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"已经建立的基线",
"局部能量下界与诱导膨胀",
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"交付与判定标准",
"复核边界与来源",
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"review_note": "推导以前文列明的构成输入与适用域为条件；正式来源状态与实验证实分别记录。",
"evidence_note": "证据按电磁实验矩阵逐项记录；本轮未新增实验拟合。",
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"missing_result": "在统一纵横场作用量中确定耦合系数，加入正二次能、梯度能和边界；求扰动谱、能量交换与稳定有限能量结构，区分自由零背景和有限幅度耦合。",
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"current_state": "书稿给出旋转标量三次耦合、相互反作用源项和加入四次项后的局部有界条件。它们仍是候选构成。"
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---
## 任务定位
**OP-EM-005 · OPEN**
以已经建立的局部变分模型为起点，继续计算归一化源、完整场与稳定有限能量构型。
## 已经建立的基线
本任务承接最终第三章第八节的非线性纵横候选，而不是从零寻找一个耦合项。将空间位移在相容边界下分为无旋 $\mathbf u_L$ 与无散 $\mathbf u_T$，定义 $\eta=-\nabla\cdot\mathbf u_L$ 和 $\boldsymbol\Omega_T=\nabla\times\mathbf u_T$；两者无量纲。$\eta>0$ 表示局域压缩，$\eta<0$ 表示膨胀。取旋转标量耦合
$g_{LT}$ 的量纲为能量密度。这一项使横向扭转强度成为纵向体应变的局域输入；它的身份是所选非线性构成，后面的变分则是该构成内可逐步检查的结果。
对于时间端点和空间边界上消失的变分，纵向部分使用 $\delta\eta=-\nabla\cdot\delta\mathbf u_L$：
$$
\delta_L S_{LT}=g_{LT}\int|\boldsymbol\Omega_T|^2\nabla\cdot\delta\mathbf u_L\,d^3x\,dt
=-g_{LT}\int\nabla|\boldsymbol\Omega_T|^2\cdot\delta\mathbf u_L\,d^3x\,dt.
$$
横向部分使用 $\delta\boldsymbol\Omega_T=\nabla\times\delta\mathbf u_T$。旋度的分部积分在此边界条件下保持符号，于是
$$
\delta_T S_{LT}=-2g_{LT}\int\eta\boldsymbol\Omega_T\cdot(\nabla\times\delta\mathbf u_T)\,d^3x\,dt
=-2g_{LT}\int\nabla\times(\eta\boldsymbol\Omega_T)\cdot\delta\mathbf u_T\,d^3x\,dt.
$$
与自由场项一起变分并移至其方程右端，得到互为反作用的两个广义力密度：
它们的单位是 N/m³，分别为纵向梯度源与横向无散源，不能解释成电流密度 $\mathbf J$。在均匀零背景下耦合是三次项，不改变最低阶自由波谱；有限背景或有限振幅下则使通道相互驱动。
## 局部能量下界与诱导膨胀
只有三次项时局部能量未必有下界。加入纵向正二次项与横向四次项，以常数 $K_L>0$、$\lambda_T$ 表示相应系数，两者都具有压力量纲。配方得到
充分条件为 $\lambda_T\geq2g_{LT}^2/K_L$。在固定横向强度的局部平衡中，对 $\eta$ 求导得到
$$
K_L\eta+g_{LT}|\boldsymbol\Omega_T|^2=0,
\qquad \eta_{\rm eq}=-\frac{g_{LT}}{K_L}|\boldsymbol\Omega_T|^2.
$$
因此 $g_{LT}>0$ 时横向扭转诱导膨胀与密度亏损。局部应变变量 $\eta$ 与外部长程引力尺度势不是同一个量；把这项响应接到长程源，还需纵向构成、完整梯度能及边界解。
在封闭保守体系中，把耦合势能一并纳入，总能量保持守恒；一个通道失去能量可以对应另一个通道获得能量。允许交换不等于不可逆耗散，局部有下界也不等于已经得到完整稳定粒子解。
## 需要完成的工作
以已有局部候选和两侧变分为起点，在统一纵横作用量中确定系数和归一化，补齐完整梯度、源与边界条件；求有限能量构型、非零背景扰动谱、通道交换及动力学稳定性。下一份成果应把假定参数、导出系数及可计算的响应函数分开列出。
## 交付与判定标准
交付同一作用量的完整方程、边界条件、量纲和守恒核查、一个可复算的构型或谱问题，以及从局部密度响应到长程源的明确映射。若只检验局部配方，应明确它已经证明的下界范围，不将其扩大为电子、质子质量或全部扰动方向稳定性。
## 复核边界与来源
依据最终合订稿第三章式（3.74）—（3.77）、第九节第四项。候选耦合的变量、双向源项、局部能量下界和诱导膨胀已经给出；微观系数与完整稳定局域结构继续研究，原问题 ID 保持 OP-EM-005。
## Space Dynamics 的正式承接
本任务由[纵横耦合页](/theory/space-dynamics/longitudinal-transverse-coupling)、[OP-SD-001](/open-problems/space-dynamics/unified-dynamics)及 [OP-SD-006](/open-problems/space-dynamics/source-coupling)承接。它们应继续计算同一组候选的源、场与稳定结构，不把已经完成的局部变分重新写成尚无任何动力学。