TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/tfpt-unified-source-derivation-20261029/aps_source_review/F5-REVERSE-HOM-FORM-TRACE-ADDENDUM.txt
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000001 | CANONICAL TRACE-CLASS REALIZATION OF THE GIVEN L1 REVERSE-HOM FORM
000002 | 30 October 2026. Addendum to F5-QP-REVERSE-HOM-KERNEL-REVIEW.txt.
000003 | The preceding report remains unchanged. No cutoff, state or reader is added.
000004 | 
000005 | 1. Use the actual source functions and the already specified form
000006 | Let X_q=(2cosh(ell h_q))^-1 and X_p=(2cosh(ell h_p))^-1. The actual
000007 | geometric map has j h_q j*=-h_p, hence j X_q j*=X_p. Both X are positive
000008 | and smoothing on the existing compact one-dimensional boundary source.
000009 | Let m be the original mu coefficient AFTER its given tangent/Clifford
000010 | contraction and the actual spin/coefficient pairing. At the ramification
000011 | marks m behaves at worst as |theta-theta_j|^-2/3, so m is L1. This does not
000012 | make raw multiplication M_m, or M_m X_q, an everywhere-defined L2 operator.
000013 | 
000014 | There is nevertheless a source-determined realization of exactly its
000015 | diagonal form pairing:
000016 |   F_m=X_p^(1/2) M_m X_q^(1/2) j*:H_p -> H_p.          (1)
000017 | The two positive roots are the spectral roots of the EXISTING crossing
000018 | kernels. They are not a newly chosen ultraviolet cutoff or state.
000019 | 
000020 | 2. Trace-class proof with the singular coefficient retained
000021 | Use the pointwise polar decomposition m=U|m|, with U the partial isometry
000022 | between the actual q and p coefficient fibers. The factors
000023 |   A=X_p^(1/2) U |m|^(1/2),
000024 |   B=|m|^(1/2) X_q^(1/2)
000025 | are Hilbert-Schmidt. For example
000026 |   ||B||_HS^2=integral tr[|m|(theta) X_q(theta,theta)] dtheta<infinity.
000027 | The diagonal of the positive smoothing kernel is bounded; the L1 norm of
000028 | m controls the integral. The same argument with U*X_p(theta,theta)U proves
000029 | the assertion for A. These formulas first define the factors on smooth
000030 | test fields and give their unique Hilbert-Schmidt extensions. Therefore
000031 | F_m=ABj* is trace class, even when the unsandwiched product is not an L2
000032 | operator. No extra pointwise vanishing at the four marks is required.
000033 | 
000034 | By cyclicity of Hilbert-Schmidt products and
000035 |   j* X_p^(1/2)=X_q^(1/2)j*,
000036 | its ordinary trace is exactly
000037 |   Tr F_m=integral tr[m(theta)(X_q j*)(theta,theta)] dtheta. (2)
000038 | Thus this realizes the SAME original finite L1 form pairing, with its
000039 | correct endpoint order. It neither removes the singularity from m nor
000040 | equates the raw M_m X_q j* with an unproved trace-class L2 product.
000041 | 
000042 | All statements are transported by the existing W maps with their given
000043 | metric, half-forms, connection and corner domains. The full Hom deck test
000044 | of the preceding review applies to (1) as well, since the positive roots
000045 | are equivariant and no new reader character has been inserted.
000046 | 
000047 | 3. Physical and sewing types remain precise
000048 | Equation (1) is an actual source-response operator and (2) is its form
000049 | trace. Declaring the many-body observable dGamma(F_m) would be a further
000050 | operator assertion. In particular its state expectation, when defined,
000051 | would be Tr(G_p F_m), not automatically Tr(F_m). This addendum does not
000052 | replace the original insertion or phase functional by a new response
000053 | algebra or by that invented expectation rule.
000054 | 
000055 | The original U_Gamma=M_z matching and the geometric R transport also have
000056 | different roles. R is already a covariance/automorphism of that matching
000057 | under endpoint exchange. It does not have to EQUAL the gluing operator.
000058 | The prior report's trace formula (11) is a general diagram for comparing
000059 | actual endpoint maps; it does not impose b_Gamma=R_x as a new requirement.
000060 | A sewing diagram may keep R as a symmetry and U_Gamma as the matching.
000061 | The next original-source test is the actual insertion/trace consumer of
000062 | that diagram, retaining the existing whole graph and state.