Key quotes / one-liners from research (paraphrase-safe)

For legal/traceability prefer opening the parent files. These are compressed restatements used in agent context.

1. Wrong target: exact recovery of all scores forces Θ(m d) parameters (17 Thm0).
2. Right target: ε-approximate A(q)=softmax(Kᵀq/T)V.
3. Covering: N(M_A,ε) scales as (C/ε)^{r_Σ}, independent of m under stated bounds.
4. A0: winner value error decays with score gap exp(−Δ).
5. Hybrid: exact recent + manifold far past is the practical architecture.
6. Empiric: some early Qwen heads r_Σ≈4–7; deeper heads much less compressible.
7. Feasibility: full high-d chirotope on 128k context is not the MVP.
8. Novelty neighbors: Tropical Attention (trained kernel), KV-Fold (still linear KV), KV-Direct (residual→K/V) — different objects.
9. Honesty: reconstruction error ≠ e2e generation quality.
10. Gate: measure r_Σ; do not force AOC on high-rank heads.