[
  {
    "claim": 1,
    "text": "The paper's First-Order Rejection Sampling (FORS) meta-algorithm (Theorem 3.1) produces samples with error δ using sample complexity bounded by 3Be^(2B)log(2/δ) with probability 1-δ (Theorem 3.1)."
  },
  {
    "claim": 2,
    "text": "Under only a finite second-moment assumption (minimal assumptions), the diffusion sampler achieves query complexity O(d·log²(1/δ) + log³(1/δ)), giving polylog(1/δ) dependence rather than the poly(1/δ) of prior work (Theorem 4.1, Section 4)."
  },
  {
    "claim": 3,
    "text": "Under a non-uniform Lipschitz condition on the score (Assumption 4.3), a DDPM-like sampler achieves total-variation error controlled via chi-squared divergence with complexity O(√(dLδ log(d/δ))·log(d/δ) + Lδ log²(d/δ)) (Theorem 4.4)."
  },
  {
    "claim": 4,
    "text": "For distributions with low intrinsic dimension d★, an adaptive-step-size method attains complexity O(d★·log²((d+M₂²)/δ²)), replacing the ambient dimension d with d★ (Theorem 4.6)."
  },
  {
    "claim": 5,
    "text": "Section 5 extends the FORS framework to sample from general log-concave distributions using only gradient evaluations (no density evaluations), giving the first polylog(1/δ) sampler in this setting (Section 5)."
  }
]