Sharp Root Anti-Concentration via Projective Incidence and Ordered Root Laws

This paper answers the one-dimensional local root anti-concentration questions posed by Balcan, Pegden, and Sharma in the context of online optimization of piecewise-Lipschitz functions. For a homogeneous feature curve and coefficients whose density relative to the uniform law on a symmetric convex body $K$ is bounded by $A$, we show that the worst-case interval-hitting constant equals $A$ times a section-averaged projective incidence speed. For cube-supported coefficients, this speed is equivalent, up to universal constants, to the projective Lipschitz constant. This yields a sharp, dimension-free characterization and removes the previous $\sqrt N$ loss. For monic degree-$d$ polynomials under arbitrary coefficient laws, we prove that the interval-hitting constant is finite if and only if the ordered real-root laws have bounded densities, with a factor-$d$ comparison that is sharp. Conditional and joint coefficient-space area formulas, together with a two-chart certificate, make this criterion verifiable for dependent and singular coefficient laws. We also give two graph-learning applications that complete the transition-to-regret chain. A cost-sensitive Gaussian-RBF harmonic classifier uses the projective incidence theorem and achieves expected regret $\widetilde O((An^2D e^{BD}/\ell+1)\sqrt T)$. A common-offset polynomial-kernel model uses rigid translation of the ordered roots and achieves $\widetilde O((qn^2κ+1)\sqrt T)$ regret, even when the induced coefficient law is singular in the ambient coefficient space.

R3-REC: Reasoning-Driven Recommendation via Retrieval-Augmented LLMs over Multi-Granular Interest Signals

This paper addresses two persistent challenges in sequential recommendation: (i) evidence insufficiency-cold-start sparsity together with noisy, length-varying item texts; and (ii) opaque modeling of dynamic, multi-faceted intents across long/short horizons. We propose R3-REC (Reasoning-Retrieval-Recommendation), a prompt-centric, retrieval-augmented framework that unifies Multi-level User Intent Reasoning, Item Semantic Extraction, Long-Short Interest Polarity Mining, Similar User Collaborative Enhancement, and Reasoning-based Interest Matching and Scoring. Across ML-1M, Games, and Bundle, R3-REC consistently surpasses strong neural and LLM baselines, yielding improvements up to +10.2% (HR@1) and +6.4% (HR@5) with manageable end-to-end latency. Ablations corroborate complementary gains of all modules.