The Lipschitz isomorphism problem for separable Banach spaces

Theorem 1.1: there are separable real Banach spaces $X,Y$ and a bijection $\Psi:X\to Y$ with $\frac{4}{21}\|s-t\|\le\|\Psi(s)-\Psi(t)\|\le\frac{76}{25}\|s-t\|$, where $X$ contains a linearly isometric copy of $c_0(\ell_2)$ and $Y$ contains no subspace isomorphic to $c_0(\ell_2)$, so $X$ and $Y$ are not linearly isomorphic. The construction uses a near-isometric bi-Lipschitz change of variables between Hilbert spaces with a local orthogonality property, Lipschitz-free spaces, and a triangular graph-space construction. Real scalars only; it asserts no reflexive or complex analogue and does not say $Y$ lacks ordinary $c_0$. The companion on $c_0$ absorption gives a second, independent counterexample and is a separate entry.

Theorem 1.1: there are separable real Banach spaces $X,Y$ and a bijection $\Psi:X\to Y$ with $\frac{4}{21}\|s-t\|\le\|\Psi(s)-\Psi(t)\|\le\frac{76}{25}\|s-t\|$, where $X$ contains a linearly isometric copy of $c_0(\ell_2)$ and $Y$ contains no subspace isomorphic to $c_0(\ell_2)$, so $X$ and $Y$ are not linearly isomorphic. The construction uses a near-isometric bi-Lipschitz change of variables between Hilbert spaces with a local orthogonality property, Lipschitz-free spaces, and a triangular graph-space construction. Real scalars only; it asserts no reflexive or complex analogue and does not say $Y$ lacks ordinary $c_0$. The companion on $c_0$ absorption gives a second, independent counterexample and is a separate entry.