Positive-scale activity

For completeness, all seven retained row contractions remain nonzero at a common sufficiently small positive scale. At a non-tip null point $x=\alpha {\ell }_{\mathbf{n}}$, direct temporal cone integration gives

Here $P_{\ast }^{\prime \prime }(0)<0$. Consequently, for positive $\alpha ,\lambda$,

These limits follow by expanding the translated non-tip cone formula for $A_{\delta }$; compact profile support permits differentiation and dominated convergence.

At regular independent cone points, the highest normal polynomials of the two-line kernels are proportional to

The response rows also have nonzero highest line or incidence jets, as their separate adjoints show. The multiplier is nonvanishing near a central aligned point, and each regular seed stratum meets that neighborhood. Multiplication by a smooth nonvanishing function is locally invertible on distributions. Thus the nonzero internal row jets give nonzero contractions. A common scale exists because there are finitely many rows.

A complete kinetic witness is obtained in the first component by choosing

with normal cutoffs constant near $N$, where $\zeta =1$. It has

The mass-vertex contribution is bounded. Hence the complete kinetic contraction is nonzero for all sufficiently small scales.