Manifold Hypothesis: Conditional Certificates and Consistent Intrinsic Dimension Estimation

Abstract

We study what noisy data can justify before a manifold and its intrinsic dimension are assumed. For repeated observations with independent isotropic Gaussian errors of unknown variance, we compose a support confidence event with a sound geometric verifier to certify a nearby smooth witness of arbitrary compact latent support. Whole-support inference requires an external lower mass condition; independent validation instead certifies proximity of probability mass. Separate all-scale hypotheses yield true-support structure and dimension recovery: a credited Reifenberg--Dini criterion and positive continuous density give local $C^1$ structure and consistent dimension estimation, while two-sided Ahlfors regularity and positive reach give quantitative covariance separation under the stated sampling and observation-transfer conditions. An exact-arithmetic implementation verifies restricted witness families. A further matrix check certifies linear coordinates on a geometrically verified affine span containing the witness and propagates approximation and decoding errors. This restricted construction neither implements the general minimal-dimension Whitney theorem nor infers exact manifold support from finite approximation. Executed torus and Julia-set experiments test the observation-to-witness-to-coordinate route, including singular-support controls. Dynamical systems and chaos are the primary application context; state-identification and finite-horizon prediction results are conditional on observation embeddings and transition validation. Takens reconstruction is an optional scalar-delay application requiring additional smoothness, dynamics and observable conditions. Image comparisons are auxiliary reconstruction diagnostics. The finite-data verifier establishes neither those dynamical conditions nor the all-scale geometric premises for true-dimension recovery.

arXiv comment139 pages, 15 figures; 4-page numerical supplement included as an ancillary file