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          "adopted_scope": "Original ID and formula retained; compare frequencies and wavelengths in a common reference.",
          "conditions": [
            "Fixed G₀; isolated, static, spherically symmetric, uncharged exterior.",
            "Areal radius, actual spatial length and reference optical path are defined separately.",
            "Weak-field perturbation, slow-rotation far field and boundary statistics each have separate conditions."
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      "source_id": "ASD-BOOK-ddbade997b45",
      "chapter": "第二章",
      "reference": "最终合订稿 · 第二章 §一 (2.1)—(2.2a)；同一参考下的波长频率比",
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        "固定G₀；孤立静态球对称无电荷外部场",
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        "弱场摄动、远区慢旋转和边界统计各自有独立条件"
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        "微观源匹配、动态强场及边界态完整动力学仍开放"
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        "note": "The compatibility symbol ψL here denotes the source-free linear density channel σ=-∇·uL. Homogeneous long-wave regime and compatible boundaries; retain its 1/k² kinetic kernel rather than introducing a new scalar constitutive law.",
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        "title": "∂²t ψL − c²∇²ψL = 0 ; ω² = c²|k|²",
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            "The compatibility symbol ψL here denotes the source-free linear density channel σ=-∇·uL. Homogeneous long-wave regime and compatible boundaries; retain its 1/k² kinetic kernel rather than introducing a new scalar constitutive law."
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    "source_reference": "Ch. 5 (5.4.6), source-free density channel; (5.4.8)–(5.4.10)",
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      "en": {
        "note": "Compatibility formula for transverse displacement at constant coefficients, long wavelength and zero source after common-speed matching; retain the divergence-free constraint. It does not replace nonlinear matter or tensor branches.",
        "source_reference": "Ch. 4 (4.8)–(4.9); Ch. 5 (5.1.9), (5.4.1)–(5.4.2)",
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        "title": "∂²t uT − c²∇²uT = 0 ; ∇·uT = 0",
        "plain": "∂²t uT − c²∇²uT = 0 ; ∇·uT = 0",
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        "note": "Geometric second moment of twelve neighbours; a₀ is the conventional cell edge and a₀/√2 the neighbour spacing. Equal vector-sector speeds additionally require link dynamics and constitutive matching.",
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  },
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    "id": "ASD-WEB-QM-002",
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    "plain": "ρ = |ψ|²",
    "latex": "\\rho=|\\psi|^2",
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    "note": "保留历史 OPEN 身份；最终书稿已有线性吸收、条件事件率及位置归一化结果。普适统计、任意初态弛豫与一次一记录机制仍待闭合。",
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        "note": "Historical OPEN classification retained. Linear absorption, conditional event rates and normalized position distributions are established in the final model; universal statistics, relaxation and exclusive clicks remain open.",
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        "title": "Elapsed time along the measurement direction",
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            "A1–A5; tt′≠0; 0≤v<c",
            "Directed effective length is not an arbitrary Euclidean vector norm"
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      "adopted_scope": "公式及其量义、适用条件逐项核对；原ID和科学状态保留",
      "conditions": [
        "φ 为观察者相对速度与所指定光传播／测量方向之间的角。由沿方向的投影关系得到，不能替代未说明同步规则的一般坐标时间变换。"
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        "title": "Components for motion along x",
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        "note": "Frequency follows from wavelength change at the same c. Match emission/reception frames and angle definitions before comparison with another theory.",
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    "plain": "纵向两程历时相加为2γL/c",
    "latex": "\\begin{aligned}t_1^{\\prime}&=\\frac Lc\\gamma(1-\\beta),&t_2^{\\prime}&=\\frac Lc\\gamma(1+\\beta)\\\\T_{\\parallel}^{\\prime}&=t_1^{\\prime}+t_2^{\\prime}=\\frac{2\\gamma L}{c}\\end{aligned}",
    "path": "/theory/lve/tests",
    "note": "理想等臂构型，忽略实验装置非理想项；比较的是两臂差分。",
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    ],
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    "source_reference": "Ch. 4 (4.6)",
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    "role": "constitutive-input",
    "translations": {
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        "title": "Quadratic continuum action",
        "plain": "Quadratic continuum action",
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        "path": "/en/theory/space-dynamics/wave-propagation",
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "均匀、小振幅、常系数且 ρs,KL,KT>0。"
      ],
      "unresolved": [],
      "migration_status": "reviewed",
      "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
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    "id": "ASD-WEB-SD-006",
    "key": "sd-field-equation",
    "title": "状态场变分方程",
    "plain": "State-field variational equation",
    "latex": "\\rho_s\\partial_t^2\\mathbf u=K_L\\nabla(\\nabla\\cdot\\mathbf u)-K_T\\nabla\\times(\\nabla\\times\\mathbf u)",
    "path": "/theory/space-dynamics/wave-propagation",
    "note": "边界变分为零或由相容条件消去。",
    "canonical": true,
    "version": "1.0.0",
    "status": "DERIVED",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-005",
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    ],
    "source_tier": "canonical",
    "source_reference": "Ch. 4 (4.7)",
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    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "State-field variational equation",
        "plain": "State-field variational equation",
        "note": "Boundary variations vanish or are removed by compatible conditions.",
        "path": "/en/theory/space-dynamics/wave-propagation",
        "source_reference": "Ch. 4 (4.7)",
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          "locators": [
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
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            "Boundary variations vanish or are removed by compatible conditions."
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    },
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        "p1322"
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    "id": "ASD-WEB-SD-007",
    "key": "sd-speeds",
    "title": "纵横传播与速度",
    "plain": "Sector propagation and speeds",
    "latex": "(\\partial_t^2-c_A^2\\nabla^2)\\mathbf u_A=0,\\quad c_L^2=K_L/\\rho_s,\\quad c_T^2=K_T/\\rho_s,\\quad A=L,T",
    "path": "/theory/space-dynamics/wave-propagation",
    "note": "完整常系数方程的投影结果。",
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    "status": "DERIVED",
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        "id": "ASD-WEB-SD-006",
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        "id": "ASD-WEB-SD-002",
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    ],
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    "source_reference": "Ch. 4 (4.8)",
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    "role": "mathematical-result",
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        "title": "Sector propagation and speeds",
        "plain": "Sector propagation and speeds",
        "note": "Projection of the complete constant-coefficient equation.",
        "path": "/en/theory/space-dynamics/wave-propagation",
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        "title": "Common vacuum-speed matching",
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    "path": "/theory/space-dynamics/continuum-dynamics",
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        "title": "Nonnegative free-field energy",
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        "note": "Positive coefficients give nonnegative quadratic energy; zero modes and boundaries still need specification.",
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    "note": "ZA>0；保留投影约束。与第四章相同归一化时 ZA=ρs。",
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    "role": "constitutive-input",
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        "title": "Constrained sector action",
        "plain": "Constrained sector action",
        "note": "ZA>0 with projection constraints; matching Chapter 4’s normalization uses ZA=ρs.",
        "path": "/en/theory/space-dynamics/continuum-dynamics",
        "source_reference": "Ch. 5 (5.4.1)",
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    "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json",
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        "kind": "final-combined-manuscript",
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    ]
  },
  "sd-plane-waves": {
    "id": "ASD-WEB-SD-011",
    "key": "sd-plane-waves",
    "title": "平面波色散与偏振",
    "plain": "Plane-wave dispersion and polarization",
    "latex": "\\omega_A^2=c_A^2k^2,\\quad\\mathbf k\\times\\mathbf U_L=0,\\quad\\mathbf k\\cdot\\mathbf U_T=0",
    "path": "/theory/space-dynamics/wave-propagation",
    "note": "几何偏振分类不自动指定粒子种类。",
    "canonical": true,
    "version": "1.0.0",
    "status": "DERIVED",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-007",
        "kind": "algebra"
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    ],
    "source_tier": "canonical",
    "source_reference": "Ch. 5 (5.4.2)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Plane-wave dispersion and polarization",
        "plain": "Plane-wave dispersion and polarization",
        "note": "Geometric polarization does not automatically identify particle species.",
        "path": "/en/theory/space-dynamics/wave-propagation",
        "source_reference": "Ch. 5 (5.4.2)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 5 (5.4.2)",
          "locators": [
            "p1324",
            "p2020"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "Geometric polarization does not automatically identify particle species."
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          "unresolved": [],
          "migration_status": "reviewed",
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    },
    "final_source": {
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      "reference": "Ch. 5 (5.4.2)",
      "locators": [
        "p1324",
        "p2020"
      ],
      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "几何偏振分类不自动指定粒子种类。"
      ],
      "unresolved": [],
      "migration_status": "reviewed",
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    "previous_source": {
      "source_reference": "Ch. 5 (5.4.2)",
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    "source_chapter": "第四章；第五章",
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    ]
  },
  "sd-energy-flux": {
    "id": "ASD-WEB-SD-012",
    "key": "sd-energy-flux",
    "title": "能流与功平衡",
    "plain": "Energy flux and work balance",
    "latex": "\\begin{aligned}\\mathcal E_A&=\\tfrac12Z_A(|\\partial_t\\mathbf u_A|^2+c^2|\\nabla\\mathbf u_A|^2),\\\\S_{A,i}&=-Z_Ac^2\\partial_tu_{A,a}\\partial_i u_{A,a},\\\\\\partial_t\\mathcal E_A+\\nabla\\cdot\\mathbf S_A&=\\mathbf J_A\\cdot\\partial_t\\mathbf u_A.\\end{aligned}",
    "path": "/theory/space-dynamics/longitudinal-waves",
    "note": "分量指标求和，cA=c；无源守恒不排除几何扩散。",
    "canonical": true,
    "version": "1.0.0",
    "status": "DERIVED",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-010",
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        "id": "ASD-WEB-SD-008",
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    ],
    "source_tier": "canonical",
    "source_reference": "Ch. 5 (5.4.3)–(5.4.4)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Energy flux and work balance",
        "plain": "Energy flux and work balance",
        "note": "Component indices are summed, cA=c; conservation permits geometric spreading.",
        "path": "/en/theory/space-dynamics/longitudinal-waves",
        "source_reference": "Ch. 5 (5.4.3)–(5.4.4)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
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          "reference": "Ch. 5 (5.4.3)–(5.4.4)",
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            "p2023–2025"
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "Component indices are summed, cA=c; conservation permits geometric spreading."
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          "unresolved": [],
          "migration_status": "reviewed",
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    "final_source": {
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      "reference": "Ch. 5 (5.4.3)–(5.4.4)",
      "locators": [
        "p2023–2025"
      ],
      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "分量指标求和，cA=c；无源守恒不排除几何扩散。"
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      "unresolved": [],
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  },
  "sd-density-wave": {
    "id": "ASD-WEB-SD-013",
    "key": "sd-density-wave",
    "title": "密度波方程与源",
    "plain": "Density wave equation and source",
    "latex": "(\\partial_t^2-c^2\\nabla^2)\\sigma=s_\\sigma,\\quad s_\\sigma=-Z_L^{-1}\\nabla\\cdot\\mathbf J_L",
    "path": "/theory/space-dynamics/longitudinal-field",
    "note": "取位移方程的负散度；源不等于任意指定的质量密度。",
    "canonical": true,
    "version": "1.0.0",
    "status": "DERIVED",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-003",
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    ],
    "source_tier": "canonical",
    "source_reference": "Ch. 5 (5.4.6)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Density wave equation and source",
        "plain": "Density wave equation and source",
        "note": "Minus the divergence of displacement dynamics; the source is not an arbitrarily specified mass density.",
        "path": "/en/theory/space-dynamics/longitudinal-field",
        "source_reference": "Ch. 5 (5.4.6)",
        "final_source": {
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "取位移方程的负散度；源不等于任意指定的质量密度。"
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      "unresolved": [],
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  },
  "sd-density-energy": {
    "id": "ASD-WEB-SD-014",
    "key": "sd-density-energy",
    "title": "密度变量的非局域能量",
    "plain": "Nonlocal density energy",
    "latex": "\\mathbf u_L(\\mathbf k)=\\frac{i\\mathbf k}{k^2}\\sigma,\\quad E_L=\\frac{Z_L}{2}\\int\\frac{d^3k}{(2\\pi)^3}\\left[\\frac{|\\partial_t\\sigma|^2}{k^2}+c^2|\\sigma|^2\\right]",
    "path": "/theory/space-dynamics/longitudinal-field",
    "note": "k≠0；实空间动能含逆拉普拉斯。",
    "canonical": true,
    "version": "1.0.0",
    "status": "DERIVED",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-003",
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    "source_tier": "canonical",
    "source_reference": "Ch. 5 (5.4.8)–(5.4.10)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Nonlocal density energy",
        "plain": "Nonlocal density energy",
        "note": "k≠0; real-space kinetic energy contains the inverse Laplacian.",
        "path": "/en/theory/space-dynamics/longitudinal-field",
        "source_reference": "Ch. 5 (5.4.8)–(5.4.10)",
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
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        "k≠0；实空间动能含逆拉普拉斯。"
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    "key": "sd-em-map",
    "title": "自由电磁通道表示",
    "plain": "Free electromagnetic representation",
    "latex": "\\mathbf A_T=N_T\\mathbf u_T,\\quad\\mathbf E_T=-\\partial_t\\mathbf A_T,\\quad\\mathbf B_{\\rm em}=\\nabla\\times\\mathbf A_T",
    "path": "/theory/space-dynamics/transverse-field",
    "note": "无源辐射区、规定规范与归一化；有源时另需标势与源耦合。",
    "canonical": true,
    "version": "1.0.0",
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    "source_tier": "canonical",
    "source_reference": "Ch. 5 §4.3, (5.4.11)",
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    "role": "normalization",
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      "en": {
        "title": "Free electromagnetic representation",
        "plain": "Free electromagnetic representation",
        "note": "Source-free radiation zone with specified gauge and normalization; sources also require a scalar potential and coupling.",
        "path": "/en/theory/space-dynamics/transverse-field",
        "source_reference": "Ch. 5 §4.3, (5.4.11)",
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    "note": "σ与ΩT无量纲，gLT单位J/m³；不是唯一构成。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
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    ],
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    "role": "constitutive-input",
    "translations": {
      "en": {
        "title": "Existing L–T coupling candidate",
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        "path": "/en/theory/space-dynamics/longitudinal-transverse-coupling",
        "source_reference": "Ch. 5 (5.4.19)",
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
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    "latex": "\\mathbf J_L^{\\rm nl}=-g_{LT}\\nabla|\\boldsymbol\\Omega_T|^2,\\quad\\mathbf J_T^{\\rm nl}=-2g_{LT}\\nabla\\times(\\sigma\\boldsymbol\\Omega_T)",
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    "note": "J是广义力密度，不是电流；变分限该候选。",
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        "title": "Reciprocal coupling sources",
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    "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json",
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  },
  "sd-order-parameter": {
    "id": "ASD-WEB-SD-022",
    "key": "sd-order-parameter",
    "title": "候选非线性取向变量",
    "plain": "Candidate nonlinear orientation",
    "latex": "\\Psi=\\chi Q,\\quad Q\\in SU(2),\\quad Q^\\dagger Q=I,\\quad\\det Q=1,\\quad\\chi^2=\\tfrac12\\operatorname{Tr}(\\Psi^\\dagger\\Psi)",
    "path": "/theory/space-dynamics/degrees-of-freedom",
    "note": "χ实幅度，Q紧致取向；不是已解粒子或规范场。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [],
    "source_tier": "research-notes",
    "source_reference": "Ch. 4 (4.16)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "constitutive-input",
    "translations": {
      "en": {
        "title": "Candidate nonlinear orientation",
        "plain": "Candidate nonlinear orientation",
        "note": "χ is a real amplitude and Q compact orientation; this is not a solved particle or gauge field.",
        "path": "/en/theory/space-dynamics/degrees-of-freedom",
        "source_reference": "Ch. 4 (4.16)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 4 (4.16)",
          "locators": [
            "p1351"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "χ is a real amplitude and Q compact orientation; this is not a solved particle or gauge field."
          ],
          "unresolved": [],
          "migration_status": "reviewed",
          "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
          "source_manifest": "final-manuscript-source-manifest.json"
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    },
    "final_source": {
      "source_id": "ASD-BOOK-ddbade997b45",
      "chapter": "第四章；第五章",
      "reference": "Ch. 4 (4.16)",
      "locators": [
        "p1351"
      ],
      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "χ实幅度，Q紧致取向；不是已解粒子或规范场。"
      ],
      "unresolved": [],
      "migration_status": "reviewed",
      "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
      "source_manifest": "final-manuscript-source-manifest.json"
    },
    "previous_source": {
      "source_reference": "Ch. 4 (4.16)",
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    "source_chapter": "第四章；第五章",
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    "source_kind": "manifest",
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    "source_manifest_url": "/downloads/ASD_Final_Manuscript_Source_Manifest.json",
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  },
  "sd-dual-action": {
    "id": "ASD-WEB-SD-023",
    "key": "sd-dual-action",
    "title": "书稿局部双场候选泛函",
    "plain": "Manuscript local dual-field functional",
    "latex": "\\begin{aligned}\\mathcal L_{u\\Theta}&=\\tfrac12Z_u|\\partial_t\\mathbf u|^2+\\tfrac12I_\\Theta\\sum_A(\\partial_t\\Theta_A)^2-\\mathcal E_0+\\mathbf J_{\\rm ext}\\cdot\\mathbf u,\\\\\\mathcal E_0&=\\tfrac12K_u(s^2+|\\mathbf w|^2)+\\tfrac12A_\\Theta\\sum_A|\\nabla\\Theta_A|^2+W(\\Theta)\\\\&\\quad+g_LsF(\\Theta)+g_T\\mathbf w\\cdot\\mathbf T+\\mathcal E_{\\rm stab}+\\mathcal E_{\\rm nl},\\\\F&=\\sum_A\\Theta_A^2-v_0^2,\\quad s=\\nabla\\cdot\\mathbf u,\\quad\\mathbf w=\\nabla\\times\\mathbf u,\\quad K_u=Z_uc_u^2.\\end{aligned}",
    "path": "/theory/space-dynamics/master-equation",
    "note": "已给出的局部双场作用量；Θ实分量，W扣真空。稳定项、本构参数及目标空间需具体选择；完整微观统一仍待联合求解。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [],
    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.3)–(5.6.5)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "constitutive-input",
    "translations": {
      "en": {
        "title": "Manuscript local dual-field functional",
        "plain": "Manuscript local dual-field functional",
        "note": "An explicit local dual-field action; Θ has real components and W subtracts vacuum energy. Specify stabilizers, constitutive parameters and target space; complete microscopic unification still needs joint solution.",
        "path": "/en/theory/space-dynamics/master-equation",
        "source_reference": "Ch. 5 (5.6.3)–(5.6.5)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 5 (5.6.3)–(5.6.5)",
          "locators": [
            "p2220–2227"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; source-grounded scope note synchronized",
          "conditions": [
            "An explicit local dual-field action; Θ has real components and W subtracts vacuum energy. Specify stabilizers, constitutive parameters and target space; complete microscopic unification still needs joint solution."
          ],
          "unresolved": [],
          "migration_status": "reviewed",
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          "source_manifest": "final-manuscript-source-manifest.json"
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    },
    "final_source": {
      "source_id": "ASD-BOOK-ddbade997b45",
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      "reference": "Ch. 5 (5.6.3)–(5.6.5)",
      "locators": [
        "p2220–2227"
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      "adopted_scope": "现有登记公式逐式核对；source-grounded scope note synchronized",
      "conditions": [
        "已给出的局部双场作用量；Θ实分量，W扣真空。稳定项、本构参数及目标空间需具体选择；完整微观统一仍待联合求解。"
      ],
      "unresolved": [],
      "migration_status": "reviewed",
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    },
    "previous_source": {
      "source_reference": "Ch. 5 (5.6.3)–(5.6.5)",
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    "source_chapter": "第四章；第五章",
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    ]
  },
  "sd-dual-u": {
    "id": "ASD-WEB-SD-024",
    "key": "sd-dual-u",
    "title": "双场候选的位移方程",
    "plain": "Candidate displacement equation",
    "latex": "Z_u(\\partial_t^2-c_u^2\\nabla^2)\\mathbf u=g_L\\nabla F-g_T\\nabla\\times\\mathbf T+\\mathbf J_{\\rm add}+\\mathbf J_{\\rm ext}",
    "path": "/theory/space-dynamics/master-equation",
    "note": "Jadd=−δEadd/δu；源与反作用来自同一泛函。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-023",
        "kind": "algebra"
      }
    ],
    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.8)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Candidate displacement equation",
        "plain": "Candidate displacement equation",
        "note": "Jadd=−δEadd/δu; sources and feedback come from one functional.",
        "path": "/en/theory/space-dynamics/master-equation",
        "source_reference": "Ch. 5 (5.6.8)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 5 (5.6.8)",
          "locators": [
            "p2232–2233"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "Jadd=−δEadd/δu; sources and feedback come from one functional."
          ],
          "unresolved": [],
          "migration_status": "reviewed",
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    },
    "final_source": {
      "source_id": "ASD-BOOK-ddbade997b45",
      "chapter": "第四章；第五章",
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      "locators": [
        "p2232–2233"
      ],
      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "Jadd=−δEadd/δu；源与反作用来自同一泛函。"
      ],
      "unresolved": [],
      "migration_status": "reviewed",
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    "previous_source": {
      "source_reference": "Ch. 5 (5.6.8)",
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    ]
  },
  "sd-dual-theta": {
    "id": "ASD-WEB-SD-025",
    "key": "sd-dual-theta",
    "title": "序参量反馈方程",
    "plain": "Order-parameter feedback",
    "latex": "\\begin{aligned}I_\\Theta\\partial_t^2\\Theta_A-A_\\Theta\\nabla^2\\Theta_A+\\frac{\\partial W}{\\partial\\Theta_A}+2g_Ls\\Theta_A&+g_T\\left[\\frac{\\partial\\mathcal C}{\\partial\\Theta_A}-\\partial_i\\frac{\\partial\\mathcal C}{\\partial(\\partial_i\\Theta_A)}\\right]+\\frac{\\delta E_{\\rm add}}{\\delta\\Theta_A}=0,\\\\\\mathcal C&=\\mathbf w\\cdot\\mathbf T.\\end{aligned}",
    "path": "/theory/space-dynamics/master-equation",
    "note": "保持反演时T须为轴矢量；内部对称性须指定。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-023",
        "kind": "algebra"
      }
    ],
    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.9)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Order-parameter feedback",
        "plain": "Order-parameter feedback",
        "note": "Parity invariance requires axial T; its internal symmetry must be specified.",
        "path": "/en/theory/space-dynamics/master-equation",
        "source_reference": "Ch. 5 (5.6.9)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 5 (5.6.9)",
          "locators": [
            "p2235–2236"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "Parity invariance requires axial T; its internal symmetry must be specified."
          ],
          "unresolved": [],
          "migration_status": "reviewed",
          "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
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    "final_source": {
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      "chapter": "第四章；第五章",
      "reference": "Ch. 5 (5.6.9)",
      "locators": [
        "p2235–2236"
      ],
      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "保持反演时T须为轴矢量；内部对称性须指定。"
      ],
      "unresolved": [],
      "migration_status": "reviewed",
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      "source_reference": "Ch. 5 (5.6.9)",
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      }
    ]
  },
  "sd-mixed-spectrum": {
    "id": "ASD-WEB-SD-026",
    "key": "sd-mixed-spectrum",
    "title": "有序真空的混合谱",
    "plain": "Mixed ordered-vacuum spectrum",
    "latex": "(c_u^2k^2-\\omega^2)(\\Omega_\\Theta^2+c_\\Theta^2k^2-\\omega^2)-\\frac{\\lambda^2k^2}{Z_uI_\\Theta}=0,\\quad\\lambda=2g_Lv_0",
    "path": "/theory/space-dynamics/wave-propagation",
    "note": "ΩΘ²>0；完整线性化谱决定分支。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-024",
        "kind": "algebra"
      },
      {
        "id": "ASD-WEB-SD-025",
        "kind": "algebra"
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    ],
    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.12)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Mixed ordered-vacuum spectrum",
        "plain": "Mixed ordered-vacuum spectrum",
        "note": "ΩΘ²>0; the full linearized spectrum determines the branches.",
        "path": "/en/theory/space-dynamics/wave-propagation",
        "source_reference": "Ch. 5 (5.6.12)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 5 (5.6.12)",
          "locators": [
            "p2241–2242"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "ΩΘ²>0; the full linearized spectrum determines the branches."
          ],
          "unresolved": [],
          "migration_status": "reviewed",
          "manuscript_sha256": "ddbade997b45913248e588953b1d27ccb9c7c206c3c8d992d3aef2364a64836e",
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "ΩΘ²>0；完整线性化谱决定分支。"
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      "unresolved": [],
      "migration_status": "reviewed",
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  "sd-effective-speeds": {
    "id": "ASD-WEB-SD-027",
    "key": "sd-effective-speeds",
    "title": "反馈后低波数速度",
    "plain": "Low-k speeds after feedback",
    "latex": "c_{L,\\rm eff}^2=c_u^2-\\frac{\\lambda^2}{Z_uI_\\Theta\\Omega_\\Theta^2},\\quad c_{T,\\rm eff}^2=c_u^2",
    "path": "/theory/space-dynamics/wave-propagation",
    "note": "限该候选的低波数无隙支；裸同速不保证有效同速。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [
      {
        "id": "ASD-WEB-SD-026",
        "kind": "algebra"
      }
    ],
    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.13)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Low-k speeds after feedback",
        "plain": "Low-k speeds after feedback",
        "note": "For this candidate’s low-k gapless branch; equal bare speeds do not ensure equal effective speeds.",
        "path": "/en/theory/space-dynamics/wave-propagation",
        "source_reference": "Ch. 5 (5.6.13)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
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            "p2243–2245"
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "For this candidate’s low-k gapless branch; equal bare speeds do not ensure equal effective speeds."
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "限该候选的低波数无隙支；裸同速不保证有效同速。"
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    "id": "ASD-WEB-SD-028",
    "key": "sd-static-density",
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    "plain": "Static response to a local source",
    "latex": "\\nabla^2(K_us+g_LF)=0\\quad\\Longrightarrow\\quad s=-\\frac{g_L}{K_u}F,\\quad\\sigma=\\frac{g_L}{K_u}F",
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    "note": "全空间光滑、局域源，无奇异调和项，无穷远零边界。",
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    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
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    "source_reference": "Ch. 5 (5.6.14)–(5.6.15)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Static response to a local source",
        "plain": "Static response to a local source",
        "note": "Smooth fields on all space, localized source, no singular harmonic term and zero boundary at infinity.",
        "path": "/en/theory/space-dynamics/longitudinal-field",
        "source_reference": "Ch. 5 (5.6.14)–(5.6.15)",
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            "p2247–2252"
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
          "conditions": [
            "Smooth fields on all space, localized source, no singular harmonic term and zero boundary at infinity."
          ],
          "unresolved": [],
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
      "conditions": [
        "全空间光滑、局域源，无奇异调和项，无穷远零边界。"
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    "id": "ASD-WEB-SD-029",
    "key": "sd-stretch-action",
    "title": "独立拉伸的候选扩展",
    "plain": "Independent stretch candidate",
    "latex": "\\mathcal L_\\varphi=\\tfrac12Z_\\varphi[(\\partial_t\\varphi)^2-c^2|\\nabla\\varphi|^2]+\\varphi\\mathcal J_g[\\Theta,\\nabla\\Theta]",
    "path": "/theory/space-dynamics/master-equation",
    "note": "不同于位移散度的能量核；共同保留需约束或尺度分离。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
    "dependencies": [],
    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.16)",
    "source_version": "ASD-BOOK-ddbade997b45",
    "role": "constitutive-input",
    "translations": {
      "en": {
        "title": "Independent stretch candidate",
        "plain": "Independent stretch candidate",
        "note": "Its energy kernel differs from displacement divergence; keeping both requires constraints or scale separation.",
        "path": "/en/theory/space-dynamics/master-equation",
        "source_reference": "Ch. 5 (5.6.16)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
          "reference": "Ch. 5 (5.6.16)",
          "locators": [
            "p2254–2257"
          ],
          "adopted_scope": "Registered formula checked individually against the final manuscript; semantic potential phi→varphi",
          "conditions": [
            "Its energy kernel differs from displacement divergence; keeping both requires constraints or scale separation."
          ],
          "unresolved": [],
          "migration_status": "reviewed",
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      "locators": [
        "p2254–2257"
      ],
      "adopted_scope": "现有登记公式逐式核对；semantic potential phi→varphi",
      "conditions": [
        "不同于位移散度的能量核；共同保留需约束或尺度分离。"
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      "unresolved": [],
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    "id": "ASD-WEB-SD-030",
    "key": "sd-stretch-farfield",
    "title": "拉伸候选的静态远场",
    "plain": "Static stretch far field",
    "latex": "Z_\\varphi(\\partial_t^2-c^2\\nabla^2)\\varphi=\\mathcal J_g,\\quad\\varphi(\\mathbf r)=\\frac{1}{4\\pi Z_\\varphi c^2}\\int\\frac{\\mathcal J_g(\\mathbf r')}{|\\mathbf r-\\mathbf r'|}\\,d^3r'",
    "path": "/theory/space-dynamics/longitudinal-field",
    "note": "无量纲varphi；右式为局域源与无穷远零边界下的静态Green积分。球对称紧支撑源有精确核外1/r，只有可积尾部时为远场渐近；源及反作用由物质模型确定。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
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      {
        "id": "ASD-WEB-SD-029",
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    "source_tier": "research-notes",
    "source_reference": "Ch. 5 (5.6.17)",
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    "role": "mathematical-result",
    "translations": {
      "en": {
        "title": "Static stretch far field",
        "plain": "Static stretch far field",
        "note": "Dimensionless varphi; the right side is the static Green integral for a localized source and zero far boundary. Spherical compact support gives exact exterior 1/r; an integrable tail gives an asymptote. Matter fixes the source and feedback.",
        "path": "/en/theory/space-dynamics/longitudinal-field",
        "source_reference": "Ch. 5 (5.6.17)",
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          "adopted_scope": "Registered formula checked individually against the final manuscript; semantic potential phi→varphi; source-grounded scope note synchronized",
          "conditions": [
            "Dimensionless varphi; the right side is the static Green integral for a localized source and zero far boundary. Spherical compact support gives exact exterior 1/r; an integrable tail gives an asymptote. Matter fixes the source and feedback."
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      "adopted_scope": "现有登记公式逐式核对；semantic potential phi→varphi; source-grounded scope note synchronized",
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    "title": "取向稳定项候选",
    "plain": "Orientation stabilizer candidate",
    "latex": "E_Q=\\int\\left[\\frac{A_2}{2}\\sum_i\\|L_i\\|^2+\\frac{A_4}{4}\\sum_{i<j}\\|[L_i,L_j]\\|^2+W(Q,\\chi)\\right]d^3x,\\quad L_i=Q^{-1}\\partial_iQ",
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    "note": "A2,A4>0，范数平方Tr(X†X)/2；存在性与完整稳定谱待求。",
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      {
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    "source_tier": "research-notes",
    "source_reference": "Ch. 4 (4.21)",
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    "role": "constitutive-input",
    "translations": {
      "en": {
        "title": "Orientation stabilizer candidate",
        "plain": "Orientation stabilizer candidate",
        "note": "A2,A4>0 and norm squared=Tr(X†X)/2; existence and the full stable spectrum remain open.",
        "path": "/en/theory/space-dynamics/master-equation",
        "source_reference": "Ch. 4 (4.21)",
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          "adopted_scope": "Registered formula checked individually against the final manuscript; The mathematical content agrees; the existing expression and its derivation conditions are retained.",
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      "adopted_scope": "现有登记公式逐式核对；数学内容相符，保留既有公式及推导条件",
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    "plain": "Three-dimensional scaling and analytic stationary radius",
    "latex": "\\begin{aligned}Q_\\lambda(\\mathbf x)&=Q(\\mathbf x/\\lambda),\\\\E(\\lambda)&=\\lambda E_2+\\lambda^{-1}E_4+\\lambda^3E_V,\\\\E_2-E_4+3E_V&=0\\quad(\\lambda=1),\\\\E(R)&=A_2R+A_4/R+A_0R^3,\\\\E'(R)&=A_2-A_4/R^2+3A_0R^2,\\\\R_*^2&=\\frac{\\sqrt{A_2^2+12A_0A_4}-A_2}{6A_0}\\quad(A_0>0),\\\\&=\\frac{2A_4}{A_2+\\sqrt{A_2^2+12A_0A_4}}\\quad(A_0\\ge0),\\\\R_*^2&=A_4/A_2\\quad(A_0=0),\\\\E''(R_*)&=\\frac{2A_4}{R_*^3}+6A_0R_*>0.\\end{aligned}",
    "path": "/theory/space-dynamics/master-equation",
    "note": "保留第四章λ增大代表放大构型的约定；R为真实长度。A₂,A₄>0、A₀≥0；含6A₀分母的式子限A₀>0，有理式含A₀=0。A₂:J/m、A₄:J·m、A₀:J/m³。解析径向极小值不是完整三维解或全扰动稳定性。",
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    ],
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        "plain": "Three-dimensional scaling and analytic stationary radius",
        "note": "Retains Chapter 4’s convention that increasing λ enlarges the profile; R is a physical length. A₂,A₄>0, A₀≥0. The 6A₀ denominator requires A₀>0; the rationalized expression includes A₀=0. Units: A₂ J/m, A₄ J·m, A₀ J/m³. The analytic radial minimum is not a complete three-dimensional solution or full perturbation stability.",
        "path": "/en/theory/space-dynamics/master-equation",
        "source_reference": "Ch. 4 (4.20)–(4.22); Ch. 5 (5.6.23)–(5.6.25)",
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          "adopted_scope": "Registered formula checked individually against the final manuscript; analytic radius, rationalization, zero-potential limit and positive radial second derivative; source-grounded scope note synchronized",
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    "previous_source": {
      "source_reference": "Ch. 4 (4.22), elementary differentiation with EV=0",
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    "id": "ASD-WEB-MAT-009",
    "key": "mat-rest-mass",
    "title": "完整静止能量与能量质量",
    "plain": "Complete rest energy and energy mass",
    "latex": "E_{\\rm rest}=\\int(\\mathcal H[Q,\\eta,\\chi,\\varphi]-\\mathcal H_{\\rm vac})\\,d^3x,\\qquad m_E=E_{\\rm rest}/c^2",
    "path": "/theory/matter/mass",
    "note": "包括核、伴随场与相互作用；mE是定义，不能替代对惯性与引力质量的计算。",
    "canonical": true,
    "version": "1.0.0",
    "status": "FOUNDATION",
    "dependencies": [
      {
        "id": "ASD-WEB-MAT-001",
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    ],
    "source_tier": "canonical",
    "source_reference": "Ch. 4 (4.34)",
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    "role": "definition",
    "translations": {
      "en": {
        "title": "Complete rest energy and energy mass",
        "plain": "Complete rest energy and energy mass",
        "note": "Includes core, accompanying fields and interactions; defining mE does not calculate inertia or gravitational mass.",
        "path": "/en/theory/matter/mass",
        "source_reference": "Ch. 4 (4.34)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
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          "locators": [
            "p1413–p1415"
          ],
          "adopted_scope": "Registered equation, symbols and scope checked against the final manuscript; original ID and scientific status preserved.",
          "conditions": [
            "Includes core, accompanying fields and interactions; defining mE does not calculate inertia or gravitational mass."
          ],
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      "conditions": [
        "包括核、伴随场与相互作用；mE是定义，不能替代对惯性与引力质量的计算。"
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    "id": "ASD-WEB-MAT-010",
    "key": "mat-collective-mass",
    "title": "平移集体坐标与惯性",
    "plain": "Translational collective coordinate and inertia",
    "latex": "L_{\\rm coll}=-E_{\\rm rest}+\\tfrac12M_{ij}\\dot X_i\\dot X_j+\\cdots,\\qquad M_{ij}=m_I\\delta_{ij}\\ \\text{in an isotropic limit}",
    "path": "/theory/matter/mass",
    "note": "须把实际可平移构型代入同一作用量，并处理随速形变；各向同性是额外条件。",
    "canonical": false,
    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
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    "source_reference": "Ch. 4 (4.35); Ch. 5 (5.6.26)",
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    "role": "definition",
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      "en": {
        "title": "Translational collective coordinate and inertia",
        "plain": "Translational collective coordinate and inertia",
        "note": "Insert an actual moving family into the same action, including velocity-induced deformation; isotropy is an additional condition.",
        "path": "/en/theory/matter/mass",
        "source_reference": "Ch. 4 (4.35); Ch. 5 (5.6.26)",
        "final_source": {
          "source_id": "ASD-BOOK-ddbade997b45",
          "chapter": "Chapters 4 and 5",
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            "p1416–p1418",
            "p2283–p2288"
          ],
          "adopted_scope": "Registered equation, symbols and scope checked against the final manuscript; original ID and scientific status preserved.",
          "conditions": [
            "Insert an actual moving family into the same action, including velocity-induced deformation; isotropy is an additional condition."
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        "p1416–p1418",
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      "adopted_scope": "最终书稿中本登记公式、符号与适用条件；保留原ID及科学状态。",
      "conditions": [
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    "id": "ASD-WEB-MAT-011",
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    "title": "主动与被动引力质量",
    "plain": "Active and passive gravitational mass",
    "latex": "\\varphi(r)\\sim\\frac{Gm_G}{c^2r},\\quad\\Phi_g^{\\rm ext}=-c^2\\varphi^{\\rm ext},\\quad U_{\\rm ext}=m_P\\Phi_g^{\\rm ext},\\quad\\mathbf F=-m_P\\nabla\\Phi_g^{\\rm ext}",
    "path": "/theory/matter/mass",
    "note": "孤立源远场、弱外场、缓变背景；mP须由构型的外场响应求出。",
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    "version": "1.0.0",
    "status": "CANDIDATE MODEL",
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    "source_tier": "research-notes",
    "source_reference": "Ch. 4 (4.36)–(4.38); Ch. 5 (5.6.28) and following text",
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    "role": "definition",
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      "en": {
        "title": "Active and passive gravitational mass",
        "plain": "Active and passive gravitational mass",
        "note": "Isolated-source far field and weak slowly varying external field; mP must be obtained from the configuration’s response.",
        "path": "/en/theory/matter/mass",
        "source_reference": "Ch. 4 (4.36)–(4.38); Ch. 5 (5.6.28) and following text",
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          "adopted_scope": "Registered equation, symbols and scope checked against the final manuscript; original ID and scientific status preserved.",
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            "Isolated-source far field and weak slowly varying external field; mP must be obtained from the configuration’s response."
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    "id": "ASD-WEB-MAT-012",
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    "title": "四种质量的匹配任务",
    "plain": "Matching four mass definitions",
    "latex": "\\boxed{m_E=m_I=m_G=m_P=m}",
    "path": "/theory/matter/mass",
    "note": "最终书稿（4.36）的共同匹配要求：同一解、参数和归一化须独立给出四种质量并实现相等；等号不把未闭合的微观计算升级为已证明。",
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    "role": "normalization",
    "translations": {
      "en": {
        "title": "Matching four mass definitions",
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    "id": "ASD-WEB-MAT-014",
    "key": "mat-fr",
    "title": "有条件的 FR 量子化选择",
    "plain": "Conditional FR quantization choice",
    "latex": "\\psi(-A)=-\\psi(A)\\quad\\Longrightarrow\\quad j\\in\\{\\tfrac12,\\tfrac32,\\ldots\\}",
    "path": "/theory/matter/spin",
    "note": "须证明完整构型空间的非平凡旋转环路并指定该量子化选择。并不独自选定j=1/2或证明交换统计。",
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        "title": "Conditional FR quantization choice",
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        "note": "Requires a nontrivial rotation loop in the full configuration space and this quantization choice; it alone fixes neither j=1/2 nor exchange statistics.",
        "path": "/en/theory/matter/spin",
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    "latex": "\\oint_C d\\theta=-2\\pi\\ (e^-),\\qquad\\oint_C d\\theta=+2\\pi\\ (e^+)",
    "path": "/theory/matter/electron",
    "note": "相位约定不是完整场解，亦不自动等于NQ；电荷识别需完整映射。",
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        "note": "A phase convention is not a full field solution or automatically NQ; charge requires the complete mapping.",
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    "title": "质子的内部绕转候选",
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    "latex": "\\oint_C d\\theta=6\\pi,\\quad N_{\\rm int}=3,\\qquad N_{\\rm em}=+1,\\quad Q_p=+e",
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        "title": "Candidate internal proton winding",
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        "note": "Internal winding and EM degree are distinct; 6π fixes neither a trefoil knot nor +3e.",
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    "latex": "\\mathbf n=\\tfrac1{\\sqrt3}(1,1,1)^T,\\quad P_{A_1}=\\mathbf n\\mathbf n^T,\\quad P_E=I-P_{A_1},\\quad\\widehat Q_p=e(q_E P_E+q_A P_{A_1})",
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        "title": "Net charge in a pair process",
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    "key": "qm-fresnel",
    "title": "一维横向孔径的 Fresnel 传播",
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    "latex": "\\psi(L,Y)=\\frac1{\\sqrt{i\\lambda L}}\\int_{-\\infty}^{\\infty}T(y)\\psi_{\\rm in}(y)\\exp\\!\\left[\\frac{ik(Y-y)^2}{2L}\\right]dy,\\qquad\\lambda=2\\pi/k",
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    "note": "B=I、近单能、薄孔径和傍轴近似；省略共同的纵向及时间相位。",
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        "title": "Fresnel propagation of a one-dimensional transverse aperture",
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        "source_reference": "Ch. 5 (5.3.1)–(5.3.3)",
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    "note": "两路径主要速度相同；重力式限弱均匀场与指定路径面积。MP 为被动引力质量。",
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    "note": "位置二阶矩与一阶弱导数平方可积；标准差依照这些数学权重定义。",
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        "title": "Fourier convention for a normalized envelope",
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        "note": "Finite second position moment and square-integrable first weak derivative; standard deviations refer to these mathematical weights.",
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    "note": "σx>0；宽度为标准差，不能直接换成狭缝第一零点或半高宽。",
    "canonical": true,
    "version": "1.0.0",
    "status": "DERIVED",
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      "en": {
        "title": "Gaussian envelope saturating the width bound",
        "plain": "Gaussian envelope saturating the width bound",
        "note": "σx>0; widths are standard deviations, not slit first-zero or half-maximum widths.",
        "path": "/en/theory/quantum/uncertainty-probability",
        "source_reference": "Ch. 5 (5.3.26)",
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          "reference": "Ch. 5 (5.3.26)",
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            "p1967",
            "p1971",
            "p1974",
            "p1979",
            "p1980"
          ],
          "adopted_scope": "σx>0; widths are standard deviations, not slit first-zero or half-maximum widths.",
          "conditions": [
            "Selected channel and stated initial/boundary conditions; action/energy matching retained",
            "Continuum, slow-envelope and detector approximations kept at their respective levels"
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        "p1967",
        "p1971",
        "p1974",
        "p1979",
        "p1980"
      ],
      "adopted_scope": "保留宽度证明；补全矩形时间窗第一零点精确恒等式",
      "conditions": [
        "所选通道与已声明初边值；作用量及能量匹配保留",
        "连续、慢包络和探测近似分别保留在对应层级"
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    "latex": "M_D(\\ddot q_D+2\\gamma_D\\dot q_D+\\omega_D^2q_D)=g_D\\Phi_{\\rm real}+\\xi_D,\\qquad\\chi_D(\\omega)=\\frac1{M_D(\\omega_D^2-\\omega^2-2i\\gamma_D\\omega)}",
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    "note": "采用 e⁻ⁱωᵗ 约定，γD>0；这是线性材料响应模型，不是普适测量公设。",
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        "title": "Average absorbed power and conditional event rate",
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    "title": "相位流重写与额外核心运动律",
    "plain": "Phase-flow rewriting and an additional core law",
    "latex": "\\Psi=\\sqrt n\\,e^{iS/\\hbar},\\quad v_i=\\frac{B_{ij}\\partial_jS}{m_0},\\quad\\partial_t n+\\partial_i(nv_i)=0;\\qquad\\dot X_i\\stackrel{\\rm candidate}{=}v_i(\\mathbf X,t)",
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    "note": "前三项限无源包络、n>0；最后一项是额外核心—场识别。等变性只保持已给初态分布。",
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    "latex": "-\\frac{\\hbar^2}{2m_0}\\psi''+V\\psi=E\\psi,\\quad k=\\sqrt{2m_0E}/\\hbar,\\quad\\kappa=\\sqrt{2m_0(V_0-E)}/\\hbar,\\quad\\psi_{\\rm II}=Ae^{\\kappa x}+Be^{-\\kappa x}",
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    "note": "0<E<V₀、同一质量、B=I、势垒宽 a；ψ 与 ψ′ 在有限阶跃连续。",
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