Predicting How Transformers Attend Analytic Power-Law Theory, Phase Transitions, and Practical Compression Tools

⚠ ERRATUM (2026-05-02): §5.2 Theorem 5.2 coefficient corrected from (log N)²/4 to (log N)²/12. See [https://zenodo.org/records/19986818] A first-principles explanation of the ubiquitous power-law decay of attention weights in transformer LLMs. The RoPE positional encoding imposes a log-distance constraint on the attention score; the maximum-entropy distribution compatible with that constraint is a power law A(d) ∝ d^(-γ) with closed-form exponent γ = (2θ - T_eval √2) / (2θ + T_eval √2) (the [1,1] Padé approximant of e^(-z)). Validated on 30+ models from Pythia-70M to Qwen2.5-7B, median MAE 4.3% (n=9 non-anomalous subset, n=56 full panel) on the geometric centroid; corpus / architecture / induction-head phase contribute the residual variance via a five-axis decomposition (R²=0.44 on n=23). Three operational consequences: a regime diagram (γ1) classifying long-context use, a closed-form KV-cache compression window D_f predicting the operating point that empirical methods (SnapKV, PyramidKV, BLASST) calibrate by sweep, and a closed-form NTK base scaling α_opt for zero-shot context extension — Pareto-dominant on n=4 Pythia models against the unscaled baseline which collapses to chance retrieval at L > T_train. A controlled-θ pretraining pilot at θ ∈ {10⁴, 10⁵, 10⁶} confirms quantitative agreement (max 5.07% relative error vs Padé) under causal isolation. Higher-order predictions empirically validated: power law beats exponential 54/56 measurements; per-layer γ stability CV