TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/boundary_origin_20261027/ETA-CONVENTION-AND-BOUNDARY-FACTOR.txt
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Inclusive original lines: 1-73
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000001 | ETA PHASES DO NOT BY THEMSELVES IDENTIFY THE SOURCE BOUNDARY LINE
000002 | 2026-10-27
000003 | 
000004 | Scope: a convention and type check, not a no-go for a TFPT anomaly theory.
000005 | The new positive Hardy cable construction is a separate result. This note
000006 | prevents a generic eta theorem from being used as its missing physical
000007 | operator identification.
000008 | 
000009 | 1. Fix lambda(E)=det H0(E) tensor det H1(E)^(-1). The positive theta
000010 |    determinant line is lambda(E)^(-1), not lambda(E).
000011 | 
000012 | 2. Dai and Freed define their graded determinant as det(V-) tensor
000013 |    det(V+)^(-1), their equation (2.7). For a Dolbeault boundary operator
000014 |    this is lambda(E)^(-1). Their exponentiated invariant
000015 | 
000016 |        tau=exp(2 pi i xi),   xi=(eta+dim ker D)/2,
000017 | 
000018 |    takes values in the inverse of that determinant line, hence in lambda(E)
000019 |    in the present convention. Merely reading the word 'inverse' without
000020 |    translating conventions reverses the conclusion.
000021 | 
000022 | 3. The original primitive Euclidean action includes +(i pi/2) Delta eta.
000023 |    Its contribution to exp(-S) is therefore
000024 | 
000025 |        a=exp(-i pi Delta eta/2).
000026 | 
000027 |    Even when zero modes vanish and the compared eta invariants are those
000028 |    in the Dai-Freed setting, the numerical relation is
000029 | 
000030 |        a^2=tau_rel^(-1),
000031 | 
000032 |    not a=tau_rel. If zero modes are retained, the missing factor is
000033 |    exp(i pi Delta dim ker D). Any square-root determinant or Pfaffian
000034 |    interpretation needs its actual real structure and gluing data.
000035 | 
000036 | 4. As a conditional diagnostic, if the boundary operator pair were
000037 |    E(-Dq) relative to E, its cohomology determinant ratio would be Bq^(-1).
000038 |    Under precisely that dictionary the square a^2 would be in Bq, not
000039 |    automatically in Bq^(-2) as a=Nq^(-1) would require. This comparison is
000040 |    not a claim that the original D_rel/D_ref is this Dolbeault pair.
000041 | 
000042 | 5. The original D_ref and D_rel are even and odd parts of a given operator
000043 |    under the geometric involution. This definition supplies neither the
000044 |    boundary Dolbeault operator nor the bundle pair (S0 tensor L,L) required
000045 |    by the relative Hardy norm construction. A genuine application must
000046 |    identify the boundary family, coefficient bundles, orientation, kernel
000047 |    correction and determinant/Pfaffian convention on the actual source.
000048 | 
000049 | Verdict: no eta-only shortcut closes that physical identification. The
000050 | finite clock phase and the correctly constructed Hardy norm line remain
000051 | valid. This note makes no claim that a suitably derived real or relative
000052 | anomaly theory cannot realize the needed line.
000053 | 
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000063 | 
000064 | Original source checked through the current Atlas:
000065 | paper-source:8027564739689dd02e02f11c
000066 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/
000067 | sections/001-section-primitive-core-and-carrier-interface-handoff.tex
000068 | lines 302-318 (primitive Euclidean action and source-net distinction).
000069 | SHA256 7d6e673d99c29d950dc601283cd51c1cc0b113567eeb79c8396fec9c82a02c79
000070 | 
000071 | The same source's general identification of the geometric deck reflection
000072 | with Calderon polarization is not adopted; the existing scoped counterexample
000073 | in seam-full-collar-reconstruction-20261015 remains authoritative.