TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/primitive-seam-operator-reconstruction-20261025/OS-SOURCE-REPAIR.txt
Original file SHA-256: 50bc771398e8a363a617c677f503f556c9f2adccf6e40cfcb460ca4eb9f55ba6
Inclusive original lines: 1-80
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000001 | OS SOURCE REPAIR: FULL MOMENTS, GRADED WORDS, AND THE PRIMITIVE SEAM BOUNDARY
000002 | ================================================================================
000003 | 
000004 | Scope and verdict
000005 | -----------------
000006 | 
000007 | This note repairs one precise mathematical defect in the archived TFPT
000008 | Osterwalder--Schrader argument.  It does not construct a new physical source.
000009 | 
000010 | PROVED here:
000011 | 
000012 | 1. The connected or truncated distributions
000013 | 
000014 |        kappa_n = delta^n log Z |_{0}
000015 | 
000016 |    cannot themselves be used as the full Schwinger moment functional in the OS
000017 |    Gram form.  A two-variable Gaussian already gives a 2 by 2 counterexample.
000018 | 
000019 | 2. The correct input is the full normalized moment functional
000020 | 
000021 |        S_n = Z[0]^{-1} delta^n Z |_{0},
000022 | 
000023 |    equivalently the graded moment--cumulant expansion of exp(log Z).
000024 | 
000025 | 3. A unital, reflection-intertwining boundary word map which preserves the
000026 |    complete original Z-functional on all reflected word products induces an OS
000027 |    isometry.  This is a sufficient repair theorem.  It does not require odd CAR
000028 |    fields to belong to the physical observable algebra.
000029 | 
000030 | OPEN after the repair:
000031 | 
000032 | The cited TFPT source does not construct Z_adm from D_+, B_Sigma, the Calderon
000033 | projector, or the relative spectral/Chern--Simons expressions.  It also does not
000034 | provide the state-preserving boundary word map required below.  Thus this note
000035 | repairs the logical form of the OS step but does not close the primitive source
000036 | gap.
000037 | 
000038 | 
000039 | 1. The defect in the cited source
000040 | ---------------------------------
000041 | 
000042 | The relevant original passages are:
000043 | 
000044 | * documents/archive--tfpt-45--source_extracts--04_qft_source/sections/
000045 |   005-subsection-admissible-schwinger-functions-minkowski-loca.tex:144-161
000046 |   defines W_rel = log Z_rel and explicitly calls
000047 | 
000048 |        S_n^T = delta^n W_rel |_{0}
000049 | 
000050 |   the truncated admissible Schwinger distributions.
000051 | 
000052 | * The same file, lines 163-184, then applies the OS axioms directly to
000053 |   {S_n^T} and says that the reconstructed theory has these truncated
000054 |   distributions as its Schwinger functions.
000055 | 
000056 | * documents/archive--tfpt-45--source_extracts--04_qft_source/sections/
000057 |   008-section-relative-reflection-positivity-with-fermions.tex:181-196 and
000058 |   217-254 repeats the same typing: it proves distributional properties of the
000059 |   truncated family and supplies that family directly to the stated OS/CAR
000060 |   reconstruction theorem.
000061 | 
000062 | This is not the standard OS input.  Reflection positivity is a positivity
000063 | condition on the full moment functional, or equivalently on the normalized
000064 | generating functional Z/Z[0].  It is nonlinear under passage from moments to
000065 | connected cumulants.  Positivity of a reflected full Gram matrix is therefore
000066 | not positivity of a matrix formed by replacing its entries with cumulants.
000067 | 
000068 | 
000069 | 2. Small exact counterexample
000070 | -----------------------------
000071 | 
000072 | Let X be a centered real Gaussian variable smeared at positive Euclidean time,
000073 | and write Y = Theta X for its reflected copy.  Choose the covariance
000074 | 
000075 |        E[X^2] = E[Y^2] = a > 0,
000076 |        E[YX] = c > 0,
000077 | 
000078 | with 0 < c <= a, so that the elementary covariance matrix
000079 | 
000080 |        [ a  c ]