Non-coordinate residual extraction

The asymptotic rank of the small Coppersmith--Winograd tensor $P=\mathrm{cw}_2$
over $\mathbb C$ satisfies $\widetilde R(P)<3.876919161$. The new ingredient is
a three-dimensional isotropic space of matrices that is not obtained by
selecting coordinates. It lets a literal copy of $P$ survive inside a tensor
retained from two Coppersmith--Winograd residuals: for positive odd
$\ell_1,\ell_2$ the retained tensor is $P\otimes Z_{\ell_1,\ell_2}$, where
$Z_{\ell_1,\ell_2}$ is the symmetric tensor of a complete bipartite graph, and
every point $\phi$ of the asymptotic spectrum satisfies
$\phi(P\otimes Z_{\ell_1,\ell_2})\ge3\phi(P)(\ell_1\ell_2)^{2/3}$. Three
squaring steps of the resulting extraction theorem, from a Fourier-exchange
seed, give the bound through Strassen's spectral duality and exact integer
cube-root enclosures. A fully stored degeneration
$P^{\otimes2}\oplus\mathrm{cw}_{10}\trianglelefteq\mathrm{cw}_4^{\otimes2}$,
with rational coefficients in paired coordinates, illustrates the construction. A separate theorem shows that the scalar
recurrence tests used, over every finite binary tree from the same seed, cannot
exclude $\phi(P)=3.652$.