---
slug: the-edge-of-the-bow
passed: 2026-10-14
by: the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex and a Claude agent
checked: 161
confirmed: 97
wrong: 62
unverifiable: 0
observed: 0
stale: 2
---

This was the 2026-10-14 directed repair pass for the layer read at f5e5f84f34. I re-read the evidence for all 64 triage items, retained each item separately even where the two checkers reported the same error, and carried forward their distinct CONFIRMED source checks with their stated qualifications. The 24 MINOR items count toward wrong for this gate; they have findings but no public correction lines. The 38 WRONG items are covered by 12 correction groups: 36 are fixed, while X17 and K12 retain an open copy in another layer. The exact-42° group is fixed in page text, authoring sources and the SVG text; share card rebuilt at landing from the corrected dek. The existing PNG and thumbnail are left to landing, not rendered in this pass. The two STALE metadata items use the triage's explicit no-line exceptions. Historical quotations and source-record claim text remain evidence of the old page, not current assertions.

## Claims

- MINOR | [X1] “it is always 42° away” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Book I, Part II, Proposition IX distinguishes the red and violet limiting angles. The model at n = 1.333 gives 42.0781°, not a universal exact 42°.
- WRONG | [X2] “Plot the deflection against where the ray hit” | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- WRONG | [X3] “The bottom of the curve is flat” | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- WRONG | [X4] “Deflection vs impact parameter.” | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- WRONG | [X5] “the shaded sliver shows how wide a band of rays crowds” | public/strata/the-edge-of-the-bow/index.html | The old code assigned a fillStyle and began an empty path; later paths replaced it without drawing a band. The only curve-panel fill was the steered dot.
- MINOR | [X6] “numerically hunting the minimum of the traced deflection” | research/the-edge-of-the-bow/verify.mjs | numericExtremum minimizes deviation(), the analytic formula. A separate test compares traced exit directions with that formula; the optimizer does not minimize traced directions.
- WRONG | [X7] “fire the rain and look at the tally” | public/strata/the-edge-of-the-bow/index.html; research/the-edge-of-the-bow/artifact.json | Rain of rays draws 90 paths and displays four angles, with no histogram. The offline histogram bins 400,000 rays: 13,352 / 950.1187648456057 = 14.053 times the mean occupied-bin count.
- MINOR | [X8] “double-reflection light only outside 51°” | research/the-edge-of-the-bow/artifact.json | The red secondary edge is 50.36511801705433°, while the yellow secondary is 50.89075776990069°.
- MINOR | [X9] “it has a name older than any of the geometry” | https://www.atoptics.org.uk/rainbows/adband.htm | The opening paragraph dates Alexander's description to about 200 AD. It does not date the modern name.
- WRONG | [X10] “every angle is fed by exactly two rays” | research/the-edge-of-the-bow/verify.mjs | Re-executing the original root functions gives one root at 10°, two at 41°, and none at 43° for n = 1.333. The grazing limit is 14.4265°. The old check sampled only 1° inside the edge.
- MINOR | [X11] “Thomas Young saw this in 1804” | https://www.philiplaven.com/Publications/AO-56-19-G104a.pdf | Laven names the 1803 Bakerian Lecture, while reference 1 gives Philosophical Transactions 94, 1-16 (1804).
- MINOR | [X12] “the full wave theory” | https://www.philiplaven.com/p8a.html | Figures 1-3 compare the Airy approximation with Mie and Debye results and show errors in some fringe positions.
- MINOR | [X13] “better than a millionth of a degree” | research/the-edge-of-the-bow/verify.mjs; research/the-edge-of-the-bow/artifact.json | The reported maximum error is 1.21e-6°, with a 1e-5° pass threshold. This does not support better than 1e-6°.
- WRONG | [X14] “violet 1.344” | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Visually re-read Table 5, printed p. 446 (PDF p. 30), from the retained scan: at 20 °C, 4046.6 angstroms has prefix 1.342 and residual 7422, giving 1.3427422. Three-decimal input: 1.343. Recomputed violet primary 40.645880523217556°, secondary 53.47776353180757°, width 1.7239606812711372° with red 1.331.
- MINOR | [X15] “Everything else is traced from those three numbers.” | public/strata/the-edge-of-the-bow/index.html | The sky uses five additional intermediate indices as well as the red and violet endpoints. Those values are not derived from the three-colour inputs.
- STALE | [X16] “Last verified 19 June 2026, when it was written.” | git show f5e5f84f34:research/the-edge-of-the-bow/artifact.json | The old artifact is dated 2026-09-07 with 23/23 checks passing, later than the placard's 19 June date. This pass reran the corrected verifier successfully on 2026-10-14.
- WRONG | [X17] “built from refraction alone” | public/strata/the-edge-of-the-bow/index.html; src/content/strata/light-leans-toward-the-cold.md; scripts/nearby-layers.mjs | traceDraw performs internal reflection between entry and exit refractions. The incoming note is authored in another layer. nearby-layers prioritizes this layer's own relationships before reverse edges.
- WRONG | [X18] “flat bottom ⇒ the pile-up” | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- WRONG | [X19] “violet · 405 nm 1.344” | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Visually re-read Table 5, printed p. 446 (PDF p. 30), from the retained scan: at 20 °C, 4046.6 angstroms has prefix 1.342 and residual 7422, giving 1.3427422. Three-decimal input: 1.343. Recomputed violet primary 40.645880523217556°, secondary 53.47776353180757°, width 1.7239606812711372° with red 1.331.
- MINOR | [X20] “it sits at the same 42° every time” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Book I, Part II, Proposition IX distinguishes the red and violet limiting angles. The model at n = 1.333 gives 42.0781°, not a universal exact 42°.
- WRONG | [X21] “the angle of the minimum is its radius” | research/the-edge-of-the-bow/verify.mjs; research/the-edge-of-the-bow/artifact.json | The primary minimum deflection is 137.92189261975645°, while the apparent radius is its supplement, 42.07810738024352°.
- WRONG | [X22] “a strip of sky no ray can reach” | https://www.philiplaven.com/p8a.html | The discussion of Figures 7-9 identifies externally reflected p = 0 light in Alexander's dark band.
- WRONG | [X23] “the deflection curve whose flat bottom is the rainbow” | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- WRONG | [X24] “supernumeraries show only in fine mist” | https://www.atoptics.org.uk/rainbows/supers.htm | The photograph caption records a bow after a heavy shower; the accompanying simulation uses a 0.7 mm mean diameter and a narrow size spread.
- MINOR | [X25] “a rainbow always sits at 42 degrees” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Book I, Part II, Proposition IX distinguishes the red and violet limiting angles. The model at n = 1.333 gives 42.0781°, not a universal exact 42°.
- WRONG | [X26] “watch the deflection curve bottom out” | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- MINOR | [X27] “42° from the point opposite the sun, every time.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Book I, Part II, Proposition IX distinguishes the red and violet limiting angles. The model at n = 1.333 gives 42.0781°, not a universal exact 42°.
- MINOR | [X28] “named around 200 AD” | https://www.atoptics.org.uk/rainbows/adband.htm | The opening paragraph dates Alexander's description to about 200 AD. It does not date the modern name.
- WRONG | [X29] “the two rays that reach each angle inside the bow” | research/the-edge-of-the-bow/verify.mjs | Re-executing the original root functions gives one root at 10°, two at 41°, and none at 43° for n = 1.333. The grazing limit is 14.4265°. The old check sampled only 1° inside the edge.
- WRONG | [X30] “supernumeraries appear only in fine mist” | https://www.atoptics.org.uk/rainbows/supers.htm | The photograph caption records a bow after a heavy shower; the accompanying simulation uses a 0.7 mm mean diameter and a narrow size spread.
- MINOR | [X31] “both turn on an extremum the shape forces” | src/content/strata/egregium.md | The linked layer explains curvature preserved by isometries, not an extremum argument.
- WRONG | [X32] “every hue of the rainbow is a single wavelength” | https://www.atoptics.org.uk/rainbows/supcols.htm | The diagram sums the wavelength components; the accompanying account describes mixed, non-spectral pinks and purples in supernumeraries.
- WRONG | [X33] “the colour the rainbow can never contain” | https://www.atoptics.org.uk/rainbows/supcols.htm | The diagram sums the wavelength components; the accompanying account describes mixed, non-spectral pinks and purples in supernumeraries.
- WRONG | [X34] “The rainbow is exactly the set of colours that ARE wavelengths” | https://www.atoptics.org.uk/rainbows/supcols.htm | The diagram sums the wavelength components; the accompanying account describes mixed, non-spectral pinks and purples in supernumeraries.
- WRONG | [X35] “the retina's own machinery at the threshold of darkness” | src/content/strata/seeing-in-the-dark.md | The linked page's dek and seoTitle describe the Elitzur-Vaidman bomb tester and interaction-free measurement with an interferometer.
- WRONG | [X36] “supernumeraries show only in fine mist” | https://www.atoptics.org.uk/rainbows/supers.htm | The photograph caption records a bow after a heavy shower; the accompanying simulation uses a 0.7 mm mean diameter and a narrow size spread.
- WRONG | [X37] “supernumeraries appear only in fine mist” | https://www.atoptics.org.uk/rainbows/supers.htm | The photograph caption records a bow after a heavy shower; the accompanying simulation uses a 0.7 mm mean diameter and a narrow size spread.
- WRONG | [X38] “the Answer to Everything in 1979” | https://www.panmacmillan.com/blogs/science-fiction-and-fantasy/hitchhikers-guide-to-the-galaxy-book-series | The publisher places the radio story and its answer in 1978 and the novel in 1979.
- WRONG | [X39] “the Answer to Everything in 1979” | https://www.panmacmillan.com/blogs/science-fiction-and-fantasy/hitchhikers-guide-to-the-galaxy-book-series | The publisher places the radio story and its answer in 1978 and the novel in 1979.
- STALE | [X40] (added) “"dateModified":"2026-06-19"” | git log f5e5f84f34 -- public/strata/the-edge-of-the-bow/index.html | The page had later edits, including 2026-10-01, while dateModified still said 2026-06-19.
- WRONG | [K1] Deflection vs impact parameter. | public/strata/the-edge-of-the-bow/index.html | viewAngle converts the deflection before samp.push; the vertical scale increases upwards. The primary has a maximum of 42.0781°, the secondary a minimum of 50.8908°.
- WRONG | [K2] The horizontal dashed line marks the extremum -- the rainbow angle -- and the shaded sliver shows how wide a band of rays crowds within a fraction of a degree of it. | public/strata/the-edge-of-the-bow/index.html | The old code assigned a fillStyle and began an empty path; later paths replaced it without drawing a band. The only curve-panel fill was the steered dot.
- MINOR | [K3] The verifier finds the same angle two independent ways: by this formula, and by numerically hunting the minimum of the traced deflection. | research/the-edge-of-the-bow/verify.mjs | numericExtremum minimizes deviation(), the analytic formula. A separate test compares traced exit directions with that formula; the optimizer does not minimize traced directions.
- WRONG | [K4] In the instrument above, fire the rain and look at the tally: of the rays binned by exit angle, the fullest sliver near 42° holds about fourteen times as many as a typical one. | public/strata/the-edge-of-the-bow/index.html; research/the-edge-of-the-bow/artifact.json | Rain of rays draws 90 paths and displays four angles, with no histogram. The offline histogram bins 400,000 rays: 13,352 / 950.1187648456057 = 14.053 times the mean occupied-bin count.
- MINOR | [K5] Descartes got the geometry and the brightness exactly right -- and got the colour entirely wrong, blaming a spin imparted to light particles. | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Les Meteores VIII gives ray crowding and observed colour order as well as the mistaken spinning-particle theory. It does not establish exact brightness or make every colour observation wrong.
- MINOR | [K6] The primary bow runs from violet at its inner edge to red on the outside -- a fan only about 1.9° wide. | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Visually re-read Table 5, printed p. 446 (PDF p. 30), from the retained scan: at 20 °C, 4046.6 angstroms has prefix 1.342 and residual 7422, giving 1.3427422. Three-decimal input: 1.343. Recomputed violet primary 40.645880523217556°, secondary 53.47776353180757°, width 1.7239606812711372° with red 1.331.
- MINOR | [K7] It is measurably darker than the sky on either side, and it has a name older than any of the geometry: Alexander's dark band, after Alexander of Aphrodisias, who wrote it down around 200 AD. | https://www.atoptics.org.uk/rainbows/adband.htm | The opening paragraph dates Alexander's description to about 200 AD. It does not date the modern name.
- WRONG | [K8] Inside the bow, every angle is fed by exactly two rays -- one from each side of the minimum (the instrument in §II shows the curve crossing any horizontal line twice). | research/the-edge-of-the-bow/verify.mjs | Re-executing the original root functions gives one root at 10°, two at 41°, and none at 43° for n = 1.333. The grazing limit is 14.4265°. The old check sampled only 1° inside the edge.
- MINOR | [K9] Every angle on this page is recomputed from scratch by one offline verifier -- none is quoted from memory -- by tracing rays through a sphere of water and reading the geometry off the traces. | research/the-edge-of-the-bow/verify.mjs | numericExtremum minimizes deviation(), the analytic formula. A separate test compares traced exit directions with that formula; the optimizer does not minimize traced directions.
- MINOR | [K10] The verifier builds a vector ray-tracer (refract → reflect → refract), checks it against the closed-form deflection to better than a millionth of a degree, then derives the rainbow angle as the extremum two independent ways, reproduces Descartes' ray-census pile-up, places every colour edge of both bows, proves Alexander's dark band is empty of rays, shows the higher-order bows sit around the sun (why the 3rd and 4th are nearly unseeable -- first convincingly photographed only in 2011), and confirms the two-ray interference and its (λ⁄a)2/3 scaling. | research/the-edge-of-the-bow/verify.mjs; research/the-edge-of-the-bow/artifact.json | The reported maximum error is 1.21e-6°, with a 1e-5° pass threshold. This does not support better than 1e-6°.
- WRONG | [K11] Inputs that are measured, not derived: the refractive index of water at three wavelengths (red 1.331, yellow 1.333, violet 1.344), standard textbook values near 20 °C. | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Visually re-read Table 5, printed p. 446 (PDF p. 30), from the retained scan: at 20 °C, 4046.6 angstroms has prefix 1.342 and residual 7422, giving 1.3427422. Three-decimal input: 1.343. Recomputed violet primary 40.645880523217556°, secondary 53.47776353180757°, width 1.7239606812711372° with red 1.331.
- WRONG | [K12] Two layers of atmospheric optics built from refraction alone. | public/strata/the-edge-of-the-bow/index.html; src/content/strata/light-leans-toward-the-cold.md; scripts/nearby-layers.mjs | traceDraw performs internal reflection between entry and exit refractions. The incoming note is authored in another layer. nearby-layers prioritizes this layer's own relationships before reverse edges.
- MINOR | [K13] Descartes found it in 1637 by tracing rays and counting where they crowded (no calculus) and got the geometry exactly right while getting colour entirely wrong (Newton fixed that: violet bends a hair more than red, so there is one rainbow angle per colour and the bow is the thin fan between them, red outside). | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Les Meteores VIII gives ray crowding and observed colour order as well as the mistaken spinning-particle theory. It does not establish exact brightness or make every colour observation wrong.
- MINOR | [K14] The second bow, with two internal bounces, reverses the colours and lands at 51°; between the two lies a strip of sky no ray can reach: Alexander's dark band, named c. 200 AD. | https://www.atoptics.org.uk/rainbows/adband.htm | The opening paragraph dates Alexander's description to about 200 AD. It does not date the modern name.
- MINOR | [K15] Newton supplied colour: water bends violet slightly more than red, so each colour has its own rainbow angle and the bow is the thin fan (~1.9° wide) between violet inside and **red outside**. | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Visually re-read Table 5, printed p. 446 (PDF p. 30), from the retained scan: at 20 °C, 4046.6 angstroms has prefix 1.342 and residual 7422, giving 1.3427422. Three-decimal input: 1.343. Recomputed violet primary 40.645880523217556°, secondary 53.47776353180757°, width 1.7239606812711372° with red 1.331.
- MINOR | [K16] A second internal bounce makes the **secondary bow** at 51° with the colours reversed, and between the two lies **Alexander's dark band** -- a strip of sky no single- or double-bounced ray can reach, named around 200 AD. | https://www.atoptics.org.uk/rainbows/adband.htm | The opening paragraph dates Alexander's description to about 200 AD. It does not date the modern name.
- MINOR | [K17] The page makes all of this runnable in three live instruments and checks every angle offline (`research/the-edge-of-the-bow/verify.mjs`, 23/23) by building a vector ray-tracer and reading the geometry off the traces. | research/the-edge-of-the-bow/verify.mjs | numericExtremum minimizes deviation(), the analytic formula. A separate test compares traced exit directions with that formula; the optimizer does not minimize traced directions.
- WRONG | [K18] This page lays out the spectrum the way a drop sorts it -- every hue of the rainbow is a single wavelength bent by its own refractive index. | https://www.atoptics.org.uk/rainbows/supcols.htm | The diagram sums the wavelength components; the accompanying account describes mixed, non-spectral pinks and purples in supernumeraries.
- WRONG | [K19] That page names the colour the rainbow can never contain: magenta is non-spectral, invented by the eye from the two ends at once. | https://www.atoptics.org.uk/rainbows/supcols.htm | The diagram sums the wavelength components; the accompanying account describes mixed, non-spectral pinks and purples in supernumeraries.
- WRONG | [K20] The rainbow is exactly the set of colours that ARE wavelengths; magenta is the proof the set is incomplete. | https://www.atoptics.org.uk/rainbows/supcols.htm | The diagram sums the wavelength components; the accompanying account describes mixed, non-spectral pinks and purples in supernumeraries.
- WRONG | [K21] Both are about how light reaches the eye and what the eye then makes of it -- there, the retina's own machinery at the threshold of darkness; here, the geometry a sky full of drops performs before the light ever arrives. Two layers of the same path from photon to perception. | src/content/strata/seeing-in-the-dark.md | The linked page's dek and seoTitle describe the Elitzur-Vaidman bomb tester and interaction-free measurement with an interferometer.
- WRONG | [K22] supernumeraries show only in fine mist | https://www.atoptics.org.uk/rainbows/supers.htm | The photograph caption records a bow after a heavy shower; the accompanying simulation uses a 0.7 mm mean diameter and a narrow size spread.
- WRONG | [K23] supernumeraries appear only in fine mist | https://www.atoptics.org.uk/rainbows/supers.htm | The photograph caption records a bow after a heavy shower; the accompanying simulation uses a 0.7 mm mean diameter and a narrow size spread.
- WRONG | [K24] the same 42 Douglas Adams made the Answer to Everything in 1979 | https://www.panmacmillan.com/blogs/science-fiction-and-fantasy/hitchhikers-guide-to-the-galaxy-book-series | The publisher places the radio story and its answer in 1978 and the novel in 1979.
- CONFIRMED | [html-01-02] “All three facts are one piece of geometry” [...] “a minimum in the bending of light through a drop, where the rays pile up into an edge.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [html-03-01] “A raindrop is, to a good approximation, a tiny sphere of water.” | https://www.weather.gov/fgz/Rainbow | Retained checker result, with dependent updates where stated: NWS describes “a single round raindrop”. Sphericity is the stated approximation for the geometric model, not a claim that all large falling drops are exact spheres.
- CONFIRMED | [html-03-02] “Sunlight that strikes it bends going in (refraction), can bounce off the far inside wall (reflection), and bends again coming out.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “two Refractions of the Sun's Light, and one Reflexion between them”; the exterior bow has two internal reflections. Read all of Proposition IX for the color order and limiting angles.
- CONFIRMED | [html-03-03] “The total turn the ray makes is its deflection.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, p. 3: “the deviation undergone by the ray from its original direction”. This agrees with the definition of total deflection in the page.
- CONFIRMED | [html-04-02] “The angle it has turned through is the deflection.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, p. 3: “the deviation undergone by the ray from its original direction”. This agrees with the definition of total deflection in the page.
- CONFIRMED | [html-04-04] “That edge is the rainbow.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [html-05-02] “The deflection swings” [...] “but it does not swing freely.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “a far greater number of the Rays” emerge “in the Limits of these Angles, than in any other Inclinations.” This supports concentration near the stationary deflection, including its locally flat tangent.
- CONFIRMED | [html-05-03] “It runs down to a lowest value and then turns back up.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “a far greater number of the Rays” emerge “in the Limits of these Angles, than in any other Inclinations.” This supports concentration near the stationary deflection, including its locally flat tangent.
- CONFIRMED | [html-05-04] “There is a minimum deflection, and the rays that happen to hit near that special height all leave at nearly the same angle.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “a far greater number of the Rays” emerge “in the Limits of these Angles, than in any other Inclinations.” This supports concentration near the stationary deflection, including its locally flat tangent.
- CONFIRMED | [html-05-05] “Uniformly spaced going in, they come out bunched.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “a far greater number of the Rays” emerge “in the Limits of these Angles, than in any other Inclinations.” This supports concentration near the stationary deflection, including its locally flat tangent.
- CONFIRMED | [html-05-06] “Bunched rays mean concentrated light.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [html-05-07] “The rainbow is that bunch.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [html-06-01] “The bow is not painted on the sky at 42°.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton, Book I, Part II, Proposition IX: “42 Degrees and 2 Minutes” for the least refrangible rays, “40 Degrees and 17 Minutes” for the most refrangible. These are wavelength-dependent limiting angles. The independent calculation with the stipulated yellow index gives 42.07810738024352 degrees.
- CONFIRMED | [html-06-02] “It is the one angle the drop cannot avoid sending light to.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton, Book I, Part II, Proposition IX: “42 Degrees and 2 Minutes” for the least refrangible rays, “40 Degrees and 17 Minutes” for the most refrangible. These are wavelength-dependent limiting angles. The independent calculation with the stipulated yellow index gives 42.07810738024352 degrees.
- CONFIRMED | [html-08-03] “That flatness is the pile-up, made visible.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [html-10-01] “There is a clean closed form for where the bottom sits.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, equations (1)-(2), pp. 3-4: i_c = arccos(sqrt((n^2-1)/(p^2-1))), with p-1 internal reflections. Substituting p=k+1 gives the displayed formula. It applies when the stationary incidence exists.
- CONFIRMED | [html-10-02] “For a ray that bounces k times inside, the minimum is at the incidence angle where” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, equations (1)-(2), pp. 3-4: i_c = arccos(sqrt((n^2-1)/(p^2-1))), with p-1 internal reflections. Substituting p=k+1 gives the displayed formula. It applies when the stationary incidence exists.
- CONFIRMED | [html-11-01] “cos²i = (n² − 1) / k(k+2)” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, equations (1)-(2), pp. 3-4: i_c = arccos(sqrt((n^2-1)/(p^2-1))), with p-1 internal reflections. Substituting p=k+1 gives the displayed formula. It applies when the stationary incidence exists.
- CONFIRMED | [html-12-01] “with n the refractive index of water, about 1.333.” | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Retained checker result, with dependent updates where stated: NBS, Table 5, printed p. 446, PDF page 30, read visually: 20 C gives 1.3311512 at 6562.8 angstroms, 1.3329877 at 5892.6, and 1.3427422 at 4046.6. The yellow value rounds to 1.333. The source record calls this Table 3 incorrectly.
- CONFIRMED | [html-12-04] “For one bounce that puts the bottom at an incidence of 59.4° and a deflection of 137.9°” [...] “and a deflection of 137.9° means the light comes back 180° − 137.9° = 42.1° from the point directly opposite the sun.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton, Book I, Part II, Proposition IX: “42 Degrees and 2 Minutes” for the least refrangible rays, “40 Degrees and 17 Minutes” for the most refrangible. These are wavelength-dependent limiting angles. The independent calculation with the stipulated yellow index gives 42.07810738024352 degrees.
- CONFIRMED | [html-12-05] “That is the radius of the bow.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton, Book I, Part II, Proposition IX: “42 Degrees and 2 Minutes” for the least refrangible rays, “40 Degrees and 17 Minutes” for the most refrangible. These are wavelength-dependent limiting angles. The independent calculation with the stipulated yellow index gives 42.07810738024352 degrees.
- CONFIRMED | [html-14-01] “In Les Météores, René Descartes did exactly what the Rain of rays button does” [...] “only with arithmetic, no calculus and no computer.” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes, Les Meteores VIII, 1637: “ayant pris la plume et calculé” and “41 à 42 degrés”, followed by “51 à 52 degrés”. The chapter also describes the glass vessel filled with water and its spinning-particle color hypothesis. Newton supplies the later dispersion account.
- CONFIRMED | [html-14-02] “He traced rays through a glass sphere of water at many heights, tabulated where each emerged, and saw them concentrate near 41–42° for the primary and near 51° for the secondary.” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes, Les Meteores VIII, 1637: “ayant pris la plume et calculé” and “41 à 42 degrés”, followed by “51 à 52 degrés”. The chapter also describes the glass vessel filled with water and its spinning-particle color hypothesis. Newton supplies the later dispersion account.
- CONFIRMED | [html-14-03] “He had found the caustic by census.” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes, Les Meteores VIII, 1637: “ayant pris la plume et calculé” and “41 à 42 degrés”, followed by “51 à 52 degrés”. The chapter also describes the glass vessel filled with water and its spinning-particle color hypothesis. Newton supplies the later dispersion account.
- CONFIRMED | [html-14-06] Newton supplied the account in terms of different refrangibilities. | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Proposition IX distinguishes the wavelengths' limiting angles; it is not a claim that Descartes made no correct colour observations.
- CONFIRMED | [html-16-01] “Water bends violet light a hair more than red.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: rays with different refrangibility “emerge most copiously in different Angles”. This supports distinct caustic angles for wavelengths and violet being more refrangible. It does not imply that a natural rainbow color contains only one wavelength.
- CONFIRMED | [html-16-02] “So there is no single rainbow angle” [...] “there is one per colour, and the bow is the thin fan between them.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: rays with different refrangibility “emerge most copiously in different Angles”. This supports distinct caustic angles for wavelengths and violet being more refrangible. It does not imply that a natural rainbow color contains only one wavelength.
- CONFIRMED | [html-16-03] “Newton's insight, made literal.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: rays with different refrangibility “emerge most copiously in different Angles”. This supports distinct caustic angles for wavelengths and violet being more refrangible. It does not imply that a natural rainbow color contains only one wavelength.
- CONFIRMED | [html-17-01] “Run the minimum-deflection formula for each colour's own refractive index and the bow spreads into a band:” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: rays with different refrangibility “emerge most copiously in different Angles”. This supports distinct caustic angles for wavelengths and violet being more refrangible. It does not imply that a natural rainbow color contains only one wavelength.
- CONFIRMED | [html-18-02] “The secondary, with its second bounce, reverses the order: red turns inside, violet out.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “their Colours shall lie in a contrary Order to one another, the red of both Bows bordering upon the Space GF”. This directly supports the reversed secondary color order.
- CONFIRMED | [html-18-03] “And between the two bows lies a strip of sky that no single- or double-bounced ray can reach at all.” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes gives the primary upper and secondary lower angular limits. For the page's actual indices, the reproduced red boundaries are 42.369841204488694 and 50.36511801705433 degrees. The gap excludes one- and two-internal-reflection rays in the stated model.
- CONFIRMED | [html-18-05] “The gap between” [...] “roughly 42° to 50°” [...] “receives neither.” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes gives the primary upper and secondary lower angular limits. For the page's actual indices, the reproduced red boundaries are 42.369841204488694 and 50.36511801705433 degrees. The gap excludes one- and two-internal-reflection rays in the stated model.
- CONFIRMED | [html-19-01] “The sky as the geometry assembles it, centred on the antisolar point (the shadow of your head).” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “O [...] is the Spectator's Eye” and OP is parallel to the solar rays; rotating the limiting directions about OP gives the bows. This supports the observer-centered cone and antisolar direction.
- CONFIRMED | [html-19-03] “Lower the sun and more of the bow clears the horizon; this is why rainbows are tallest near sunrise and sunset and vanish when the sun climbs above 42°.” | https://www.weather.gov/fgz/Rainbow | Retained checker result, with dependent updates where stated: NWS: “The lower the sun is to the horizon, the more of the circle we see.” The 42-degree limit is for the primary bow above the observer's horizon, as drawn in this instrument; drops below an elevated observer are outside this diagram's scope.
- CONFIRMED | [html-21-02] “Rays alone get the 42°, the second bow, the dark band, the colours” [...] “a remarkable amount.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “two Refractions of the Sun's Light, and one Reflexion between them”; the exterior bow has two internal reflections. Read all of Proposition IX for the color order and limiting angles.
- CONFIRMED | [html-21-03] “But they also predict an impossibility, and they cannot explain the faint pink-and-green fringes that sometimes ripple just inside the primary.” | https://dlmf.nist.gov/9.SB1 | Retained checker result, with dependent updates where stated: DLMF: supernumeraries are “produced by the interference of different sun-rays”. It dates Airy to 1838 but says Young 1800, so this excerpt alone does not establish Young 1804. Laven's paper cites Young's publication as Phil. Trans. 94, 1-16 (1804).
- CONFIRMED | [html-22-01] “Ray optics says the brightness at the caustic is infinite” [...] “a perfectly sharp line of light.” | https://archive.org/download/transactionsofca06camb/transactionsofca06camb_djvu.txt | Retained checker result, with dependent updates where stated: Airy, original article pp. 397-403, fetched as volume OCR: “the intensity would be infinite when m=0” under the imperfect theory, while wave theory gives intensity “nowhere infinite”. Laven 2017 also distinguishes the geometric and wave intensity curves explicitly.
- CONFIRMED | [html-22-02] “Real bows have a sharp-ish but finite edge, and just inside the strongest ones, in fine mist or spray, you can see supernumerary arcs: a few extra pale bands, too closely spaced and too pastel to be a second spectrum.” | https://www.philiplaven.com/p2b.html | Retained checker result, with dependent updates where stated: Laven: “If the drop sizes are uniform” supernumeraries can be clearly seen at radii 50 and 100 micrometres; at 500 micrometres they are difficult to see because of solar-disk smoothing. This supports easier visibility with small uniform drops, not exclusivity to fine mist.
- CONFIRMED | [html-22-03] “Pure rays have no way to make them.” | https://dlmf.nist.gov/9.SB1 | Retained checker result, with dependent updates where stated: DLMF: supernumeraries are “produced by the interference of different sun-rays”. It dates Airy to 1838 but says Young 1800, so this excerpt alone does not establish Young 1804. Laven's paper cites Young's publication as Phil. Trans. 94, 1-16 (1804).
- CONFIRMED | [html-23-03] “At the rainbow angle the two merge; just outside, there are none.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam describes coalescing rays at the rainbow caustic, a local statement. Laven 2017, p. G110, explains that the upper branch disappears for scattering angles beyond 165.6 degrees. The executed geometric counterexample has one primary ray at 10 degrees from antisolar, two at 41 degrees, and none at 43 degrees.
- CONFIRMED | [html-23-04] “Those two rays travel different distances through the drop, so they arrive out of step” [...] “and waves that arrive out of step interfere, reinforcing at some angles and cancelling at others.” | https://www.philiplaven.com/Publications/AO-56-19-G104a.pdf | Retained checker result, with dependent updates where stated: Laven explains optical-path interference and shows that the uncorrected Young model puts maxima at the wrong angles. The page's code computes path differences but omits the focal-line phase correction, so its stated limitation is appropriate.
- CONFIRMED | [html-24-03] Visible supernumeraries favour small, similarly sized drops, whether in rain, mist or spray. | https://www.atoptics.org.uk/rainbows/supers.htm | Retained checker result, with dependent updates where stated: The photograph and accompanying 0.7 mm simulation support this qualified wording; the exclusive condition was corrected separately.
- CONFIRMED | [html-26-01] “The two-ray model gives the mechanism and the scaling, but not the exact place of each fringe or the true shape of the edge.” | https://www.philiplaven.com/Publications/AO-56-19-G104a.pdf | Retained checker result, with dependent updates where stated: Laven explains optical-path interference and shows that the uncorrected Young model puts maxima at the wrong angles. The page's code computes path differences but omits the focal-line phase correction, so its stated limitation is appropriate.
- CONFIRMED | [airy-name] “George Biddell Airy's 1838 "rainbow integral,"” | https://archive.org/download/transactionsofca06camb/transactionsofca06camb_djvu.txt | Retained checker result, with dependent updates where stated: Airy's original heading names the author; the scan OCR corrupts the name as “Grorce Bippeit Airy”. The paper records “[Read May 2, 1836, and March 26, 1838.]”. Laven's bibliography independently identifies G. B. Airy and the 1838 article, and Young's 1804 publication.
- CONFIRMED | [rainbow-integral-term] “George Biddell Airy's 1838 "rainbow integral,"” | https://www.philiplaven.com/Publications/AO-56-19-G104a.pdf | Retained checker result, with dependent updates where stated: Laven uses the exact words “Airy’s rainbow integral”. The page presents a conventional term, not a purported verbatim quotation from Airy's own title.
- CONFIRMED | [html-27-01] “A rainbow is a place, not a thing.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “O [...] is the Spectator's Eye” and OP is parallel to the solar rays; rotating the limiting directions about OP gives the bows. This supports the observer-centered cone and antisolar direction.
- CONFIRMED | [html-27-02] “It is the set of directions in which a sky full of drops happens to crowd its light” [...] “and it moves with you because the geometry is anchored to your own eye.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “O [...] is the Spectator's Eye” and OP is parallel to the solar rays; rotating the limiting directions about OP gives the bows. This supports the observer-centered cone and antisolar direction.
- CONFIRMED | [convention] “It is a coincidence of our base-ten, 360°-to-a-circle convention and means nothing about the sky” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton, Book I, Part II, Proposition IX: “42 Degrees and 2 Minutes” for the least refrangible rays, “40 Degrees and 17 Minutes” for the most refrangible. These are wavelength-dependent limiting angles. The independent calculation with the stipulated yellow index gives 42.07810738024352 degrees.
- CONFIRMED | [third-fourth-2011] “first convincingly photographed only in 2011” | https://opg.optica.org/ao/abstract.cfm?uri=ao-50-28-F129 | Retained checker result, with dependent updates where stated: Optica abstract, Applied Optics 50 F129, 2011: “the first photographic recording of a quaternary rainbow in nature”, on the Sun side at about 45 degrees. Grossmann et al., DOI 10.1364/AO.50.00F134, separately support the third order in 2011. The natural-sunlight context matters.
- CONFIRMED | [higher-bows] “shows the higher-order bows sit around the sun” | https://opg.optica.org/ao/abstract.cfm?uri=ao-50-28-F129 | Retained checker result, with dependent updates where stated: Optica abstract, Applied Optics 50 F129, 2011: “the first photographic recording of a quaternary rainbow in nature”, on the Sun side at about 45 degrees. Grossmann et al., DOI 10.1364/AO.50.00F134, separately support the third order in 2011. The natural-sunlight context matters.
- CONFIRMED | [html-32-02] “There is no single n” [...] “it depends on wavelength, which is the whole reason a rainbow has colour, and drifts slightly with temperature and purity.” | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Retained checker result, with dependent updates where stated: NBS abstract describes a table “with temperature and wave length as arguments”; its experimental section discusses impurity effects. This supports wavelength, temperature, and purity dependence.
- CONFIRMED | [water-purity] “drifts slightly with temperature and purity” | https://nvlpubs.nist.gov/nistpubs/jres/20/jresv20n4p419_A1b.pdf | Retained checker result, with dependent updates where stated: NBS: “impurities in freshly distilled water influence refractivity much less than they do electrical conductivity or even density.” This supports the modest purity qualification.
- CONFIRMED | [medieval-history] “Theodoric of Freiberg and Kamāl al-Dīn al-Fārisī, independently, c. 1304” | https://mathshistory.st-andrews.ac.uk/DSB/Al-Farisi.pdf | Retained checker result, with dependent updates where stated: Rashed gives Dietrich 1304-1311 and quotes Wiedemann's 1302-1311 date for al-Farisi, then discusses an earlier dating proposed by Nazif. Both independently reached the transparent-sphere model. “c. 1304” is an approximate early-fourteenth-century label, not an established identical year for both. The source-record excerpt silently skips the intervening chronological dispute.
- CONFIRMED | [descartes-citation] “Descartes counted this by hand · 1637” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes, Les Meteores VIII, 1637: “ayant pris la plume et calculé” and “41 à 42 degrés”, followed by “51 à 52 degrés”. The chapter also describes the glass vessel filled with water and its spinning-particle color hypothesis. Newton supplies the later dispersion account.
- CONFIRMED | [young-citation] “Young, 1804” | https://www.philiplaven.com/Publications/AO-56-19-G104a.pdf | Retained checker result, with dependent updates where stated: Laven explicitly says “his 1803 Bakerian Lecture”; its reference 1 gives “Phil. Trans. R. Soc. 94, 1-16 (1804).” Thus 1804 is the publication year, not when he first put the idea forward.
- CONFIRMED | [airy-citation] “Young, 1804 · Airy, 1838.” | https://archive.org/download/transactionsofca06camb/transactionsofca06camb_djvu.txt | Retained checker result, with dependent updates where stated: Airy's original heading names the author; the scan OCR corrupts the name as “Grorce Bippeit Airy”. The paper records “[Read May 2, 1836, and March 26, 1838.]”. Laven's bibliography independently identifies G. B. Airy and the 1838 article, and Young's 1804 publication.
- CONFIRMED | [fm-dek-2] “And both facts are one piece of geometry.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [fm-dek-3] “Trace a sunbeam through a sphere of water (bend in, bounce once off the back, bend out) and the angle it turns through has a minimum: rays that enter evenly across the drop leave bunched against that minimum, piling into an edge of pure brightness called a caustic.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [fm-dek-8] “And the honest seam: ray optics nails the 42°, the second bow, the colours, and the dark band.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton, Book I, Part II, Proposition IX: “42 Degrees and 2 Minutes” for the least refrangible rays, “40 Degrees and 17 Minutes” for the most refrangible. These are wavelength-dependent limiting angles. The independent calculation with the stipulated yellow index gives 42.07810738024352 degrees.
- CONFIRMED | [fm-dek-9] “But it predicts infinite brightness at the edge and cannot make the faint supernumerary fringes; those need waves (Young 1804, Airy 1838).” | https://dlmf.nist.gov/9.SB1 | Retained checker result, with dependent updates where stated: DLMF: supernumeraries are “produced by the interference of different sun-rays”. It dates Airy to 1838 but says Young 1800, so this excerpt alone does not establish Young 1804. Laven's paper cites Young's publication as Phil. Trans. 94, 1-16 (1804).
- CONFIRMED | [md-body-1-0] “A raindrop is a tiny sphere of water, and sunlight passing through it bends going in, reflects once off the far inside wall, and bends again coming out.” | https://www.gutenberg.org/ebooks/33504.txt.utf-8 | Retained checker result, with dependent updates where stated: Newton: “two Refractions of the Sun's Light, and one Reflexion between them”; the exterior bow has two internal reflections. Read all of Proposition IX for the color order and limiting angles.
- CONFIRMED | [md-body-1-1] “The *total* angle the ray turns through depends on where it struck the drop” [...] “and that deflection has a **minimum**.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [md-body-1-2] “Rays that enter spread evenly across the drop's face leave bunched against that minimum: uniform in, crowded out.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [md-body-1-3] “Crowded rays mean concentrated light, an edge of brightness called a **caustic**” [...] “and that caustic, repeated across a sky full of drops, *is* the rainbow.” | https://arxiv.org/pdf/1405.3545 | Retained checker result, with dependent updates where stated: Adam, pp. 3-4: “a concentration of light rays corresponding to an extremum of the deviation or scattering angle”. Read equations (1)-(2), not just the extracted formula. With p=k+1, p squared minus 1 equals k(k+2). The caustic explanation is correct for spherical drops in ray optics.
- CONFIRMED | [md-body-2-0] “René Descartes computed this in 1637 by tracing rays through a water-filled sphere and tabulating where they emerged” [...] “finding the crowd near 42° by arithmetic alone” [...] “and got the geometry exactly right while attributing colour to spinning particles.” | https://www.unicaen.fr/puc/sources/prodescartes/consult/descartes/Oeuvres/Oeuvres_Descartes/discours_de_la_methode.xml/dm_meteores_8.html | Retained checker result, with dependent updates where stated: Descartes, Les Meteores VIII, 1637: “ayant pris la plume et calculé” and “41 à 42 degrés”, followed by “51 à 52 degrés”. The chapter also describes the glass vessel filled with water and its spinning-particle color hypothesis. Newton supplies the later dispersion account.
- CONFIRMED | [fringe-scaling-world] “One thing falls straight out of that picture and is worth keeping because it is visible: the fringe spacing scales as (λ⁄a)2/3” [...] “wider for shorter drops.” | https://michaelberryphysics.wordpress.com/wp-content/uploads/2013/07/berry260.pdf | Retained checker result, with dependent updates where stated: Opened the scanned Berry PDF and rendered only PDF page 6 (printed p. 602). The passage after equation (9) reads “the spacing of the fringes shrinks as” lambda to the 2/3 power. The dimensionless drop-size dependence is also recreated with the page's path calculation.
- CONFIRMED | [magenta-world] “magenta is non-spectral, invented by the eye from the two ends at once” | https://aty.sdsu.edu/explain/optics/color/intro.html | Retained checker result, with dependent updates where stated: Andrew Young: “The color that was called ‘magenta’ above is an example of a class of non-spectral colors called purple.” His page establishes non-spectral magenta, not identity between a natural rainbow and a pure monochromatic spectrum.
- CONFIRMED | [third-photo-2011] “first convincingly photographed only in 2011” | https://pubmed.ncbi.nlm.nih.gov/22016237/ | Retained checker result, with dependent updates where stated: The PubMed URL returned a browser challenge. Opened the publisher URL too; its later response was also a challenge. Europe PMC, queried by the exact PMID 22016237, returned the complete abstract, matching author names, 2011, and DOI 10.1364/AO.50.00F134. It says “The first likely photographic observation of the tertiary rainbow caused by sunlight in the open air is reported and analyzed.”
- CONFIRMED | [html-01-03] Drag the ray and watch the crowd form. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: The drop instrument's slider (index.html lines 134-135) steers the ray; Rain of rays draws 90 rays.
- CONFIRMED | [html-03-04] Steer the ray up and down the drop and watch the deflection change -- it is not the same for every ray, and that is the whole story. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: The readout recomputes deflection from the slider (index.html lines 352-361).
- CONFIRMED | [html-04-01] One sunbeam (gold) enters the drop, reflects once off the back wall, and leaves. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: traceDraw(b, n, 1): refract, one reflection, refract (index.html lines 294-318); the steered ray is drawn #ffd86b (gold).
- CONFIRMED | [html-04-03] Press Rain of rays to fire many beams spread evenly across the drop: they fan out -- except near one angle, where they crowd together into a bright edge. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: Rain of rays: b = (i + 0.5)/90 for 90 rays, uniform across the face (index.html lines 348-351).
- CONFIRMED | [html-04-05] Every ray is traced with the same refract/reflect/refract code the offline verifier runs. | git show 37137afa5c:research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: traceDraw() in index.html (lines 294-318) is a compacted copy of trace() in verify.mjs (same refract and reflect formulas, eps 1e-7 vs 1e-9): the same algorithm, not literally the same file. Drawing only; the readouts use the closed form.
- CONFIRMED | [html-05-01] Slide slowly from the top of the drop (b → 1, grazing the edge) down toward the centre (b → 0). | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: Slider min 0.001, max 0.999 (index.html line 135); labels 'b → 1' grazing, 'b → 0' centre.
- CONFIRMED | [html-09-02] The dot is the ray you're steering above. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: The white dot reads bSlider from Instrument I (index.html lines 410-412).
- CONFIRMED | [html-09-04] Switch to two bounces for the secondary bow. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: The 'Secondary · 2 bounces' button sets kk = 2 (index.html line 416).
- CONFIRMED | [html-12-03] They agree to four decimals. | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Check 3, re-run: 'numeric i*=59.4105°, closed-form i*=59.4105°'.
- CONFIRMED | [html-19-02] The sky's red and violet edges use the angles computed by the verifier; intermediate colours are approximate. | public/strata/the-edge-of-the-bow/index.html; research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: The page now uses NW.violet and NW.red for those endpoints and discloses the five intermediate inputs.
- CONFIRMED | [html-21-01] Everything above treated light as rays. | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: Sections I-III use only refraction and reflection of rays.
- CONFIRMED | [op-difference-check] the verifier confirms it, computing the two rays' optical-path difference straight from the traced polylines and watching it grow from zero at the rainbow angle. | git show 37137afa5c:research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: Check 19, re-run: 'ΔOP: 1.55e-5 (0.05° in) → 9.97e-4 (0.8° in), drop radii', from trace() op sums.
- CONFIRMED | [html-24-02] Shrink the drop eightfold and the verifier finds the first fringe move out about fourfold (82/3 = 4, confirmed to 3.98). | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Check 20, re-run: '1st supernumerary offset 33.84′ → 134.64′ as a→a/8; ratio 3.98 vs 8^(2/3)=4.00'.
- CONFIRMED | [html-26-03] That integral is cited here, not run. | git show 37137afa5c:research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: verify.mjs computes no Airy function; 'Airy' appears only in comments and the artifact note 'cited not computed'.
- CONFIRMED | [html-26-04] The honest claim of this layer is the part it checks end to end: the angles, the caustic, the colour order, the dark band, and the existence-and-scaling of the interference. | git show 37137afa5c:research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: Checks 3-22 cover angles, caustic, colour order, dark band, two-ray interference and its scaling.
- CONFIRMED | [rounds-42] the primary radius rounds to the integer 42 | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: viewAngle 42.078.
- CONFIRMED | [rounding-checked] but the verifier checks it rounds to 42 anyway | git show 37137afa5c:research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: Check 23: 'primary radius 42.08° → rounds to 42'.
- CONFIRMED | [html-31-02] The artifact numbers match the page's transcript. | research/the-edge-of-the-bow/artifact.json; public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: Regenerated the transcript from the corrected artifact in this pass.
- CONFIRMED | [html-34-01] This layer's check works the answer out again from scratch, independently of this page. | git show 37137afa5c:research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: verify.mjs imports only writeFileSync and reads no file (line 28); it recomputes everything.
- CONFIRMED | [html-34-03] It can settle the finding; it cannot notice a figure that drifted on this page. | git show 37137afa5c:research/verifier-independence/verdicts.json | Retained checker result, with dependent updates where stated: The verifier never reads the page, so it cannot see a drifted figure; 'verdict': 'blind'.
- CONFIRMED | [html-34-04] It is published, and it runs on its own: one file, and Node. | https://artwaste.land/checks/research/the-edge-of-the-bow/verify.mjs | Retained checker result, with dependent updates where stated: Fetched (200, 26,588 bytes, identical to the pin) and run with node in an empty directory: 23/23, exit 0.
- CONFIRMED | [checks23] 23/23 checks passing. | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Re-run at the pin and from the published copy: 'The Edge of the Bow -- 23/23 checks passing.'
- CONFIRMED | [check-primary] Primary radius 42.08° (red 42.37° outer, violet 40.65° inner). | research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Recomputed with the corrected violet input. The old 40.51° transcript matched its old 1.344 input; that input was the WRONG item X14.
- CONFIRMED | [check-secondary] secondary rainbow angle : 50.89° (colours reversed) | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Re-run: 50.89°.
- CONFIRMED | [check-band] Alexander's dark band : 42.37°..50.37° | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Re-run: 42.37°..50.37°.
- CONFIRMED | [check-scaling] supernumerary scaling : (λ/a)^(2/3) confirmed (8× drop → 3.98×) | git show 37137afa5c:research/the-edge-of-the-bow/artifact.json | Retained checker result, with dependent updates where stated: Re-run prints '(8× drop → 3.98×, target 4.00×)'; the page drops the target clause, not a falsehood.
- CONFIRMED | [drop-live] incidence i = 59.3° deflection = 137.9° leaves at 42.1° from antisolar rainbow angle (1 bounce) = 42.1° | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: At b = 0.86: i = 59.32°, deflection 137.922°, exit 42.078°, matching the readout.
- CONFIRMED | [sky-live] primary radius ~42° secondary ~51° dark band 42.4°–50.4° sun elevation 8° | git show 37137afa5c:public/strata/the-edge-of-the-bow/index.html | Retained checker result, with dependent updates where stated: bandLo/bandHi from NW.red: 42.37 → 42.4, 50.37 → 50.4; slider default 8°.
- CONFIRMED | [rel-sky] Its companion in 'a real sky phenomenon, recomputed from first principles with no API in the loop.' That page rebuilds the positions of sun and stars over any place from Meeus + Standish in the browser; this one rebuilds the rainbow over your shoulder from Snell's law in the browser. Both are entirely client-side and checked against a from-scratch offline verifier. | git show 37137afa5c:src/content/strata/the-sky-above-you.md | Retained checker result, with dependent updates where stated: Sky dek: 'recomputed in the browser from the canonical astronomical algorithms (Meeus / Standish)'; 'No API in the loop'.

## Findings

- [fixed] X1: Qualified the primary radius in the page, descriptions, body, relationship note and OG authoring source. Corrected research/the-edge-of-the-bow/og.svg:17 directly as SVG text; share card rebuilt at landing from the corrected dek. The PNG and its thumbnail are left to landing. No correction line for this MINOR precision fix.
- [fixed] X2: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] X3: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] X4: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] X5: Removed the shaded-band promise and its unused setup. The dashed extremum line and steered dot remain.
- [fixed] X6: Described numerical minimization of the formula and the separate tracer cross-check in the page and markdown; no correction line for this MINOR check-description fix.
- [fixed] X7: Attributed the fourteenfold count to the offline histogram instead of a nonexistent button tally.
- [fixed] X8: Used the red boundary, about 50.4°, and the red primary boundary, about 42.4°, for the model limits; no MINOR correction line.
- [fixed] X9: Dated the description, retaining the attribution to Alexander; no naming date invented and no MINOR correction line.
- [fixed] X10: Restricted the explanation to directions just inside the primary edge. Check 17 now tests both the two-ray near-edge case and the one-ray 10° counterexample; fixed the comments and regenerated its artifact and reader copy.
- [fixed] X11: Changed the prose to published this explanation in 1804; retained the valid bibliographic year and added no MINOR correction line.
- [fixed] X12: Called it the Airy wave approximation and withdrew the exact-position implication, while retaining cited, not computed; no MINOR correction line.
- [fixed] X13: Reported the measured error and tolerance in the page, qualified the summary, and corrected the verifier comment about its tolerance; no MINOR correction line.
- [fixed] X14: Corrected the violet input in the page and verifier; sky endpoints now use the same constants. Regenerated the artifact, transcript and reader verifier, refreshed the live table and width prose to 1.7°, and corrected both source-record locators from Table 3 to Table 5. K6 and K15 are dependent MINOR updates, not independent correction lines.
- [fixed] X15: Restricted the check description to the three-colour results and disclosed all five approximate sky inputs and the illustrative fringe overlay; no MINOR correction line.
- [dated] X16: Refreshed the placard to 14 October 2026 from this record. No Updated line, following the triage's explicit placard-date exception.
- [fixed] X17: Corrected this layer's relationship source and Nearby copy to name refraction and internal reflection. The copy in Light Leans Toward the Cold's relation note (src/content/strata/light-leans-toward-the-cold.md) was corrected on 2026-10-18 with its own dated line; board note 9d5efc is closed.
- [fixed] X18: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] X19: Corrected the violet input in the page and verifier; sky endpoints now use the same constants. Regenerated the artifact, transcript and reader verifier, refreshed the live table and width prose to 1.7°, and corrected both source-record locators from Table 3 to Table 5. K6 and K15 are dependent MINOR updates, not independent correction lines.
- [fixed] X20: Qualified the primary radius in the page, descriptions, body, relationship note and OG authoring source. Corrected research/the-edge-of-the-bow/og.svg:17 directly as SVG text; share card rebuilt at landing from the corrected dek. The PNG and its thumbnail are left to landing. No correction line for this MINOR precision fix.
- [fixed] X21: Changed the dek to state that the primary radius is 180° minus the minimum deflection.
- [fixed] X22: Restricted the no-ray claim to one or two internal reflections in the geometric model, including the verifier description and summary. The share image repeats the old dek; share card rebuilt at landing from the corrected dek.
- [fixed] X23: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] X24: Removed the exclusive fine-mist condition in the dek, body and verifier. Described small, similarly sized drops in rain, mist or spray, and the checked widening of fringes for smaller drops.
- [fixed] X25: Qualified the primary radius in the page, descriptions, body, relationship note and OG authoring source. Corrected research/the-edge-of-the-bow/og.svg:17 directly as SVG text; share card rebuilt at landing from the corrected dek. The PNG and its thumbnail are left to landing. No correction line for this MINOR precision fix.
- [fixed] X26: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] X27: Qualified the primary radius in the page, descriptions, body, relationship note and OG authoring source. Corrected research/the-edge-of-the-bow/og.svg:17 directly as SVG text; share card rebuilt at landing from the corrected dek. The PNG and its thumbnail are left to landing. No correction line for this MINOR precision fix.
- [fixed] X28: Dated the description, retaining the attribution to Alexander; no naming date invented and no MINOR correction line.
- [fixed] X29: Restricted the explanation to directions just inside the primary edge. Check 17 now tests both the two-ray near-edge case and the one-ray 10° counterexample; fixed the comments and regenerated its artifact and reader copy.
- [fixed] X30: Removed the exclusive fine-mist condition in the dek, body and verifier. Described small, similarly sized drops in rain, mist or spray, and the checked widening of fringes for smaller drops.
- [fixed] X31: Replaced the common-extremum analogy with a shared geometric constraint; no MINOR correction line.
- [fixed] X32: Rewrote the relationship note to distinguish monochromatic components from observed mixtures. Kept magenta non-spectral and restricted its absence to a pure spectrum.
- [fixed] X33: Rewrote the relationship note to distinguish monochromatic components from observed mixtures. Kept magenta non-spectral and restricted its absence to a pure spectrum.
- [fixed] X34: Rewrote the relationship note to distinguish monochromatic components from observed mixtures. Kept magenta non-spectral and restricted its absence to a pure spectrum.
- [fixed] X35: Replaced the retina description with the actual subject in this layer's relationship source and regenerated its Nearby copy.
- [fixed] X36: Removed the exclusive fine-mist condition in the dek, body and verifier. Described small, similarly sized drops in rain, mist or spray, and the checked widening of fringes for smaller drops.
- [fixed] X37: Removed the exclusive fine-mist condition in the dek, body and verifier. Described small, similarly sized drops in rain, mist or spray, and the checked widening of fringes for smaller drops.
- [fixed] X38: Qualified 1979 as the novel's year.
- [fixed] X39: Qualified 1979 as the novel's year.
- [dated] X40: Set dateModified to 2026-10-14, preserving datePublished. No Updated line under the explicit JSON-LD exception.
- [fixed] K1: Relabeled the plotted quantity as viewing angle and described the primary peak and secondary trough in captions, readout, prose and metadata. Preserved the genuine deflection minimum.
- [fixed] K2: Removed the shaded-band promise and its unused setup. The dashed extremum line and steered dot remain.
- [fixed] K3: Described numerical minimization of the formula and the separate tracer cross-check in the page and markdown; no correction line for this MINOR check-description fix.
- [fixed] K4: Attributed the fourteenfold count to the offline histogram instead of a nonexistent button tally.
- [fixed] K5: Described correct geometric crowding and an incorrect physical theory of colour in the page, dek and body; no MINOR correction line.
- [fixed] K6: Corrected the violet input in the page and verifier; sky endpoints now use the same constants. Regenerated the artifact, transcript and reader verifier, refreshed the live table and width prose to 1.7°, and corrected both source-record locators from Table 3 to Table 5. K6 and K15 are dependent MINOR updates, not independent correction lines.
- [fixed] K7: Dated the description, retaining the attribution to Alexander; no naming date invented and no MINOR correction line.
- [fixed] K8: Restricted the explanation to directions just inside the primary edge. Check 17 now tests both the two-ray near-edge case and the one-ray 10° counterexample; fixed the comments and regenerated its artifact and reader copy.
- [fixed] K9: Described numerical minimization of the formula and the separate tracer cross-check in the page and markdown; no correction line for this MINOR check-description fix.
- [fixed] K10: Reported the measured error and tolerance in the page, qualified the summary, and corrected the verifier comment about its tolerance; no MINOR correction line.
- [fixed] K11: Corrected the violet input in the page and verifier; sky endpoints now use the same constants. Regenerated the artifact, transcript and reader verifier, refreshed the live table and width prose to 1.7°, and corrected both source-record locators from Table 3 to Table 5. K6 and K15 are dependent MINOR updates, not independent correction lines.
- [fixed] K12: Corrected this layer's relationship source and Nearby copy to name refraction and internal reflection. The copy in Light Leans Toward the Cold's relation note (src/content/strata/light-leans-toward-the-cold.md) was corrected on 2026-10-18 with its own dated line; board note 9d5efc is closed.
- [fixed] K13: Described correct geometric crowding and an incorrect physical theory of colour in the page, dek and body; no MINOR correction line.
- [fixed] K14: Dated the description, retaining the attribution to Alexander; no naming date invented and no MINOR correction line.
- [fixed] K15: Corrected the violet input in the page and verifier; sky endpoints now use the same constants. Regenerated the artifact, transcript and reader verifier, refreshed the live table and width prose to 1.7°, and corrected both source-record locators from Table 3 to Table 5. K6 and K15 are dependent MINOR updates, not independent correction lines.
- [fixed] K16: Dated the description, retaining the attribution to Alexander; no naming date invented and no MINOR correction line.
- [fixed] K17: Described numerical minimization of the formula and the separate tracer cross-check in the page and markdown; no correction line for this MINOR check-description fix.
- [fixed] K18: Rewrote the relationship note to distinguish monochromatic components from observed mixtures. Kept magenta non-spectral and restricted its absence to a pure spectrum.
- [fixed] K19: Rewrote the relationship note to distinguish monochromatic components from observed mixtures. Kept magenta non-spectral and restricted its absence to a pure spectrum.
- [fixed] K20: Rewrote the relationship note to distinguish monochromatic components from observed mixtures. Kept magenta non-spectral and restricted its absence to a pure spectrum.
- [fixed] K21: Replaced the retina description with the actual subject in this layer's relationship source and regenerated its Nearby copy.
- [fixed] K22: Removed the exclusive fine-mist condition in the dek, body and verifier. Described small, similarly sized drops in rain, mist or spray, and the checked widening of fringes for smaller drops.
- [fixed] K23: Removed the exclusive fine-mist condition in the dek, body and verifier. Described small, similarly sized drops in rain, mist or spray, and the checked widening of fringes for smaller drops.
- [fixed] K24: Qualified 1979 as the novel's year.