正有理数上的倒数代价
Reciprocal Cost on the Positive Rationals
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中文总结 AI 辅助
该研究分析正有理数上倒数代价律的非负解，通过代换将其约化为达朗贝尔函数方程，确定解的结构与正则延拓的条件，还推广到实数的非平凡子群并给出规范倒数代价。
中文摘要 AI 辅助
我们研究正有理数上倒数代价律的非负解，确定其中哪些解可正则延拓到正实数上。利用代换 $H=1+F$，我们将倒数合成律约化为达朗贝尔（d'Alembert）函数方程。我们证明，$\boldsymbol{\text{Q}}_{>0}$ 上的每个非负解由每个素数 $p$ 对应的一个实权重 $\boldsymbol{\text{α}}_p$ 决定，仅存在一个全局符号识别，因此有理解空间是无限维的。我们证明，每个非负有理解都有到 $\boldsymbol{\text{R}}_{>0}$ 的代数延拓，但正则延拓恰好存在当且仅当素数权重满足 $\boldsymbol{\text{α}}_p=\boldsymbol{\text{λ}}\boldsymbol{\text{log}}p$（其中 $\boldsymbol{\text{λ}}\boldsymbol{\text{∈}}\boldsymbol{\text{R}}$）。此情况下延拓唯一，属于单参数族 $\boldsymbol{\text{F}}_{\boldsymbol{\text{λ}}}(\boldsymbol{\text{x}})=\boldsymbol{\text{cosh}}(\boldsymbol{\text{λ}}\boldsymbol{\text{log}}\boldsymbol{\text{x}})-1$；否则，有理解在 $\boldsymbol{\text{Q}}_{>0}$ 的每个非空开子集上无界，且正则轨迹是闭的、无处稠密的。我们还将结果推广到 $\boldsymbol{\text{R}}_{>0}$ 的任意非平凡子群，其中二分由有理秩决定。最后，我们证明单位对数曲率选出规范倒数代价 $\boldsymbol{\text{J}}(\boldsymbol{\text{x}})=(\boldsymbol{\text{x}}+\boldsymbol{\text{x}}^{-1})/2-1$，而对于对数在 $\boldsymbol{\text{R}}$ 中稠密的载体 $\boldsymbol{\text{G}}$，单渐近条件 $\boldsymbol{\text{F}}(\boldsymbol{\text{e}}^{\boldsymbol{\text{s}}})\boldsymbol{\text{∼}}\boldsymbol{\text{s}}^2/2$ 同时蕴含正则性和校准性。
英文摘要
We study nonnegative solutions of the reciprocal cost law on the positive rationals and determine which of them admit regular extensions to the positive reals. Using the substitution $H=1+F$, we reduce the reciprocal composition law to d'Alembert's functional equation. We prove that every nonnegative solution on $\mathbb{Q}_{>0}$ is determined by one real weight $α_p$ for each prime $p$, with only a global sign identification. Thus the rational solution space is infinite dimensional. We prove that every nonnegative rational solution has an algebraic extension to $\mathbb{R}_{>0}$, but a regular extension exists exactly when the prime weights satisfy $α_p=λ\log p$ for some $λ\in\mathbb{R}$. In this case the extension is unique and belongs to the one-parameter family $F_λ(x)=\cosh(λ\log x)-1$. Otherwise the rational solution is unbounded on every nonempty open subset of $\mathbb{Q}_{>0}$, and the regular locus is closed and nowhere dense. We also extend the result to arbitrary nontrivial subgroups of $\mathbb{R}_{>0}$, where the alternative is governed by rational rank. Finally, we show that unit logarithmic curvature selects the canonical reciprocal cost $J(x)=(x+x^{-1})/2-1$, while, for a carrier $G$ whose logarithm is dense in $\mathbb{R}$, the single asymptotic condition $F(e^s)\sim s^2/2$ implies both regularity and calibration.
