Topo-Consynch: Topological Nyquist Limit and Jiangqiao Topological Extraction Criterion for Discrete-Continuous Multimodal Consistency

This paper addresses the inherent structural incompatibility between continuous visual manifolds and discrete LiDAR simplicial complexes in cross-modal perception fusion. Traditional multimodal fusion methods perform full-scale geometric matching, which overfits discretization noise and generates spurious topological artifacts. Starting from Riemannian manifold geometry, persistent homology stability and generalized sampling theory, the work rigorously derives the Topological Nyquist Limit (TNL) theorem, proving that only zero-order and first-order topological invariants maintain stable cross-modal consistency under finite sampling for ego-centric 3D perception, with the maximum valid Taylor expansion order bounded as K_{max}\equiv1. It further proposes the Jiangqiao Topological Extraction Criterion (JTEC), a principled rule to suppress high-order aliasing topological components beyond the sampling critical scale. Built on TNL and JTEC, the Topo-Consynch multimodal consensus framework dynamically adjusts effective coupling coefficient C_{eff} to calibrate valid topological bandwidth and eliminates unphysical high-order topological fluctuations via de Rham curl regularization L_{curl}=d\circ d. The proposed paradigm abandons empirical topological filtering and microscopic numerical matching, relying on macroscopic topological equilibrium to resolve discrete-continuous cross-modal aliasing, with universal applicability in autonomous driving, remote sensing, medical multimodal imaging and robotic perception.