Function patch_isometry_flow_reparameterization 

pub fn patch_isometry_flow_reparameterization(
    evaluator: &dyn SaeBasisEvaluator,
    decoder: ArrayView2<'_, f64>,
    row_coords: ArrayView2<'_, f64>,
) -> Result<Option<PatchIsometryFlowReparameterization>, String>

Compute the minimum-isometry-defect flow reparameterization of a fitted d = 2 free/patch atom: the canonical per-row coordinates t̃_i = φ_θ(t_i) and the exactly-recomposed decoder.

This is the #1019 unblocked free-chart case the issue charter calls out: for a manifold patch (a contractible Euclidean-patch atom) a global truncated flow basis DOES exist (no hairy-ball obstruction — FreePatchFlowBasis), so the defect is genuinely MINIMIZED (not merely measured as on the sphere). The reference metric is the flat g_ref = I, so the canonical chart is the uniform-speed (minimum-anisotropy) one. Everything else — the scale-invariant isometry defect E(θ) = Σ_i ‖A_iᵀA_i − c·Ĝ_i‖²_F, the analytic profiled scale c, the exact Gauss–Newton, the det Dφ > δ diffeomorphism guard, and the exact-LS decoder transport with the CHART_RECOMPOSITION_REL_TOL honesty gate — is the SHARED machinery the torus path uses ([minimize_isometry_defect_flow], [recompose_decoder_exact_ls]); see torus_isometry_flow_reparameterization for the full derivation.

The residual chart freedom after pinning is the finite isometry group of the flat patch with the reference uniform metric: O(2) ⋉ ℝ² (rotation + reflection + translation) — reported on the certificate as the PinnedByCanonicalization residual gauge.

Honest refusals (Ok(None), never a lossy or folded swap)

• degenerate chart: empty rows/basis, non-finite coordinates, a collapsed coordinate box on some axis, or a rank-deficient pullback metric anywhere;
• the optimizer finds no strict improvement over the identity flow (the fitted chart is already minimum-defect within the flow family);
• every improving candidate violates the diffeomorphism guard;
• the basis cannot absorb γ ∘ φ⁻¹ within CHART_RECOMPOSITION_REL_TOL on the audit grid (shared gate).