Predicting How Transformers Attend, Part II: A Six-Axis Decomposition with the Learned Imprint ν = -1/(2π), Sink-Dominated Precision Boundaries, Bimodal Phase Structure, and Honest Revisions

Companion paper to "Predicting How Transformers Attend" https://zenodo.org/records/20314038 Which introduced the Thermodynamic Attention Framework (TAF) and the closed-form predictor γ_Padé(θ, T) = (2θ - T√2)/(2θ + T√2) for the attention-decay exponent γ. This paper presents a phenomenological extension of TAF with five constructive contributions (ordered by empirical strength):

(1) A precision-direction rule for 4-bit NF4 inference on full multi-head attention: the R² of the bf16 power-law fit predicts the sign of Δγ_{4-bit − bf16}. Sign-correct on 5/5 paired bf16/4-bit measurements (DeepSeek-7B-base, DeepSeek-7B-chat, Pythia-2.8B, Pythia-1B, Llama-3-8B, Qwen2.5-7B-Instruct) — a deployment heuristic for practitioners serving 4-bit inference.

(2) A learned-imprint axis with slope ν ≈ -1/(2π), supported by three convergent arguments and a random-init falsifier (Pythia 70M/410M/1B at random init, p = 0.44). Honest caveats: bootstrap CI is wide ([-0.260, -0.008]) and a Pythia-70M trajectory across 9 checkpoints does not monotonically converge.

(3) An algebraic decomposition of the Cardy-like entropy anomaly: ΔH_Padé(γ) = log(z/2) + 2·arctanh(γ), linearising empirically with slope ≈ 5 across the panel.

(4) A bimodal phase structure of γ_text across the panel, with ~36% of measured LLMs sitting at γ ≥ 1 (Hagedorn zone), reframed as an industrial GQA-design correlate rather than a phase attractor.

(5) Machine-verified algebraic backbone: all 15 algebraic identities of the framework verified by both Sage Groebner basis and Lean Mathlib4, including a previously-unstated quadratic identity D-SAGE-1: 2η² + η·γ_χ + 1 = 0. To our knowledge, this is the first transformer-attention paper with end-to-end dual-tool machine verification of its algebraic content.

A separate correlation finding sits below the constructive contributions: the architectural concentration relation γ_text ≈ γ_Padé − 0.012·n_kv reproduces an in-sample R² = 0.30 (vs R² = 0.02 for Padé alone) but fails out-of-sample (median 70/30 hold-out R² ≈ -0.09 over 1000 random splits; family-leave-out aggregate R² = -0.027). The n_kv coefficient is statistically significant (bootstrap CI excludes zero) and the relative improvement over Padé alone is robust (+0.20 R² family-LOO), but absolute predictive power is essentially nil. We report it as a cross-panel correlation structure, not a predictive law — a down-grade documented internally rather than discovered by reviewers.

A symmetric set of honest revisions accompanies the constructive material: seven claims from paper I or paper II drafts are withdrawn or demoted (R_c* ≈ 1.68 as a sharp boundary, γ = 1 - 1/φ as a code-tuning attractor, the Mittag-Leffler prefactor 1/Γ(1-γ), the universal soft-decay KV truncation rule, the "0.3% match" framing for ν, the d_horizon "law" which we show is algebraically ≡ T when γ matches Padé, and the κ·N_sem topological invariant). The framework emerges leaner and more honest.

The accompanying public dataset (karlexmarin/taf-attention-decay on HuggingFace, 79 records across 33 models, CC-BY-4.0) and the diagnostic tool (karlexmarin/taf-agent on HuggingFace Spaces) provide reproducible measurements and operational recipes.

This release includes both the English version (73 pages) and the Spanish version (76 pages) of the paper.