[
  "Under finite drift energy assumptions, the intra-generation KL divergence between the generated distribution and the training target is bounded above by (1/2)ε̂ᵢ², the learned-path score error energy (Proposition 3.1).",
  "A matching lower bound on the chi-squared divergence, χ²(p̂^(i+1) ∥ qᵢ) ≥ (1/4)ηᵢε²⋆,ᵢ − C·ε⁴⋆,ᵢ, holds for small score errors (ε²⋆,ᵢ ≤ 1), where ηᵢ ∈ [0,1] is the observability coefficient (Proposition 3.3, Definition 3.2).",
  "Combining the upper and lower bounds yields a two-sided equivalence χ²(p̂^(i+1) ∥ qᵢ) ≍ ε²⋆,ᵢ in the perturbative regime (Theorem 3.4).",
  "When the score-error series ∑ᵢ ε²⋆,ᵢ diverges, the accumulated divergence across generations cannot vanish and has a non-zero floor limsup Dᵢ ≥ αη̄ε̄ / [16(1+(1-α)²)] (Proposition 4.1).",
  "When ∑ᵢ ε²⋆,ᵢ converges, accumulated divergence across generations satisfies D_{N+1} + C_bias ≍ Σᵢ(1-α)^{2(N-i)}ε²⋆,ᵢ + (1-α)^{2(N+1-i₀)}D_{i₀}, decaying geometrically at rate (1-α)² per generation, where α is the fraction of fresh data mixed in each round (Theorem 4.2).",
  "The α-dependent tradeoff between drift and stability is empirically confirmed on a 10D Gaussian mixture and on Fashion-MNIST/CIFAR-10, with low α producing collapse-driven drift and high α maintaining stability (Figure 1)."
]