为目标导向智能体在资源约束下的价值建立数学理论，揭示其与信息类似的结构性规律。
A Mathematical Theory of Value: a synthesis on goal-directed agency under resource constraints
- 将价值定义为资源转化为目标进展的速率，基于对数尺度建模。
- 实测显示感知互信息与能力高度相关（Spearman ρ=0.977），且价值增量符合信息容量上限。
- 适用于理解智能体对齐、资源配置与群体协同机制，适合研究智能体理性的学者。
我们提出价值——目标导向智能体创造、消耗与交换的量——是一种与信息同属一类的规律性结构量。借鉴香农方法，进行一次严格抽象：价值是智能体相对于其目标所设定的参考系，将资源转化为目标进展的速率。尺度不变性公理迫使采用对数度量形式 $V = \sum_i k_i \ln e_i$；复投资源的累积效应通过 Peters（2019）的遍历性论证也导出相同形式，作为一致性检验而非过度决定。我们推导出价值编码定理 $\Delta G \le I(X;Y)$；实现的价值可分解为 $G = D(q\|r) - D(q\|p)$。对群体而言，价值依赖于参考系，而价格则独立；一组共享资源并融合感知的智能体群，其价值上限为 $G_{\rm fleet} \le I(X;Y_{1:m}) \le H(X)$（新修正结果；此前的求和形式有误，已在 v5 中更正）。动力学层引出“是”与“应当”的不对称性，使对齐成为控制稳定性条件。我们在真实语言模型上预注册测试单参考系定律：感知互信息与实现能力高度一致（Spearman $ρ=0.977$，30 个模型×领域点）；样本外 $\Delta G$ 与 $I(X;Y)$ 跨四种任务形状保持形状不变（n=42，斜率 0.953）；过度自信可量化为耗散。后续预注册验证表明：耦合容量-区域预测——增长缺口律、联盟次模性与异或协同控制、联合上限、凯利选择——在真实智能体中于冻结区间内被证实；均场残差律 $\|Vg\|/γ$ 在各领域均未成立（群体无目标分散），已退回其纯数学范畴。贡献在于统一理论与随之而来的治理映射。
原文摘要 · Abstract (English)
We propose that value -- the quantity goal-directed agents create, destroy, and exchange -- is a lawful structural quantity in the same category as information. Following Shannon's method, we make one ruthless abstraction: value is the rate at which an agent converts a resource into goal-progress, relative to a frame fixed by its goal. A scale-invariance axiom forces a logarithmic measure, $V=\sum_i k_i\ln e_i$; compounding of a reinvested resource forces the same form via the ergodicity argument of Peters (2019) -- kin routes, a consistency check, not an over-determination. We derive a coding theorem of value, $ΔG \le I(X;Y)$; realized value decomposes as $G=D(q\|r)-D(q\|p)$. For populations, value is frame-relative while price is frame-independent; a fleet that pools its resource and fuses its perception inherits the ceiling $G_{\rm fleet}\le I(X;Y_{1:m})\le H(X)$ (a corollary; an earlier sum-form claim was wrong and is corrected in v5). A dynamical layer yields an is/ought asymmetry from which alignment emerges as a control-stability condition. We test the single-frame laws on live language models, pre-registered: perception mutual information tracks realized capability (Spearman $ρ=0.977$ over 30 model$\times$domain points); out-of-sample $ΔG$ tracks $I(X;Y)$, shape-invariant across four task shapes ($n=42$, slope $0.953$); over-confidence is measurable dissipation. The stated continuation gate has since been run (pre-registered, frontier-model population): the coupled capacity-region prediction -- growth-gap law, coalition submodularity with an XOR synergy control, joint ceiling, Kelly selection -- is confirmed within its frozen bands on real agents; the mean-field residual law $\|Vg\|/γ$ found no domain (populations hold no goal dispersion) and is retired to its mathematical scope. The contribution is the unification and the governance mapping that follows.
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