LoRA fine-tuning can create intruder dimensions: new leading singular vectors of the updated weight matrix $W+BA$ that are nearly orthogonal to all pretrained singular vectors and that drive catastrophic forgetting. Since their discovery, no theory has predicted, layer by layer on measured spectra, when they appear. We derive a per-layer critical update strength $s^\ast=\barθ/(γσ_1(BA))$, computed from the measured spectrum of $W$ alone through the rectangular spiked-deformation transform, together with an exact secular-equation characterization of the updated spectrum, with no fitted parameters. In a pre-specified study spanning four dense Transformer families, a state-space model, a mixture-of-experts model, and an encoder-decoder (18 adapters, 9{,}840 layer scans), the law localizes the empirical threshold within a factor of two on $82\%$ of layers, separates intruder-bearing from intruder-free layers at deployment with a mean AUC of $0.89$, holds unchanged on six third-party adapters, and predicts where WikiText-2 perplexity begins to degrade; a combination of the two pre-specified edge evaluations reaches $98\%$ and is confirmed out-of-bag on the external adapters ($0.997$). Full fine-tuning disperses its update far below the threshold of every layer, which resolves the asymmetry between LoRA and full fine-tuning. Norm-matched interventions confirm that threshold-crossing layers, rather than update magnitude, carry the forgetting, and a spike-budget rule derived from the thresholds, requiring one SVD and no validation sweeps, reduces forgetting by $62\%$ on the most fragile model at no task cost.
Conventional uniform convergence bounds and empirical risk minimization break down in massive over-parameterized models, such as large language transformers and biological sequence networks. With near-infinite unconstrained internal degrees of freedom, their optimization landscapes develop flat vertical gauge valleys, rendering classical generalization metrics vacuous and inducing severe pathologies, specifically 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 ($\mathcal{M}, \mathcal{B}, π, \mathcal{V}, \mathcal{H}, ω$), 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 (Structural Internal Directions, or SID) and isolates learning trajectories along the strictly non-degenerate horizontal distribution (Statistical Variational Directions, or SVD$χ$). 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 $\mathcal{B}$, establishing 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 the total non-asymptotic elimination of catastrophic forgetting. SMG replaces empirical fine-tuning heuristics with coordinate-free topological constraints, bridging advanced differential geometry with structural reliability in AI.