消元几何
Elimination Geometry
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中文总结 AI 辅助
本专著提出消元几何（EG）框架，用于研究局部最优对象的共享部署规则实现问题，整合多学科工具，其应用涵盖模型选择、预测等领域，可指导架构修复。
中文摘要 AI 辅助
本专著开发了消元几何（EG），这是一种面向类型、原生损失、审计导向的框架，用于研究局部最优对象何时可通过共享部署规则实现。消元与压缩可能会抹去预测、推理、控制或表示所需的区分。EG询问哪些区分会丢失、诱导的缺陷是否对声明的任务可见，以及改变信息、架构、动作空间或部署域是否能修复它。EG区分局部可解性、全局可实现性和有限样本可认证性。它从原始目标中推导原生缺陷，并区分架构阻碍与模型近似、泛化及实现误差。本专著整合了几何、优化、信息论、统计学和机器学习的工具，形成可积性、表示可容许性、资源约束、观测重叠和共同部署的接口。形式化结果涉及规则、协调、奇异、组合和资源受限机制，具有明确的前因和声明边界。应用包括稀疏模型选择、无分布预测、观测治疗策略、路由专家与检索系统以及学习得分场。阻碍感知学习与推理将结构诊断与有限数据授权、机制匹配干预及独立验证关联。可复现的合成与真实数据研究阐明了证书如何指导架构修复，同时记录失效门和未解决案例。该框架要求在将持续性能下限归因于架构之前，先确定部署契约、原生端点、竞争解释、信息与计算预算及验证规则。
英文摘要
This monograph develops elimination geometry (EG), a typed, native-loss, audit-oriented framework for studying when locally optimal objects can be realized by a shared deployment rule. Elimination and compression may erase distinctions required by prediction, inference, control, or representation. EG asks which distinctions are lost, whether the induced defect is visible to the declared task, and whether changing information, architecture, action space, or deployment domain can repair it. EG separates local solvability, global realizability, and finite-sample certifiability. It derives native defects from the original objective and distinguishes architecture obstruction from model approximation, generalization, and implementation error. The monograph synthesizes tools from geometry, optimization, information theory, statistics, and machine learning into interfaces for integrability, representation admissibility, resource constraints, observational overlap, and common deployment. Formal results address regular, coordination, singular, compositional, and resource-limited mechanisms with explicit antecedents and claim boundaries. Applications include sparse model selection, distribution-free prediction, observational treatment policies, routed expert and retrieval systems, and learned score fields. Obstruction-Aware Learning and Inference links structural diagnosis to finite-data authorization, mechanism-matched intervention, and independent validation. Reproducible synthetic and real-data studies illustrate how certificates can guide architecture repair while recording failed gates and unresolved cases. The framework requires the deployment contract, native endpoint, competing explanations, information and compute budgets, and validation rule to be fixed before a persistent performance floor is attributed to architecture.
