arXiv:2610.34494v1 Announce Type: new Abstract: Eigensolvers return orthonormal bases of invariant subspaces, but each basis is determined only up to a unitary factor, so bases computed at different parameter values cannot, in general, be interpolated directly. Discrete parallel transport resolves this ambiguity by aligning consecutive bases through the polar factors of their overlaps, but approximates the continuous parallel-transported frame only to second order. We show that the resulting frames can nevertheless be interpolated with spectral accuracy. For analytic families of invariant subspaces sampled at $N+1$ Chebyshev nodes, the computed frames are exact samples of an analytic frame related to the parallel-transported frame by a gauge that remains within $O(N^{-2})$ of the identity throughout a fixed complex neighborhood, uniformly in $N$. Consequently, Chebyshev interpolation approximates this analytic frame with geometric convergence. Moreover, the continuous parallel-transported frame can be recovered to spectral accuracy by solving a small matrix differential equation, without additional eigensolves or derivatives of the underlying matrix family. The results do not require the family to be Hermitian and remain valid through eigenvalue crossings and branch points, provided the selected invariant subspace depends analytically on the parameter. Numerical examples confirm the predicted rates.