Fourier: frequency selection and reconstruction error

Question

How do the term budget and frequency-selection rule affect discrete reconstruction error?

Mathematical background

The forward complex DFT uses normalization 1/N and the inverse is unscaled. The input contour is resampled at equal arc-length positions without duplicating the endpoint.

Discrete orthogonality predicts that sample MSE equals the energy of the omitted coefficients. These are evaluated through separate sample and coefficient calculations.

\frac1N\sum_{j=0}^{N-1}e^{2\pi i(k-\ell)j/N}=\begin{cases}1,&k\equiv\ell\pmod N,\\0,&\text{otherwise}.\end{cases}

Sources: [M2]

c_k=\frac1N\sum_{j=0}^{N-1}z_je^{-2\pi ikj/N},\qquad z_j=\sum_{k=0}^{N-1}c_ke^{2\pi ikj/N}

Sources: [M2]

\operatorname{MSE}=\frac1N\sum_{j=0}^{N-1}|z_j-\widehat z_j|^2=\sum_{k\notin S}|c_k|^2

Sources: [M2]

Read the mathematical derivations.

Hypothesis or expected result

Theoretical expectations

With DC retained and counted in the budget, selecting the remaining largest coefficients minimizes this discrete squared error for the specified budget.

Complete reconstruction is expected at the sample positions. It is not a claim that the interpolated curve equals the original contour between samples.

Parameters and assumptions

The circle, triangle and smooth polar contour are fixed reproducible presets.
The run matrix uses N = 64,128,256,512, both selection rules and term budgets 4,16,64,N, with duplicate budgets removed.
DC is fixed in every comparison. The original preset and resampling specification remain part of the input.

Each case’s complete parameter object appears below and in its JSON file. Display rounding is used only in this article; the saved data retain full numeric precision.

Method and numerical settings

Compute every case with the same shared DFT kernel. Save coefficients, selected frequencies, sample reconstruction and numerical error outputs.

Compare strategies at the same contour, N and term budget. Treat changes in N as sampling comparisons rather than as pure truncation comparisons.

Methods reported by the executed kernel

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Software identifier: 0.1.0+0e6017e6a747.dirty.02665ba9098c
Git revision: 0e6017e6a747d3c610b956612b4125e232283001
Uncommitted worktree state: Present when the batch was generated
Source SHA-256: 02665ba9098c7b8fe2a4b728fbd2b3749b1724fe9999f830d8fa90cb5d8b209b
Runtime: v26.5.1

Results from an actual run

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.

Download summary CSV Download complete summary JSON

Circle · saved N = 256 cases

Vertical: Sample MSE (logarithmic scale)

Band-limitedLargest coefficients

Triangle · saved N = 256 cases

Vertical: Sample MSE (logarithmic scale)

Band-limitedLargest coefficients

Smooth · saved N = 256 cases

Vertical: Sample MSE (logarithmic scale)

Band-limitedLargest coefficients

Lines connect the recorded cases to aid comparison. Intermediate settings are not additional experiments. Zero has no logarithm and is omitted from logarithmic plots; all values remain in the table and downloads. Tiny nonzero errors for the circle and full reconstruction reflect floating-point arithmetic. Their plotted differences do not establish an advantage in approximation quality.

Actual case outputs. JSON retains every observable and the complete computed data.
Case	Selected measurements	Residual and tolerance	Data
1. circle · N=64 · K=4 · band	Samples N: 64 Retained terms K: 4 Sample MSE: 4.38103e-29 Discarded coefficient energy: 4.96366e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 7.27596e-12	JSON · CSV
2. circle · N=64 · K=16 · band	Samples N: 64 Retained terms K: 16 Sample MSE: 7.09423e-30 Discarded coefficient energy: 1.29169e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 7.27596e-12	JSON · CSV
3. circle · N=64 · K=64 · band	Samples N: 64 Retained terms K: 64 Sample MSE: 7.16245e-30 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 7.27596e-12	JSON · CSV
4. circle · N=64 · K=4 · largest	Samples N: 64 Retained terms K: 4 Sample MSE: 4.38103e-29 Discarded coefficient energy: 4.96366e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 7.27596e-12	JSON · CSV
5. circle · N=64 · K=16 · largest	Samples N: 64 Retained terms K: 16 Sample MSE: 1.05440e-29 Discarded coefficient energy: 1.11576e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 7.27596e-12	JSON · CSV
6. circle · N=64 · K=64 · largest	Samples N: 64 Retained terms K: 64 Sample MSE: 7.16245e-30 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 7.27596e-12	JSON · CSV
7. circle · N=128 · K=4 · band	Samples N: 128 Retained terms K: 4 Sample MSE: 4.45955e-29 Discarded coefficient energy: 8.29056e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
8. circle · N=128 · K=16 · band	Samples N: 128 Retained terms K: 16 Sample MSE: 7.43494e-30 Discarded coefficient energy: 4.59641e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
9. circle · N=128 · K=64 · band	Samples N: 128 Retained terms K: 64 Sample MSE: 4.54004e-30 Discarded coefficient energy: 3.72278e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
10. circle · N=128 · K=128 · band	Samples N: 128 Retained terms K: 128 Sample MSE: 3.82943e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
11. circle · N=128 · K=4 · largest	Samples N: 128 Retained terms K: 4 Sample MSE: 4.45955e-29 Discarded coefficient energy: 8.29056e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
12. circle · N=128 · K=16 · largest	Samples N: 128 Retained terms K: 16 Sample MSE: 3.12866e-29 Discarded coefficient energy: 2.77897e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
13. circle · N=128 · K=64 · largest	Samples N: 128 Retained terms K: 64 Sample MSE: 3.85222e-29 Discarded coefficient energy: 6.18529e-30 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
14. circle · N=128 · K=128 · largest	Samples N: 128 Retained terms K: 128 Sample MSE: 3.82943e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 1.45519e-11	JSON · CSV
15. circle · N=256 · K=4 · band	Samples N: 256 Retained terms K: 4 Sample MSE: 4.50920e-29 Discarded coefficient energy: 1.50004e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
16. circle · N=256 · K=16 · band	Samples N: 256 Retained terms K: 16 Sample MSE: 7.47492e-30 Discarded coefficient energy: 1.12639e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
17. circle · N=256 · K=64 · band	Samples N: 256 Retained terms K: 64 Sample MSE: 2.86082e-30 Discarded coefficient energy: 1.05887e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
18. circle · N=256 · K=256 · band	Samples N: 256 Retained terms K: 256 Sample MSE: 1.04962e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
19. circle · N=256 · K=4 · largest	Samples N: 256 Retained terms K: 4 Sample MSE: 4.50920e-29 Discarded coefficient energy: 1.50004e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
20. circle · N=256 · K=16 · largest	Samples N: 256 Retained terms K: 16 Sample MSE: 6.98544e-29 Discarded coefficient energy: 7.90257e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
21. circle · N=256 · K=64 · largest	Samples N: 256 Retained terms K: 64 Sample MSE: 9.32163e-29 Discarded coefficient energy: 2.56101e-29 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
22. circle · N=256 · K=256 · largest	Samples N: 256 Retained terms K: 256 Sample MSE: 1.04962e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 2.91038e-11	JSON · CSV
23. circle · N=512 · K=4 · band	Samples N: 512 Retained terms K: 4 Sample MSE: 4.52604e-29 Discarded coefficient energy: 6.73784e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
24. circle · N=512 · K=16 · band	Samples N: 512 Retained terms K: 16 Sample MSE: 7.48327e-30 Discarded coefficient energy: 6.36286e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
25. circle · N=512 · K=64 · band	Samples N: 512 Retained terms K: 64 Sample MSE: 2.62836e-30 Discarded coefficient energy: 6.30149e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
26. circle · N=512 · K=512 · band	Samples N: 512 Retained terms K: 512 Sample MSE: 6.31689e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
27. circle · N=512 · K=4 · largest	Samples N: 512 Retained terms K: 4 Sample MSE: 2.35558e-28 Discarded coefficient energy: 6.05949e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
28. circle · N=512 · K=16 · largest	Samples N: 512 Retained terms K: 16 Sample MSE: 4.10928e-28 Discarded coefficient energy: 2.63519e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
29. circle · N=512 · K=64 · largest	Samples N: 512 Retained terms K: 64 Sample MSE: 5.33248e-28 Discarded coefficient energy: 1.22056e-28 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
30. circle · N=512 · K=512 · largest	Samples N: 512 Retained terms K: 512 Sample MSE: 6.31689e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 4.61873e-14	1.11022e-16 Tolerance: 5.82077e-11	JSON · CSV
31. triangle · N=64 · K=4 · band	Samples N: 64 Retained terms K: 4 Sample MSE: 0.04650419 Discarded coefficient energy: 0.04650419 Dense reference MSE (all N): 1.05846e-5	1.11022e-16 Tolerance: 1.04774e-11	JSON · CSV
32. triangle · N=64 · K=16 · band	Samples N: 64 Retained terms K: 16 Sample MSE: 0.0003723756 Discarded coefficient energy: 0.0003723756 Dense reference MSE (all N): 1.05846e-5	1.11022e-16 Tolerance: 1.04774e-11	JSON · CSV
33. triangle · N=64 · K=64 · band	Samples N: 64 Retained terms K: 64 Sample MSE: 5.60081e-30 Discarded coefficient energy: 0 Dense reference MSE (all N): 1.05846e-5	1.11022e-16 Tolerance: 1.04774e-11	JSON · CSV
34. triangle · N=64 · K=4 · largest	Samples N: 64 Retained terms K: 4 Sample MSE: 0.001738773 Discarded coefficient energy: 0.001738773 Dense reference MSE (all N): 1.05846e-5	1.11022e-16 Tolerance: 1.04774e-11	JSON · CSV
35. triangle · N=64 · K=16 · largest	Samples N: 64 Retained terms K: 16 Sample MSE: 1.64773e-5 Discarded coefficient energy: 1.64773e-5 Dense reference MSE (all N): 1.05846e-5	1.11022e-16 Tolerance: 1.04774e-11	JSON · CSV
36. triangle · N=64 · K=64 · largest	Samples N: 64 Retained terms K: 64 Sample MSE: 5.61022e-30 Discarded coefficient energy: 0 Dense reference MSE (all N): 1.05846e-5	1.11022e-16 Tolerance: 1.04774e-11	JSON · CSV
37. triangle · N=128 · K=4 · band	Samples N: 128 Retained terms K: 4 Sample MSE: 0.04646278 Discarded coefficient energy: 0.04646278 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
38. triangle · N=128 · K=16 · band	Samples N: 128 Retained terms K: 16 Sample MSE: 0.0003653104 Discarded coefficient energy: 0.0003653104 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
39. triangle · N=128 · K=64 · band	Samples N: 128 Retained terms K: 64 Sample MSE: 5.15514e-6 Discarded coefficient energy: 5.15514e-6 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
40. triangle · N=128 · K=128 · band	Samples N: 128 Retained terms K: 128 Sample MSE: 3.25463e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
41. triangle · N=128 · K=4 · largest	Samples N: 128 Retained terms K: 4 Sample MSE: 0.001725392 Discarded coefficient energy: 0.001725392 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
42. triangle · N=128 · K=16 · largest	Samples N: 128 Retained terms K: 16 Sample MSE: 1.40530e-5 Discarded coefficient energy: 1.40530e-5 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
43. triangle · N=128 · K=64 · largest	Samples N: 128 Retained terms K: 64 Sample MSE: 3.34894e-7 Discarded coefficient energy: 3.34894e-7 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
44. triangle · N=128 · K=128 · largest	Samples N: 128 Retained terms K: 128 Sample MSE: 3.25322e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 1.49372e-6	1.11022e-16 Tolerance: 2.09548e-11	JSON · CSV
45. triangle · N=256 · K=4 · band	Samples N: 256 Retained terms K: 4 Sample MSE: 0.04645238 Discarded coefficient energy: 0.04645238 Dense reference MSE (all N): 1.74442e-7	3.46945e-17 Tolerance: 4.19095e-11	JSON · CSV
46. triangle · N=256 · K=16 · band	Samples N: 256 Retained terms K: 16 Sample MSE: 0.0003633574 Discarded coefficient energy: 0.0003633574 Dense reference MSE (all N): 1.74442e-7	3.57787e-18 Tolerance: 4.19095e-11	JSON · CSV
47. triangle · N=256 · K=64 · band	Samples N: 256 Retained terms K: 64 Sample MSE: 4.93913e-6 Discarded coefficient energy: 4.93913e-6 Dense reference MSE (all N): 1.74442e-7	1.56701e-18 Tolerance: 4.19095e-11	JSON · CSV
48. triangle · N=256 · K=256 · band	Samples N: 256 Retained terms K: 256 Sample MSE: 8.66129e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 1.74442e-7	8.66129e-29 Tolerance: 4.19095e-11	JSON · CSV
49. triangle · N=256 · K=4 · largest	Samples N: 256 Retained terms K: 4 Sample MSE: 0.001722026 Discarded coefficient energy: 0.001722026 Dense reference MSE (all N): 1.74442e-7	7.37257e-18 Tolerance: 4.19095e-11	JSON · CSV
50. triangle · N=256 · K=16 · largest	Samples N: 256 Retained terms K: 16 Sample MSE: 1.34003e-5 Discarded coefficient energy: 1.34003e-5 Dense reference MSE (all N): 1.74442e-7	7.72494e-19 Tolerance: 4.19095e-11	JSON · CSV
51. triangle · N=256 · K=64 · largest	Samples N: 256 Retained terms K: 64 Sample MSE: 2.27122e-7 Discarded coefficient energy: 2.27122e-7 Dense reference MSE (all N): 1.74442e-7	3.36749e-19 Tolerance: 4.19095e-11	JSON · CSV
52. triangle · N=256 · K=256 · largest	Samples N: 256 Retained terms K: 256 Sample MSE: 8.66652e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 1.74442e-7	8.66652e-29 Tolerance: 4.19095e-11	JSON · CSV
53. triangle · N=512 · K=4 · band	Samples N: 512 Retained terms K: 4 Sample MSE: 0.04644977 Discarded coefficient energy: 0.04644977 Dense reference MSE (all N): 2.17994e-8	3.46945e-17 Tolerance: 8.38190e-11	JSON · CSV
54. triangle · N=512 · K=16 · band	Samples N: 512 Retained terms K: 16 Sample MSE: 0.0003628581 Discarded coefficient energy: 0.0003628581 Dense reference MSE (all N): 2.17994e-8	6.07153e-18 Tolerance: 8.38190e-11	JSON · CSV
55. triangle · N=512 · K=64 · band	Samples N: 512 Retained terms K: 64 Sample MSE: 4.83475e-6 Discarded coefficient energy: 4.83475e-6 Dense reference MSE (all N): 2.17994e-8	4.70103e-19 Tolerance: 8.38190e-11	JSON · CSV
56. triangle · N=512 · K=512 · band	Samples N: 512 Retained terms K: 512 Sample MSE: 4.51653e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 2.17994e-8	4.51653e-28 Tolerance: 8.38190e-11	JSON · CSV
57. triangle · N=512 · K=4 · largest	Samples N: 512 Retained terms K: 4 Sample MSE: 0.001721183 Discarded coefficient energy: 0.001721183 Dense reference MSE (all N): 2.17994e-8	8.67362e-18 Tolerance: 8.38190e-11	JSON · CSV
58. triangle · N=512 · K=16 · largest	Samples N: 512 Retained terms K: 16 Sample MSE: 1.32342e-5 Discarded coefficient energy: 1.32342e-5 Dense reference MSE (all N): 2.17994e-8	2.60886e-19 Tolerance: 8.38190e-11	JSON · CSV
59. triangle · N=512 · K=64 · largest	Samples N: 512 Retained terms K: 64 Sample MSE: 1.90442e-7 Discarded coefficient energy: 1.90442e-7 Dense reference MSE (all N): 2.17994e-8	1.64880e-19 Tolerance: 8.38190e-11	JSON · CSV
60. triangle · N=512 · K=512 · largest	Samples N: 512 Retained terms K: 512 Sample MSE: 4.51648e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 2.17994e-8	4.51648e-28 Tolerance: 8.38190e-11	JSON · CSV
61. smooth · N=64 · K=4 · band	Samples N: 64 Retained terms K: 4 Sample MSE: 0.02070733 Discarded coefficient energy: 0.02070733 Dense reference MSE (all N): 7.00419e-10	1.73472e-17 Tolerance: 1.04774e-11	JSON · CSV
62. smooth · N=64 · K=16 · band	Samples N: 64 Retained terms K: 16 Sample MSE: 1.72857e-5 Discarded coefficient energy: 1.72857e-5 Dense reference MSE (all N): 7.00419e-10	6.97955e-19 Tolerance: 1.04774e-11	JSON · CSV
63. smooth · N=64 · K=64 · band	Samples N: 64 Retained terms K: 64 Sample MSE: 7.88697e-30 Discarded coefficient energy: 0 Dense reference MSE (all N): 7.00419e-10	7.88697e-30 Tolerance: 1.04774e-11	JSON · CSV
64. smooth · N=64 · K=4 · largest	Samples N: 64 Retained terms K: 4 Sample MSE: 0.0001994884 Discarded coefficient energy: 0.0001994884 Dense reference MSE (all N): 7.00419e-10	5.69206e-19 Tolerance: 1.04774e-11	JSON · CSV
65. smooth · N=64 · K=16 · largest	Samples N: 64 Retained terms K: 16 Sample MSE: 8.39441e-10 Discarded coefficient energy: 8.39441e-10 Dense reference MSE (all N): 7.00419e-10	3.23586e-20 Tolerance: 1.04774e-11	JSON · CSV
66. smooth · N=64 · K=64 · largest	Samples N: 64 Retained terms K: 64 Sample MSE: 7.89791e-30 Discarded coefficient energy: 0 Dense reference MSE (all N): 7.00419e-10	7.89791e-30 Tolerance: 1.04774e-11	JSON · CSV
67. smooth · N=128 · K=4 · band	Samples N: 128 Retained terms K: 4 Sample MSE: 0.02070715 Discarded coefficient energy: 0.02070715 Dense reference MSE (all N): 6.97552e-10	2.42861e-17 Tolerance: 2.09548e-11	JSON · CSV
68. smooth · N=128 · K=16 · band	Samples N: 128 Retained terms K: 16 Sample MSE: 1.72798e-5 Discarded coefficient energy: 1.72798e-5 Dense reference MSE (all N): 6.97552e-10	5.14996e-19 Tolerance: 2.09548e-11	JSON · CSV
69. smooth · N=128 · K=64 · band	Samples N: 128 Retained terms K: 64 Sample MSE: 1.27957e-10 Discarded coefficient energy: 1.27957e-10 Dense reference MSE (all N): 6.97552e-10	2.81428e-21 Tolerance: 2.09548e-11	JSON · CSV
70. smooth · N=128 · K=128 · band	Samples N: 128 Retained terms K: 128 Sample MSE: 4.31467e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 6.97552e-10	4.31467e-29 Tolerance: 2.09548e-11	JSON · CSV
71. smooth · N=128 · K=4 · largest	Samples N: 128 Retained terms K: 4 Sample MSE: 0.0001994798 Discarded coefficient energy: 0.0001994798 Dense reference MSE (all N): 6.97552e-10	4.06576e-19 Tolerance: 2.09548e-11	JSON · CSV
72. smooth · N=128 · K=16 · largest	Samples N: 128 Retained terms K: 16 Sample MSE: 7.97000e-10 Discarded coefficient energy: 7.97000e-10 Dense reference MSE (all N): 6.97552e-10	1.30695e-22 Tolerance: 2.09548e-11	JSON · CSV
73. smooth · N=128 · K=64 · largest	Samples N: 128 Retained terms K: 64 Sample MSE: 3.60882e-12 Discarded coefficient energy: 3.60882e-12 Dense reference MSE (all N): 6.97552e-10	3.64454e-21 Tolerance: 2.09548e-11	JSON · CSV
74. smooth · N=128 · K=128 · largest	Samples N: 128 Retained terms K: 128 Sample MSE: 4.31063e-29 Discarded coefficient energy: 0 Dense reference MSE (all N): 6.97552e-10	4.31063e-29 Tolerance: 2.09548e-11	JSON · CSV
75. smooth · N=256 · K=4 · band	Samples N: 256 Retained terms K: 4 Sample MSE: 0.02070709 Discarded coefficient energy: 0.02070709 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
76. smooth · N=256 · K=16 · band	Samples N: 256 Retained terms K: 16 Sample MSE: 1.72793e-5 Discarded coefficient energy: 1.72793e-5 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
77. smooth · N=256 · K=64 · band	Samples N: 256 Retained terms K: 64 Sample MSE: 1.27451e-10 Discarded coefficient energy: 1.27451e-10 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
78. smooth · N=256 · K=256 · band	Samples N: 256 Retained terms K: 256 Sample MSE: 1.19334e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
79. smooth · N=256 · K=4 · largest	Samples N: 256 Retained terms K: 4 Sample MSE: 0.0001994762 Discarded coefficient energy: 0.0001994762 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
80. smooth · N=256 · K=16 · largest	Samples N: 256 Retained terms K: 16 Sample MSE: 7.79110e-10 Discarded coefficient energy: 7.79110e-10 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
81. smooth · N=256 · K=64 · largest	Samples N: 256 Retained terms K: 64 Sample MSE: 3.70935e-12 Discarded coefficient energy: 3.70935e-12 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
82. smooth · N=256 · K=256 · largest	Samples N: 256 Retained terms K: 256 Sample MSE: 1.19325e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 6.97832e-10	2.22045e-16 Tolerance: 4.19095e-11	JSON · CSV
83. smooth · N=512 · K=4 · band	Samples N: 512 Retained terms K: 4 Sample MSE: 0.02070708 Discarded coefficient energy: 0.02070708 Dense reference MSE (all N): 6.97822e-10	2.08167e-17 Tolerance: 8.38190e-11	JSON · CSV
84. smooth · N=512 · K=16 · band	Samples N: 512 Retained terms K: 16 Sample MSE: 1.72792e-5 Discarded coefficient energy: 1.72792e-5 Dense reference MSE (all N): 6.97822e-10	2.60886e-19 Tolerance: 8.38190e-11	JSON · CSV
85. smooth · N=512 · K=64 · band	Samples N: 512 Retained terms K: 64 Sample MSE: 1.26440e-10 Discarded coefficient energy: 1.26440e-10 Dense reference MSE (all N): 6.97822e-10	8.62938e-21 Tolerance: 8.38190e-11	JSON · CSV
86. smooth · N=512 · K=512 · band	Samples N: 512 Retained terms K: 512 Sample MSE: 6.60712e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 6.97822e-10	6.60712e-28 Tolerance: 8.38190e-11	JSON · CSV
87. smooth · N=512 · K=4 · largest	Samples N: 512 Retained terms K: 4 Sample MSE: 0.0001994752 Discarded coefficient energy: 0.0001994752 Dense reference MSE (all N): 6.97822e-10	8.67362e-19 Tolerance: 8.38190e-11	JSON · CSV
88. smooth · N=512 · K=16 · largest	Samples N: 512 Retained terms K: 16 Sample MSE: 7.76479e-10 Discarded coefficient energy: 7.76479e-10 Dense reference MSE (all N): 6.97822e-10	1.06411e-20 Tolerance: 8.38190e-11	JSON · CSV
89. smooth · N=512 · K=64 · largest	Samples N: 512 Retained terms K: 64 Sample MSE: 3.44549e-12 Discarded coefficient energy: 3.44549e-12 Dense reference MSE (all N): 6.97822e-10	1.07197e-21 Tolerance: 8.38190e-11	JSON · CSV
90. smooth · N=512 · K=512 · largest	Samples N: 512 Retained terms K: 512 Sample MSE: 6.60690e-28 Discarded coefficient energy: 0 Dense reference MSE (all N): 6.97822e-10	6.60690e-28 Tolerance: 8.38190e-11	JSON · CSV

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.

Limitations and possible errors

Finite samples do not preserve unlimited detail. Differences against a continuous or dense reference depend on the independently specified parameterization.
Near-zero coefficient phases are numerically indeterminate. The study uses no stochastic contour or external image.

The kernel reported no warnings for this batch. This is a recorded software status, not an independent review.

Reproduce this study

Choose the saved preset, N, term budget, selection rule and fixed-DC setting in Fourier.
Import a run JSON or use the corresponding CSV to inspect the kernel’s coefficients and reconstruction data.

Open fourier

1. circle · N=64 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 64,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.381e-29
Discarded coefficient energy: 4.964e-29
MSE identity residual: 5.826e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

2. circle · N=64 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 64,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.094e-30
Discarded coefficient energy: 1.292e-29
MSE identity residual: 5.823e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

3. circle · N=64 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 64,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 7.162e-30
Discarded coefficient energy: 0
MSE identity residual: 7.162e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

4. circle · N=64 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 64,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.381e-29
Discarded coefficient energy: 4.964e-29
MSE identity residual: 5.826e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

5. circle · N=64 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 64,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.054e-29
Discarded coefficient energy: 1.116e-29
MSE identity residual: 6.137e-31
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

6. circle · N=64 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 64,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 7.162e-30
Discarded coefficient energy: 0
MSE identity residual: 7.162e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

7. circle · N=128 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.460e-29
Discarded coefficient energy: 8.291e-29
MSE identity residual: 3.831e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

8. circle · N=128 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.435e-30
Discarded coefficient energy: 4.596e-29
MSE identity residual: 3.853e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

9. circle · N=128 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 4.540e-30
Discarded coefficient energy: 3.723e-29
MSE identity residual: 3.269e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

10. circle · N=128 · K=128 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 128,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 128
DC retained: Yes; counted in K
Sample MSE: 3.829e-29
Discarded coefficient energy: 0
MSE identity residual: 3.829e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

11. circle · N=128 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.460e-29
Discarded coefficient energy: 8.291e-29
MSE identity residual: 3.831e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

12. circle · N=128 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 3.129e-29
Discarded coefficient energy: 2.779e-29
MSE identity residual: 3.497e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

13. circle · N=128 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 3.852e-29
Discarded coefficient energy: 6.185e-30
MSE identity residual: 3.234e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

14. circle · N=128 · K=128 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 128,
  "terms": 128,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 128
DC retained: Yes; counted in K
Sample MSE: 3.829e-29
Discarded coefficient energy: 0
MSE identity residual: 3.829e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

15. circle · N=256 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.509e-29
Discarded coefficient energy: 1.500e-28
MSE identity residual: 1.049e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

16. circle · N=256 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.475e-30
Discarded coefficient energy: 1.126e-28
MSE identity residual: 1.052e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

17. circle · N=256 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 2.861e-30
Discarded coefficient energy: 1.059e-28
MSE identity residual: 1.030e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

18. circle · N=256 · K=256 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 256,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 256
DC retained: Yes; counted in K
Sample MSE: 1.050e-28
Discarded coefficient energy: 0
MSE identity residual: 1.050e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

19. circle · N=256 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.509e-29
Discarded coefficient energy: 1.500e-28
MSE identity residual: 1.049e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

20. circle · N=256 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 6.985e-29
Discarded coefficient energy: 7.903e-29
MSE identity residual: 9.171e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

21. circle · N=256 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 9.322e-29
Discarded coefficient energy: 2.561e-29
MSE identity residual: 6.761e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

22. circle · N=256 · K=256 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 256,
  "terms": 256,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 256
DC retained: Yes; counted in K
Sample MSE: 1.050e-28
Discarded coefficient energy: 0
MSE identity residual: 1.050e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

23. circle · N=512 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 4.526e-29
Discarded coefficient energy: 6.738e-28
MSE identity residual: 6.285e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

Download complete case JSON · Download case CSV

24. circle · N=512 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.483e-30
Discarded coefficient energy: 6.363e-28
MSE identity residual: 6.288e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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25. circle · N=512 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 2.628e-30
Discarded coefficient energy: 6.301e-28
MSE identity residual: 6.275e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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26. circle · N=512 · K=512 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 512,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 512
DC retained: Yes; counted in K
Sample MSE: 6.317e-28
Discarded coefficient energy: 0
MSE identity residual: 6.317e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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27. circle · N=512 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 2.356e-28
Discarded coefficient energy: 6.059e-28
MSE identity residual: 3.704e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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28. circle · N=512 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 4.109e-28
Discarded coefficient energy: 2.635e-28
MSE identity residual: 1.474e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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29. circle · N=512 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 5.332e-28
Discarded coefficient energy: 1.221e-28
MSE identity residual: 4.112e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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30. circle · N=512 · K=512 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "circle",
  "sampleCount": 512,
  "terms": 512,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 512
DC retained: Yes; counted in K
Sample MSE: 6.317e-28
Discarded coefficient energy: 0
MSE identity residual: 6.317e-28
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 4.619e-14
Dense reference MSE (all N): 4.619e-14Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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31. triangle · N=64 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 64 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 64,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.04650419238
Discarded coefficient energy: 0.04650419238
MSE identity residual: 3.469e-17
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 0.04644922475
Dense reference MSE (all N): 1.058e-5Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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32. triangle · N=64 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 64 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 64,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 0.0003723756412
Discarded coefficient energy: 0.0003723756412
MSE identity residual: 1.409e-18
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 0.0003637123105
Dense reference MSE (all N): 1.058e-5Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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33. triangle · N=64 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 64 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 64,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 5.601e-30
Discarded coefficient energy: 0
MSE identity residual: 5.601e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.058e-5
Dense reference MSE (all N): 1.058e-5Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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34. triangle · N=64 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 64 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 64,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.001738773119
Discarded coefficient energy: 0.001738773119
MSE identity residual: 8.674e-19
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 0.001721021127
Dense reference MSE (all N): 1.058e-5Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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35. triangle · N=64 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 64 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 64,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.648e-5
Discarded coefficient energy: 1.648e-5
MSE identity residual: 1.033e-18
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.335e-5
Dense reference MSE (all N): 1.058e-5Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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36. triangle · N=64 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 64 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 64,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 5.610e-30
Discarded coefficient energy: 0
MSE identity residual: 5.610e-30
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.058e-5
Dense reference MSE (all N): 1.058e-5Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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37. triangle · N=128 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.04646277785
Discarded coefficient energy: 0.04646277785
MSE identity residual: 3.469e-17
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 0.04644897474
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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38. triangle · N=128 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 0.0003653103674
Discarded coefficient energy: 0.0003653103674
MSE identity residual: 7.156e-18
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 0.0003627615238
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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39. triangle · N=128 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 5.155e-6
Discarded coefficient energy: 5.155e-6
MSE identity residual: 7.962e-20
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 5.065e-6
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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40. triangle · N=128 · K=128 · band

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 128,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 128
DC retained: Yes; counted in K
Sample MSE: 3.255e-29
Discarded coefficient energy: 0
MSE identity residual: 3.255e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.494e-6
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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41. triangle · N=128 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.00172539238
Discarded coefficient energy: 0.00172539238
MSE identity residual: 1.106e-17
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 0.001720925711
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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42. triangle · N=128 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.405e-5
Discarded coefficient energy: 1.405e-5
MSE identity residual: 1.188e-18
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.319e-5
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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43. triangle · N=128 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 3.349e-7
Discarded coefficient energy: 3.349e-7
MSE identity residual: 6.330e-19
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.161e-6
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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44. triangle · N=128 · K=128 · largest

Recorded 17 September 2026 at 00:39 UTC. 128 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 128,
  "terms": 128,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 128
DC retained: Yes; counted in K
Sample MSE: 3.253e-29
Discarded coefficient energy: 0
MSE identity residual: 3.253e-29
Parseval energy residual: 1.110e-16
Dense reference MSE (selected K): 1.494e-6
Dense reference MSE (all N): 1.494e-6Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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45. triangle · N=256 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.0464523758
Discarded coefficient energy: 0.0464523758
MSE identity residual: 3.469e-17
Parseval energy residual: 0
Dense reference MSE (selected K): 0.04644895953
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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46. triangle · N=256 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 0.0003633573709
Discarded coefficient energy: 0.0003633573709
MSE identity residual: 3.578e-18
Parseval energy residual: 0
Dense reference MSE (selected K): 0.0003627048334
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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47. triangle · N=256 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 4.939e-6
Discarded coefficient energy: 4.939e-6
MSE identity residual: 1.567e-18
Parseval energy residual: 0
Dense reference MSE (selected K): 4.813e-6
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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48. triangle · N=256 · K=256 · band

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 256,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 256
DC retained: Yes; counted in K
Sample MSE: 8.661e-29
Discarded coefficient energy: 0
MSE identity residual: 8.661e-29
Parseval energy residual: 0
Dense reference MSE (selected K): 1.744e-7
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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49. triangle · N=256 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.00172202588
Discarded coefficient energy: 0.00172202588
MSE identity residual: 7.373e-18
Parseval energy residual: 0
Dense reference MSE (selected K): 0.001720919776
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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50. triangle · N=256 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.340e-5
Discarded coefficient energy: 1.340e-5
MSE identity residual: 7.725e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.318e-5
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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51. triangle · N=256 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 2.271e-7
Discarded coefficient energy: 2.271e-7
MSE identity residual: 3.367e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.799e-7
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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52. triangle · N=256 · K=256 · largest

Recorded 17 September 2026 at 00:39 UTC. 256 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 256,
  "terms": 256,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 256
DC retained: Yes; counted in K
Sample MSE: 8.667e-29
Discarded coefficient energy: 0
MSE identity residual: 8.667e-29
Parseval energy residual: 0
Dense reference MSE (selected K): 1.744e-7
Dense reference MSE (all N): 1.744e-7Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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53. triangle · N=512 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.04644977235
Discarded coefficient energy: 0.04644977235
MSE identity residual: 3.469e-17
Parseval energy residual: 0
Dense reference MSE (selected K): 0.04644895859
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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54. triangle · N=512 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 0.0003628580795
Discarded coefficient energy: 0.0003628580795
MSE identity residual: 6.072e-18
Parseval energy residual: 0
Dense reference MSE (selected K): 0.0003627013496
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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55. triangle · N=512 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 4.835e-6
Discarded coefficient energy: 4.835e-6
MSE identity residual: 4.701e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 4.799e-6
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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56. triangle · N=512 · K=512 · band

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 512,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 512
DC retained: Yes; counted in K
Sample MSE: 4.517e-28
Discarded coefficient energy: 0
MSE identity residual: 4.517e-28
Parseval energy residual: 0
Dense reference MSE (selected K): 2.180e-8
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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57. triangle · N=512 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.001721182923
Discarded coefficient energy: 0.001721182923
MSE identity residual: 8.674e-18
Parseval energy residual: 0
Dense reference MSE (selected K): 0.001720919412
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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58. triangle · N=512 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.323e-5
Discarded coefficient energy: 1.323e-5
MSE identity residual: 2.609e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.318e-5
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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59. triangle · N=512 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 1.904e-7
Discarded coefficient energy: 1.904e-7
MSE identity residual: 1.649e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.783e-7
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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60. triangle · N=512 · K=512 · largest

Recorded 17 September 2026 at 00:39 UTC. 512 exported data rows.

{
  "preset": "triangle",
  "sampleCount": 512,
  "terms": 512,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 512
DC retained: Yes; counted in K
Sample MSE: 4.516e-28
Discarded coefficient energy: 0
MSE identity residual: 4.516e-28
Parseval energy residual: 0
Dense reference MSE (selected K): 2.180e-8
Dense reference MSE (all N): 2.180e-8Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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61. smooth · N=64 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 64,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.02070732704
Discarded coefficient energy: 0.02070732704
MSE identity residual: 1.735e-17
Parseval energy residual: 0
Dense reference MSE (selected K): 0.02071143596
Dense reference MSE (all N): 7.004e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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62. smooth · N=64 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 64,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.729e-5
Discarded coefficient energy: 1.729e-5
MSE identity residual: 6.980e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.729e-5
Dense reference MSE (all N): 7.004e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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63. smooth · N=64 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 64,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 7.887e-30
Discarded coefficient energy: 0
MSE identity residual: 7.887e-30
Parseval energy residual: 0
Dense reference MSE (selected K): 7.004e-10
Dense reference MSE (all N): 7.004e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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64. smooth · N=64 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 64,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.0001994884112
Discarded coefficient energy: 0.0001994884112
MSE identity residual: 5.692e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 0.0001995880136
Dense reference MSE (all N): 7.004e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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65. smooth · N=64 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 64,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 8.394e-10
Discarded coefficient energy: 8.394e-10
MSE identity residual: 3.236e-20
Parseval energy residual: 0
Dense reference MSE (selected K): 1.193e-9
Dense reference MSE (all N): 7.004e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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66. smooth · N=64 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 64,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 64
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 7.898e-30
Discarded coefficient energy: 0
MSE identity residual: 7.898e-30
Parseval energy residual: 0
Dense reference MSE (selected K): 7.004e-10
Dense reference MSE (all N): 7.004e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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67. smooth · N=128 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.02070714653
Discarded coefficient energy: 0.02070714653
MSE identity residual: 2.429e-17
Parseval energy residual: 0
Dense reference MSE (selected K): 0.02071143594
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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68. smooth · N=128 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.728e-5
Discarded coefficient energy: 1.728e-5
MSE identity residual: 5.150e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.729e-5
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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69. smooth · N=128 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 1.280e-10
Discarded coefficient energy: 1.280e-10
MSE identity residual: 2.814e-21
Parseval energy residual: 0
Dense reference MSE (selected K): 5.987e-10
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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70. smooth · N=128 · K=128 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 128,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 128
DC retained: Yes; counted in K
Sample MSE: 4.315e-29
Discarded coefficient energy: 0
MSE identity residual: 4.315e-29
Parseval energy residual: 0
Dense reference MSE (selected K): 6.976e-10
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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71. smooth · N=128 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.0001994798381
Discarded coefficient energy: 0.0001994798381
MSE identity residual: 4.066e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 0.00019958802
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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72. smooth · N=128 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.970e-10
Discarded coefficient energy: 7.970e-10
MSE identity residual: 1.307e-22
Parseval energy residual: 0
Dense reference MSE (selected K): 1.200e-9
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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73. smooth · N=128 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 3.609e-12
Discarded coefficient energy: 3.609e-12
MSE identity residual: 3.645e-21
Parseval energy residual: 0
Dense reference MSE (selected K): 6.947e-10
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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74. smooth · N=128 · K=128 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 128,
  "terms": 128,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 128
Retained terms K: 128
DC retained: Yes; counted in K
Sample MSE: 4.311e-29
Discarded coefficient energy: 0
MSE identity residual: 4.311e-29
Parseval energy residual: 0
Dense reference MSE (selected K): 6.976e-10
Dense reference MSE (all N): 6.976e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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75. smooth · N=256 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.02070709102
Discarded coefficient energy: 0.02070709102
MSE identity residual: 1.388e-17
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 0.02071143595
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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76. smooth · N=256 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.728e-5
Discarded coefficient energy: 1.728e-5
MSE identity residual: 9.487e-20
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 1.729e-5
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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77. smooth · N=256 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 1.275e-10
Discarded coefficient energy: 1.275e-10
MSE identity residual: 9.965e-22
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 5.985e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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78. smooth · N=256 · K=256 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 256,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 256
DC retained: Yes; counted in K
Sample MSE: 1.193e-28
Discarded coefficient energy: 0
MSE identity residual: 1.193e-28
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 6.978e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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79. smooth · N=256 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.0001994761788
Discarded coefficient energy: 0.0001994761788
MSE identity residual: 1.409e-18
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 0.0001995880293
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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80. smooth · N=256 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.791e-10
Discarded coefficient energy: 7.791e-10
MSE identity residual: 1.509e-22
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 1.211e-9
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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81. smooth · N=256 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 3.709e-12
Discarded coefficient energy: 3.709e-12
MSE identity residual: 9.397e-22
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 6.946e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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82. smooth · N=256 · K=256 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 256,
  "terms": 256,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 256
Retained terms K: 256
DC retained: Yes; counted in K
Sample MSE: 1.193e-28
Discarded coefficient energy: 0
MSE identity residual: 1.193e-28
Parseval energy residual: 2.220e-16
Dense reference MSE (selected K): 6.978e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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83. smooth · N=512 · K=4 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 4,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.020707082
Discarded coefficient energy: 0.020707082
MSE identity residual: 2.082e-17
Parseval energy residual: 0
Dense reference MSE (selected K): 0.02071143595
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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84. smooth · N=512 · K=16 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 16,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 1.728e-5
Discarded coefficient energy: 1.728e-5
MSE identity residual: 2.609e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 1.729e-5
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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85. smooth · N=512 · K=64 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 64,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 1.264e-10
Discarded coefficient energy: 1.264e-10
MSE identity residual: 8.629e-21
Parseval energy residual: 0
Dense reference MSE (selected K): 5.994e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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86. smooth · N=512 · K=512 · band

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 512,
  "strategy": "band",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 512
DC retained: Yes; counted in K
Sample MSE: 6.607e-28
Discarded coefficient energy: 0
MSE identity residual: 6.607e-28
Parseval energy residual: 0
Dense reference MSE (selected K): 6.978e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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87. smooth · N=512 · K=4 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 4,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 4
DC retained: Yes; counted in K
Sample MSE: 0.0001994752386
Discarded coefficient energy: 0.0001994752386
MSE identity residual: 8.674e-19
Parseval energy residual: 0
Dense reference MSE (selected K): 0.0001995880303
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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88. smooth · N=512 · K=16 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 16,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 16
DC retained: Yes; counted in K
Sample MSE: 7.765e-10
Discarded coefficient energy: 7.765e-10
MSE identity residual: 1.064e-20
Parseval energy residual: 0
Dense reference MSE (selected K): 1.212e-9
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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89. smooth · N=512 · K=64 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 64,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 64
DC retained: Yes; counted in K
Sample MSE: 3.445e-12
Discarded coefficient energy: 3.445e-12
MSE identity residual: 1.072e-21
Parseval energy residual: 0
Dense reference MSE (selected K): 6.949e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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90. smooth · N=512 · K=512 · largest

Recorded 17 September 2026 at 00:39 UTC. 513 exported data rows.

{
  "preset": "smooth",
  "sampleCount": 512,
  "terms": 512,
  "strategy": "largest",
  "fixDC": true,
  "input": []
}

Model version: fourier-1.0.0

Direct Float64 complex DFT; forward 1/N, inverse unscaled; closed piecewise-linear equal-arclength resampling; signed Nyquist −N/2; independent sample MSE and omitted coefficient energy.

Samples N: 512
Retained terms K: 512
DC retained: Yes; counted in K
Sample MSE: 6.607e-28
Discarded coefficient energy: 0
MSE identity residual: 6.607e-28
Parseval energy residual: 0
Dense reference MSE (selected K): 6.978e-10
Dense reference MSE (all N): 6.978e-10Comparison at 2,048 arclength positions against the saved dense contour reference; this is separate from sample truncation error.

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Each run’s download address is available in the summary JSON. The printed article preserves the methods, measurements and limitations.

References and review status

Project attribution: BIPH Math Society. Exposition and implementation were prepared with tool assistance.

Review status: Not reviewed. Independent teacher or member review has not yet been recorded.

[M2] Discrete Fourier Transform — NumPy reference — NumPy contributors
Definitions, signed-frequency conventions and normalization options. This project uses a forward factor of 1/N and an unscaled inverse, corresponding to norm="forward" rather than the library default. The browser implementation does not depend on NumPy. Source checked 16 September 2026.
[M3] But what is the Fourier Transform? — Grant Sanderson / 3Blue1Brown
An author-created explanation connecting complex rotation and frequency. Used as a reference for mathematical exposition, not as a source of copied animation or as a substitute for the finite DFT derivation. Source checked 16 September 2026.