Manuscript XIV-H@1 · XIV-H.tex · section · 695c7c3a

H3: common representation not entered

H3: common representation not entered · lines 1683–1751

The exact regrouping (eq:regroup) continues to hold for the already defined smooth-slot tree functionals. The new kernel formulas refine three pieces of that identity with $z$ retained.

They do not construct one characteristic operator representation of (eq:target).

The pieces remain:

$C_{G}$ after the exact cancellation of the explicit second derivative of the scalar test;
one divergence of the complete tensor $B^{α μ} [G; h]$ ;
the complete scalar root (eq:root).

The $B_{6}$ numerator is not substituted for the second piece or for the complete sum.

Piece	Established internal information	Still required
$K_{5}$	Exact two-placement tensor pullbacks and regular relation charts	Uniform amplitude description at the internal characteristic diagonal.
$K_{8}$	Exact retarded-convolution reduction and regular conormal representation	Compatibility with a common representation of all retained pieces and the grouped cross-lines.
$K_{9}$	Complete root, including the mass term, and regular conormal representation	The corresponding common-composition theorem.
$K_{4}, K_{6}, K_{7}$	Inherited exact tree kernels and regrouping	Their participation in that same justified common representation.

No ordinary product theorem is asserted for multiplying these kernels by the grouped cross-line difference at $N$ . The copied XIV-G audit already prevents an automatic factorwise argument. It is not retried.

A generalized composition would have to be proved for its own actual relation and amplitude hypotheses. No such theorem is supplied here.

No equality

U_{can} = W ∖ N_{att}

is asserted. The pre-existing canonical domain remains only the domain on which the complete specified products have already been proved.

Displayed formulas in this passage

unlabeled · line 1744
$U_{can} = W ∖ N_{att}$