{
  "schema_version": "1.1.0",
  "volume_id": "navier_stokes",
  "title": "The River and the Storm",
  "subtitle": "A guided journey through fluid motion and the three-dimensional smoothness problem",
  "status": "Open Millennium Prize Problem; parent-challenge documentary; no smoothness proof claim",
  "claim_boundary": "Browser-native, source-normalized exposition of the three-dimensional incompressible Navier–Stokes existence-and-smoothness problem. Energy estimates, Leray–Hopf weak existence, two-dimensional regularity, small-data theory, continuation criteria, partial regularity, numerical simulations, and NS-CI-001 progress are not promoted to global smoothness or finite-time breakdown.",
  "asset_base": "docs_root",
  "math_rendering": {
    "provider": "MathJax",
    "version": "3.2.2",
    "delivery": "version_pinned_cdn",
    "script_url": "https://cdn.jsdelivr.net/npm/mathjax@3.2.2/es5/tex-mml-chtml.js",
    "archival_role": "enhancement_only",
    "fallback": "source_tex_remains_readable_without_javascript"
  },
  "palette": {
    "navy": "#071D2B",
    "parchment": "#F2EAD7",
    "gold": "#B99043",
    "ink": "#17252D"
  },
  "plates": [
    {
      "id": "cover",
      "label": "Cover",
      "title": "The River and the Storm",
      "asset": "assets/documentaries/navier_stokes/cover.svg",
      "alt": "A dark illuminated cover shows streamlines, a vortex, and the incompressible Navier–Stokes equation.",
      "authority": "pedagogical_orientation_only"
    },
    {
      "id": "field",
      "label": "Plate I",
      "title": "A fluid has a velocity everywhere",
      "asset": "assets/documentaries/navier_stokes/field.svg",
      "alt": "A vector field flows through a channel while pressure and viscosity act across the moving fluid.",
      "authority": "pedagogical_orientation_only"
    },
    {
      "id": "vorticity",
      "label": "Plate II",
      "title": "Energy guards; vorticity stretches",
      "asset": "assets/documentaries/navier_stokes/vorticity.svg",
      "alt": "Energy cascades downward while a vortex tube lengthens and thins under three-dimensional stretching.",
      "authority": "pedagogical_orientation_only"
    },
    {
      "id": "classes",
      "label": "Plate III",
      "title": "Weak existence is not smooth existence",
      "asset": "assets/documentaries/navier_stokes/classes.svg",
      "alt": "Nested solution classes distinguish Leray–Hopf weak solutions, strong solutions, and globally smooth solutions.",
      "authority": "pedagogical_orientation_only"
    },
    {
      "id": "frontier",
      "label": "Plate IV",
      "title": "The critical frontier",
      "asset": "assets/documentaries/navier_stokes/frontier.svg",
      "alt": "A scaling ladder and a singular-set diagram separate continuation criteria, partial regularity, and the unresolved endpoint.",
      "authority": "pedagogical_orientation_only"
    }
  ],
  "chapters": [
    {
      "id": "reader-note",
      "title": "How to Read the River",
      "source_heading": "A Note to the Reader",
      "claim_classes": [
        "exposition",
        "guardrail",
        "open"
      ]
    },
    {
      "id": "field-equation",
      "title": "The Language of a Fluid",
      "source_heading": "The Language of a Fluid",
      "claim_classes": [
        "established",
        "guardrail"
      ]
    },
    {
      "id": "energy",
      "title": "Energy: The First Guardian",
      "source_heading": "Energy: The First Guardian",
      "claim_classes": [
        "established",
        "guardrail"
      ]
    },
    {
      "id": "vorticity",
      "title": "The Three-Dimensional Difficulty",
      "source_heading": "Vorticity and Vortex Stretching",
      "claim_classes": [
        "established",
        "open",
        "guardrail"
      ]
    },
    {
      "id": "solution-classes",
      "title": "Weak, Strong, and Smooth",
      "source_heading": "Weak, Strong, and Smooth",
      "claim_classes": [
        "established",
        "imported",
        "guardrail"
      ]
    },
    {
      "id": "millennium-frontier",
      "title": "The Millennium Frontier",
      "source_heading": "The Millennium Question, Exactly",
      "claim_classes": [
        "open",
        "conditional",
        "guardrail"
      ]
    }
  ],
  "appendices": [
    {
      "id": "appendix-formulations",
      "title": "Whole-Space and Periodic Formulations",
      "source_heading": "Technical Appendix: Equations and Normalizations",
      "claim_classes": [
        "established",
        "guardrail"
      ]
    },
    {
      "id": "appendix-scaling",
      "title": "Scaling and Critical Spaces",
      "source_heading": "Technical Appendix: Scaling and Function Spaces",
      "claim_classes": [
        "established",
        "conditional",
        "guardrail"
      ]
    },
    {
      "id": "appendix-regularity",
      "title": "Continuation and Partial Regularity",
      "source_heading": "Technical Appendix: Continuation and Partial Regularity",
      "claim_classes": [
        "imported",
        "conditional",
        "guardrail"
      ]
    },
    {
      "id": "appendix-trust",
      "title": "Claim-Level Trust Matrix",
      "source_heading": "Technical Appendix: Claim-Level Trust Matrix",
      "claim_classes": [
        "guardrail",
        "open"
      ]
    }
  ],
  "sources": [
    "https://www.claymath.org/millennium/navier-stokes-equation/",
    "https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf",
    "https://doi.org/10.1007/BF02547754",
    "https://doi.org/10.1002/cpa.3160350604",
    "https://doi.org/10.1070/RM2003v058n02ABEH000608"
  ]
}