Linear capacity rigidity ($c(TE)=c(E)\text{ for all ellipsoids}⟹T^{\ast }\omega =\pm \omega$)

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 $0<|\omega (T^{T}u,T^{T}v)|<1$, 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.