{
  "title": "The forced Navier–Stokes blowup theorem and computation in viscous flows",
  "sources": [
    {
      "source_id": "OAI-NS-166-original:0e779481c4da40bd28d1e642e1d8ca57447d129610df28dfa5a11e9af8ae228f",
      "author": "OpenAI",
      "title": "Finite Time Blowup for Navier–Stokes",
      "url": "https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf",
      "sha256": "0e779481c4da40bd28d1e642e1d8ca57447d129610df28dfa5a11e9af8ae228f",
      "locators": "Theorem 1.1, Sections 3–10, Corollary 10.6 and the cited appendices."
    },
    {
      "source_id": "ARXIV:2609.16470v1",
      "authors": [
        "Levent Alpöge",
        "Tristan Buckmaster",
        "Matei P. Coiculescu"
      ],
      "title": "Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing",
      "url": "https://arxiv.org/abs/2609.16470v1",
      "author_source_url": "https://arxiv.org/src/2609.16470v1",
      "archive_sha256": "1f5494876f93ffe8cc030b34d01bb1f027b03d0c8ba38a895662dfe89b82fed4",
      "source_kind": "Original author TeX",
      "locators": "Introduction and construction overview; historical and mathematical context."
    },
    {
      "source_id": "OAI-NS-166-transcription:e672c8021820af39df19eb0bb1fdfa5ceb990751:pp007-012.tex",
      "attributed_source_author": "OpenAI",
      "source_kind": "Independent transcription; not author TeX; bounded comparisons only",
      "source_url": "https://github.com/KokunoYumeto/yang-mills-interacting-workbench/blob/e672c8021820af39df19eb0bb1fdfa5ceb990751/navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp007-012.tex",
      "revision": "e672c8021820af39df19eb0bb1fdfa5ceb990751",
      "file": "navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp007-012.tex",
      "sha256": "077c50a30e68490736f531773ccc5bfdf168ca03df5703087a18d7ba94c2fc12",
      "receiving_lesson": "NS-BLOWUP N6.1"
    },
    {
      "source_id": "OAI-NS-166-transcription:e672c8021820af39df19eb0bb1fdfa5ceb990751:pp055-060.tex",
      "attributed_source_author": "OpenAI",
      "source_kind": "Independent transcription; not author TeX; bounded comparisons only",
      "source_url": "https://github.com/KokunoYumeto/yang-mills-interacting-workbench/blob/e672c8021820af39df19eb0bb1fdfa5ceb990751/navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp055-060.tex",
      "revision": "e672c8021820af39df19eb0bb1fdfa5ceb990751",
      "file": "navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp055-060.tex",
      "sha256": "5d5f23647375e6c69e60d866ddf6802d40f1e7fbc7e4fe45f27516d05b7a6864",
      "receiving_lesson": "NS-BLOWUP N6.1"
    },
    {
      "source_id": "OAI-NS-166-transcription:e672c8021820af39df19eb0bb1fdfa5ceb990751:pp103-108.tex",
      "attributed_source_author": "OpenAI",
      "source_kind": "Independent transcription; not author TeX; bounded comparisons only",
      "source_url": "https://github.com/KokunoYumeto/yang-mills-interacting-workbench/blob/e672c8021820af39df19eb0bb1fdfa5ceb990751/navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp103-108.tex",
      "revision": "e672c8021820af39df19eb0bb1fdfa5ceb990751",
      "file": "navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp103-108.tex",
      "sha256": "ca19ef475da18d2a1e15d18af7e2d66181c0fb3a3920838a199933c8ebf10ffe",
      "receiving_lesson": "NS-BLOWUP N6.1"
    },
    {
      "source_id": "OAI-NS-166-transcription:e672c8021820af39df19eb0bb1fdfa5ceb990751:pp109-114.tex",
      "attributed_source_author": "OpenAI",
      "source_kind": "Independent transcription; not author TeX; bounded comparisons only",
      "source_url": "https://github.com/KokunoYumeto/yang-mills-interacting-workbench/blob/e672c8021820af39df19eb0bb1fdfa5ceb990751/navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp109-114.tex",
      "revision": "e672c8021820af39df19eb0bb1fdfa5ceb990751",
      "file": "navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp109-114.tex",
      "sha256": "7e8094ab1cb8144eec7c9e0e746935f921f3307ff83a78a37996ee83ee4199c1",
      "receiving_lesson": "NS-BLOWUP N6.1"
    },
    {
      "source_id": "OAI-NS-166-transcription:e672c8021820af39df19eb0bb1fdfa5ceb990751:pp115-120.tex",
      "attributed_source_author": "OpenAI",
      "source_kind": "Independent transcription; not author TeX; bounded comparisons only",
      "source_url": "https://github.com/KokunoYumeto/yang-mills-interacting-workbench/blob/e672c8021820af39df19eb0bb1fdfa5ceb990751/navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp115-120.tex",
      "revision": "e672c8021820af39df19eb0bb1fdfa5ceb990751",
      "file": "navier-stokes/sources/openai-source-faithful-20260920/upstream/reconstruction/sections/pp115-120.tex",
      "sha256": "349578662f5c4b29f1021d09199731a9aa764e5c2703d8c4c75b9bce7bca3726",
      "receiving_lesson": "NS-BLOWUP N6.1"
    },
    {
      "source_id": "OAI-COMP:adc7f1241b42e322a6451854ab7e4b4c146bf78a:preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/build/main.tex",
      "attributed_source_author": "OpenAI",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/build/main.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/build/main.tex",
      "sha256": "cd3b7ba8ed771536d52e8adbe918bff5ffe9ac20158c548541139ef3de376e03",
      "git_blob": "cfb8c3d315b164d3cf00231720ff1e076981ac65",
      "receiving_lesson": "NS-BLOWUP-N8-1"
    },
    {
      "source_id": "OAI-COMP:adc7f1241b42e322a6451854ab7e4b4c146bf78a:preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/build/references.bib",
      "attributed_source_author": "OpenAI",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/build/references.bib",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "preprints/Universal-Computation-with-Eventually-Stationary-Navier-Stokes-Forcing-September-27-2026/build/references.bib",
      "sha256": "1484b92d1ee358b29c11c7c5fae5e4415083bb2f5ebcd304ad3dc75cadd95fe3",
      "git_blob": "06d405fb45a73a00dd8990ec25e9b91d04017fa7",
      "receiving_lesson": "NS-BLOWUP-N8-1"
    },
    {
      "author": "OpenAI",
      "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/build/main.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "main.tex",
      "receiving_lesson": "NS-BLOWUP-N8-2"
    },
    {
      "author": "OpenAI",
      "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/build/addressed.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "addressed.tex",
      "receiving_lesson": "NS-BLOWUP-N8-2"
    },
    {
      "author": "OpenAI",
      "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/build/alternating.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "alternating.tex",
      "receiving_lesson": "NS-BLOWUP-N8-2"
    },
    {
      "author": "OpenAI",
      "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/build/lattice.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "lattice.tex",
      "receiving_lesson": "NS-BLOWUP-N8-2"
    },
    {
      "author": "OpenAI",
      "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026/build/references.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "references.tex",
      "receiving_lesson": "NS-BLOWUP-N8-2"
    },
    {
      "author": "OpenAI",
      "title": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/build/main.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "main.tex",
      "receiving_lesson": "NS-BLOWUP-N8-3"
    },
    {
      "author": "OpenAI",
      "title": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/build/compact.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "compact.tex",
      "receiving_lesson": "NS-BLOWUP-N8-3"
    },
    {
      "author": "OpenAI",
      "title": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/build/bounded.tex",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "bounded.tex",
      "receiving_lesson": "NS-BLOWUP-N8-3"
    },
    {
      "author": "OpenAI",
      "title": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows",
      "source_kind": "Original author LaTeX or bibliography",
      "source_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026/build/references.bib",
      "revision": "adc7f1241b42e322a6451854ab7e4b4c146bf78a",
      "file": "references.bib",
      "receiving_lesson": "NS-BLOWUP-N8-3"
    }
  ],
  "lessons": [
    {
      "id": "NS-BLOWUP-N1",
      "file": "src/the-forced-blowup-theorem-and-its-geometry.md",
      "title": "The forced blowup theorem and its geometry",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N2-1",
      "file": "src/from-velocity-profiles-to-stress-and-moments.md",
      "title": "From velocity profiles to stress and five exact moments",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N2-2",
      "file": "src/constructing-the-outer-profile-and-heat-exterior.md",
      "title": "Constructing the outer profile and its heat exterior",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N2-3",
      "file": "src/constructing-the-analytic-axis-profile.md",
      "title": "Constructing the analytic axis profile",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N2-4",
      "file": "src/joining-the-analytic-axis-to-the-outer-field.md",
      "title": "Joining the analytic axis to the outer field",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N2-5",
      "file": "src/realizing-the-admissible-shear.md",
      "title": "Realizing the admissible shear and completing the leading profile",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N3-1",
      "file": "src/correcting-coefficients-on-the-axis.md",
      "title": "Correcting the coefficients on one fixed axis interval",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N3-2",
      "file": "src/extending-coefficients-and-cancelling-moments.md",
      "title": "Extending every coefficient and cancelling its five moments",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N3-3",
      "file": "src/constructing-the-actual-base-field.md",
      "title": "Constructing the actual base field",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N4-1",
      "file": "src/separating-the-oscillatory-supports.md",
      "title": "Separating the oscillatory supports",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N4-2",
      "file": "src/constructing-the-pulses-and-their-velocities.md",
      "title": "Constructing the pulses and their actual velocities",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N4-3",
      "file": "src/realizing-stress-and-controlling-interactions.md",
      "title": "Realizing the stress and controlling nonlinear interactions",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N5-1",
      "file": "src/radial-primitives-and-the-mean-equations.md",
      "title": "Radial primitives and the complete mean equations",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N5-2",
      "file": "src/cancelling-the-nonconstant-auxiliary-means.md",
      "title": "Cancelling the nonconstant auxiliary means",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N5-3",
      "file": "src/correcting-the-three-defects-with-five-profiles.md",
      "title": "Correcting the three defects with five compact profiles",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N6-1",
      "file": "src/summing-the-correction-fields.md",
      "title": "Summing the correction fields without losing the equation",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N6-2",
      "file": "src/initializing-and-repeating-the-correction-cycle.md",
      "title": "Initializing and repeating the complete correction cycle",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N6-3",
      "file": "src/constructing-the-actual-local-field.md",
      "title": "Constructing the actual local velocity and pressure",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N7-1",
      "file": "src/localizing-the-field-and-extending-the-force.md",
      "title": "Localizing the field and extending its smooth force",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N7-2",
      "file": "src/energy-comparison-and-the-final-blowup-conclusions.md",
      "title": "Energy, comparison and the final blowup conclusions",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N8-1",
      "file": "src/computation-with-eventually-stationary-forcing.md",
      "title": "Computation with a force that becomes stationary",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N8-2",
      "file": "src/computation-with-rapidly-decreasing-forcing.md",
      "title": "Computation with rapidly decreasing forcing",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-N8-3",
      "file": "src/detecting-computation-in-the-velocity.md",
      "title": "Detecting computation in the velocity of a viscous fluid",
      "worked_exercises": 5
    }
  ],
  "source_relationship": "These lessons are separate mathematical exposition. Original works retain their attribution and rights.",
  "rights": "Authored text and figures in the twenty-three main lessons: CC0. Third-party sources, software and fonts retain their own notices. Appendix source files retain their own attribution and notices.",
  "appendices": [
    {
      "id": "NS-BLOWUP-APPENDIX-A",
      "title": "Exact vacuum constraint data from a velocity field",
      "file": "appendices/vacuum-gravity/src/velocity-fields-and-vacuum-constraints.md",
      "worked_exercises": 5
    },
    {
      "id": "NS-BLOWUP-APPENDIX-B",
      "title": "The gravitational shear response and its fluid limit",
      "file": "appendices/vacuum-gravity/src/gravitational-shear-and-the-fluid-limit.md",
      "worked_exercises": 5
    }
  ],
  "appendix_sources": "appendices/vacuum-gravity/SOURCES.json"
}