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  "volume_id": "yang_mills",
  "title": "The Geometry of Force and Silence",
  "subtitle": "A guided journey through gauge fields, quantum existence, and the mass gap",
  "status": "Open Millennium Prize Problem; constructive quantum-field obligation; no mass-gap proof claim",
  "claim_boundary": "Browser-native, source-normalized exposition of four-dimensional quantum Yang–Mills existence and mass gap. Classical solutions, instantons, perturbation theory, asymptotic freedom, finite-lattice models, strong-coupling expansions, numerical glueball spectra, supersymmetric theories, and confinement heuristics are not substituted for an axiomatic nontrivial continuum theory with a positive spectral gap.",
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      "id": "cover",
      "label": "Cover",
      "title": "The Geometry of Force and Silence",
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      "alt": "An illuminated gauge connection curves across a dark field above a spectral gap.",
      "authority": "pedagogical_orientation_only"
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    {
      "id": "gauge",
      "label": "Plate I",
      "title": "Symmetry that changes from point to point",
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      "alt": "Local gauge frames rotate across space while the connection records how neighbouring frames compare.",
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      "label": "Plate II",
      "title": "Connection, curvature, and holonomy",
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      "alt": "A loop measures holonomy while curvature marks the infinitesimal failure to return unchanged.",
      "authority": "pedagogical_orientation_only"
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    {
      "id": "quantum",
      "label": "Plate III",
      "title": "From classical action to quantum theory",
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      "alt": "A classical Yang–Mills field passes through regularization, positivity, reconstruction, and continuum-limit obligations.",
      "authority": "pedagogical_orientation_only"
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    {
      "id": "gap",
      "label": "Plate IV",
      "title": "The silence above the vacuum",
      "asset": "assets/documentaries/yang_mills/gap.svg",
      "alt": "A Hamiltonian spectrum shows a vacuum at zero and the required positive interval before the first excitation.",
      "authority": "pedagogical_orientation_only"
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    {
      "id": "reader-note",
      "title": "How to Read the Field",
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    {
      "id": "gauge-symmetry",
      "title": "Symmetry That Changes from Point to Point",
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      "id": "connection-curvature",
      "title": "Connection, Curvature, and Holonomy",
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      "id": "classical",
      "title": "The Classical Equation",
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      "id": "quantum-existence",
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      "title": "The Silence Above the Vacuum",
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  "appendices": [
    {
      "id": "appendix-geometry",
      "title": "Bundles, Connections, and Action",
      "source_heading": "Technical Appendix: Gauge Geometry",
      "claim_classes": [
        "established"
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    },
    {
      "id": "appendix-axioms",
      "title": "Wightman and Osterwalder–Schrader Obligations",
      "source_heading": "Technical Appendix: Axiomatic Quantum Field Theory",
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      "id": "appendix-spectrum",
      "title": "Spectral Gap and Correlation Decay",
      "source_heading": "Technical Appendix: Spectrum and Clustering",
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    {
      "id": "appendix-lattice",
      "title": "Lattice, Renormalization, and Trust Matrix",
      "source_heading": "Technical Appendix: Lattice and Continuum Obligations",
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  "sources": [
    "https://www.claymath.org/millennium/yang-mills-the-maths-gap/",
    "https://www.claymath.org/wp-content/uploads/2022/05/yangmills.pdf",
    "https://doi.org/10.1007/BF01645738",
    "https://doi.org/10.1103/PhysRevLett.30.1343",
    "https://doi.org/10.1103/PhysRevD.10.2445"
  ]
}