Habish Dhakal

Pratyush Acharya, Nuraj Rimal, Habish Dhakal

We test whether the causal inner product of \citet{park2024linear} -- defined by the unembedding covariance $Σ$ -- enables cross-lingual concept transport. Across 17 models and 4 language pairs, a matched-spectrum randomization test finds that Whitened Causal Alignment is indistinguishable from spectral regularization alone ($p = 0.95$). However, this failure reveals a broader phenomenon: anti-concentration is observed in residual-stream difference-of-means vectors across five architecture families ($p < 10^{-33}$) and supported by SAE features (e.g., $p = 4.5 \times 10^{-19}$) and linear probes on Gemma and Llama. We discover a \emph{dual geometry}: activation-space concept directions anti-concentrate in the spectral tail, while static unembedding-row contrasts \emph{concentrate} in high-variance directions ($p < 10^{-4}$). Split-injection causal interventions support the functional basis on Gemma and Llama (Cohen's $d$ up to $1.80$), and POS-tag probing across 8 models shows syntax preferentially encodes in the high-variance subspace in 6 of 8 architectures ($p < 0.013$), with the Qwen~2.5 family showing a significant reversal consistent with architecture-specific spectral structure. These results suggest transformers may rotate semantic content into spectrally quiet regions during contextualized processing, encoding concepts where they can be manipulated with reduced grammatical disruption.

Pratyush Acharya, Habish Dhakal

Standard optimization theories struggle to explain grokking, where generalization occurs long after training convergence. While geometric studies attribute this to slow drift, they often overlook the interaction between the optimizer's noise structure and landscape curvature. This work analyzes AdamW dynamics on modular arithmetic tasks, revealing a ``Spectral Gating'' mechanism that regulates the transition from memorization to generalization. We find that AdamW operates as a variance-gated stochastic system. Grokking is constrained by a stability condition: the generalizing solution resides in a sharp basin ($λ_{max}^H$) initially inaccessible under low-variance regimes. The ``delayed'' phase represents the accumulation of gradient variance required to lift the effective stability ceiling, permitting entry into this sharp manifold. Our ablation studies identify three complexity regimes: (1) \textbf{Capacity Collapse} ($P < 23$), where rank-deficiency prevents structural learning; (2) \textbf{The Variance-Limited Regime} ($P \approx 41$), where generalization waits for the spectral gate to open; and (3) \textbf{Stability Override} ($P > 67$), where memorization becomes dimensionally unstable. Furthermore, we challenge the "Flat Minima" hypothesis for algorithmic tasks, showing that isotropic noise injection fails to induce grokking. Generalization requires the \textit{anisotropic rectification} unique to adaptive optimizers, which directs noise into the tangent space of the solution manifold.