用可验证的有限尺寸实验，证明了模型泛化延迟现象具有相变特征。
Grokking as a Falsifiable Finite-Size Transition
- 将群阶p作为广延变量，用谱头尾对比作序参数分析
- 宾德交叉与敏感度对比显示存在共享的有限尺寸边界
- 为理解模型泛化延迟提供了可检验的物理类比框架
Grokking——即在早期记忆后出现的延迟泛化现象——常被描述为相变，但缺乏可验证的有限尺寸输入。本文通过将 $\mathbb{Z}_p$ 的群阶 $p$ 视为可接受的广延变量，并以保留的谱头尾对比作为表征级序参数，采用凝聚态物理风格的诊断链对粗网格扫描和密集临界区域审计进行分析。宾德式交叉揭示了共享的有限尺寸边界，而灵敏度比较强烈否定平滑过渡解释（近临界审计中 $Δ\mathrm{AIC}=16.8$）。因此，grocking 中的相变语言可作为定量有限尺寸假设进行检验，而非仅作类比，尽管当前仍无法确定相变阶数。
原文摘要 · Abstract (English)
Grokking -- the delayed onset of generalization after early memorization -- is often described with phase-transition language, but that claim has lacked falsifiable finite-size inputs. Here we supply those inputs by treating the group order $p$ of $\mathbb{Z}_p$ as an admissible extensive variable and a held-out spectral head-tail contrast as a representation-level order parameter, then apply a condensed-matter-style diagnostic chain to coarse-grid sweeps and a dense near-critical addition audit. Binder-like crossings reveal a shared finite-size boundary, and susceptibility comparison strongly disfavors a smooth-crossover interpretation ($Δ\mathrm{AIC}=16.8$ in the near-critical audit). Phase-transition language in grokking can therefore be tested as a quantitative finite-size claim rather than invoked as analogy alone, although the transition order remains unresolved at present.
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