Manuscript XV-A@1 · XV-A.tex · section · f53f3515

The precomposition sum and the missing operation

Internal kernels are not the joint quantum operation › The precomposition sum and the missing operation · lines 1020–1086

XV-A@1#eq:sum, XV-A@1#eq:supported, XV-A@1#eq:K47, XV-A@1#eq:K6, XV-A@1#eq:map

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The copied common presentation is

K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9} .

Here $K_{47}$ groups branches $B_{4} + B_{7}$ . Thus five named summands represent the retained six-branch sum $B_{4}, \dots, B_{9}$ ; this notation does not delete a branch. The supported sector is only

K_{supp} = K_{6} + K_{8} + K_{9} .

The source formulas include

K_{47}^{I, J} K_{6}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],

with the inherited bilinear tensor map determined by

a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J} .

These are copied internal formulas, not new joint composition formulas Iamp.

The unresolved operation has the form

M : K_{int} ⟶ D^{'} (W; E_{res}),

where $E_{res}$ denotes the residual-species bundle. Its required value on (eq:sum) must retain the prescribed grouped quantum cross-lines, vertex Hessian, and tensor coefficients.

The supplied I3 statement requires: agreement with canonical composition on its proved domain; definition on tests meeting $N$ ; retention of the grouped factors and coefficients; and an appropriate complete-graph wavefront bound Ijoint. Internal reconstruction alone proves none of these requirements for the new cross-line operation.

In particular, XV-A does not identify $U_{can}$ with $W ∖ N$ , or with another entire complement of an attachment locus. The direct I3 excerpt explicitly warns that an internal representation does not establish such a domain equality.

Displayed formulas in this passage

XV-A@1#eq:sum · line 1023
$K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9} .$
XV-A@1#eq:supported · line 1031
$K_{supp} = K_{6} + K_{8} + K_{9} .$
XV-A@1#eq:K47 · line 1037
$K_{47}^{I, J} K_{6}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-A@1#eq:K6 · line 1037
$K_{47}^{I, J} K_{6}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)],$
XV-A@1#eq:map · line 1065
$M : K_{int} ⟶ D^{'} (W; E_{res}),$
unlabeled · line 1057
$a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J} .$