Floquet Fibre Geometry and Higher-Order Reduced Coordinates for Off-Manifold Transients near Nonlinear Aeroelastic Flutter

Assigning reduced coordinates to states near an attracting limit cycle requires the correct invariant-fibre geometry. The classical first-order phase-isostable chart obtained from adjoint Floquet modes projects along the strong-stable quotient fibre, whereas a metric-orthogonal complement of the retained slow bundle generally does not. We prove locally that a chart satisfying the linearised semiconjugacy relation leaves an O(delta^2) invariance residual, while projection along a non-invariant complement generically leaves an O(delta) term. For a nonlinear aeroelastic limit cycle, the metric-normal and strong-stable directions differ by 48.5 to 71.7 degrees, and metric-normal perturbations contain first-order retained phase and slow-amplitude components. Replacing the metric normal by the strong-stable fibre changes the measured residual scaling from delta^1.01 to delta^1.87 without fitted parameters. We then test learned higher-order corrections whose linearisation is pinned to the adjoint-Floquet chart, whose symmetry is exact, and whose reduced flow is fixed. Although they reduce the registered fixed-normalisation latent residual, post-hoc amplitude recalibration and adjoint-Floquet-targeted future consistency move or reverse the ranking. Because the learned maps already share the baseline's first-order gauge and the future target is supplied by the baseline chart, these diagnostics establish neither an independent positive nor negative higher-order result. Correct first-order Floquet geometry is therefore necessary in this benchmark, while the additional predictive value of the learned correction remains unidentified by the available representation-dependent diagnostics.