镜像视界：作为有界反射度量的可行路径熵

中文摘要 AI 辅助

镜像理论提出，研究智能系统不仅应依据其表示的内容，还应考虑在重复反射下它能维持的连贯延续。我们通过可行路径熵（VPE）使这一主张可操作，它是一种验证延续能力的有限预算度量。给定镜像状态、展开协议、验证器和模式映射，VPE将有界能力分解为两部分：到达可行延续的概率和成功展开中达到的已验证延续模式的多样性。本文恢复了该度量背后的完整理论框架，包括直觉作为局部欠定约束、偏好作为不变选择压力、反射作为由偏好引导的欠定解决方式以及几何作为使未来反射稳定的学习结构。然后在GSM8K的语言模型推理实验中实例化该理论。在Qwen2.5-Instruct模型、每个问题32次采样展开以及两个反射视界的情况下，将令牌预算从96增加到160可大幅扩展已验证可达性、降低零可达性、增加已验证模式熵并改善平滑VPE。在160个令牌时，Qwen2.5-1.5B在测试模型中实现了最强的镜像视界，这表明镜像视界不是参数数量，而是有界反射协议下可访问的已验证延续能力。结果支持镜像理论作为一种度量层面的解释：能力是可达可行延续的结构，而非仅仅是一次性准确率或通过率@k。

英文摘要

Mirror Theory proposes that an intelligent system should be studied not only by what it represents, but by what coherent continuations it can sustain under repeated reflection. We make this claim operational through \emph{viable path entropy} (VPE), a finite-budget measure of verified continuation capacity. Given a mirror state, a rollout protocol, a verifier, and a mode map, VPE decomposes bounded capability into two parts: the probability of reaching a viable continuation and the diversity of verified continuation modes reached among successful rollouts. This paper restores the full theoretical scaffold behind the measure: intuition as local underdetermining constraint, taste as invariant-selecting pressure, reflection as taste-guided resolution of underdetermination, and geometry as the learned structure that makes future reflection stable. We then instantiate the theory in language-model reasoning experiments on GSM8K. Across Qwen2.5-Instruct models, 32 sampled rollouts per problem, and two reflection horizons, increasing the token budget from 96 to 160 substantially expands verified reachability, reduces zero-reachability, increases verified-mode entropy, and improves smoothed VPE. At 160 tokens, Qwen2.5-1.5B realizes the strongest mirror horizon among the tested models, even though Qwen2.5-3B has more parameters. This shows that mirror horizon is not parameter count, but accessible verified continuation capacity under a bounded reflection protocol. The result supports Mirror Theory as a measure-level account: capability is the structure of viable continuations made reachable, not merely one-shot accuracy or pass@k.

发表机构

• Columbia University（哥伦比亚大学）

机构由 AI 辅助整理，请以论文原文为准。