Manuscript IX-B@2 · IX-B-v2.tex · section · 8ce915d6

Continuity reconciliation

Involution and the three continuity statements › Continuity reconciliation · lines 1319–1369

Topology or operation	Statement and source
VII-C normal stages	The unchanged kernels $b_{f, ε}$ converge to $b_{f}$ , while the unchanged coefficient $G_{ε}$ diverges for separated witness tests. Agreement with the old product and sequential continuity are incompatible.
IX-A admission topology $τ_{Σ}$	For every nonzero real $f$ , $B_{f, ε} \neq \to B_{f}$ . The additional boundary-jet coordinates prevent the normal-stage admission convergence. This is a different topology, not a subtraction.
VII-D numerical subtraction	$T_{ε}^{ρ} - lo g ((μ ε)^{- 1}) r \to u_{μ}^{ρ}$ strongly in $D^{'} (X)$ . IX-B transfers this changed numerical operation to $a D$ ; it does not infer a sharp product.

The first row is VII-C's coefficient obstruction. Its algebra consequence assumes all of (E1)–(E4): retention of the old bulk Wick algebra, exact sharp admission, normal-topology sequential admission continuity agreeing on regular kernels, and bounded-set hypocontinuous multiplication with continuous field-independent coefficient extraction VIIC (ass:extension, cor:noalgebra). No relaxation of those requirements is silently introduced here.

The second row is IX-A's explicit nonconvergence and compatibility boundary IXA (thm:IX1, cor:compatibility). It is consistent with the different-topology possibility that VII-C explicitly does not exclude. No claim is made that $τ_{Σ}$ is globally finer than every normal topology on every polynomial stage, or that it supports a hypocontinuous quantum product.

The third row is Assumption ass:VIID and Theorem thm:descendants. As VII-D's scope audit states, this numerical subtraction leaves the VII-C obstruction unchanged. A limit in the strong topology of joint-test distributions is not admission continuity for a sharp binary algebra.