Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

The comparison requirements

E5: VII-C and the non-transferred analytic problems › The comparison requirements · lines 2167–2224

VII-C's obstruction concerns its specified sharp mixed insertion and Wick matrix VIIC. For comparison, its extension requires:

a Hausdorff locally convex formal algebra extending its fixed multiplet Wick algebra;
an admitted insertion agreeing with the unchanged sharp functional on all smooth configurations;
sequentially continuous admission of its regular approximants in its specified mixed-kernel normal topology;
bounded-set hypocontinuous multiplication and continuous extraction of the field-independent order- $ℏ^{3}$ coefficient of the required word.

Its detecting word is $B_{f} ⋆_{H} W_{g} ⋆_{H} B_{f^{'}}$ , with $W_{g} = \int g h_{00}^{2}$ . The supplied logarithmic coefficient is a conclusion for that package, not a universal obstruction to all fixed-width free-polynomial constructions.

{}1.12

Item	VII-C requires	Present construction
(E1)	Its exact de Donder multiplet and fixed Wick matrix	A free scalar and free gauge-fixed tensor with the normalization $- 4 P_{tr} H_{0}$ ; this block is not absent, but is not identified with VII-C's block
(E2)	Its unchanged sharp zero-Cauchy mixed insertion	A completed fixed-width observable and its isolated scalar-sourced cubic
(E3)	The stated normal-topology approximating family	No width-removal family for the mixed insertion
(E4)	The mandatory triple word and its fully contracted $ℏ^{3}$ extraction	An isolated retarded tree coefficient; no $W_{g}$ word or corresponding extraction

VII-C consequently does not apply and is not contradicted. Unlike X-D-ST, this paper does contain a free graviton block, so that part of the audit has been redone rather than copied as “absent.” Its exact kernel, topology, normalization, and mandatory word still do not match.