TFPT scientific source excerpt
Original repository path: verification/v311_gap_decoupled_measure.py
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000001 | """v311 -- the physical QG measure as a gap-decoupled, exponentially-clustering object.
000002 | 
000003 | Attack point 4 of the TOE roadmap: the nonperturbative ambient measure QG.AMB.01 /
000004 | G_metric.  Rather than (over-)claim a 4D interacting measure, this module proves the
000005 | honest, useful statement: the PHYSICAL (admissible) sector is a well-defined,
000006 | convergent object on its own, GAP-DECOUPLED from the un-built ambient measure.  The
000007 | spectral-action shadow (a2 = Einstein-Hilbert, a4 = R^2/72; v36) and the KMS beta=1
000008 | cutoff (v259) give the low-energy readout; this module adds the convergence side.
000009 | 
000010 |   [E] 1. CLUSTERING: the seam transfer operator T has spectrum {1, (2/3)^6, (1/3)^6}
000011 |         (v221/v302); on the deviation (trace-zero) subspace the connected correlator
000012 |         decays geometrically, <O_0 O_n>_c ~ (2/3)^{6n}, so correlations cluster
000013 |         exponentially with correlation length xi = 1/Delta, Delta = 6 ln(3/2).
000014 |   [E] 2. FINITE SUSCEPTIBILITY: sum_n (2/3)^{6n} = 1/(1-(2/3)^6) = 729/665 < infinity
000015 |         -- the admissible-sector connected sum converges (a well-defined thermodynamic
000016 |         limit / unique vacuum on the admissible sector).
000017 |   [E] 3. GAP-DECOUPLING (v76): the margin Delta - 31/(4 pi^2) = 6 ln(3/2) - 31/(4 pi^2)
000018 |         ~ 1.648 > 0 means the gap SURVIVES metric dressing, so the physical sector is
000019 |         decoupled from the (un-built) ambient measure -- QG.AMB is math completeness,
000020 |         not a physical input.
000021 |   [E] 4. NEG CONTROL: an ungapped/non-primitive transfer (second eigenvalue -> 1)
000022 |         sends xi -> infinity and the susceptibility sum -> infinity (no clustering,
000023 |         no well-defined sector) -- the convergence is the gap, not generic.
000024 | 
000025 | HONEST SCOPE: [E] the clustering / convergence / margin (the admissible-sector
000026 | measure is a convergent object); [C] the spectral-action low-energy dictionary
000027 | (v36/v259); [O] the FULL ambient measure QG.AMB.01 is NOT constructed -- it is
000028 | gap-decoupled and inert, so the physical sector does not need it.  Python-only.
000029 | """
000030 | import numpy as np
000031 | 
000032 | from tfpt_constants import check, summary, reset, N_fam
000033 | 
000034 | 
000035 | def transfer():
000036 |     """The gapped boundary transport (v221): spectrum {1,(2/3)^6,(1/3)^6}."""
000037 |     lam2, lam3 = (2 / 3) ** 6, (1 / 3) ** 6
000038 |     u2 = np.array([1.0, -1.0, 0.0]); u2 /= np.linalg.norm(u2)
000039 |     u3 = np.array([1.0, 1.0, -2.0]); u3 /= np.linalg.norm(u3)
000040 |     J = np.ones((3, 3)) / 3.0
000041 |     T = J + lam2 * np.outer(u2, u2) + lam3 * np.outer(u3, u3)
000042 |     return T, lam2, lam3
000043 | 
000044 | 
000045 | def run():
000046 |     reset()
000047 |     print("v311  gap-decoupled, exponentially-clustering physical QG measure")
000048 | 
000049 |     T, lam2, lam3 = transfer()
000050 |     Delta = 6 * np.log(1.5)                       # = 2 N_fam ln(3/2)
000051 |     check("transfer spectrum = {1, (2/3)^6, (1/3)^6} = {1, 64/729, 1/729}; "