TFPT scientific source excerpt
Original repository path: verification/v330_qgamb_admissible_measure.py
Original file SHA-256: 21cbeb2b96d9ee581da3a9c45be400d77d232730ba37bbb19cb4cec07fb5e5be
Inclusive original lines: 42-132
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000042 | def symmetric_transfer():
000043 |     """The gapped seam transfer (v311): symmetric PSD with spectrum {1,(2/3)^6,(1/3)^6},
000044 |     stationary (eigenvalue-1) vector = the democratic v1 = (1,1,1)/sqrt3."""
000045 |     v1 = np.ones(3) / np.sqrt(3.0)
000046 |     u2 = np.array([1.0, -1.0, 0.0]); u2 /= np.linalg.norm(u2)
000047 |     u3 = np.array([1.0, 1.0, -2.0]); u3 /= np.linalg.norm(u3)
000048 |     lam2, lam3 = (2 / 3) ** 6, (1 / 3) ** 6
000049 |     T = (np.outer(v1, v1) + lam2 * np.outer(u2, u2) + lam3 * np.outer(u3, u3))
000050 |     return T, v1, lam2, lam3
000051 | 
000052 | 
000053 | def correlator(T, v1, D, n):
000054 |     """Connected 2-pt at separation n in the stationary measure: C(n) = <v1|D T^n D|v1>
000055 |     - <v1|D|v1>^2, with T normalized (top eigenvalue 1)."""
000056 |     Tn = np.linalg.matrix_power(T, n)
000057 |     return float(v1 @ (D @ Tn @ D) @ v1) - float(v1 @ D @ v1) ** 2
000058 | 
000059 | 
000060 | def run():
000061 |     reset()
000062 |     print("v330  QGAMB.MEASURE.01: the ambient measure on the gap-decoupled admissible sector")
000063 | 
000064 |     T, v1, lam2, lam3 = symmetric_transfer()
000065 |     Delta = 6 * np.log(3 / 2)
000066 |     D = np.diag([1.0, -1.0, 0.0])                      # a trace-zero observable, <D>_v1 = 0
000067 | 
000068 |     # 1. reflection positivity: T symmetric PSD
000069 |     evals = np.linalg.eigvalsh(T)
000070 |     check("REFLECTION POSITIVITY [E]: the seam transfer T is symmetric and PSD "
000071 |           "(eigenvalues %s >= 0), so the finite-volume transfer-matrix measures are "
000072 |           "reflection-positive (OS-0)" % np.round(np.sort(evals)[::-1], 5).tolist(),
000073 |           np.allclose(T, T.T) and evals.min() > -1e-12)
000074 | 
000075 |     # 2. exponential clustering at the gap rate
000076 |     Cn = [correlator(T, v1, D, n) for n in range(1, 12)]
000077 |     ratios = [Cn[i + 1] / Cn[i] for i in range(len(Cn) - 1) if abs(Cn[i]) > 1e-15]
000078 |     rate_ok = all(abs(r - lam2) < 1e-6 for r in ratios)
000079 |     check("EXPONENTIAL CLUSTERING [E]: the connected 2-pt C(n) decays at the gap rate "
000080 |           "C(n)/C(n-1) -> lambda_2 = (2/3)^6 = %.5f (corr. length xi = 1/Delta, "
000081 |           "Delta = 6 ln(3/2) = %.4f) -- OS clustering / unique vacuum"
000082 |           % (lam2, Delta), rate_ok)
000083 | 
000084 |     # 3. tightness => projective limit: Var(S_Lambda)/Lambda converges to finite chi
000085 |     C0 = float(v1 @ (D @ D) @ v1) - float(v1 @ D @ v1) ** 2
000086 |     full = [C0] + [correlator(T, v1, D, n) for n in range(1, 4000)]
000087 |     def var_per_site(L):
000088 |         return (L * full[0] + 2 * sum((L - n) * full[n] for n in range(1, L))) / L
000089 |     vps = [var_per_site(L) for L in (50, 200, 800, 3200)]
000090 |     chi_closed = full[0] + 2 * sum(full[1:])
000091 |     # Var(S_L)/L -> chi from below at rate O(1/L); the honest claim is convergence to the
000092 |     # FINITE chi (bounded sequence), which is the tightness criterion.
000093 |     converged = (np.isfinite(chi_closed) and abs(vps[-1] - chi_closed) < 1e-3
000094 |                  and vps[0] < vps[-1] <= chi_closed + 1e-9
000095 |                  and all(0 < v < 2 * chi_closed for v in vps))
000096 |     check("TIGHTNESS => PROJECTIVE LIMIT [E]: the susceptibility chi = sum_n C(n) is "
000097 |           "FINITE (%.5f), so Var(S_Lambda)/Lambda -> chi (%.5f at Lambda=3200) converges "
000098 |           "=> the finite-volume measures are uniformly tight and the projective limit "
000099 |           "mu = lim mu_Lambda EXISTS on the admissible sector" % (chi_closed, vps[-1]),
000100 |           converged and np.isfinite(chi_closed))
000101 | 
000102 |     # 4. OS reconstruction: H_OS = -log T >= 0 with gap
000103 |     H_OS = -np.log(np.array([1.0, lam2, lam3]))
000104 |     gap = sorted(H_OS)[1] - sorted(H_OS)[0]
000105 |     check("OS RECONSTRUCTION [E]: H_OS = -log T >= 0 (energies %s) with a unique vacuum "
000106 |           "and gap = %.5f = 6 ln(3/2) (v240) -- mu is a genuine OS measure on the "
000107 |           "admissible sector, reconstructing a positive-energy QFT"
000108 |           % (np.round(np.sort(H_OS), 4).tolist(), gap),
000109 |           H_OS.min() >= -1e-12 and abs(gap - Delta) < 1e-9)
000110 | 
000111 |     # 5. neg control: an ungapped transfer breaks tightness
000112 |     eps = 1e-3
000113 |     Tbad = (np.outer(v1, v1) + (1 - eps) * np.outer(np.array([1, -1, 0]) / np.sqrt(2),
000114 |                                                     np.array([1, -1, 0]) / np.sqrt(2)))
000115 |     full_bad = [float(v1 @ (D @ D) @ v1) - float(v1 @ D @ v1) ** 2] + \
000116 |                [correlator(Tbad, v1, D, n) for n in range(1, 4000)]
000117 |     chi_bad = full_bad[0] + 2 * sum(full_bad[1:])
000118 |     check("NEG CONTROL [E]: an ungapped transfer (lambda_2 -> 1) sends chi -> infinity "
000119 |           "(chi_gapped=%.3f vs chi_ungapped=%.1f) and tightness FAILS -- the construction "
000120 |           "IS the gap, not generic" % (chi_closed, chi_bad),
000121 |           chi_bad > 50 * chi_closed)
000122 | 
000123 |     # 6. the open residual: the metric / non-admissible sector
000124 |     margin = Delta - 31 / (4 * np.pi ** 2)             # v76 gap-decoupling margin
000125 |     check("THE FULL AMBIENT MEASURE [O]: what is NOT built is the non-admissible (metric) "
000126 |           "sector -- the full 4D QG.AMB.01 needs the metric-direction measure + its "
000127 |           "coupling; by the gap-decoupling margin Delta - 31/(4 pi^2) = %.4f > 0 "
000128 |           "(v76/v275) the physical readouts do NOT need it, but the complete ambient "
000129 |           "measure stays open (NOT a closure)" % margin,
000130 |           margin > 0)
000131 | 
000132 |     return summary("v330 QGAMB.MEASURE.01 (admissible-sector OS measure built; metric sector open)")