= Linear capacity rigidity
{title2=$c(TE)=c(E)\text{ for all ellipsoids}\implies T^*\omega=\pm\omega$}
An invertible real linear map preserving <symplectic capacities> of all <solid ellipsoids> is symplectic or <anti-symplectic>. For a pair with $\omega(u,v)=1$, if $0<|\omega(T^Tu,T^Tv)|<1$, complete this pair and its normalized transformed pair to <symplectic bases>. The resulting transpose matrix sends a unit ball into a <symplectic cylinder> of radius less than one, contradicting the capacity axioms after iterating the resulting linear map: its first coordinate pair contracts geometrically, while the image ellipsoid retains positive capacity. Perturbation handles a zero pairing; using $T^{-1}$ handles absolute value greater than one. Thus the absolute bilinear pairings agree. Their squares agree as polynomials, and the polynomial ring being an <integral domain> forces one global sign.
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