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经典算法学习理论极限识别直接用可计算性定义什么是可学习的，本文用算法信息提供一种新定义。对任意自然数集S以及它的覆盖C，当S的特征序列的算法信息下界大于等于覆盖元素特征序列的算法信息下界之和时，称S对C满足算法信息下界可加性，C是S的信息下界解释。S是可学习的，当其仅当任意它的信息下界解释是有穷集。在这个定义下，本文证明了1-随机集是不可学习的，它的算法信息下界可以由可数无穷个子集的算法信息下界累加而成。算法独立性讨论字符串之间是否相互不存在算法信息增益，本文认为它可能是与算法信息下界有关联的概念。设sᵢ是C的某个元素，当任意{C/sᵢ}的子集对sᵢ没有算法信息增益时，称sᵢ具备1-≤|C|-1算法独立性或是集合C的算法信息孤岛，本文证明了存在1-随机的有限覆盖和无限覆盖，1-随机对二者满足算法信息下界可加性，且任意覆盖元素是信息孤岛。

研究目的：刻画大视觉Transformer（ViT）在小样本工业计算机断层扫描（CT）缺陷识别任务中，采用参数高效微调（PEFT）时出现的静默训练崩溃失效模式，并找出会掩盖该失效模式的评估方案。 研究方法：基于包含673张图像的肺部CT数据集，使用LoRA对DINOv2 ViT‑L/14（参数量3.04亿）开展微调；通过单变量消融实验定位根本成因；在LIDC‑IDRI数据集（6691张切片）上复现该失效特征；采用2×2匹配学习率对照组，检验平均绝对误差（MAE）域自适应出现的表观性能下降现象。 研究结果：模型发生静默崩溃，退化为不具备学习能力的多数类预测器（准确率32.4%，交叉熵1.370，略低于均匀基准值ln(4)≈1.3863）。该崩溃现象被最优准确率追踪机制掩盖：最优轮次快照准确率为51.11%，掩盖了最终模型仅达到32.4%的先验基准水平。仅将学习率从10⁻³下调至2×10⁻⁴即可避免崩溃（冻结归一化层的ViT‑L模型准确率可达94.20%）；文献中报道的LayerNorm带来的11.15%表观性能优势，实际缩减为无统计学意义的‑1.19%。在LIDC‑IDRI数据集上，ViT‑B/14在三分之二的交叉验证折上发生崩溃（准确率62.24%，F1值为0，AUC为0.5），修正评估方案后模型性能恢复至96.85%。匹配学习率对照实验表明，文献报道的MAE指标下降28.7个百分点属于学习率导致的伪现象；采用稳定学习率后，基于MAE自适应的LoRA模型准确率可达92.6%。 研究局限性：ln(N)基准阈值仅适用于分类任务；本研究仅测试了LoRA方法；所用数据集样本规模为数百至数千级别；工业场景验证依赖合成仿真数据。 研究结论：公开报道的归一化效果是训练不稳定造成的伪现象，并非LayerNorm带来的真实增益。研究人员应当同时报告最终轮次指标与最优轮次指标；将损失收敛至ln(N)视作训练崩溃的判断依据；并使用适配模型规模的学习率。

作业负担治理成效易反弹，是我国基础教育领域亟待破解的现实难题。既有研究大多聚焦制度设计、升学竞争等外部因素解读作业负担的形成机制，较少从儿童认知发展的内在规律出发开展深层次学理分析。本文以核心认知内表型——执行功能为分析抓手，整合认知神经科学、教育学、心理学相关理论与国内大样本调研数据，系统阐释作业负担产生与控后反弹的内在机理。研究表明，现阶段形成了民众生活水平快速提升与优质家庭教育发展不充分之间的矛盾，造成儿童青少年群体执行功能发育出现结构性缺口。受此影响，低成本作业训练，成为家庭与学校弥补儿童认知发展短板的现实选择，进而陷入恶性循环，带来作业负担膨胀、减负治理成效难以巩固的困境。受出生人口波动影响，教育竞争格局将发生结构性翻转，从短期负担外溢到长期群体内卷分化，多重风险交织叠加，还会对减负治理构成复合冲击，凸显开展长期性、体系化防控的必要性。究其本质，作业负担是儿童青少年执行功能发展需求与现实育人供给错配的外在显现，仅靠行政手段压缩作业总量，无法达成长效减负。应立足我国本土教育实际，坚守五育并举顶层育人逻辑，立足五育融合实践路径，聚焦智育现实问题，搭建家-校-社协同联动的一体化执行功能培育支撑体系，充分挖掘德、智、体、美、劳各领域的认知培育价值，同步构建涵盖学业质量、视力健康、体质健康、心理健康的多维互补评价监测体系，健全从幼儿园到高级中学的基础教育全过程评估框架，秉持循序渐进的改革实施策略，从源头上消解作业负担的生成动因，推动作业减负与五育育人提质协同共进，为我国基础教育长效减负治理提供学理支撑与政策启示。

Approximate caches for large language models reuse a stored answer when a new prompt is semantically close to a cached one,yet similarity alone does not establish that the evidence supporting the stored answer is still available. This paper introduces proof-validbenchmarking (PVB): a cached answer may be served only with a certificate whose premises survive a controlled erasure stress test.We map class–response pairs to response-conditioned evidence-support graphs, derive the exact residual-premise law andthe exact price of transparent, shared-module, and coded protection. General semantic-module selection is NP-complete, but a greedyrule retains a logarithmic guarantee; a certificate-based converse shows that query-only coded caching must protect a proof certificaterather than an embedding key; and a hybrid theorem under a common erasure realization gives the exact transparent–coded trade-off.Experiments close a three-layer loop: a trace-conditioned stress test on 40,000 prompts is consistent with the exact law; a grounded auditon 7,405 HotpotQA questions confirms it with measured exponents and quantifies proxy bias; and an instrumented pipeline constructscertificates at 38 μs and verifies them at 1.6 μs per question, predicting measured availability to within 0.0016. Correlated and burstyerasure delimit the i.i.d. assumption quantitatively, and PVB serving dominates the error–throughput region reachable by thresholdtuning alone. A real-dynamics audit on five monthly Wikipedia and four Wikidata snapshots measures the erasure process in vivo andaudits the serving gate end to end against statement-level ground truth.

We determine the nonlinear large-time profile of the incompressibleNavier--Stokes--Coriolis equations with horizontal viscosity on$\mathbb T^2\times\mathbb R$ for an arbitrary constant rotation rate. Writing$u=\bar u+\widetilde u$ and denoting by $R_\Omega(t)$ the planar inertialrotation, we prove that\[ R_\Omega(-t)\bar u_h(t)\longrightarrow A_\infty =\bar u_h(0)-\int_0^\infty R_\Omega(-s)\partial_3   \overline{\widetilde u_3\widetilde u_h}(s)\,\mathrm ds,\]and that the full solution converges exponentially to the inertial orbit$(R_\Omega(t)A_\infty,0)$. Thus the limiting profile is selected by theaccumulated rotated Reynolds stress and is not, in general, determined by theinitial horizontal mean. For zero-horizontal-mean data $u_0=af$, the profilemap has the quadratic expansion$A_\infty=a^2\mathcal B_\Omega(f)+O_{H^{m-2}}(a^3)$ for every integer$m\ge2$. For each viscosity and rotation rate, we also construct arbitrarilysmall smooth data for which $A_\infty\ne0$, showing that interactions amongdamped horizontal modes can generate a persistent inertial oscillation fromzero initial mean. These asymptotic results are built on a global stabilitytheory at the $H^2$ level: sufficiently small divergence-free $H^m$ datagenerate a unique global solution, with a smallness threshold independent ofthe rotation rate, and the nonzero horizontal modes decay exponentially in$H^{m-1}$. The analysis uses the horizontal spectral gap, anisotropic productestimates, and mean--oscillation cancellations, without dispersive estimatesor a fast-rotation assumption. Fourier pseudo-spectral computations within anexact $x_2$-independent invariant subspace quantitatively support theReynolds-stress representation and the quadratic scaling of the selectedamplitude, and illustrate the dependence of its leading-order profile on therotation rate and horizontal viscosity.