Reproduce the polynomial-averaging figure
The complete averaging lesson contains the entire learner, all three examples, six complete solutions and the full PA1–PA16 proof.
Download the unchanged original files
The seven original files below are unchanged, including both complete Markdown sources and CREDITS.txt. CURRENT-SCOPE.txt records the current qualified status. LICENSE.txt identifies the original-work dedication and retained component terms.
- polynomial-averaging-working-proof.md — 10314 bytes; SHA-256
4A6075023A7C4AABE773575575EBDFEC9E06C8E74B27C9BCD37BBEFCB0A739FE. - polynomial-averaging-learner.md — 15886 bytes; SHA-256
10F9FA7A9F534EE74F089A49E3A4E6B306D9939EE881AF325F273AE28CB2D8FB. - make_figures.py — 5316 bytes; SHA-256
D32C98136C6C6B2D5DA58116DBDC74DD512FB1CD338C69762C91BB405903F675. - polynomial-averaging-annulus.png — 196470 bytes; SHA-256
2B199B9986D6A9C2E070597DFB570B973E98967A314D1DB1C789285406E992A1. - polynomial-averaging-annulus.svg — 96720 bytes; SHA-256
6F24F68629AAB720C21F1C564D6C790BB08521A0B38244F414CCD5DA2C8E365A. - polynomial-averaging.geometry.json — 825 bytes; SHA-256
1D0B8D15247125FDDB25F28D8E235AE050E9103CEE639E19FF8C83A6B3C82588. - CREDITS.txt — 638 bytes; SHA-256
8370EBF024CCE76A28AA9C58544F681B84B3D2EC1DC734048223D7AAA7A430F7. - LICENSE.txt — 393 bytes; SHA-256
D7603F058784D1874463FEA43C8DEEB48863E630214DA069CF5563E562B45CF0. - CURRENT-SCOPE.txt — 928 bytes; SHA-256
516054B983CDFEF4797F770DF6DE6C7935704E7A0B8F2437000902683C0A65EE.
Run the portable renderer
Keep make_figures.py at the root of a fresh scratch directory. Keep the figure and geometry downloads in figures/; both Markdown sources and text notices remain at the root and are downloads, not renderer outputs. Run:
python -X utf8 make_figures.py
The program creates figures/ and writes the PNG, SVG and geometry JSON. A fresh run with Python 3.13.9, NumPy 2.4.4 and Matplotlib 3.10.9 using its bundled DejaVu Sans font matched all three supplied outputs byte for byte and the full decoded PNG RGBA pixels. No SVG date or identifier normalization was performed. This byte identity is qualified by that observed environment; other software/font versions can change rendered bytes. The exact geometry JSON specifies the objects independently.
Exact diagram, proof locators and credits
The left panel shows the annulus 1/3 ≤ |z| ≤ 5/12, roots ±1/4, phase circle |z|=3/8 and unit-radius boundary for q(z)=z²−1/16. Shading is an available support region; the actual smooth density has compact support strictly inside the open annulus. The right panel is its exact phase-circle image, centred at −1/16 with radius 9/64, traversed twice, and separated from zero by 5/64. The annular lower bound is 7/144 and the normalized coefficient bound is 7/(9√1025). The full native caption, learner section 5/exercise 4 and PA036-3–4 locate every bound; samples do not prove nonvanishing. Original diagram and reproducible geometry/renderer are GPT-6.1 Sol (OpenAI), Ultra work, October 2026, CC0.
Historical mathematical credit remains Lars Hörmander, Bernard Malgrange and Leon Ehrenpreis; the primary archive and precise comparison limits are retained in the full proof/learner and CREDITS.txt. Original proof pages and exact original lemma numbers have not been compared. The independently written argument requires no unavailable source proof. All declared scalar prerequisites, the n=0 exception and broader incomplete scopes remain explicit.