Non-Identity Calculus: Resolving the Curse of Dimensionality via Topological Spectral Collapse
Year: 2026
Type: preprint
Abstract: The "curse of dimensionality" poses a fundamental computational barrier in numerical analysis, quantum field theory, and high-dimensional optimization, scaling exponentially as O(N^D). In this paper, we introduce Non-Identity Calculus, a novel mathematical paradigm that completely bypasses this limitation. By defining a parameterized Non-Identity Operator N̂_β and enforcing a Non-complex measure constraint, we project the D-dimensional integration space onto an invariant k-dimensional sub-manifold (k << D). We rigorously prove that the residual error bounded by the uncollapsed dimensions decays exponentially, while the computational complexity collapses to polynomial time O(N^k). We apply this framework to infinite-dimensional Feynman Path Integrals, demonstrating natural ultraviolet (UV) regularization without ad-hoc cutoffs. Finally, we map the tensor structures to modern GPU architectures (NVIDIA RTX 4090, sm_89), leveraging L2 cache persistence to evaluate 100-dimensional integrals in sub-millisecond execution times. This theoretical breakthrough establishes an exact numerical foundation for holographic principles and ultra-high-dimensional topology.
Author: Lee, Seonggil
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208715563
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