[
  {"text":"Theorem 2 establishes an in-sample MSE bound for each task j that guarantees safety, ℰⱼⁱⁿ(θ̂ⱼ) ≲ q²(d/n)ζ regardless of the balancedness constant B, outlier fraction ε, or heterogeneity δ (Section 5.1, Theorem 2).","status":"unverified"},
  {"text":"Theorem 2 also shows a transfer guarantee for inlier tasks when B ≲ min(1/ε, m): ℰⱼⁱⁿ(θ̂ⱼ) ≲ (Bd/mn + min(Bδ², d/n) + B²ε²d/n)ζ, achieved without knowing ε, δ, or the inlier set S (Section 5.1, Theorem 2).","status":"unverified"},
  {"text":"Assumption 1 (Balancedness) replaces the classical Lower Boundedness of Second Moments condition ρI ⪯ Σⱼ with the one-sided condition Σⱼ ⪯ B·Σ_S, accommodating rank-deficient or decaying covariate spectra where prior eigenvalue-lower-bound approaches (e.g. Duan & Wang 2023, depending on 1/ρ²) fail (Section 4, Assumption 1).","status":"unverified"},
  {"text":"Theorem 3 extends the in-sample MSE guarantees of Theorem 2 to population risk via an empirical-to-population comparability constant νⱼ, retaining an intrinsic-dimension fallback for the safety guarantee (Section 5.2, Theorem 3).","status":"unverified"},
  {"text":"Theorem 4 extends the same adaptive safety/transfer MSE guarantees to generalized linear models under bounded-domain assumptions (Section 6, Theorem 4).","status":"unverified"},
  {"text":"Algorithm 1 solves a joint convex objective ℒ(Θ)=Σⱼ wⱼ(fⱼ(θⱼ)+λⱼ‖θⱼ-β‖_{Σⱼ}) that penalizes disagreement in prediction space (via task-specific norms) rather than raw parameter space (Section 3, Algorithm 1).","status":"unverified"}
]