发表机构
Pebble ML（Pebble ML）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过群组合测试平台，证明梯度下降在矩阵记忆中招募的秩等于任务所需的最小忠实表示维数，并验证了该秩定律的因果性及可解/不可解群的等价性。
AI 中文摘要
矩阵值记忆使秩成为学习表示的自然预算：状态所跨越的独立方向数限制了其能够绑定、组合和跟踪的内容。我们在一个群组合测试平台上报告了因果证据，该平台在硬单状态瓶颈下训练，并配有无法清洗秩的固定解码器，结果表明梯度下降恰好招募了任务代数所需的秩。配套论文 [Larson, 2026a] 在 K 对关联绑定测试平台上建立了类似的招募和因果必要性模式，其中精确恢复可证明要求状态秩至少为 K；本文继承了该工具，并将秩定律从标量容量界限扩展为表示论界限。我们针对嵌入在更大矩阵中的选定最小忠实参考表示进行训练。在跨越可解/不可解鸿沟的五个有限群的群组合状态跟踪中，招募的秩等于群的最小忠实实表示维数 $d_{\min}$（Spearman $\rho = 0.9747$，设计中的并列上限），维度匹配的可解/不可解对 $S_4$/$A_5$ 在预注册检验下统计等价，并且预注册的强制秩检验将保证的相似性上限与目标维数下的经验恢复区分开来：在 $d_{\min}$ 以下一个秩时，余弦相似度被目标的相关单位谱限制在 $\sqrt{(d_{\min}{-}1)/d_{\min}} \le 0.894$，在构造上低于每个组的 $0.9$ 阈值，观测单元达到该上限的 86-95%（平均 91%）；在 $d_{\min}$ 时，尽管先验上不保证，恢复在所有五个组的每组四个种子下均超过预注册的锚定相对标准。在此测试平台内，测量的有效秩跟踪表示维数；匹配维数的 $S_4$/$A_5$ 比较在预注册容差内建立了等价性。
英文摘要
Matrix-valued memories make rank the natural budget of a learned representation: the number of independent directions a state spans bounds what it can bind, compose, and track. We report causal evidence, on a group-composition testbed trained under a hard single-state bottleneck with a fixed decoder that cannot launder rank, that gradient descent recruits precisely the rank the task's algebra demands. A companion paper [Larson, 2026a] establishes the analogous recruitment and causal necessity pattern on a $K$-pair associative-binding testbed, where exact recovery provably requires state rank at least $K$; this paper inherits that instrument and extends the rank law from a scalar capacity bound to a representation-theoretic one. We train toward chosen minimal faithful reference representations embedded in larger matrices. On group-composition state tracking over five finite groups spanning the solvable/non-solvable divide, the recruited rank equals the group's minimal faithful real representation dimension $d_{\min}$ (Spearman $ρ= 0.9747$, the design's tie-capped maximum), the dimension-matched solvable/non-solvable pair $S_4$/$A_5$ is statistically equivalent under a pre-registered test, and a pre-registered force-rank test separates a guaranteed similarity ceiling from empirical recovery at the target dimension: one rank below $d_{\min}$, cosine similarity is capped by the target's tied unit spectrum at $\sqrt{(d_{\min}{-}1)/d_{\min}} \le 0.894$, below the $0.9$ threshold in every group by construction, with observed cells at 86-95% (mean 91%) of that ceiling; at $d_{\min}$, not guaranteed a priori, recovery clears the pre-registered anchor-relative bar at four seeds per group in all five groups. Within this testbed, measured effective rank tracks representation dimension; the matched-dimension $S_4$/$A_5$ comparison establishes equivalence within the pre-registered tolerance.
Comments11 pages, 2 figures
