The SMG-Yau-Yau Filter and Lossless Data Assimilation: Resolving Infinite Lie Algebras via Statistical Fiber Theory
Abstract
Classical continuous-time non-linear filtering fails in generic non-linear state spaces due to an infinite Lie algebraic derivative explosion (dim(ℰ) = ∞), leading to filter divergence. To resolve this four-decade crisis, we introduce the SMG-Yau-Yau filter and lossless data assimilation framework built on Statistical Fiber Theory over an Orlicz manifold ℳ. By equipping ℳ with a Riemannian submersion and an Ehresmann connection, the unconstrained Duncan-Mortensen-Zakai score velocity field is orthogonally decomposed into Statistically Verifiable Directions (SVDχf) and Structural Internal Directions (SIDf). System non-linearities and unclosed Lie commutators are orthogonally quarantined in SIDf, protecting macroscopic base parameters from spatial derivative pollution while preserving total score variance energy. We unify the asymptotics through a Dual-Axis Collapse Mechanism: proving our model identically recovers classical Yau-Yau dynamics when dim(ℰ) < ∞, while large-sample limits (N → ∞) induce a thermodynamic quench that flattens infinite-dimensional geometry under dim(ℰ) = ∞. Finally, we formulate the Active Acausal Tension (AAT) functional to monitor accumulated model misspecification online. Exceeding a topological capacity threshold triggers Gauge Symmetry Breaking (GSB), which dynamically expands base coordinates (d → d+1) to ensure non-asymptotic stability and convergence to the exact state density.
Related
- Non-isomorphism of q-Gaussian von Neumann Algebras in Infinite Variables
- Higher Fomin-Kirillov algebras are infinite-dimensional
- On a fundamental obstruction in the Derscariu Programme: Failure of Dixmier trace factorisation on infinite tensor products of Toeplitz algebras
- Discrete C*-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
- Just-Infinite Loops and Loop Algebras
BibTeX
@misc{cheng2026smgyauyau,
title = {{The SMG-Yau-Yau Filter and Lossless Data Assimilation: Resolving Infinite Lie Algebras via Statistical Fiber Theory}},
author = {Bing Cheng and Yi-Shuai Niu and Howell Tong and Shing-Tung Yau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28062},
eprint = {2609.28062},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28062},
}
RIS
TY - UNPB
TI - The SMG-Yau-Yau Filter and Lossless Data Assimilation: Resolving Infinite Lie Algebras via Statistical Fiber Theory
AU - Bing Cheng
AU - Yi-Shuai Niu
AU - Howell Tong
AU - Shing-Tung Yau
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28062
UR - https://arxiv.org/abs/2609.28062
AB - Classical continuous-time non-linear filtering fails in generic non-linear state spaces due to an infinite Lie algebraic derivative explosion (dim(ℰ) = ∞), leading to filter divergence. To resolve this four-decade crisis, we introduce the SMG-Yau-Yau filter and lossless data assimilation framework built on Statistical Fiber Theory over an Orlicz manifold ℳ. By equipping ℳ with a Riemannian submersion and an Ehresmann connection, the unconstrained Duncan-Mortensen-Zakai score velocity field is orthogonally decomposed into Statistically Verifiable Directions (SVDχ_f) and Structural Internal Directions (SID_f). System non-linearities and unclosed Lie commutators are orthogonally quarantined in SID_f, protecting macroscopic base parameters from spatial derivative pollution while preserving total score variance energy. We unify the asymptotics through a Dual-Axis Collapse Mechanism: proving our model identically recovers classical Yau-Yau dynamics when dim(ℰ) < ∞, while large-sample limits (N → ∞) induce a thermodynamic quench that flattens infinite-dimensional geometry under dim(ℰ) = ∞. Finally, we formulate the Active Acausal Tension (AAT) functional to monitor accumulated model misspecification online. Exceeding a topological capacity threshold triggers Gauge Symmetry Breaking (GSB), which dynamically expands base coordinates (d → d+1) to ensure non-asymptotic stability and convergence to the exact state density.
ER -