Paragraph Boundaries Are Not White Space:Compression Depth as the Signature of Hierarchical Structure

Positional encoding (PE) underpins how permutation-invariant Transformers represent sequence order, yet how positional information is processed and stored remains poorly understood. Modern PE methods such as RoPE still struggle on tasks such as long-context understanding or retrieval \cite{chen-etal-2025-hope}. Hence, a better understanding of the internal positional mechanism could help design better PE. Building on evidence that positional and semantic signals occupy nearly orthogonal subspaces in trained Transformers, we modify an encoder Transformer to process three explicitly disentangled streams: semantic, absolute positional (AP) and relative positional (RP), and confine the masked-language-modeling (MLM) objective to the semantic stream. This decoupling enables a clean mechanistic study and yields three take-aways. (1) The isolated AP subspace spontaneously collapses into a low-frequency two-dimensional manifold that captures the structure of the document; (2) Attention heads specialize into structure and semantic-oriented groups, with RP exclusively supporting the latter; (3) Standard positional encodings do not robustly retain macroscopic structure: RoPE and RP only weakly encode it, and entangled AP loses it in the final layers under MLM pressure. The disentangled approach preserves positional encoding, which improves linguistic representation on 49 of the 65 linguistic phenomena of the Flash-Holmes probing benchmark.

Rotary Positional Encodings (RoPE) are currently the most popular positional encodings used in modern language models. RoPE rotates two-dimensional chunks of query and key vectors, operating as a function of their relative positional offset. The position-wise rates of rotation in RoPE typically follow a geometric sequence specified by a fixed base-frequency hyperparameter. Prior work has improved performance by either increasing this parameter to slow rotation or by applying RoPE to only a subset of QK dimensions. In this work we modify RoPE by learning a scalar per frequency, treating frequencies as learnable parameters rather than hyperparameters. We validate Learned RoPE by training a ladder of language models from scratch, ranging from 52M to 2.5B parameters. We observe and analyze the emergence of a high-norm, positional LeRoPE band. LeRoPE consistently outperforms RoPE and partial RoPE across all scales, with RoPE requiring 3.4% more compute (FLOPs) to match LeRoPE at the largest scale.

Rotary position embedding (RoPE) uses each token's integer position to determine the rotation applied inside attention. This works well for local token order, but increasing context length creates a positional train-test mismatch: RoPE produces relative rotations at offsets not seen during training. Methods that rescale, interpolate, randomize, or bias positions specify how attention handles those offsets, but still derive positional information from a growing token counter. We instead divide a token stream into units, retain ordinary RoPE positions within each unit, and assign every completed unit an address computed from its content. Adding units then applies the same learned map to new content rather than extending a positional range or an identifier table. We prove that this construction preserves local RoPE exactly, leaves the attention comparison between two fixed tokens unchanged when other units are inserted or reordered, and does not create new relative rotations merely because more units are added. In a character-level Tiny Shakespeare diagnostic, all-token validation perplexity remains approximately constant from contexts of 256 to 4096 characters. A second diagnostic shows that content-based addressing can retrieve and use information from multiple serialized facts. These are controlled shallow experiments, not scale benchmarks, but they support a direct prescription: use position to address locally and content to address across units.