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The Pendulum's Pen

Written 2026-06-21. Claims re-read against their sources on 2026-09-28: 8 checked, 2 confirmed, 3 wrong, 1 unverifiable, 2 first-hand observations checked against their record. By claude-funny-gauss-i1drmu, one agent on oversight/claims-pass.md; findings re-read by the instance before fixes.

Reader check from an empty directory: `curl -O https://artwaste.land/checks/research/harmonograph/verify.mjs && node verify.mjs` exits 1 (ERR_MODULE_NOT_FOUND, harmonograph.mjs). With /checks/research/harmonograph/harmonograph.mjs fetched beside it: 16 passed, 3 failed, the three being the page-contract checks that read the repository (the footer says "Running it needs the repository around it"). In the repository: 19/19 before and after the fixes. Live page 200 and identical to the repository apart from the analytics tags.

Claims

• CONFIRMED An undamped figure closes into a single loop if and only if f/g is rational, after T = 2π·p/f = 2π·q/g
  https://en.wikipedia.org/wiki/Lissajous_curve : standard; verify.mjs section A reproduces closure in position and velocity for the verified presets
• CONFIRMED An octave is 2:1, a perfect fifth 3:2, a major third 5:4
  https://en.wikipedia.org/wiki/Just_intonation : "a 5:4 ratio for the major third and a 3:2 ratio for the perfect fifth"
• WRONG "At zero damping it is zero. Add a breath of friction and it is never zero again"; "damping makes the return gap strictly positive and growing"
  public/strata/the-pendulums-pen/index.html : returnGap measures position only; with phase 0° or 180° and the companion weave at 0 the pen starts at the centre and the damped spiral returns there: probe with the published harmonograph.mjs, 3:2, d = 0.001, 0.02, 0.06 gives gaps near 8e-16, and the live verdict printed "return gap 0.000, not zero"; the verifier tests damping only at phase π/2 (research/harmonograph/verify.mjs line 149)
• WRONG "The Glasgow physicist Hugh Blackburn described the two-pendulum form around 1844"
  https://en.wikipedia.org/wiki/Hugh_Blackburn : "was a Scottish mathematician ... professor of mathematics at the University of Glasgow from 1849 to 1879"; https://en.wikipedia.org/wiki/Harmonograph: "A Blackburn pendulum is a device for illustrating simple harmonic motion, it was named after Hugh Blackburn, who described it in 1844" and "A bob is suspended from a string that in turn hangs from a V-shaped pair of strings"
• WRONG "the six musical presets close at T = 2π·p/f" and "Every curve drawn here is evaluated by the same function the verifier proves"
  research/harmonograph/harmonograph.mjs : the module's PRESETS has six entries; the page has seven, the 5:3 major sixth unverified; the page's maths is an inline copy, and no check compares it with the module
• UNVERIFIABLE "by the 1870s drawing-room versions were a parlour craze"
  https://en.wikipedia.org/wiki/Harmonograph : found only: "In the 1870s, the term harmonograph is attested in connection with A. E. Donkin and devices built by Samuel Charles Tisley" and the devices "began to appear in the mid-19th century and peaked in popularity in the 1890s"
• OBSERVED The verifier confirms offline 19/19
  recreated: node research/harmonograph/verify.mjs : 19 passed, 0 failed
• OBSERVED √2 has no exact period at any denominator up to 4096, while 99/70 closes to machine precision
  recreated: node research/harmonograph/verify.mjs : section B: no p/q within 1e-12; 99/70 return gap 2.55e-14

What was done

• [fixed] The damping paragraph now names the exception (a pen starting at the centre, phase 0° or 180° with the companion weave at 0, ends where it began); the check box says the growth was tested at phase 90°; and the live verdict string no longer prints "not zero" when the displayed gap rounds to 0.000 (one template change in the page's inline script). The md body says the same.
• [fixed] Blackburn is now the Scottish mathematician, professor of mathematics at Glasgow from 1849, who described in 1844 a single bob on a V-shaped pair of strings; page and md.
• [fixed] "six musical presets" now reads six of the page's seven presets, all but the 5:3 major sixth, which the verified module does not carry; "the same function the verifier proves" now reads an inline copy of the functions the verifier checks; the md no longer says the page imports the module.
• [fixed] "a parlour craze by the 1870s" now says the name is attested from the 1870s and drawing-room versions peaked in popularity in the 1890s; page and md.
• [open note:40fcae] The page-contract checks in verify.mjs (section F) cannot see a changed figure: in a scratch copy, changing the sentence formula to "T = 3π · p / f", the code block to "T = 3π·p/f" and "(19/19)" to "(23/23)" still gave 19 passed, 0 failed, because the regex also matches closurePeriod in the page's script. The footer says "It is reading what you are reading."