隐藏学习状态动力学的纤维指纹
Fiber Fingerprints of Hidden Learning-State Dynamics
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中文总结 AI 辅助
该研究提出纤维指纹对学习系统隐藏学习状态动力学进行形式化，结合相关模型与算法，通过Qwen2.5-7B等模型的实验验证，表明当前行为并非未来学习的充分统计量。
中文摘要 AI 辅助
学习系统可占据在所有已声明的当前行为读出下无法区分但对未来训练响应不同的执行状态。我们通过纤维指纹对这一现象进行形式化：纤维指纹是受限于当前行为等价类的受控未来学习响应规律。前缀兼容的有限探针诱导出预测商函子，这是一种Nerode型的最小递归充分表示，且是一个规范的集合级预测纤维，无需假设平滑性、可逆性、有限秩或流形。在显式有限维希尔伯特实现下，响应可分解为可见、可见模式复用及不可约新扇区；历史可达性桥梁仅保留自然训练历史产生的区分。随后，条件机制结果确定了图霍奇时序分解，其正则切换类具有均方根尺度√pη^(3/2)和有限尺度修正，以及一个精确的Adam矩截面，其直接自适应场为常数，而常见的未来梯度可揭示隐藏的矩差异。针对Qwen2.5-7B和Mistral-7B-v0.3的冻结Transformer-LoRA-AdamW研究支持局部作用主干、更长视界的首次返回非闭合、以及具有输出范围复用和低秩不可约扇区的新鲜可见相对完成。更强的主张受预注册的阴性或混合结果限制：重新锚定的传输在其测量底限之上未得到解决；严格的有限网格霍奇-3/2结合虽有预期收缩但未满足；Qwen可访问性在冻结原始矩图表中未确立；Mistral的揭示依赖于未来上下文而非银行不变性。在这些由支持、尺度、度量和上下文解析的边界内，当前行为并非已声明未来学习的充分统计量。
英文摘要
A learning system can occupy execution states that are indistinguishable under every declared present-behavior readout yet respond differently to future training. We formalize this through fiber fingerprints: controlled future-learning response laws restricted to present-behavior equivalence classes. Prefix-compatible finite probes induce a predictive quotient functor, a Nerode-type minimal recursively sufficient representation, and a canonical set-level predictive fiber without assuming smoothness, reversibility, finite rank, or a manifold. Under an explicit finite-dimensional Hilbert realization, response decomposes into visible, visible-mode-reuse, and irreducible-new sectors; a history-reachability bridge retains only distinctions generated by natural training histories. Conditional mechanism results then identify a graph-Hodge chronology decomposition, a regular switching class with root-mean-square scale $\sqrt{p}η^{3/2}$ and finite-scale corrections, and an exact Adam moment section whose immediate adaptive field is constant while common future gradients can reveal hidden moment differences. Frozen Transformer--LoRA--AdamW studies with Qwen2.5-7B and Mistral-7B-v0.3 support a local action backbone, longer-horizon first-return non-closure, and fresh visible-relative completion with output-range reuse and a low-rank irreducible sector. Stronger claims remain bounded by preregistered negative or mixed results: re-anchored transport is unresolved above its measurement floor; the strict finite-grid Hodge--$3/2$ conjunction is unmet despite prospective contraction; Qwen accessibility is not established in the frozen raw moment chart; and Mistral revelation is future-context dependent rather than bank invariant. Within these support-, scale-, metric-, and context-resolved boundaries, present behavior is not a sufficient statistic for declared future learning.
