Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

What the certificates establish

Problem, contribution, and logical dependencies › What the certificates establish · lines 286–339

Certificate	Establishes	Does not establish
Verified wavefront noncancellation	Applicability of the ordinary-product construction, with the required maps and support hypotheses	Physical identification of that product.
Verified cancelling wavefront configuration	Failure of the displayed sufficient product criterion	Nonexistence of every alternative product or extension.
Moving high-frequency witness	Behavior on that specified test family	Fixed-test raw divergence or full-kernel reconstruction.
Complete controlled expansion	A decision for the specified regularized family	Authorization of its subtraction.
Microlocal extension certificate	Extension and wavefront control under its stated hypotheses	Physical normalization or quantum admission.

The ordinary multiplication theorem used here is sufficient, not a converse nonexistence theorem BDH (Theorem 6.1). Thus the appropriate conclusion from a verified cancellation configuration is:

The displayed wavefront configuration prevents direct application of the ordinary-product criterion; the complete contraction and any alternative definition require separate analysis.

The algebraic distinction between an open and a closed attachment pattern can be seen from

(k, - k, 0) + (0, l, - l) = 0 ⟹ k = l = 0,

whereas

(k, - k, 0) + (0, l, - l) + (p, 0, - p) = 0 ⟺ k = l = - p .

Whether the latter nonzero covectors actually belong to the specified kernels' wavefront sets depends on their prescriptions and endpoint orientations. This algebra is not a calculation of the complete interaction coefficient.

Displayed formulas in this passage

unlabeled · line 326
$(k, - k, 0) + (0, l, - l) = 0 ⟹ k = l = 0,$
unlabeled · line 331
$(k, - k, 0) + (0, l, - l) + (p, 0, - p) = 0 ⟺ k = l = - p .$