---
slug: the-quickest-way-down
passed: 2026-11-01
by: the assay line (research/assay-line/claude-1101/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5)
checked: 68
confirmed: 60          # 53 CONFIRMED + 7 MINOR
wrong: 2
unverifiable: 0
observed: 6
stale: 0
---

This pass was the Assay Office's assay line, batch claude-1101: a Claude agent (claude-sonnet-5-5) wrote a source record for every claim on /strata/the-quickest-way-down/ (research/assay-line/claude-1101/the-quickest-way-down/N1-sources.json), a second Claude agent (claude-opus-5-5) re-read each against its source and re-ran both verifiers (N2-claude.md, whose claim lines are carried here), and the fixer (claude-opus-5-5) decided every WRONG, MINOR and STALE line in triage.json, re-reading the sources for both WRONG items before editing. Two WRONG claims were fixed on the page with dated Corrected lines (the June 1696 challenge's address, which belongs to the 1 January 1697 Groningen sheet; and a Galileo "hedge" that is not in Two New Sciences). Seven MINOR lines are recorded and declined under the tier C rule, which lists MINOR items rather than editing them. Both verifiers pass after the edits; neither reads the page.

## Claims
- MINOR | Every time on screen is integrated from the constrained-bead equation of motion by the same code the verifier runs | research/brachistochrone/verify.mjs | [engine-same-code] Not the same code. The page runs its own port (index.html:448 buildRamp, dt=5e-5, NP=900 points per ramp) and the verifier runs descentTime (dt=1e-5, N=6000); research/verifier-independence/census-pairs.tsv says the verifier "touches nothing that /strata/the-quickest-way-down/ ships". Same method (RK4 on ds/dt=v, dv/dt=g dy/ds) and the same ramp equations, and the page's own engine run in node at the default endpoints gives cycloid 0.80138, arc 0.82000, line 0.93178 s (verifier 0.93179), so its 4-decimal legend agrees. The claim the reader sees is index.html:370, "The physics in this page is the same code that the verifier runs", then "what you race is what is checked". Fix: "a port of the verifier's integrator (same method, coarser step)".
- CONFIRMED | The page races a bead down three ramps between the same two points | research/brachistochrone/verify.mjs | [race-three-ramps] verify.mjs ramps: straight line, circular arc (Galileo), cycloid; the page's RAMPS array (index.html:525) has the same three; rerun lists all three.
- CONFIRMED | and the cycloid wins | research/brachistochrone/verify.mjs | [cycloid-wins] rerun at the pin (repository copy and the served /checks/ copy, identical): BRACHISTOCHRONE claim: winner = "cycloid (brachistochrone)", CONFIRMED.
- CONFIRMED | its true descent time is infinite | research/brachistochrone/verify.mjs | [parabola-infinite] rerun prints "T_parabola = ∞ (a bead at a horizontal tangent, at rest, never leaves)"; analytically a bead at rest where the tangent is horizontal feels no tangential force, so it never starts.
- CONFIRMED | A second instrument shows the tautochrone: on one cycloid, beads released from five different heights reach the bottom together, in π√(a/g). | research/brachistochrone/verify.mjs | [tauto-five-heights] the page instrument is five beads ("Five beads, five heights", index.html:706 "All five arrive at"); the verifier releases six heights, all T=1.003210 s against the closed form 1.003205 s.
- MINOR | is checked constant (to sixteen digits) along the exact curve | research/brachistochrone/verify.mjs | [sixteen-digits] The rerun prints "(spread 3.33e-16)" on a value of 0.19029542: agreement to 15 decimal places, about 15 significant figures (relative 1.75e-15), not sixteen; and the assertion only requires (smax-smin) < 1e-9 (verify.mjs:225), so the verifier "pins" nine digits. Rendered twice: index.html:231 "the verifier pins it to sixteen digits along the exact curve" and index.html:324 "is constant to sixteen digits". The 2026-06-29 log records "constant to **3.3e-16**", which the commit message turned into "16 digits". Fix: "constant to machine precision (a spread of 3e-16)".
- CONFIRMED | the layered ray's descent time converges to the cycloid's from below | research/brachistochrone/verify.mjs | [layered-from-below] rerun, N=2 to 128: 0.74627, 0.76186, 0.77750, 0.78887, 0.79580, 0.79930, 0.80074 s, all below and rising toward 0.80138 s; the assertion at verify.mjs:255 passes.
- CONFIRMED | cycloid (brachistochrone) 0.80138 s 2.604 m; circular arc (Galileo's guess) 0.82000 s 2.861 m; straight line (the short path) 0.93179 s 2.441 m; parabola (flat start) ∞ (diverges) 2.533 m | research/brachistochrone/verify.mjs | [table-rows] rerun: cycloid T=0.80138 arc=2.6041, circular arc T=0.82000 arc=2.8610, straight line T=0.93179 arc=2.4413. The parabola length is not printed; parabolaPts is y=(Y/X^2)x^2 = 0.35x^2 (verify.mjs:75) and its closed-form arc length from x=0 to 2 is 2.53331 m (my computation), so 2.533 m is right.
- CONFIRMED | It is longer than the straight line, yet it arrives first. | research/brachistochrone/verify.mjs | [longer-yet-first] 2.6041 m > 2.4413 m and 0.80138 s < 0.93179 s (rerun).
- CONFIRMED | The cycloid wins by 14.0% over the straight line and by 2.3% over Galileo's circle. | research/brachistochrone/verify.mjs | [wins-by] rerun: "Cycloid beats the straight line by 14.00%"; (0.82000-0.80138)/0.82000 = 2.27%, which rounds to 2.3%.
- CONFIRMED | the simulated value keeps climbing as the mesh is refined (3.50 → 4.23 → 4.69 → 4.95 → 5.22 s at 900 → 80,000 points) | research/brachistochrone/verify.mjs | [mesh-divergence] rerun: M=900 T=3.50330, 6000 4.22734, 20000 4.68686, 40000 4.95142, 80000 5.21597 s.
- CONFIRMED | every bead that does arrive reaches B at the identical speed √(2gY)=5.2401 m/s | research/brachistochrone/verify.mjs | [energy-check] rerun: v(B)=5.2401 m/s for all three ramps; sqrt(2 x 9.80665 x 1.4) = 5.24010.
- CONFIRMED | beads released from six different heights all reach the bottom in 1.00321 s, matching the closed form π√(a/g)=1.003205 s to within 0.01 ms | research/brachistochrone/verify.mjs | [tauto-time] rerun: six heights, each T=1.003210 s, Δ 0.005 ms; π/sqrt(9.80665) = 1.0032046.
- CONFIRMED | the cusp→θ descent law T=√(a/g)·θ matches the simulation to under 0.01 ms | research/brachistochrone/verify.mjs | [cusp-law] rerun: Δ = 0.008, 0.005, 0.004 ms.
- CONFIRMED | at 64 layers it misses the finish by just 1.1 cm and its time (0.79930 s) closes on the cycloid's 0.80138 s from below | research/brachistochrone/verify.mjs | [optics-64] rerun: "N= 64 lands x=1.9886 m (gap 1.141 cm) time 0.79930 s"; cycloid 0.80138 s.
- CONFIRMED | equal to 1/(2√(g·a)) = 0.1903 s/m | research/brachistochrone/verify.mjs | [optics-invariant] rerun: 0.190295 s/m at a=0.703984 m; 1/(2 sqrt(9.80665 x 0.703984)) = 0.1902954.
- CONFIRMED | Default endpoints A=(0, 0) to B=(2, 1.4) m, g=9.80665 m/s². | research/brachistochrone/verify.mjs | [endpoints] "const X = 2.0, Y = 1.4;" and "const g = 9.80665;"; the page's engine uses G = 9.80665 (index.html:440).
- MINOR | Every word and number on screen is verbatim from this race's verifier (19/19) | research/cycloid/verify.mjs | [film-verifier] 19/19 is right (rerun: "RESULT: 19/19 checks passed.") and the film's numbers are in that output, but its words are not: captions in research/cycloid/film/film.html such as "So beautiful, and the cause of so many quarrels", "A theoretical triumph that lost in practice", "Five beads, five heights", "Huygens cut the curve into a clock" have 0 matches in verify.mjs. The film's own build script claims less: "every number and quote on screen is verbatim from research/cycloid/verify.mjs (19/19) or a cited primary source" (research/cycloid/film/build.sh:10). Fix (index.html:150): "Every number on screen is verbatim from this race's verifier (19/19)".
- CONFIRMED | cycloid 0.63866 s · arc 0.64559 s · line 0.67833 s | research/cycloid/verify.mjs | [film-times] rerun: times (s) cycloid 0.63866, parabola 0.64003, arc 0.64559, line 0.67833 (the film's own endpoints, not the page race's).
- CONFIRMED | the music's beat is the time a bead takes to fall to the bottom, π√(R/g) | research/cycloid/film/make-audio.mjs | [film-beat] make-audio.mjs:36 sets the beat T_DROP = Math.PI * Math.sqrt(R / g), and line 101 reads "Each beat is the time a bead takes to reach the bottom."
- CONFIRMED | The Newton-overnight anecdote is flagged as the story told, not documented fact | research/cycloid/film/film.html | [film-newton-flag] film card: "Newton, the story goes, solved it overnight", cite "the Newton-overnight anecdote is the story told, not documented fact (Conduitt; Whitman 1943)".
- OBSERVED | Companion film ... 2:30 | recreated: git show 60cbd1924f:public/strata/the-quickest-way-down/film.mp4 > film.mp4; ffprobe -show_entries format=duration | [film-length] duration=150.000000 s, which is 2:30.
- CONFIRMED | The cycloid as the Helen of geometers | https://en.wikipedia.org/wiki/Cycloid | [film-helen] lead: 'The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians'.
- CONFIRMED | In June 1696 Johann Bernoulli printed a challenge in the journal Acta Eruditorum | https://archive.org/download/s1id13206630/s1id13206630_djvu.txt | [challenge-june-1696] the June 1696 issue itself (running head "MENSIS JUNII A, M DC XCVI."): "Problema novum ad cujus folutionemMathcmatici invitantur. Datis in plano verticali duobus punblis A & B" (OCR; p. 269 per Bernoulli's Opera); MacTutor: "posed by Johann Bernoulli in Acta Eruditorum ... in June 1696".
- WRONG | In June 1696 Johann Bernoulli printed a challenge in the journal Acta Eruditorum "to the most acute mathematicians of the whole world" | https://archive.org/download/bub_gb_JIyE5FgkdscC/bub_gb_JIyE5FgkdscC_djvu.txt | [challenge-quote] The address is real but belongs to a different document printed six months later. The June 1696 Acta notice (https://archive.org/download/s1id13206630/s1id13206630_djvu.txt) has no salutation: it goes from its title "Problema novum ad cujus solutionem Mathematici invitantur" straight to "Datis in plano verticali duobus punctis A & B". The page's words translate the heading of Bernoulli's Groningen broadsheet of 1 January 1697, reprinted in his Opera omnia I (1742) as No. XXXIII: "PROGRAMMA Editum GRaoN1NG& 2f. 1697. AAcutiffmis qui tto. Orbe florent. Mathematicis S. P. D. JOHANNES BERNOULLI" (OCR of "Acutissimis qui toto Orbe florent Mathematicis"), signed "Dabam Groningz, ipfis Cal. Jan. 1697."; the Royal Society's copy, which Newton received on 29 January 1697, is "Printed at Groningen in 1697" (https://catalogues.royalsociety.org/CalmView/rdf/catalog/CLP/1/30). MacTutor and Wikipedia make the same conflation with a looser translation. The N1 record found no source; this is it. Fixer re-read both (Acta June 1696 p. 269; Opera omnia I No. XXXIII, "Acutissimis qui toto Orbe florent Mathematicis", "Dabam Groningae, ipsis Cal. Jan. 1697", "propositionem hic repeto") and moved the quoted words, unchanged, to the 1 January 1697 Groningen sheet; Corrected line added.
- CONFIRMED | He gave six months; at Leibniz's urging he extended the deadline so foreign mathematicians could join. | https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ | [six-months] MacTutor: "Leibniz persuaded Johann Bernoulli to allow a longer time for solutions to be produced than the six months he had originally intended so that foreign mathematicians would also have a chance to solve the problem." The primary agrees (Programma 1697, Opera I No. XXXIII, OCR): "Sex menfium fpatium a prima publicationis die Geometris conceffum eft" and Leibniz asked "ut praftitutum limitem ad proximum Pafcha extendi paterer; quo interea apud Gallos, Italofque, idem illud publicari poffet".
- CONFIRMED | In the end five solutions arrived, from Bernoulli himself, his brother Jacob, Leibniz, the Marquis de l'Hôpital, and Isaac Newton. | https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ | [five-solutions] "Five solutions were obtained, Newton, Jacob Bernoulli, Leibniz and de L'Hôpital solving the problem in addition to Johann Bernoulli." This is Johann's own count too (MacTutor quotes him: "my elder brother made up the fourth of these"). See the next line for Tschirnhaus.
- MINOR | In the end five solutions arrived, from Bernoulli himself, his brother Jacob, Leibniz, the Marquis de l'Hôpital, and Isaac Newton. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [five-solutions-wiki] This entry's own source does not support the list as written: "In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus, and Guillaume de l'Hôpital. Four of the solutions (excluding L'Hôpital's) were published in the same edition of the journal as Bernoulli's." So a sixth responder, Tschirnhaus, appeared in the same May 1697 Acta; the page follows MacTutor and Bernoulli's own count, which leave him out. Near-miss: "five solutions, by Bernoulli's own count" or name Tschirnhaus (index.html:267-268).
- CONFIRMED | Newton, then Warden of the Mint and largely retired from mathematics | https://mathshistory.st-andrews.ac.uk/Biographies/Newton/ | [newton-warden] "becoming Warden of the Royal Mint in 1696 and Master in 1699"; and for the half the N1 record could not source: "After suffering a second nervous breakdown in 1693, Newton retired from research." and his third period "saw Newton as a highly paid government official in London with little further interest in mathematical research."
- CONFIRMED | His niece's husband, John Conduitt | https://en.wikipedia.org/wiki/John_Conduitt | [niece-husband] "He was married to granddaughter of Hannah Ayscough and half-niece of Sir Isaac Newton"; "his half-niece Catherine Barton". MacTutor and Wikipedia's Brachistochrone article call her Newton's niece; a half-niece is a niece.
- CONFIRMED | recorded the story decades later | https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00168 | [conduitt-recorded] John Conduitt, 'Miscellanea', Keynes Ms. 130.05: "When the problem in 1697 was sent by Bernoulli ... did not sleep till he had solved it which was by 4 in the morning C. C."; the same manuscript has entries dated "13. May 1730", about 33 years on.
- CONFIRMED | Newton came home tired from the Mint at four in the afternoon | https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00168 | [four-afternoon] "did not come home till four from the Tower very much tired" (the Mint was in the Tower); MacTutor's quotation supplies "(in the afternoon)", and Wikipedia has "At 4 p.m. on 29 January 1697, when he arrived home from the Royal Mint".
- CONFIRMED | "did not sleep till he had solved it, which was by four in the morning." | https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ | [conduitt-quote] MacTutor: "did not sleep till he had solved it, which was by four in the morning."; the manuscript reads "did not sleep till he had solved it which was by 4 in the morning".
- CONFIRMED | His solution was published anonymously in the Philosophical Transactions, correct, but giving no hint of his method. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [anon-transactions] "This solution, later published anonymously in the Philosophical Transactions, is correct but does not indicate the method by which Newton arrived at it."; MacTutor: "The Royal Society published Newton's solution anonymously in the Philosophical Transactions of the Royal Society in January 1697."
- CONFIRMED | tanquam ex ungue leonem, "the lion is known by its claw." | https://archive.org/download/aat0604.0002.001.umich.edu/aat0604.0002.001.umich.edu_djvu.txt | [lion-quote] Brewster 1855 vol. II p. 21 (OCR): 'yet Bernoulli recognised in it his powerful mind ; " tanquam" says he, " eoe ungue leonem" as the lion is known by his claw.' Brewster has "his claw", the page "its claw": that pronoun is all the tool's offline warning found.
- CONFIRMED | The famous Latin phrasing comes from David Brewster's 1855 life of Newton. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [brewster-latin] "The much quoted version tanquam ex ungue Leonem is due to David Brewster's 1855 book on Newton's life and work."; Brewster's own text (previous line) has the "tanquam".
- CONFIRMED | "as the lion by its claw" ( ex ungue Leonem ) | https://archive.org/download/bub_gb_JIyE5FgkdscC/bub_gb_JIyE5FgkdscC_djvu.txt | [basnage-quote] Wikipedia: '"as the lion by its claw" (in Latin, ex ungue Leonem)'. Primary: Bernoulli's letter to Basnage as printed (Histoire des Ouvrages des Savans, June 1697; Opera I No. XXXVIII), OCR: "ce feroit aífez de le connoitre par cet échantillon, comme ex wee leonem" (that is, "comme ex ungue leonem").
- CONFIRMED | In his actual letter (to Henri Basnage, March 1697) Bernoulli wrote only that he recognised the anonymous solver | https://archive.org/download/bub_gb_JIyE5FgkdscC/bub_gb_JIyE5FgkdscC_djvu.txt | [basnage-whiteside] The N1 record did not reach the letter; Opera I No. XXXVIII prints it: "Quioique. l'Au- teur de cette Conftru&ion, par un exccs de modeftie ne fe nomme pas; nous favons pourtant induliablement par plu- ficurs circonftances que c'eft le célébre Mr. NEWTON" then "comme ex ... leonem", with French "comme" and no "tanquam", as Whiteside says. Wikipedia dates it March 1697 (a Basel letter of 30 March 1697 with the same content: the direct and indirect methods, Huygens's two optics problems); the print appeared June 1697.
- CONFIRMED | meaning he knew the hand by its quality, per the historian D. T. Whiteside. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [whiteside-meaning] "Bernoulli's intention was, Whiteside argues, simply to indicate he could tell the anonymous solution was Newton's, just as it was possible to tell that an animal was a lion given its claw".
- CONFIRMED | is Brewster's 1855 wording, not Bernoulli's own | https://en.wikipedia.org/wiki/Brachistochrone_curve | [brewster-not-bernoulli] Wikipedia as above; and the two primaries differ as the page says: Brewster prints "tanquam ... ungue leonem", Bernoulli's printed letter "comme ex ungue leonem".
- CONFIRMED | Newton's "overnight" solve is Conduitt's later recollection | https://www.newtonproject.ox.ac.uk/view/texts/normalized/THEM00168 | [conduitt-later] Keynes Ms. 130.05, the entry signed "C. C." (Catherine Conduitt's information, in John Conduitt's notes), in a manuscript with entries of 1730.
- CONFIRMED | In Two New Sciences (1638) he proved that a bead drops faster along a bent two-chord path than down the single straight chord | https://archive.org/download/cu31924012322701/cu31924012322701_djvu.txt | [galileo-chords] Third Day, Scholium after Prop. XXXVI (Crew and de Salvio, p. 239): "Evidently descent along the path ADC is quicker [264] than along AC alone", resting on Prop. XXXVI.
- CONFIRMED | But then he guessed the best curve was the arc of a circle. | https://archive.org/download/cu31924012322701/cu31924012322701_djvu.txt | [galileo-circle] "From the preceding it is possible to infer that the path of quickest descent [lationem omnium velocissimam] from one point to another is not the shortest path, namely, a straight line, but the arc of a circle." Galileo calls it an inference; Wikipedia calls it "Galileo's conjecture"; "guessed" is fair, since his argument only compares the arc with polygons inscribed in it.
- WRONG | Galileo, to his credit, hedged: he suspected a "higher" mathematics might do better. | https://archive.org/download/cu31924012322701/cu31924012322701_djvu.txt | [galileo-hedged] The one "higher science" in the book (Third Day, p. 194, about 45 pages before the circular-arc Scholium) is about something else. After Sagredo's picture of particles on ever-expanding circles, Simplicio suspects "some great mystery hidden in these true and wonderful results, a mystery related to the creation of the universe (which is said to be spherical in shape), and related also to the seat of the first cause", and Salviati answers "But profound considerations of this kind belong to a higher science than ours [a pill alte dottrine che le nostre]" (OCR for "più"). That defers cosmology, not the quickest-descent curve, and suggests no better curve. The Scholium itself (pp. 239-240) has no hedge and ends "the nearer the inscribed polygon approaches a circle the shorter is the time required for descent from A to C."; the First Day (p. 96) states the arc result flatly too. Wikipedia's "Galileo warns of possible fallacies and the need for a 'higher science'" points at the same p. 192-194 passage (Sagredo's "if it involve no fallacy"). The N1 record saw the passage was "a different passage from the Scholium" but kept it as support. Fixer re-read the Crew and de Salvio text (p. 194 "belong to a higher science than ours" answers a "mystery related to the creation of the universe"; the Scholium pp. 239-240 offers no better curve) and dropped the sentence; Corrected line added.
- CONFIRMED | The tool that settled it, the calculus of variations, was still half a century away. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [calc-var-half-century] Read as 1638 to the 1696-97 solutions (58-59 years), from which the field grew: Jakob Bernoulli "developed new methods that Leonhard Euler refined into what Euler called (in 1766) the calculus of variations", and Newton "wound up pioneering the field". Loose but defensible; dated to Euler instead, it would be a century or more.
- CONFIRMED | Christiaan Huygens proved this in 1659 | https://en.wikipedia.org/wiki/Tautochrone_curve | [huygens-1659] "The tautochrone problem, the attempt to identify this curve, was solved by Christiaan Huygens in 1659."; MacTutor: "Huygens had shown in 1659, prompted by Pascal's challenge about the cycloid, that the cycloid is the solution to the tautochrone problem".
- CONFIRMED | a pendulum swinging in a cycloidal cheek keeps the same period whether it swings wide or narrow, a perfect clock, in principle | https://en.wikipedia.org/wiki/Tautochrone_curve | [huygens-cheeks] Huygens made "pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve", which "proved unhelpful" in practice; the page's "in principle" carries that. research/cycloid/verify.mjs rerun: "PASS cycloidal pendulum period = 4π√(R/g) at EVERY amplitude".
- MINOR | Huygens used this page's tautochrone to build a pendulum that keeps time at any swing. | https://en.wikipedia.org/wiki/Tautochrone_curve | [huygens-built] The record's own excerpt cuts against it: "These attempts proved unhelpful for a number of reasons."; the companion film says "A theoretical triumph that lost in practice: the friction of the cycloidal cheeks outweighed the error it removed" (Yoder 1988). The note drops the page body's "in principle". It is the relates note at src/content/strata/the-quickest-way-down.md:11; this page's Nearby layers shows only its first sentence, and the full note is in the /strata/there-here/ data (public/strata/there-here/index.html:250). Fix: "to design a pendulum that, in principle, keeps time at any swing".
- CONFIRMED | The cycloid is also the tautochrone (Greek "same time") | https://en.wikipedia.org/wiki/Tautochrone_curve | [name-tautochrone] "from Ancient Greek ταὐτό (tauto-) 'same', ἴσος (isos-) 'equal' and χρόνος (chronos) 'time'".
- CONFIRMED | brákhistos "shortest" + khrónos "time" | https://en.wikipedia.org/wiki/Brachistochrone_curve | [name-brachistochrone] "from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'".
- CONFIRMED | a cycloid, the curve traced by a point on the rim of a rolling wheel, flipped upside down. | https://en.wikipedia.org/wiki/Cycloid | [cycloid-def] "a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping"; the brachistochrone article: "the differential equation of an inverted cycloid".
- CONFIRMED | he imagined the bead as a ray of light threading layers of glass whose density changes with depth. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [bernoulli-light] "considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration"; MacTutor: his solution "divides the plane into strips" and "appeals to Fermat's principle"; Bernoulli himself (Opera I No. XXXVIII, OCR): "fi un. diaphane ... change continuellement de rareté, en méme raifon que font les vitef- fes acquifes d'un corps pefant". "Glass" is the page's gloss for his transparent medium.
- CONFIRMED | Light already solves "least time" for free, that is Fermat's principle | https://en.wikipedia.org/wiki/Fermat%27s_principle | [fermat] "the path taken by a ray between two given points is the path that can be traveled in the least time" (the article adds the strict form is "stationary").
- CONFIRMED | a frictionless bead at depth y below the start moves at v = √(2gy), whatever the path. | https://en.wikipedia.org/wiki/Brachistochrone_curve | [speed-law] "By the conservation of energy, the instantaneous speed of a body v after falling a height y in a uniform gravitational field is given by" v = sqrt(2gy); the verifier's energy check gives 5.2401 m/s on every ramp.
- CONFIRMED | Real marbles roll (which changes the effective mass and the constant, not the winning shape) | https://pmm.ipmnet.ru/en/Issues/2009/4/371 | [marbles-roll] The N1 record found no source; one exists. L. D. Akulenko, "The brachistochrone problem for a disc", J. Appl. Math. Mech. 73(4), 371-378 (2009), abstract: for rolling without slip, "it is shown that the disc centre moves along a cycloid connecting specified points." Rolling adds I/r^2 to the effective mass, so v = sqrt(2gy/(1+I/mr^2)): the same square-root law with a smaller constant. Nuance: the body's centre follows the cycloid and the track is its offset.
- CONFIRMED | J. J. O'Connor & E. F. Robertson, "The brachistochrone problem," MacTutor History of Mathematics Archive, Univ. of St Andrews | https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ | [cite-mactutor] title "The brachistochrone problem", "Written by J J O'Connor and E F Robertson"; it carries everything the page credits to it: June 1696, Conduitt's account, the five solvers, "Huygens had shown in 1659", and Galileo's "incorrect deduction".
- CONFIRMED | Galileo Galilei, Dialogues Concerning Two New Sciences (1638), Third Day | https://archive.org/download/cu31924012322701/cu31924012322701_djvu.txt | [cite-galileo] Crew and de Salvio's translation (1914): the two-chord result and the circular-arc Scholium are both in the Third Day (pp. 239-240).
- CONFIRMED | the cycloid parametrisation x=r(φ−sinφ), y=r(1−cosφ) | https://en.wikipedia.org/wiki/Brachistochrone_curve | [cite-wiki-param] "x&=r(\varphi -\sin \varphi )\\y&=r(1-\cos \varphi )", with "φ is a real parameter, corresponding to the angle through which the rolling circle has rotated".
- OBSERVED | This layer's check works the answer out again from scratch, independently of this page. | research/verifier-independence/census-pairs.tsv | [check-independent] row: the-quickest-way-down, research/brachistochrone/verify.mjs, C0, "touches nothing that /strata/the-quickest-way-down/ ships"; consistent with the source: verify.mjs builds its ramps from equations and reads no file.
- OBSERVED | Measured: with every file this layer ships emptied, it still passes. | research/verifier-independence/depends-0.json | [check-emptied] entry the-quickest-way-down: both rows (brachistochrone and cycloid verify.mjs) "exitCode": 0, "depends": false.
- MINOR | Last verified 25 June 2026, when it was written. | git show 0bf28226499 | [last-verified] 25 June is when the layer was first written (memory/log.d/2026-06-25T0635Z-the-quickest-way-down.md), but the third instrument and its figures (0.1903 s/m, "sixteen digits", 64 layers, 1.1 cm, 0.79930 s) were written and verified on 29 June: commit 0bf28226499 says "research/brachistochrone/verify.mjs extended (claim 3, Snell section), all green", and the page's JSON-LD says "dateModified":"2026-06-29". The hallmark vouches for a check of text that did not exist on that date. The line is generated (scripts/check-placard.mjs:257 prints the layer's date when no re-read exists); a record from this pass replaces it.
- OBSERVED | A third instrument (added 2026-06-29) | git show 0bf28226499 | [third-added] commit "Deepen the brachistochrone: add Bernoulli's optical instrument", 2026-06-29 06:09:47 +0000, and memory/log.d/2026-06-29T0610Z-bernoullis-beam-of-light.md (markdown body only, not rendered).
- OBSERVED | "datePublished":"2026-06-25","dateModified":"2026-06-29" | git show 0bf28226499:public/strata/the-quickest-way-down/index.html | [dates-ld] first commit cd1d4f1c66e on 2026-06-25, the deepening 0bf28226499 on 2026-06-29; later chrome commits (placards, og-alt) did not move dateModified, which I do not count.
- CONFIRMED | (added) the equal fall π√(R/g) = 1.00320 s, the period 4.01282 s, length 8R, area 3πR² | research/cycloid/verify.mjs | film caption, index.html:150. Rerun: "release at θ0=0.20 → bottom in 1.00320 s (predict 1.00320)", "period 4.01281 s (predict 4.01282)", "PASS one cycloid arch has length 8R", "PASS area under one arch = 3πR² (three rolling circles)".
- CONFIRMED | (added) Sixty years earlier, Galileo had already circled the question. In Two New Sciences (1638) | https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ | MacTutor: "Galileo in 1638 had studied the problem in his famous work Discourse on two new sciences"; 1696 minus 1638 is 58 years, "sixty" as a round number.
- CONFIRMED | (added) with v=√(2gy) from Galileo | https://mathshistory.st-andrews.ac.uk/HistTopics/Brachistochrone/ | Sources line. MacTutor: "Galileo had shown that the velocity v satisfies v = √(2gy)"; Bernoulli's letter (Opera I No. XXXVIII) ties the cycloid's identity to "l'hypothèse de Galilée".
- OBSERVED | (added) verify.mjs is published, and it runs on its own: one file, and Node. | recreated: curl -sSLO https://artwaste.land/checks/research/brachistochrone/verify.mjs (and .../cycloid/verify.mjs) into an empty directory; node each | both exit 0; output byte-identical to the repository run at 60cbd1924f; the served brachistochrone verify.mjs is byte-identical to the repository's.
- MINOR | (added) the film caption links "this race's verifier" (19/19) to the verifier | https://artwaste.land/research/cycloid/verify.mjs | The link is href="../../../research/cycloid/verify.mjs" (index.html:150), which resolves to /research/cycloid/verify.mjs: curl gives 404. The served copy is /checks/research/cycloid/verify.mjs (200, the file the placard links). Fix: point the caption link there.

## Findings
- [fixed] WRONG challenge-quote: the page put "to the most acute mathematicians of the whole world" on the June 1696 Acta Eruditorum notice, which has no address (Acta Eruditorum June 1696 p. 269, archive.org s1id13206630). The words translate the heading of Bernoulli's Groningen sheet of 1 January 1697 (Opera omnia I No. XXXIII, archive.org bub_gb_JIyE5FgkdscC), which repeated the problem and announced the extension to Easter at Leibniz's request. public/strata/the-quickest-way-down/index.html now attaches the quotation, unchanged, to that sheet; Corrected 2026-11-01 line on the page and in the markdown body.
- [fixed] WRONG galileo-hedged: "Galileo, to his credit, hedged: he suspected a "higher" mathematics might do better." is not in Two New Sciences: the one "higher science" (Third Day p. 194, Crew and de Salvio) sets aside a cosmological mystery, and the Scholium (pp. 239-240) and First Day (p. 96) offer no better curve. Sentence removed; Corrected 2026-11-01 line on the page and in the markdown body.
- [declined] MINOR engine-same-code (index.html:370 and the markdown body, "the same code that the verifier runs"): a port of the verifier's integrator at a coarser step, agreeing to the four decimals shown; tier C lists MINOR items rather than editing them.
- [declined] MINOR sixteen-digits (index.html:231, 324): a spread of 3.33e-16 on 0.1903 is about fifteen significant figures and the assertion tolerance is 1e-9; tier C lists MINOR items.
- [declined] MINOR film-verifier (index.html:150): the film's numbers are from the 19/19 verifier, its caption words are not ("every number and quote" is the build script's own claim); tier C lists MINOR items.
- [declined] MINOR five-solutions-wiki (index.html "In the end five solutions arrived"): follows MacTutor and Bernoulli's own count; Wikipedia also names Tschirnhaus; tier C lists MINOR items, and the sentence is separate from the fixed one.
- [declined] MINOR huygens-built (src/content/strata/the-quickest-way-down.md relates note): drops the body's "in principle"; tier C lists MINOR items.
- [declined] MINOR last-verified (generated placard line, scripts/check-placard.mjs): "Last verified 25 June 2026" predates the 29 June instrument; this record replaces it when the placard is rebuilt at landing.
- [declined] MINOR film-link-404 (index.html:150): the caption's verifier link resolves to /research/cycloid/verify.mjs (404) while the placard links /checks/research/cycloid/verify.mjs; tier C lists MINOR items.