= Twisted half-smash product
{title2=$X\ltimes_fE$}
For compact $X$ over the space of <linear isometries> $U\to U\prime$, choose a <subordinate flag for a family of linear isometries> $V_i,W_i$. The <vector bundle> $\xi_i$ has fiber $W_i\ominus f_x(V_i)$. Form $P(W_i)=\operatorname{Th}(\xi_i)\wedge E(V_i)$. Its structure maps use the orthogonal decomposition
$$
(W_j\ominus W_i)\oplus\xi_i\cong\xi_j\oplus f_x(V_j\ominus V_i)
$$
and the structure map of $E$. The <twisted half-smash product> is the <spectrification> $LP$, after extending from the cofinal flag. Unlike an arbitrary replacement by $X_+\wedge E$, this definition retains the specified isometry family at the point-set level.
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