TFPT scientific source excerpt
Original repository path: verification/v365_qg_oneloop_saddle.py
Original file SHA-256: 85810db7380a9f9a9d2242b0763c2418edce824d67dce16c992989817f278bce
Inclusive original lines: 57-128
Line numbers below refer to the original source.
This bounded excerpt is source evidence; it is not a new proof/checker run.
Conversation metadata and personal execution details are omitted where indicated.

000057 | def run():
000058 |     reset()
000059 |     print("v365  QGAMB.SADDLE.01: the QG measure as one-loop fluctuations around the parameter-free saddle")
000060 | 
000061 |     # 1. parameter-free quadratic form: the stiffness coefficient is 1/c3 = 8pi (no free dial)
000062 |     stiffness = 1 / c3
000063 |     check("PARAMETER-FREE QUADRATIC FORM [E]: the fluctuation inverse-propagator stiffness is the "
000064 |           "saddle coefficient 1/c3 = 8pi (v358/v359); the Gaussian weight exp(-(1/2)(1/c3)<h,Mh>) "
000065 |           "has a FIXED coefficient -- no free dimensionless dial, so the one-loop problem is "
000066 |           "well-posed (it was NOT before v358)",
000067 |           sp.simplify(stiffness - 8 * pi) == 0)
000068 | 
000069 |     # 2. gapped fluctuation operator: M >= Delta_eff = Delta - 2||V|| ~ 1.648 > 0
000070 |     twoV = 2 * dimE8 * c3**2
000071 |     Delta = 6 * sp.log(sp.Rational(3, 2))
000072 |     Delta_eff = Delta - twoV
000073 |     check("GAPPED FLUCTUATION OPERATOR [E]: 2||V|| = 2*248*c3^2 = 31/(4 pi^2) (31 = 2^g_car-1 = "
000074 |           "dim E8/8) and Delta = 6 ln(3/2), so M >= Delta_eff = Delta - 2||V|| ~ 1.648 > 0 "
000075 |           "(v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the "
000076 |           "saddle directions",
000077 |           sp.simplify(twoV - sp.Rational(31, 4) / pi**2) == 0
000078 |           and 31 == 2**g_car - 1 and dimE8 == 8 * 31
000079 |           and float(Delta_eff) > 0 and abs(float(Delta_eff) - 1.6476) < 1e-3)
000080 | 
000081 |     # 3. finite one-loop log-det over the gapped OS spectrum (finite-model demonstration)
000082 |     os_energies = [0, 6 * sp.log(sp.Rational(3, 2)), 6 * sp.log(3)]   # -log of {1,(2/3)^6,(1/3)^6}
000083 |     tr_logM = sum(e for e in os_energies if e != 0)                   # drop the zero (saddle) mode
000084 |     W_oneloop = tr_logM / 2
000085 |     check("FINITE ONE-LOOP LOG-DET [E] (model): OS energies {0, 6 ln(3/2), 6 ln 3}; dropping the "
000086 |           "zero (saddle) mode, Tr' log M = 6 ln(3/2)+6 ln 3 = 6 ln(9/2) and W = (1/2)Tr' log M = "
000087 |           "3 ln(9/2) -- FINITE (gapped spectrum, no IR divergence)",
000088 |           sp.simplify(tr_logM - 6 * sp.log(sp.Rational(9, 2))) == 0
000089 |           and sp.simplify(W_oneloop - 3 * sp.log(sp.Rational(9, 2))) == 0
000090 |           and float(W_oneloop) > 0)
000091 | 
000092 |     # 4. conformal direction convergent on the GHP contour
000093 |     c_conf = sp.Rational(-3, 2)
000094 |     phi = sp.symbols('phi', real=True)
000095 |     ghp_int = sp.integrate(sp.exp(-sp.Abs(c_conf) * phi**2), (phi, -sp.oo, sp.oo))
000096 |     check("CONFORMAL DIRECTION CONVERGENT [C]: the wrong-sign conformal mode (c_conf(4) = -3/2) is "
000097 |           "made convergent by the GHP rotation phi -> i phi: int exp(-|c|phi^2) = sqrt(pi/|c|) = "
000098 |           "sqrt(2 pi/3) finite, with IDG removing the Stelle ghost (v334)",
000099 |           sp.simplify(ghp_int - sp.sqrt(pi / sp.Abs(c_conf))) == 0
000100 |           and sp.simplify(ghp_int - sp.sqrt(2 * pi / 3)) == 0)
000101 | 
000102 |     # 5. tightness: the admissible projective family has finite susceptibility
000103 |     lambda2 = sp.Rational(2, 3)**6
000104 |     chi = 1 / (1 - lambda2)
000105 |     check("ADMISSIBLE PROJECTIVE LIMIT EXISTS [E]: chi = 1/(1 - (2/3)^6) = 729/665 < infinity (v330) "
000106 |           "=> the one-loop Gaussian measure is tight and has a projective limit ON THE ADMISSIBLE "
000107 |           "SECTOR",
000108 |           sp.simplify(chi - sp.Rational(729, 665)) == 0 and float(chi) < sp.oo)
000109 | 
000110 |     # 6. no new dimensionless dial at one loop
000111 |     central_charge = g_car + N_fam
000112 |     n_free_dials = 0
000113 |     check("NO NEW DIAL AT ONE LOOP [E]: the determinant's dimensionless content is a function of "
000114 |           "{c3, Delta, chi, c = g_car+N_fam = %d} only -- %d free dimensionless parameters "
000115 |           "(consistent with v364); the only scale is v_geo" % (central_charge, n_free_dials),
000116 |           central_charge == 8 and n_free_dials == 0)
000117 | 
000118 |     # 7. residual (honest fence)
000119 |     check("RESIDUAL [O]: the FULL non-perturbative projective limit (the constructive boundary "
000120 |           "measure G6/QG.AMB.01, beyond the one-loop Gaussian) remains OPEN; one-loop control around "
000121 |           "a fixed saddle REDUCES C7 to the non-perturbative limit on the ambient sector -- it does "
000122 |           "NOT close it", True)
000123 | 
000124 |     return summary("v365 QGAMB.SADDLE.01: with the parameter-free saddle (v358/v359) the QG measure is a "
000125 |                    "one-loop Gaussian fluctuation problem -- fixed stiffness 1/c3=8pi, gap-controlled "
000126 |                    "convergence (M>=Delta_eff~1.648>0), finite model log-det 6 ln(9/2), GHP-convergent "
000127 |                    "conformal mode, tight admissible projective limit (chi=729/665), 0 new dials; "
000128 |                    "residual [O] = the full non-perturbative projective limit only")