TFPT scientific source excerpt
Original repository path: verification/v258_dirac_covariance_induction.py
Original file SHA-256: 16c60e831dceba773f3629466895f1009654827286e5b760ae01bd755fd5c0b1
Inclusive original lines: 65-150
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000065 | def induce(C):
000066 |     """one-particle modular Hamiltonian H = log((1-C) C^{-1}) of a covariance C."""
000067 |     w, V = np.linalg.eigh(C)
000068 |     w = np.clip(w, 1e-15, 1 - 1e-15)
000069 |     return V @ np.diag(np.log((1 - w) / w)) @ V.conj().T
000070 | 
000071 | 
000072 | def kms_cov(H, mu=1.0):
000073 |     """Fermi-Dirac / KMS covariance C = (1 + e^{H/mu})^{-1} of a Hermitian H."""
000074 |     w, V = np.linalg.eigh(H)
000075 |     occ = 1.0 / (1.0 + np.exp(np.clip(w / mu, -700.0, 700.0)))   # stable Fermi function
000076 |     return V @ np.diag(occ) @ V.conj().T
000077 | 
000078 | 
000079 | def run():
000080 |     reset()
000081 |     print("v258  PS.DIRAC.03: D_F as the modular/covariance induction of the seam state")
000082 |     rng = np.random.default_rng(258)
000083 | 
000084 |     # ---- 1. inversion identity on a random Hermitian + on a Dirac-type block ----
000085 |     A = rng.normal(size=(8, 8)) + 1j * rng.normal(size=(8, 8))
000086 |     H = A + A.conj().T
000087 |     ok_rand = np.allclose(induce(kms_cov(H)), H, atol=1e-9)
000088 | 
000089 |     # a small chirality-graded Dirac block (Yukawa: L<->R off-diagonal), like v252
000090 |     Y = rng.normal(size=(3, 3)) + 1j * rng.normal(size=(3, 3))
000091 |     Dblk = np.block([[np.zeros((3, 3)), Y], [Y.conj().T, np.zeros((3, 3))]])
000092 |     ok_dirac = np.allclose(induce(kms_cov(Dblk)), Dblk, atol=1e-9)
000093 |     check("INVERSION IDENTITY [E]: log((1-C)C^-1) = H exactly for C = (1+e^H)^-1 "
000094 |           "(Araki/Casini-Huerta modular Hamiltonian = inverse of the KMS "
000095 |           "occupation) -- verified on a random Hermitian and on a chirality-graded "
000096 |           "Yukawa Dirac block; the induction is the exact inverse, not a fit",
000097 |           ok_rand and ok_dirac)
000098 | 
000099 |     # ---- 2. covariance is a legal quasi-free state (positive contraction) ----
000100 |     C = kms_cov(Dblk)
000101 |     ev = np.linalg.eigvalsh(C)
000102 |     check("COVARIANCE IS A LEGAL STATE [E]: C_F = (1+e^{D/mu})^-1 is a positive "
000103 |           "contraction, spec(C_F) in (0,1) (min %.3f, max %.3f) -- a bona fide "
000104 |           "quasi-free two-point function; the carrier Dirac data ARE the data of a "
000105 |           "boundary Gaussian (KMS) state" % (ev.min(), ev.max()),
000106 |           np.all(ev > 0) and np.all(ev < 1))
000107 | 
000108 |     # ---- 3. chirality-odd: the induced operator anticommutes with gamma ----
000109 |     gamma = np.diag([1.0, 1.0, 1.0, -1.0, -1.0, -1.0])     # +1 on L, -1 on R
000110 |     Hind = induce(C)
000111 |     odd = np.allclose(Hind @ gamma + gamma @ Hind, 0, atol=1e-9)
000112 |     check("CHIRALITY-ODD = A DIRAC OPERATOR [E]: the induced modular Hamiltonian "
000113 |           "anticommutes with the chirality grading, D_F^ind gamma = -gamma D_F^ind "
000114 |           "-- the entanglement Hamiltonian of a chirality-graded covariance is "
000115 |           "automatically an odd (Dirac-type) operator (Pi_odd is the identity here)",
000116 |           odd)
000117 | 
000118 |     # ---- 4. unitary equivalence to the v252-built Dirac (same mass spectrum) ----
000119 |     masses = np.array([0.000511, 0.0022, 0.0047, 0.095, 0.1057,
000120 |                        1.27, 1.777, 4.18, 172.76]) / 174.0     # Yukawas (v18/v252)
000121 |     Dmass = np.zeros((18, 18))
000122 |     for i, y in enumerate(masses):
000123 |         Dmass[i, 9 + i] = Dmass[9 + i, i] = y                  # L<->R block
000124 |     mu_geo = 1.0
000125 |     Drec = mu_geo * induce(kms_cov(Dmass / mu_geo, mu_geo))
000126 |     same_spec = np.allclose(np.sort(np.linalg.eigvalsh(Drec)),
000127 |                             np.sort(np.linalg.eigvalsh(Dmass)), atol=1e-9)
000128 |     check("UNITARY EQUIVALENCE TO v252 [E]: building C_F from the v252 charged "
000129 |           "spectrum (9 Yukawas) and inducing returns the SAME spectrum, "
000130 |           "spec(D_F^ind) = spec(D_F^v252) up to mu_geo -- the built finite Dirac "
000131 |           "operator and the boundary-covariance readout are unitarily equivalent",
000132 |           same_spec)
000133 | 
000134 |     # ---- 5. Majorana / seesaw is an off-diagonal covariance entry ----
000135 |     mD, MR = 100.0, 1.0e14
000136 |     Dnu = np.array([[0.0, mD], [mD, MR]])                       # nu_L<->nu_R, nu_R Majorana
000137 |     Crec = kms_cov(Dnu / 1.0)
000138 |     Dnu_rec = induce(Crec)
000139 |     seesaw_in = min(abs(np.linalg.eigvalsh(Dnu)))
000140 |     seesaw_rec = min(abs(np.linalg.eigvalsh(Dnu_rec)))
000141 |     check("MAJORANA = OFF-DIAGONAL COVARIANCE [E]: the type-I seesaw block (the "
000142 |           "particle<->antiparticle Majorana off-diagonal of C_F) induces back to "
000143 |           "the seesaw eigenvalue m_light = m_D^2/M_R (= %.2e GeV) -- the "
000144 |           "sigma/Majorana term is a covariance entry, not an extra postulate"
000145 |           % (mD ** 2 / MR),
000146 |           abs(seesaw_rec - seesaw_in) < 1e-6 * seesaw_in
000147 |           and abs(seesaw_in - mD ** 2 / MR) < 1e-3 * mD ** 2 / MR)
000148 | 
000149 |     # ---- 6. physical identification (recorded) ----
000150 |     check("PHYSICAL IDENTIFICATION [C]: C_F = P_F C_Sigma P_F -- the carrier "