""" compute_het_sg0.8.py Computation of every quantity in the Summary of results of "Spectral Geometry of Vacuum Entanglement on the Cube and Prism" (het_sg0.8). Self-contained: numpy only, no files read. Paste into one Google Colab cell and run. PHYSICS CARRIED IN THIS FILE ---------------------------- Bond: S_chi = S_BH l_P^2/A = 1/4 (Bekenstein-Hawking), vacuum entanglement S* = 2/9, residual entanglement capacity xi* = S_chi - S* = 1/36. Prism Q4 = Q3 x I: 16 vertices (x,y,z,tau) in {0,1}^4, index v = 8x+4y+2z+tau; 32 edges, each oriented from its 0-vertex to its 1-vertex in the flipped coordinate. Disentangler deformation: c_D = (5/sqrt2) xi*, x and y edges weight omega = 1 + 4 c_D, z and tau edges weight 1. Weighted vertex Laplacian L_{V,omega}; M_{V,omega} = L_{V,omega} + 1. Sigma_V = Tr M_{V,omega}^-1. Gravity (sec. 8): 1/(16 pi G_N) = Sigma_V / (12 x 16) l_P^-2, so G_N = 12/(pi Sigma_V) l_P^2. Descent (sec. 9): E_phi = m_P Theta_phi Pi_phi^p exp(-1/xi*), m_P = 1.2209e19 GeV. suppression: exp(-1/xi*) = exp(-36); m_IR = m_P exp(-1/xi*); l_c = l_P exp(1/xi*) = hbar c/m_IR. vev: Higgs mode psi = (-1)^tau, lambda = 2, G_psi = 1/3, Pi_psi = G_psi/Sigma_V, Theta_psi = 1, p = 1. string tension: sourced colour flux b_s on the z-tau face at (x,y) = (0,0), bare edge operator M_E = d0 d0^T + 1, G_bs = b^T M_E^-1 b / |b|^2, Pi_bs = G_bs/Sigma_E, Theta_bs = 3/8, p = 2. glueball: closed colour flux b_g on the same face, Pi_bg = 1/Sigma_E, Theta_bg = 1/4, p = 2. Neutrinos (sec. 10): cell (a,b) of L_{V,omega}: a excited x,y directions, b excited z,tau directions, lambda(a,b) = 2 omega a + 2 b, multiplicity C(2,a) C(2,b). D_omega = det M_{V,omega}^-1, Theta_nu = 1/2, y_nu_i = Theta_nu D_omega prod over C_i of 1/(1+lambda), C_3 = {(1,2)}, C_2 = {(1,2),(1,1)}, C_1 = {(1,2),(1,1),(2,2)}; m_nu_i = y v/sqrt2. Charged leptons (sec. 11): R_ab = 1/(m^2 + lambda(a,b)), m^2 = 1; couplings m^2 and omega; y_tau = (h0 G_psi)^-1 d_m2 R22 / d_m2 ln D_omega, h0 = 1/sqrt2; y_mu = |d y_tau / d ln y_nu3| in m^2; y_e = |d y_mu / d ln y_nu2| in omega. References: v 246.22 GeV (PDG); y_tau 9.93780e-3, y_mu 5.85042e-4, y_e 2.77713e-6 at M_Z (Antusch-Hinze-Saad); sqrt(sigma) 424.264 MeV (hadron spectrum); m_0++ 1730(50)(80) MeV (anisotropic lattice); NuFIT 6.0 dm21^2 = 7.49e-5, dm31^2 = 2.513e-3 eV^2 (dm32^2 = dm31^2 - dm21^2); DESI DR2 sum < 0.0642 eV. """ import itertools, math import numpy as np # ---------------------------------------------------------------- prism S_CHI, S_STAR = 1/4, 2/9 XI = S_CHI - S_STAR V = [tuple(int(c) for c in f"{n:04b}") for n in range(16)] # (x,y,z,tau) E = [(i, j) for i, a in enumerate(V) for j, b in enumerate(V) if sum(p != q for p, q in zip(a, b)) == 1 and sum(b) > sum(a)] # 0-vertex -> 1-vertex AXIS = [[k for k in range(4) if V[i][k] != V[j][k]][0] for i, j in E] d0 = np.zeros((len(E), 16)) for k, (i, j) in enumerate(E): d0[k, i], d0[k, j] = -1, 1 C_D = 5*XI/math.sqrt(2) OM = 1 + 4*C_D W = np.array([OM if ax in (0, 1) else 1.0 for ax in AXIS]) L_V = d0.T @ np.diag(W) @ d0 M_V = L_V + np.eye(16) SIGMA_V = np.trace(np.linalg.inv(M_V)) # ---------------------------------------------------------------- gravity G_N = 12/(math.pi*SIGMA_V) # ---------------------------------------------------------------- descent M_P = 1.2209e19 DESC = math.exp(-1/XI) # descent suppression exp(-1/xi*) M_IR = M_P*DESC # m_IR = m_P exp(-1/xi*), GeV HBARC = 1.973269804e-16 # hbar c, GeV m L_C = HBARC/M_IR # l_c = l_P exp(1/xi*) = hbar c/m_IR, m PSI = np.array([(-1)**v[3] for v in V], float) G_PSI = PSI @ np.linalg.inv(M_V) @ PSI / (PSI @ PSI) VEV = M_P*1*(G_PSI/SIGMA_V)*DESC M_E = d0 @ d0.T + np.eye(len(E)) SIGMA_E = np.trace(np.linalg.inv(M_E)) EIDX = {e: k for k, e in enumerate(E)} idx = lambda s: int(s, 2) face = [(idx("0000"), idx("0010")), (idx("0010"), idx("0011")), (idx("0001"), idx("0011")), (idx("0000"), idx("0001"))] b_s = np.zeros(len(E)); b_g = np.zeros(len(E)) for e, s, g in zip(face, (+1, -1, -1, +1), (+1, +1, -1, -1)): b_s[EIDX[e]], b_g[EIDX[e]] = s, g G_BS = b_s @ np.linalg.inv(M_E) @ b_s/(b_s @ b_s) G_BG = b_g @ np.linalg.inv(M_E) @ b_g/(b_g @ b_g) SQRT_SIGMA = M_P*(3/8)*(G_BS/SIGMA_E)**2*DESC M_GLUE = M_P*(1/4)*(G_BG/SIGMA_E)**2*DESC # ---------------------------------------------------------------- neutrinos LAM = lambda a, b: 2*OM*a + 2*b MULT = lambda a, b: math.comb(2, a)*math.comb(2, b) CELLS = [(a, b) for a in range(3) for b in range(3)] D_OM = 1/np.linalg.det(M_V) THETA_NU = 0.5 READ = {"nu1": [(1, 2), (1, 1), (2, 2)], "nu2": [(1, 2), (1, 1)], "nu3": [(1, 2)]} y_nu = {n: THETA_NU*D_OM/np.prod([1 + LAM(*c) for c in cs]) for n, cs in READ.items()} m_nu = {n: y*VEV/math.sqrt(2)*1e9 for n, y in y_nu.items()} # eV D21 = m_nu["nu2"]**2 - m_nu["nu1"]**2 D31 = m_nu["nu3"]**2 - m_nu["nu1"]**2 SUM = sum(m_nu.values()) res = lambda x, r: 100*(x/r - 1) print("Table 1. Prism quantities") print(" omega = %.12f Sigma_V = %.9f Sigma_E = %.9f (6368/315 = %.9f)" % (OM, SIGMA_V, SIGMA_E, 6368/315)) print(" check: spectrum of L_V,omega = 2 omega a + 2 b:", np.allclose(np.sort(np.linalg.eigvalsh(L_V)), sorted(LAM(a, b) for a, b in CELLS for _ in range(MULT(a, b))))) print(" G_psi = %.9f (1/3) G_bs = %.9f (127/315 = %.9f) G_bg = %.9f" % (G_PSI, G_BS, 127/315, G_BG)) print(" D_omega = %.9e" % D_OM) print(" 1/xi* = %.9f exp(-1/xi*) = %.6e m_IR = %.1f GeV l_c/l_P = %.6e l_c = %.4e m" % (1/XI, DESC, M_IR, math.exp(1/XI), L_C)) print("\nTable 2. Summary of results") print(" %-22s %-24s %-20s %s" % ("quantity", "prism value", "reference value", "residual")) print(" %-22s %-24s %-20s %+.3f%%" % ("G_N", "%.6f l_P^2" % G_N, "l_P^2", res(G_N, 1.0))) print(" %-22s %-24s %-20s %+.2f%%" % ("v", "%.6f GeV" % VEV, "246.22 GeV", res(VEV, 246.22))) print(" %-22s %-24s %-20s %+.2f%%" % ("sqrt(sigma)", "%.4f MeV" % (1e3*SQRT_SIGMA), "424.264 MeV", res(1e3*SQRT_SIGMA, 424.264))) print(" %-22s %-24s %-20s %s" % ("m_0++", "%.2f MeV" % (1e3*M_GLUE), "1730(50)(80) MeV", "---")) for n in ("nu1", "nu2", "nu3"): print(" %-22s %-24s %-20s %s" % ("m_" + n, "%.6f eV" % m_nu[n], "---", "---")) print(" %-22s %-24s %-20s %+.2f%%" % ("dm21^2", "%.6e eV^2" % D21, "7.49e-5 eV^2", res(D21, 7.49e-5))) print(" %-22s %-24s %-20s %+.3f%%" % ("dm31^2", "%.6e eV^2" % D31, "2.513e-3 eV^2", res(D31, 2.513e-3))) print(" %-22s %-24s %-20s %+.3f%%" % ("dm32^2", "%.6e eV^2" % (m_nu["nu3"]**2 - m_nu["nu2"]**2), "2.4381e-3 eV^2", res(m_nu["nu3"]**2 - m_nu["nu2"]**2, 2.513e-3 - 7.49e-5))) print(" %-22s %-24s %-20s %s" % ("sum m_nu", "%.6f eV" % SUM, "< 0.0642 eV", "---")) print("\nTable 3. Neutrino Yukawa couplings: y = Theta_nu D_omega / prod(1+lambda)") for n in ("nu1", "nu2", "nu3"): print(" %s cells %-24s product of 1+lambda %.6f y %.9e" % (n, READ[n], np.prod([1 + LAM(*c) for c in READ[n]]), y_nu[n])) # selection on the prism: every form with up to four cell factors NT = [c for c in CELLS if c != (0, 0)] FORMS = [f for k in range(5) for f in itertools.combinations_with_replacement(NT, k)] mass = lambda f: THETA_NU*D_OM/np.prod([1 + LAM(*c) for c in f])*VEV/math.sqrt(2)*1e9 lo3, hi3 = math.sqrt(2.513e-3), 0.050310 sol3 = [f for f in FORMS if lo3 <= mass(f) <= hi3] m1max = math.sqrt(m_nu["nu3"]**2 - 2.513e-3) lo2, hi2 = math.sqrt(7.49e-5), math.sqrt(m1max**2 + 7.49e-5) sol2 = [f for f in FORMS if len(f) <= 2 and lo2 <= mass(f) <= hi2] print("\nTable 4. Selection on the prism") print(" forms with up to four cell factors: %d; solutions for m_nu3 in [%.6f, %.6f] eV: %s" % (len(FORMS), lo3, hi3, sol3)) print(" forms with up to two cell factors: %d; solutions for m_nu2 in [%.6f, %.6f] eV: %s" % (sum(len(f) <= 2 for f in FORMS), lo2, hi2, sol2)) for c in NT: m1 = m_nu["nu2"]/(1 + LAM(*c)) print(" nu1 = nu2/(1+lambda%s): m_nu1 %.6f eV, dm21^2 residual %+.3f%%" % (c, m1, res(m_nu["nu2"]**2 - m1**2, 7.49e-5))) # ---------------------------------------------------------------- charged leptons H0 = 1/math.sqrt(2) def Rp(cell, p): return 1/(p[0] + 2*p[1]*cell[0] + 2*cell[1]) # p = (m^2, omega) def lnD(p): return sum(MULT(*cl)*math.log(Rp(cl, p)) for cl in CELLS) def dd(f, i, p, h=1e-3): q = lambda s: f(tuple(p[k] + (s if k == i else 0) for k in range(2))) return (-q(2*h) + 8*q(h) - 8*q(-h) + q(-2*h))/(12*h) P0 = (1.0, OM) lny = {k: (lambda cs: (lambda p: lnD(p) + sum(math.log(Rp(cl, p)) for cl in cs)))(v) for k, v in READ.items()} y_tau = lambda p: dd(lambda q: Rp((2, 2), q), 0, p)/(H0*G_PSI*dd(lnD, 0, p)) y_mu = lambda p: abs(dd(y_tau, 0, p)/dd(lny["nu3"], 0, p)) y_e = lambda p: abs(dd(y_mu, 1, p, 2e-2)/dd(lny["nu2"], 1, p, 2e-2)) R22 = Rp((2, 2), P0); R12 = Rp((1, 2), P0); R11 = Rp((1, 1), P0) S1 = lambda w, k: sum(MULT(*cl)*(1.0 if w == 0 else 2.0*cl[0])*Rp(cl, P0)**k for cl in CELLS) dD = [-S1(0, 1), -S1(1, 1)]; dDm = [S1(0, 2), S1(1, 2)]; dDmm = [-2*S1(0, 3), -2*S1(1, 3)] wgt = lambda w, cl: 1.0 if w == 0 else 2.0*cl[0] YT = R22**2/(H0*G_PSI*SIGMA_V) s_tau = [-2*wgt(w, (2, 2))*R22 - dDm[w]/dD[0] for w in (0, 1)] YM = YT*abs(s_tau[0]/(dD[0] - R12)) dN = [2*wgt(w, (2, 2))*R22**2 - (dDmm[w]*dD[0] - dDm[0]*dDm[w])/dD[0]**2 for w in (0, 1)] dP = [dDm[w] + wgt(w, (1, 2))*R12**2 for w in (0, 1)] s_mu = [s_tau[w] + dN[w]/s_tau[0] - dP[w]/(dD[0] - R12) for w in (0, 1)] YE = YM*abs(s_mu[1]/(dD[1] - 2*R12 - 2*R11)) AHS = {"tau": 9.93780e-3, "mu": 5.85042e-4, "e": 2.77713e-6} print("\nTable 5. Charged-lepton Yukawa couplings (closed form; direct numerical derivative in brackets)") print(" sensitivities d ln y_tau/d(m2, omega) = %+.9f, %+.9f; d ln y_mu/d(m2, omega) = %+.9f, %+.9f" % (s_tau[0], s_tau[1], s_mu[0], s_mu[1])) for n, v, f in (("e", YE, y_e), ("mu", YM, y_mu), ("tau", YT, y_tau)): print(" y_%-4s %.9e [%.6e] reference %.5e at M_Z residual %+.3f%%" % (n, v, f(P0), AHS[n], res(v, AHS[n])))