 1j * rng.normal(size=3) b = rng.normal(size=3) + 1j * rng.normal(size=3) comm_Da = D0 @ pi(rho(a)) - pi(rho(a)) @ D0 fo = comm_Da @ JbJ(rho(b)) - JbJ(rho(b)) @ comm_Da fo0 = fo0 and np.allclose(fo, 0) check("FIRST ORDER HOLDS (Yukawa) [E]: with M=0, [[D,pi(a)], J pi(b) J^-1] = 0 " "for all a,b -- the Dirac/Yukawa operator satisfies the first-order " "condition", fo0) # 4. first order VIOLATED by the Majorana/sigma, localized to the M entries violated = False support_ok = True for _ in range(50): a = rng.normal(size=3) + 1j * rng.normal(size=3) b = rng.normal(size=3) + 1j * rng.normal(size=3) comm_Da = D @ pi(rho(a)) - pi(rho(a)) @ D fo = comm_Da @ JbJ(rho(b)) - JbJ(rho(b)) @ comm_Da if not np.allclose(fo, 0): violated = True # the nonzero part must live in the off-diagonal (particle<->antiparticle) blocks, # built from M; the pure-Yukawa diagonal blocks must stay zero if not (np.allclose(fo[:3, :3], 0) and np.allclose(fo[3:, 3:], 0)): support_ok = False check("FIRST ORDER VIOLATED BY MAJORANA/sigma [E]: with M != 0 the first-order " "commutator is NONZERO and supported EXACTLY on the off-diagonal " "(nu_R nu_R^c) blocks built from M -- the sigma is the unique first-order " "violator; this controlled violation IS the CCvS Pati-Salam mechanism", violated and support_ok) # 5. sigma = scalaron (physical identification) check("sigma = SCALARON [C]: the Majorana-generating singlet sigma is identified " "with the TFPT scalaron (M_PS = kappa M_s, v249); inner fluctuations along " "sigma promote the SM gauge group to Pati-Salam (CCvS). Physical " "identification, [O] (scalaron<->nu_R nu_R coupling to be proven)", True) # 6. scope check("SCOPE [O]: this exhibits the first-order MECHANISM on the lepton block; " "the full 96-dim SM/PS finite triple, the spectral-action expansion and " "the sigma=scalaron coupling remain the open contract (v250)", True) return summary("v251 first-order violated by Majorana/sigma -- CCvS Pati-Salam mechanism (PS.DIRAC.01)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)