All inputs

This section decides one complete family. It does not change its density map to isolate a difference. Its degeneracy is explicit: five internal rows and the second crossed tensor product are identically zero.

Choose a symmetric-tensor basis with $e_0=\eta$, completed by the six off-diagonal symmetric tensors and the three trace-free diagonal tensors

Use its algebraic dual. The residual fiber is ${\mathbb{C}}^{10}$.

Fix

Then

Let $U$ be a sufficiently small relatively compact neighborhood of

Require disjoint closures of the $b$- and $y$-projections, the time inequalities $b^0>y^0>a^0$, $b^0>z^0$, and, in the fixed-frame coordinates below,

Thus $A<B$.

Choose fixed compact smooth ${\zeta }_b,{\zeta }_y$, equal to one near the respective endpoint projections, with disjoint supports. Set

Hence $F(b)=b^0$, $g(b)=1$, and $g(y)=y^0$ on the relevant neighborhoods.

Define the quadratic source

Its derivative is

The source array used here encodes this derivative, not its half-polarization. Its sole nonzero coefficient is

with both derivative multiindices zero.

Choose the even normalized bump

and set

For $h=A\ell +uk+X$, $X\in {\mathbb{R}}^2$, define

Since ${\mathrm{d}}^{4}h=2\mathrm{d}A\mathrm{d}u{\mathrm{d}}^{2}X$, these profiles have integral one.

The crossed inputs are

Fix any compact $\chi =1$ near $\mathfrak{f}(\overline{U})$.

Define the smooth attachment and reference domain by

The reference is zero, as proved below. Set ${\mathcal{C}}_{\mathrm{allowed}}=\{0\}$: no nonzero scale-dependent subtraction or finite addition is authorized for this package. No restrictive quantum output condition is imposed.