# Statistically Meaningful Geometry (SMG) Beyond the Euclidean Paradigm (Raw Archive)

- **arXiv**: 2607.03329v1
- **Authors**: Bing Cheng (CAS), Yi-Shuai Niu (BIMSA), Howell Tong (LSE/Tsinghua), **Shing-Tung Yau** (BIMSA/Tsinghua)
- **Published**: July 3, 2026
- **Category**: cs.LG, stat.ME
- **Length**: 96 pages

## Abstract

Conventional uniform convergence bounds and empirical risk minimization break down in massive over-parameterized models. With near-infinite unconstrained internal degrees of freedom, optimization landscapes develop flat vertical gauge valleys, rendering classical generalization metrics vacuous and inducing generative hallucination and catastrophic forgetting. We introduce the Statistically Meaningful Geometry (SMG) framework, an information-geometric paradigm lifting deterministic parametric models into infinite-dimensional non-parametric Orlicz statistical manifolds. Modeling the total state space as a differential fiber bundle (M, B, π, V, H, ω), we establish a Two-Fold Inference Paradigm. We formalize an Ehresmann connection 1-form ω as a dynamic geometric filter that strips away vertical gauge noise (SID) and isolates learning along the non-degenerate horizontal distribution (SVDχ). We prove that under connection-filtered pre-training, out-of-distribution predictive variance is strictly upper-bounded by the finite diameter of the identifiable quotient base manifold B — a hard geometric containment of generative hallucinations. By projecting downstream updates onto the orthogonal complement of the historical horizontal carriage, we formalize the SMG Sequential Adaptation Flow, proving total non-asymptotic elimination of catastrophic forgetting.

## Key Framework Elements

- **SMG Fiber Bundle**: (M, B, π, V, H, ω) — total Orlicz manifold M, base manifold B, projection π, vertical subspace V (SID), horizontal distribution H (SVDχ), Ehresmann connection ω
- **Four Core Axioms**: System Set, Environment Set, Structural Mechanism, Invariance Principle
- **Two-Fold Inference Paradigm**: Horizontal (statistical on B, classical MLE/Bayes) + Vertical (geometric, connection-driven navigation of SID)
- **Quarantining Theorem**: Orthogonal metric decomposition Tf M = Hf ⊕ Vf with Fisher-Rao metric orthogonality — vertical gauge noise cannot contaminate statistical signal
- **PAC-Bayesian Capacity Collapse Theorem**: Generalization bounded by finite metric volume of base space B despite infinite parameters
- **SMG Sequential Adaptation Flow**: Non-asymptotic catastrophic forgetting elimination via orthogonal projection onto complement of historical horizontal carriage
