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forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.", "tolerance": 8.381903171539306e-11, "residual": 6.606899310233078e-28 }, "sourceIds": [ "M2", "M3" ], "jsonPath": "/studies/fourier/run-20260917003910210-62944-090.json", "csvPath": "/studies/fourier/run-20260917003910210-62944-090.csv", "csvRows": 513, "plotSampleCounts": [ { "title": "Amplitude spectrum", "series": "All coefficients", "samples": 512 }, { "title": "Coefficient phase", "series": "Defined phases", "samples": 512 }, { "title": "Reconstruction error by term budget", "series": "Band-limited", "samples": 512 }, { "title": "Reconstruction error by term budget", "series": "Largest coefficients", "samples": 512 } ], "scalarRange": null } ], "results": [ "This saved batch contains 90 completed calculations. The table and downloads below refer to those actual runs.", "The largest recorded absolute diagnostic residual was 2.22045e-16. 90 of 90 residual-bearing cases were within their own reported numerical tolerance.", "Across 90 recorded readings, Sample MSE ranged from 2.62836e-30 to 0.04650419.", "Across 90 recorded readings, Discarded coefficient energy ranged from 0 to 0.04650419.", "Across 90 recorded readings, MSE identity residual ranged from 6.13660e-31 to 3.46945e-17.", "Across 90 recorded readings, Parseval energy residual ranged from 0 to 2.22045e-16.", "For the saved smooth contour at N = 256 and K = 4, sample MSE was 0.02070709 with the band-limited rule and 0.0001994762 with largest coefficients. These readings compare the same sampled input." ], "interpretation": [ "45 of 45 matched contour/N/K pairs had largest-coefficient MSE no greater than band-limited MSE within their reported numerical tolerances. DC was fixed in each pair.", "Compare the two MSE calculations and their residual before comparing selection strategies. The mathematical optimality statement concerns a sample-average squared norm, not every measure of visual quality." ], "summaryCsvPath": "/studies/fourier/summary-20260917003910210-62944.csv" }