Global Symplectic Equivalence Does Not Determine a Lagrangian Torus Fibration

Consider the projection of the four-torus onto its last two coordinates, equipped with a constant magnetic symplectic form with real parameter c. We compare global symplectic equivalence with equivalence preserving the Lagrangian fibration. Fibre-preserving equivalence identifies exactly parameters equal up to sign and an integer shift. Global equivalence is equality of the period subgroups \(\mathbb{Z}\)\({}+c\mathbb{Z}\): for rational parameters it is equality of reduced denominators, while for irrational parameters it coincides with fibre-preserving equivalence. The parameters 1/5 and 2/5 therefore have globally symplectomorphic total spaces but no fibre-preserving equivalence over any base diffeomorphism; an explicit integral matrix realizes the global map. This disproves the unrestricted global-versus-fibre-preserving implication recorded in Leung-Symington, Section 7.2, p.185. The magnetic construction, twisting invariant and integral alternating normal-form method are classical and credited. The contextual boundary-domain version of AMR-020-0202 is not claimed solved. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.