用最优传输理论构建信念修正的成本模型，揭示认知代价的几何本质。
A Transport-Based Geometry of Belief-Cost
- 以最优传输为基础，将信念修正成本定义为水土空间中的标量代价
- 统一定价下，认知代价与费舍尔信息呈共形关系，导致确定性代价无穷大
- 适用于受限推理者、数字孪生等场景，为认知成本提供物理可解释框架
有限智能体通过有限传感器感知一个固定且有噪声的世界，其一致输出为信念：状态上的概率密度（贝叶斯后验）。该智能体无法达到绝对确定，信念修正需付出代价。本文提出基于最优传输的信念成本框架，设定两个公理：P0（场地）——修正成本是最佳传输的标量价格，信念位于沃尔什斯坦空间；P1（统一定价）——每纳特知识代价在任意位置均相同，满足埃克奥纳尔条件。在此框架下，成本度量为费舍尔信息共形重加权的最优传输度量，即 $\tilde g_{e,U}=2(e+U) , g_{W_2}$。连续信念中，统一定价等价于 $U=cJ$。由此推导出：当信念主导费舍尔信息时，确定性处于无限代价距离，故合理推断存在发散于确定性的代价下限（必要性猜想超出幂律范围）。在位置-尺度子流形上，几何为双曲型，史塔姆不等式表明高斯分布为最弯曲者（当 $e=0$ 时）。结果以纳特为单位，具尺度不变性：成本单位变化仅缩放所有距离，不改变边界、埃克奥纳尔族、双曲性或高斯极值。全局状态单位变换在 $e=0$ 时为同构映射。内容聚焦符号、排序与比值。通过兰道尔原理（1纳特对应 $k_BT$ 能量），代价下限转化为能量下限：趋近确定性需无限能量。物理提供单位锚定但未进入定理。若任一公理被移除，则选择不再唯一。
原文摘要 · Abstract (English)
A finite agent, a machine's digital twin or any bounded reasoner, infers a fixed and noisy world through finite sensors, so its coherent output is a belief: a probability density over states (the Bayes posterior). Such an agent stops short of certainty, and revising a belief carries a cost. We propose a framework for belief costs based on optimal transport, motivated by these facts. We pose two postulates. P0 (the arena): a revision cost is a scalar price on optimal transport, so beliefs live in Wasserstein space. P1 (uniform pricing): one nat of knowledge costs the same metric length everywhere, the eikonal condition. Among conceivable pricing rules we study this one. Under P0 and P1 the cost metric is optimal transport conformally reweighted by Fisher information, $\tilde g_{e,U}=2(e+U)\,g_{W_2}$, and the Fisher family is a characterization: among continuous reliefs, uniform pricing is equivalent to $U=cJ$. Two consequences follow on the conformal class. Certainty sits at infinite cost-distance once the relief dominates the Fisher information, so a well-posed inference has a cost floor diverging at certainty (necessity conjectural beyond power laws). On location-scale leaves the geometry is hyperbolic, and the Stam bound places the Gaussian as the most curved one (at $e=0$). The results are geometric, in nats, and hold up to units: a change of cost unit rescales all distances and preserves every conclusion (boundary, eikonal family, hyperbolicity, Gaussian extremum), a gauge theorem; a global change of state units at $e=0$ is an isometry; the content lies in signs, rankings and ratios. Via Landauer (one nat worth $k_BT$) the cost floor becomes an energy floor: revising toward certainty would demand unbounded energy. Physics anchors the unit and enters no theorem. Removing either postulate leaves the selection open.
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