{ "schemaVersion": 1, "studyVersion": 1, "exhibitId": "chaos", "slug": "chaos-uncertainty", "title": "Chaos: initial perturbations and integration error", "generatedAt": "2026-09-17T00:39:16.426Z", "build": { "packageVersion": "0.1.0", "softwareVersion": "0.1.0+0e6017e6a747.dirty.02665ba9098c", "gitRevision": "0e6017e6a747d3c610b956612b4125e232283001", "gitDirty": true, "sourceDigest": "02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b", "runtime": "v26.5.1" }, "reviewStatus": "not-reviewed", "sourceIds": [ "M7", "M12" ], "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "4001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window.", "101 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately.", "8001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "201 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "16001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "401 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot." ], "methods": [ "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint." ], "runCount": 24, "runs": [ { "runId": "run-20260917003910210-62944-001", "label": "ε=0 · h=0.01 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0, "step": 0.01, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.306Z", "elapsedMs": 22.46808300000066, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 4000 complete fixed steps." }, { "label": "Initial separation", "value": 0, "precision": 8 }, { "label": "Final separation", "value": 0, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 4001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "4001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 8000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-001.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-001.csv", "csvRows": 4001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1335 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1335 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1335 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 0 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-002", "label": "ε=0 · h=0.01 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0, "step": 0.01, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.332Z", "elapsedMs": 1.6870409999992262, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 100 complete fixed steps." }, { "label": "Initial separation", "value": 0, "precision": 8 }, { "label": "Final separation", "value": 0, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 201 }, { "label": "Short-window maximum step difference", "value": 0.000749085575087332, "precision": 8, "description": "Same initial state; h versus h/2 at 101 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.00009276287013989525, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "101 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 400 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-002.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-002.csv", "csvRows": 201, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 101 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 101 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 101 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 0 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 101 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-003", "label": "ε=0 · h=0.005 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0, "step": 0.005, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.368Z", "elapsedMs": 33.73083299999962, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 8000 complete fixed steps." }, { "label": "Initial separation", "value": 0, "precision": 8 }, { "label": "Final separation", "value": 0, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 8001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "8001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 16000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-003.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-003.csv", "csvRows": 8001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1601 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1601 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1601 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 0 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-004", "label": "ε=0 · h=0.005 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0, "step": 0.005, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.412Z", "elapsedMs": 2.415791999999783, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 200 complete fixed steps." }, { "label": "Initial separation", "value": 0, "precision": 8 }, { "label": "Final separation", "value": 0, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 401 }, { "label": "Short-window maximum step difference", "value": 0.00004641730006565268, "precision": 8, "description": "Same initial state; h versus h/2 at 201 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.0000024289507761028697, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "201 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 800 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-004.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-004.csv", "csvRows": 401, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 201 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 201 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 201 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 0 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 201 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-005", "label": "ε=0 · h=0.0025 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0, "step": 0.0025, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.485Z", "elapsedMs": 69.98458399999981, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 16000 complete fixed steps." }, { "label": "Initial separation", "value": 0, "precision": 8 }, { "label": "Final separation", "value": 0, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 16001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "16001 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 32000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-005.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-005.csv", "csvRows": 16001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1779 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1779 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1779 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 0 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-006", "label": "ε=0 · h=0.0025 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0, "step": 0.0025, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.565Z", "elapsedMs": 3.903291999999965, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 400 complete fixed steps." }, { "label": "Initial separation", "value": 0, "precision": 8 }, { "label": "Final separation", "value": 0, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 801 }, { "label": "Short-window maximum step difference", "value": 0.0000028816317582099183, "precision": 8, "description": "Same initial state; h versus h/2 at 401 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 1.5881468053876506e-7, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "401 base samples have zero separation. They remain zero in the data and are omitted from the logarithmic plot.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 1600 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-006.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-006.csv", "csvRows": 801, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 401 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 401 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 401 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 0 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 401 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-007", "label": "ε=1e-8 · h=0.01 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 1e-8, "step": 0.01, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.591Z", "elapsedMs": 19.248709000000417, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 4000 complete fixed steps." }, { "label": "Initial separation", "value": 9.99999993922529e-9, "precision": 8 }, { "label": "Final separation", "value": 20.25781136674361, "precision": 8 }, { "label": "Maximum sampled separation", "value": 42.7736011287924, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 28.48, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 4001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 8000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-007.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-007.csv", "csvRows": 4001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1335 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1335 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1335 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1335 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-008", "label": "ε=1e-8 · h=0.01 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 1e-8, "step": 0.01, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.618Z", "elapsedMs": 1.2414159999998446, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 100 complete fixed steps." }, { "label": "Initial separation", "value": 9.99999993922529e-9, "precision": 8 }, { "label": "Final separation", "value": 1.116034156650338e-8, "precision": 8 }, { "label": "Maximum sampled separation", "value": 2.0107375301245932e-7, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 201 }, { "label": "Short-window maximum step difference", "value": 0.000749085575087332, "precision": 8, "description": "Same initial state; h versus h/2 at 101 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.00009276287013989525, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 400 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-008.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-008.csv", "csvRows": 201, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 101 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 101 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 101 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 101 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 101 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-009", "label": "ε=1e-8 · h=0.005 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 1e-8, "step": 0.005, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.654Z", "elapsedMs": 33.16120799999953, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 8000 complete fixed steps." }, { "label": "Initial separation", "value": 9.99999993922529e-9, "precision": 8 }, { "label": "Final separation", "value": 9.035401427770934, "precision": 8 }, { "label": "Maximum sampled separation", "value": 45.85777773649623, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 27.725, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 8001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 16000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-009.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-009.csv", "csvRows": 8001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1601 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1601 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1601 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1601 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-010", "label": "ε=1e-8 · h=0.005 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 1e-8, "step": 0.005, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.700Z", "elapsedMs": 2.1300830000000133, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 200 complete fixed steps." }, { "label": "Initial separation", "value": 9.99999993922529e-9, "precision": 8 }, { "label": "Final separation", "value": 1.1159962277629426e-8, "precision": 8 }, { "label": "Maximum sampled separation", "value": 2.0107760514667746e-7, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 401 }, { "label": "Short-window maximum step difference", "value": 0.00004641730006565268, "precision": 8, "description": "Same initial state; h versus h/2 at 201 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.0000024289507761028697, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 800 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-010.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-010.csv", "csvRows": 401, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 201 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 201 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 201 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 201 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 201 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-011", "label": "ε=1e-8 · h=0.0025 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 1e-8, "step": 0.0025, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.763Z", "elapsedMs": 59.292875000000095, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 16000 complete fixed steps." }, { "label": "Initial separation", "value": 9.99999993922529e-9, "precision": 8 }, { "label": "Final separation", "value": 9.72638401532072, "precision": 8 }, { "label": "Maximum sampled separation", "value": 48.93658619223874, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 30.44, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 16001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 32000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-011.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-011.csv", "csvRows": 16001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1779 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1779 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1779 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1779 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-012", "label": "ε=1e-8 · h=0.0025 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 1e-8, "step": 0.0025, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.850Z", "elapsedMs": 5.640667000000576, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 400 complete fixed steps." }, { "label": "Initial separation", "value": 9.99999993922529e-9, "precision": 8 }, { "label": "Final separation", "value": 1.1159947262026489e-8, "precision": 8 }, { "label": "Maximum sampled separation", "value": 2.010778207352675e-7, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 801 }, { "label": "Short-window maximum step difference", "value": 0.0000028816317582099183, "precision": 8, "description": "Same initial state; h versus h/2 at 401 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 1.5881468053876506e-7, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 1600 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-012.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-012.csv", "csvRows": 801, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 401 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 401 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 401 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 401 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 401 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-013", "label": "ε=0.00001 · h=0.01 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.00001, "step": 0.01, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.879Z", "elapsedMs": 19.001166000000012, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 4000 complete fixed steps." }, { "label": "Initial separation", "value": 0.000010000000000065512, "precision": 8 }, { "label": "Final separation", "value": 12.94597591277636, "precision": 8 }, { "label": "Maximum sampled separation", "value": 48.40676134183638, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 22.94, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 4001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 8000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-013.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-013.csv", "csvRows": 4001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1335 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1335 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1335 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1335 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-014", "label": "ε=0.00001 · h=0.01 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.00001, "step": 0.01, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.907Z", "elapsedMs": 1.1521249999996144, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 100 complete fixed steps." }, { "label": "Initial separation", "value": 0.000010000000000065512, "precision": 8 }, { "label": "Final separation", "value": 0.000011160313572386339, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0.00020107317990149677, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 201 }, { "label": "Short-window maximum step difference", "value": 0.000749085575087332, "precision": 8, "description": "Same initial state; h versus h/2 at 101 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.00009276287013989525, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 400 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-014.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-014.csv", "csvRows": 201, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 101 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 101 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 101 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 101 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 101 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-015", "label": "ε=0.00001 · h=0.005 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.00001, "step": 0.005, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:15.941Z", "elapsedMs": 31.552042000000256, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 8000 complete fixed steps." }, { "label": "Initial separation", "value": 0.000010000000000065512, "precision": 8 }, { "label": "Final separation", "value": 5.825321953835089, "precision": 8 }, { "label": "Maximum sampled separation", "value": 48.58067870025925, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 21.56, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 8001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 16000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-015.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-015.csv", "csvRows": 8001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1601 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1601 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1601 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1601 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-016", "label": "ε=0.00001 · h=0.005 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.00001, "step": 0.005, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:15.987Z", "elapsedMs": 2.1273330000003625, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 200 complete fixed steps." }, { "label": "Initial separation", "value": 0.000010000000000065512, "precision": 8 }, { "label": "Final separation", "value": 0.000011159935092818826, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0.00020107704718014255, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 401 }, { "label": "Short-window maximum step difference", "value": 0.00004641730006565268, "precision": 8, "description": "Same initial state; h versus h/2 at 201 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.0000024289507761028697, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 800 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-016.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-016.csv", "csvRows": 401, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 201 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 201 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 201 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 201 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 201 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-017", "label": "ε=0.00001 · h=0.0025 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.00001, "step": 0.0025, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:16.050Z", "elapsedMs": 59.8340419999995, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 16000 complete fixed steps." }, { "label": "Initial separation", "value": 0.000010000000000065512, "precision": 8 }, { "label": "Final separation", "value": 5.3951233284521605, "precision": 8 }, { "label": "Maximum sampled separation", "value": 49.62967823487382, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 21.5575, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 16001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 32000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-017.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-017.csv", "csvRows": 16001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1779 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1779 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1779 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1779 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-018", "label": "ε=0.00001 · h=0.0025 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.00001, "step": 0.0025, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:16.133Z", "elapsedMs": 4.124375000000327, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 400 complete fixed steps." }, { "label": "Initial separation", "value": 0.000010000000000065512, "precision": 8 }, { "label": "Final separation", "value": 0.000011159932136893657, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0.00020107727419568715, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 801 }, { "label": "Short-window maximum step difference", "value": 0.0000028816317582099183, "precision": 8, "description": "Same initial state; h versus h/2 at 401 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 1.5881468053876506e-7, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 1600 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-018.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-018.csv", "csvRows": 801, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 401 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 401 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 401 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 401 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 401 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-019", "label": "ε=0.01 · h=0.01 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.01, "step": 0.01, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:16.158Z", "elapsedMs": 17.669208000000253, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 4000 complete fixed steps." }, { "label": "Initial separation", "value": 0.010000000000000009, "precision": 8 }, { "label": "Final separation", "value": 15.659985410837464, "precision": 8 }, { "label": "Maximum sampled separation", "value": 53.872566614015525, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 14.02, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 4001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 8000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-019.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-019.csv", "csvRows": 4001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1335 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1335 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1335 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1335 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-020", "label": "ε=0.01 · h=0.01 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.01, "step": 0.01, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:16.185Z", "elapsedMs": 1.0942500000001019, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.01, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 100 complete fixed steps." }, { "label": "Initial separation", "value": 0.010000000000000009, "precision": 8 }, { "label": "Final separation", "value": 0.011132690179213707, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0.2005161555813419, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 201 }, { "label": "Short-window maximum step difference", "value": 0.000749085575087332, "precision": 8, "description": "Same initial state; h versus h/2 at 101 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.00009276287013989525, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 400 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-020.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-020.csv", "csvRows": 201, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 101 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 101 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 101 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 101 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 101 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-021", "label": "ε=0.01 · h=0.005 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.01, "step": 0.005, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:16.219Z", "elapsedMs": 31.33887499999946, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 8000 complete fixed steps." }, { "label": "Initial separation", "value": 0.010000000000000009, "precision": 8 }, { "label": "Final separation", "value": 12.196719712596018, "precision": 8 }, { "label": "Maximum sampled separation", "value": 51.959145161918734, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 14.02, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 8001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 16000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-021.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-021.csv", "csvRows": 8001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1601 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1601 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1601 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1601 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-022", "label": "ε=0.01 · h=0.005 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.01, "step": 0.005, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:16.265Z", "elapsedMs": 2.057292000000416, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.005, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 200 complete fixed steps." }, { "label": "Initial separation", "value": 0.010000000000000009, "precision": 8 }, { "label": "Final separation", "value": 0.011132312897961197, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0.2005200214742473, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 401 }, { "label": "Short-window maximum step difference", "value": 0.00004641730006565268, "precision": 8, "description": "Same initial state; h versus h/2 at 201 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 0.0000024289507761028697, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 800 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-022.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-022.csv", "csvRows": 401, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 201 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 201 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 201 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 201 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 201 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-023", "label": "ε=0.01 · h=0.0025 · T=40", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.01, "step": 0.0025, "duration": 40, "threshold": 1, "compareStep": false }, "createdAt": "2026-09-17T00:39:16.326Z", "elapsedMs": 57.84162500000002, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 40, "unit": "model time", "precision": 6, "description": "Requested T=40; 16000 complete fixed steps." }, { "label": "Initial separation", "value": 0.010000000000000009, "precision": 8 }, { "label": "Final separation", "value": 3.7741761848656044, "precision": 8 }, { "label": "Maximum sampled separation", "value": 51.97781313391188, "precision": 8 }, { "label": "First sampled threshold crossing", "value": 14.0175, "unit": "model time", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 16001 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error." ], "tolerance": 5.684341886080801e-13, "residual": 0, "iterations": 32000 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-023.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-023.csv", "csvRows": 16001, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 1779 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 1779 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 1779 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 1779 } ], "scalarRange": null }, { "runId": "run-20260917003910210-62944-024", "label": "ε=0.01 · h=0.0025 · T=1", "parameters": { "sigma": 10, "rho": 28, "beta": 2.6666666666666665, "epsilon": 0.01, "step": 0.0025, "duration": 1, "threshold": 1, "compareStep": true }, "createdAt": "2026-09-17T00:39:16.415Z", "elapsedMs": 6.0847499999999854, "modelVersion": "chaos-rk4-1.0.0", "numericalMethod": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "measurements": [ { "label": "Base integration step h", "value": 0.0025, "unit": "model time", "precision": 6 }, { "label": "Integrated duration", "value": 1, "unit": "model time", "precision": 6, "description": "Requested T=1; 400 complete fixed steps." }, { "label": "Initial separation", "value": 0.010000000000000009, "precision": 8 }, { "label": "Final separation", "value": 0.01113230996938012, "precision": 8 }, { "label": "Maximum sampled separation", "value": 0.2005202483169577, "precision": 8 }, { "label": "First sampled threshold crossing", "value": "Not observed", "precision": 6, "description": "First d(t) > 1; conditional on this run and sample grid." }, { "label": "Full exported sample rows", "value": 801 }, { "label": "Short-window maximum step difference", "value": 0.0000028816317582099183, "precision": 8, "description": "Same initial state; h versus h/2 at 401 common points through t=1." }, { "label": "Short-window endpoint difference", "value": 1.5881468053876506e-7, "precision": 8 } ], "diagnostics": { "method": "Double-precision classical RK4. Fixed h; half-step samples compared at indices i and 2i. Residual: requested versus integrated time-grid endpoint.", "warnings": [ "Finite RK4 trajectories approximate this model; separation does not prove all properties of chaos or predict actual weather.", "The diagnostic residual checks time-grid alignment only. It is not a bound on integration error.", "The separation threshold was not exceeded in this observation window.", "Step comparison fixes the initial state; its short-window difference is not a certified error estimate. Long-time divergence is recorded separately." ], "tolerance": 1.4210854715202004e-14, "residual": 0, "iterations": 1600 }, "sourceIds": [ "M7", "M12" ], "jsonPath": "/studies/chaos/run-20260917003910210-62944-024.json", "csvPath": "/studies/chaos/run-20260917003910210-62944-024.csv", "csvRows": 801, "plotSampleCounts": [ { "title": "First state coordinate", "series": "Reference x(t)", "samples": 401 }, { "title": "First state coordinate", "series": "Perturbed x(t)", "samples": 401 }, { "title": "Initial-condition separation", "series": "Same h, different initial states", "samples": 401 }, { "title": "Separation on a logarithmic axis", "series": "Positive distances; zeros omitted", "samples": 401 }, { "title": "Short-window step comparison", "series": "Same initial state, different steps", "samples": 401 } ], "scalarRange": null } ], "results": [ "This saved batch contains 24 completed calculations. The table and downloads below refer to those actual runs.", "The largest recorded absolute diagnostic residual was 0. 24 of 24 residual-bearing cases were within their own reported numerical tolerance.", "Across 24 recorded readings, Initial separation ranged from 0 to 0.01.", "Across 24 recorded readings, Final separation ranged from 0 to 20.25781.", "Across 24 recorded readings, Maximum sampled separation ranged from 0 to 53.87257.", "Across 12 recorded readings, Short-window maximum step difference ranged from 2.88163e-6 to 0.0007490856.", "6 of 6 saved zero-perturbation cases had exactly zero maximum sampled separation. Identical binary64 inputs follow identical solver operations in these runs.", "At h = 0.005 and T = 40, ε = 0: final separation 0; first sampled d(t) > 1 at Not observed. These are observations on the saved time grid.", "At h = 0.005 and T = 40, ε = 1e-8: final separation 9.035401; first sampled d(t) > 1 at 27.725. These are observations on the saved time grid.", "At h = 0.005 and T = 40, ε = 0.00001: final separation 5.825322; first sampled d(t) > 1 at 21.56. These are observations on the saved time grid.", "At h = 0.005 and T = 40, ε = 0.01: final separation 12.19672; first sampled d(t) > 1 at 14.02. These are observations on the saved time grid.", "Short-window comparison with reference initial state (1,1,1), h = 0.01: maximum difference from its own h/2 run 0.0007490856; endpoint difference 9.27629e-5.", "Short-window comparison with reference initial state (1,1,1), h = 0.005: maximum difference from its own h/2 run 4.64173e-5; endpoint difference 2.42895e-6.", "Short-window comparison with reference initial state (1,1,1), h = 0.0025: maximum difference from its own h/2 run 2.88163e-6; endpoint difference 1.58815e-7." ], "interpretation": [ "The reported diagnostic residual checks only the requested and integrated time-grid endpoints. Its small value is not an integration-error certificate. Use the short-window step differences to investigate discretization; use the fixed-step perturbation cases to investigate sensitivity.", "Compare perturbation cases at fixed h, then compare step sizes at fixed initial conditions. A reported crossing is conditional on this threshold and observation window.", "A missing crossing remains not observed. It must not be replaced by the last simulated time or interpreted as a universal prediction limit." ], "summaryCsvPath": "/studies/chaos/summary-20260917003910210-62944.csv" }