---
slug: egregium
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
checked: 100
confirmed: 81
wrong: 18
unverifiable: 0
observed: 0
stale: 1
---

This pass applies the 19 directing-instance decisions in research/assay-line/queue-1014c/egregium/triage.json to egregium at f5e5f84f34. The fixer re-read every item's evidence, reproduced the geometric and numerical findings, and made the supported repairs. The 81 CONFIRMED lines preserve the source rereads recorded by Codex in N2-codex.md, with attribution; they are not presented as 81 new source visits by the fixer. The seven MINOR lines count toward wrong, alongside eleven WRONG lines. Six dated correction lines cover the six distinct factual errors and are mirrored in the markdown body; MINOR repairs have no page correction lines. The one STALE item uses the explicit JSON-LD date exception. N2's unrelated OBSERVED and three UNVERIFIABLE entries are outside this directed work list.

Repair review by Codex, 2026-10-14: all six dated correction lines are supported. X12, X13 and X14 remain open for unannotated copies in the There, Here explorer: public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1, notes[366]. Egregium's own copies are corrected. See research/assay-line/queue-1014c/egregium/repair-report.md.

## Claims

- WRONG | X1: the excess melts to zero and the angle sum falls back to 180° | git show f5e5f84f34:public/strata/egregium/index.html | The pinned compute() derives the angles before reading Rslider. Executing the preset at 1,000 and 12,000 km keeps the sum at 270 degrees and excess at 1.5708 radians; rounded displayed areas scale by about 144. The label and caption now describe fixed angular vertices and increasing area, not an approach to 180 degrees.
- WRONG | X2: a small circle is stamped at every crossing of the grid | git show f5e5f84f34:public/strata/egregium/index.html | The longitude grid has eleven lines and five latitude rows, but the glyph loop uses six longitudes: 55 crossings and 30 ellipses. The caption now says selected crossings.
- WRONG | X3: Each ellipse is the image of a true circle on the globe. | git show f5e5f84f34:public/strata/egregium/index.html | The pinned grid uses x=longitude/pi while its Mercator y is unscaled; equirectangular y is latitude/(pi/2). Their screen derivatives disagree with the glyph axes. Both coordinates now use unit-sphere units, and finite differences of the actual screen map agree with all 30 glyphs in each mode. See research/egregium/verify-page.mjs.
- MINOR | X4: Greenland sits up near 70° North, where Mercator's area inflation is roughly sec²(latitude) ≈ 13× | https://pubs.usgs.gov/pp/1395/report.pdf | Re-read Snyder pp. 41 and 44 in the retained source text: x=R*longitude, y=R*log(tan(pi/4+latitude/2)), h=k=sec(latitude), area factor sec squared. sec squared at 70 degrees is 8.548632; the stated 60 to 84 degree band mean is 12.698731. The prose now identifies 12.7 as the uniform-band mean, alongside Africa's approximately 1.15 band mean.
- WRONG | X5: the areas they enclose are the true cited figures | git show f5e5f84f34:public/strata/egregium/index.html | The filled Africa path includes the closing edge, while the old sizing calculation omitted it: 0.5684 closed area versus 0.5434 used to size it, a 4.600662 percent excess. The sizing now sums every edge. Greenland's initially clipped outline is also kept within the canvas. Independent triangle-fan areas measure the actual filled paths.
- WRONG | X6: where the two are drawn to scale: Greenland is a fourteenth of Africa, and looks it. | git show f5e5f84f34:public/strata/egregium/index.html | The original filled true-size paths give Africa/Greenland=14.6657248142; the cited areas give 30370000/2166086=14.0206806193. After the shared polygon closure repair, the filled paths give 14.0206806193. The cited areas and rounded fourteen-to-one prose are retained.
- MINOR | X7: Every number on this page is recomputed in your browser and, independently, offline | git show f5e5f84f34:research/egregium/verify.mjs | Ran the original formula verifier: 35 checks pass, but it does not read the page or measure its drawings. The apparatus now describes instrument readouts and independent checks of curvature, triangle, projection and band-area formulas. Matching metadata and verifier header wording are narrowed as well.
- MINOR | X8: the two agree to 1×10⁻¹² across 4,000 random triangles. | git show f5e5f84f34:research/egregium/verify.mjs | Re-read the 4,000-attempt loop and its skips, then ran it: n=3990, worst 1.2e-12 radians, with acceptance worstRouteErr < 1e-9. The paragraph now gives accepted count, attempt count, observed discrepancy and pass threshold separately.
- MINOR | X9: The real coastlines would shift this by a little; | https://raw.githubusercontent.com/nvkelso/natural-earth-vector/master/geojson/ne_110m_admin_0_countries.geojson; research/assay-line/queue-1014c/egregium/N2-codex.md (Coastline control) | Re-read the retained Natural Earth selected features and the N2 integration program, then reran it. The cited-area-normalized apparent ratio is 1.05715028295, compared with the band's 0.789257 result; the rectangular-band positive control gives 12.6987306971. Removed the small-change assurance in the page and verifier comment. No coastline-derived replacement figure is presented as survey precision.
- WRONG | X10: only the areas they enclose and the inflation factor are quantitative. | git show f5e5f84f34:public/strata/egregium/index.html | The stylised-coastline qualification is valid, but the original quantitative-area assurance is contradicted by the missing closing edge. The shared polygon repair makes the filled path areas agree with cited areas and the selected band inflation. See X5 and research/egregium/verify-page.mjs.
- MINOR | X11: Every figure recomputed live and offline | git show f5e5f84f34:research/egregium/verify.mjs | The markdown dek overstates the scope of the independent 35-check formula verifier. It now names curvature, triangle, projection and band-area formulas, retaining the supported 35/35 result. The HTML description and JSON-LD description receive the same scope correction.
- WRONG | X12: 'a summary is a projection that throws away the shape' | git show f5e5f84f34:src/content/strata/the-shape-the-numbers-cant-see.md | Re-read the pinned linked markdown's dek. Its words are 'a handful of summary numbers is a projection, and a projection throws away the shape'. Restored those words in egregium's authored relates note and regenerated that note's HTML copy from the source with the existing renderer.
- WRONG | X13: Anscombe's quartet projects a dataset onto five numbers and loses the dinosaur | git show f5e5f84f34:src/content/strata/the-shape-the-numbers-cant-see.md | The pinned linked markdown separates Anscombe's quartet (1973) from the Datasaurus Dozen (2017), naming the dinosaur among the latter's shapes. The relates note now names the Datasaurus Dozen for this example.
- WRONG | X14: 'a summary is a projection that throws away the shape' | git show f5e5f84f34:src/content/strata/the-shape-the-numbers-cant-see.md | Duplicate report of X12's inaccurate quotation. The same source reread and restored quotation correct both occurrences within egregium; they share one dated correction line. The unannotated explorer copies remain open with X12.
- MINOR | X15: slide to 72° to see the standard Mercator illusion · slide to 0° to see Greenland's true size | git show f5e5f84f34:public/strata/egregium/index.html | The latitude-zero handler still averages Greenland over 0.001 to 12 degrees (factor 1.014795993) and Africa over -35 to 37 degrees (1.147553493), so it is not the exact true-size comparison. The paragraph and live hint now direct readers to Show true sizes, which sets both inflation factors to one.
- WRONG | X16: Africa is 14.0× larger, and drawn so. | git show f5e5f84f34:public/strata/egregium/index.html | The printed 14.0 is supported by the cited areas, but the original drawn ratio is 14.6657248142. The shared closed-polygon repair makes the measured drawn ratio 14.0206806193, so the retained 'and drawn so' statement is now supported.
- MINOR | X17: checks every figure offline | git show f5e5f84f34:research/egregium/verify.mjs | The markdown body now names the four kinds of formula checked offline instead of claiming every figure is checked. The original 35 passing assertions are retained; a separate page check measures drawing commands without changing that standalone verifier's scope.
- WRONG | X18: Tissot's indicatrix drawn across three map projections. | git show f5e5f84f34:public/strata/egregium/index.html | Duplicate of X3's projection/glyph mismatch. Screen-map finite differences now agree with the ellipse axes in Mercator, equirectangular and Lambert modes at all 30 displayed marks, at 900px/DPR 1 and 360px/DPR 2.
- STALE | X19: "dateModified":"2026-06-14" | git show f5e5f84f34:public/strata/egregium/index.html; git log f5e5f84f34 -- public/strata/egregium/index.html | The pinned JSON-LD says 2026-06-14, but the pinned page history includes 0fa06128f48 on 2026-08-21 and 2e60bcb5dd1 on 2026-10-01. Set dateModified to 2026-10-14 for this actual revision; datePublished stays 2026-06-14. The explicit metadata exception requires no separate Updated line because the factual correction lines date the revision.
- CONFIRMED | Latin for out of the flock, distinguished, remarkable. | https://www.oxfordlearnersdictionaries.com/us/definition/english/egregious | N2 source reread, 2026-10-14: name-latin: Oxford: "originally meaning remarkably good"; "standing out from the flock"; "The current sense probably arose as an ironical use."
- CONFIRMED | The same word, in English, soured into egregious: remarkable for being bad. | https://www.oxfordlearnersdictionaries.com/us/definition/english/egregious | N2 source reread, 2026-10-14: name-pejoration: Oxford: "originally meaning remarkably good"; "standing out from the flock"; "The current sense probably arose as an ironical use."
- CONFIRMED | In 1827 Carl Friedrich Gauss published a result he thought important enough to flag in its own name: the Theorema Egregium, the Remarkable Theorem. | https://www.gutenberg.org/files/36856/36856-pdf.pdf | N2 source reread, 2026-10-14: gauss-publication: The cited introduction says the full memoir was printed in 1828, but also identifies the published abstract as "1827. November 5." Read the abstract in the same volume: it reports the intrinsic-curvature result. Thus publishing the result in 1827 is supported, although the selected excerpt alone is inadequate.
- CONFIRMED | Gauss proved you don't need to step back. The curvature is written into the surface itself, into the distances measured along it, and a creature confined to the surface can read it off without any idea that an "outside" exists. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: gauss-intrinsic: Gauss section 11: "the general expression for the magnitude of any linear element is sufficient". Section 12 calls the resulting invariant theorem "remarkable". The 1827 date is supported by the dated abstract in the Gutenberg edition.
- CONFIRMED | An ant who never leaves a surface can still prove that surface is curved | https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: ant-proof: Todhunter section 97: "triangle ABC = (A + B + C − π)r2". For geodesic triangles the positive area gives positive excess; three right angles give 270 degrees and excess π/2. Dividing radian excess by area gives 1/R². The specified pole/equator route uses three great-circle arcs.
- CONFIRMED | and that proof is the reason no flat map of the Earth can tell the truth. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: flat-map-impossible: Gauss section 12: "the measure of curvature in each point remains unchanged" under development without changing line elements. Sphere K=1/R², plane K=0; constant map scale can be absorbed in the length unit.
- CONFIRMED | It is also, exactly, the reason every world map on every wall is wrong | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: world-maps: Snyder p. 4: "No map projection shows scale correctly throughout the map". The surrounding text distinguishes conformal, equal-area and other compromises.
- CONFIRMED | On a flat sheet of paper, the three angles of a triangle add to 180°. | https://mathcs.clarku.edu/~djoyce/elements/bookI/propI32.html | N2 source reread, 2026-10-14: flat-triangle: Euclid I.32: "the sum of the three interior angles of the triangle equals two right angles". Read the proposition and its proof.
- CONFIRMED | it is the flat plane's signature, equivalent to Euclid's parallel postulate. | https://www2.math.uu.se/~thomase/TermPaper.html | N2 source reread, 2026-10-14: parallel-postulate: Opened with curl -k after the certificate-chain failure. The university-hosted paper states that an arbitrary triangle having angle sum π "is known to be equivalent to the parallel postulate". Its Symbol-font p denotes π.
- CONFIRMED | On a sphere they add to more. | https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: spherical-triangle: Todhunter section 97: "triangle ABC = (A + B + C − π)r2". For geodesic triangles the positive area gives positive excess; three right angles give 270 degrees and excess π/2. Dividing radian excess by area gives 1/R². The specified pole/equator route uses three great-circle arcs.
- CONFIRMED | Stand at the North Pole, walk straight down to the equator, turn 90°, walk a quarter of the way round, turn 90° again, and walk straight back to the pole: you have drawn a triangle with three right angles, summing to 270°. | https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: right-angle-triangle: Todhunter section 97: "triangle ABC = (A + B + C − π)r2". For geodesic triangles the positive area gives positive excess; three right angles give 270 degrees and excess π/2. Dividing radian excess by area gives 1/R². The specified pole/equator route uses three great-circle arcs.
- CONFIRMED | The 90° of surplus is not an error. It is the curvature, made visible. | https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: surplus-curvature: Todhunter section 97: "triangle ABC = (A + B + C − π)r2". For geodesic triangles the positive area gives positive excess; three right angles give 270 degrees and excess π/2. Dividing radian excess by area gives 1/R². The specified pole/equator route uses three great-circle arcs.
- CONFIRMED | The number that matters is the last one: excess ÷ area. It does not budge. No matter the triangle's size or shape, that ratio is a constant, and that constant is the Gaussian curvature, K = 1/R². The ant carries no map of the sphere and never sees it from space. It measures a triangle, divides, and knows. | https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: ratio-invariant: Todhunter section 97: "triangle ABC = (A + B + C − π)r2". For geodesic triangles the positive area gives positive excess; three right angles give 270 degrees and excess π/2. Dividing radian excess by area gives 1/R². The specified pole/equator route uses three great-circle arcs.
- CONFIRMED | the plane is the sphere with K = 0. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: plane-limit: Read in the preceding limiting context: 1/R² tends to zero as R tends to infinity. This is not a claim that a finite-radius sphere has zero curvature. Gauss section 12 supplies the intrinsic-curvature invariant.
- CONFIRMED | This is the whole theorem in one reading. The angles and the area are intrinsic | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: theorem-in-one-reading: Gauss sections 8 and 11 give the reciprocal product of "the two extreme radii of curvature" and the intrinsic line-element expression. The triangle reading is the constant-curvature sphere example, not a proof of the theorem on every surface.
- CONFIRMED | measurable with a tape measure laid on the surface, no outside view required. Their combination delivers the curvature. Gauss's deeper result is that this curvature, defined from inside, equals the curvature you'd compute from outside (the product of the two ways the surface bends in space). | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: intrinsic-measurement: Gauss sections 8 and 11 give the reciprocal product of "the two extreme radii of curvature" and the intrinsic line-element expression. The triangle reading is the constant-curvature sphere example, not a proof of the theorem on every surface.
- CONFIRMED | Curvature is not something you see done to a surface from outside. It is something the surface knows about itself. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: intrinsic-pull-1: Gauss section 11: "the general expression for the magnitude of any linear element is sufficient". Section 12 calls the resulting invariant theorem "remarkable". The 1827 date is supported by the dated abstract in the Gutenberg edition.
- CONFIRMED | a copy that preserves every distance, so a centimetre on the map means the same number of kilometres everywhere, in every direction. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: isometry-definition: Gauss section 12: "the measure of curvature in each point remains unchanged" under development without changing line elements. Sphere K=1/R², plane K=0; constant map scale can be absorbed in the length unit.
- CONFIRMED | If such a map existed, it would have to preserve the thing distances determine: the Gaussian curvature. But the sphere's curvature is 1/R² ≠ 0 and flat paper's is 0. The two can never match. So the isometry cannot exist. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: isometry-preserves: Gauss section 12: "the measure of curvature in each point remains unchanged" under development without changing line elements. Sphere K=1/R², plane K=0; constant map scale can be absorbed in the length unit.
- CONFIRMED | Every flat map therefore distorts something. The mapmaker gets exactly one choice: which truth to keep and which to sacrifice. | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: flat-distortion: Snyder p. 4: "No map projection shows scale correctly throughout the map". The surrounding text distinguishes conformal, equal-area and other compromises.
- CONFIRMED | what cartographers call Tissot's indicatrix. On the round Earth every one of those marks is a true circle of the same size. | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: indicatrix-name: Snyder pp. 4 and 20: "An infinitely small circle on the Earth projects as an infinitely small, but perfect, ellipse"; conformal indicatrices are circles and equal-area maps preserve finite areas. Shape preservation here is local, as the next paragraph specifies.
- CONFIRMED | angles and shapes survive | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: projection-caption-2: Snyder pp. 4 and 20: "An infinitely small circle on the Earth projects as an infinitely small, but perfect, ellipse"; conformal indicatrices are circles and equal-area maps preserve finite areas. Shape preservation here is local, as the next paragraph specifies.
- CONFIRMED | by letting it grow without limit toward the poles: it pays the tax in area. Lambert's equal-area keeps every ellipse the same area | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: projection-caption-3: Snyder pp. 4, 41, 44 and 77 give Mercator h=k=sec φ and Lambert h=cos φ, k=1/h. Consequently Mercator area grows as sec² φ, Lambert area is constant, and equirectangular has h=1, k=sec φ. These theoretical statements hold; the drawing mismatch is recorded separately.
- CONFIRMED | a country's size is honest | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: projection-caption-4: Snyder pp. 4 and 20: "An infinitely small circle on the Earth projects as an infinitely small, but perfect, ellipse"; conformal indicatrices are circles and equal-area maps preserve finite areas. Shape preservation here is local, as the next paragraph specifies.
- CONFIRMED | by squashing it: it pays the tax in shape. Equirectangular, the naïve "just plot latitude and longitude," keeps neither and is merely simple. There is no fourth column where the circles stay circles and stay the same size. | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: projection-caption-5: Snyder pp. 4, 41, 44 and 77 give Mercator h=k=sec φ and Lambert h=cos φ, k=1/h. Consequently Mercator area grows as sec² φ, Lambert area is constant, and equirectangular has h=1, k=sec φ. These theoretical statements hold; the drawing mismatch is recorded separately.
- CONFIRMED | Conformal maps (Mercator) preserve angles, so shapes look right locally and a compass bearing is a straight line | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: conformal-and-bearing: Read as claims about the explicitly named Mercator projection. Snyder p. 38 says "Conformal", "Loxodromes (rhumb lines) are straight lines", "Used for navigation", and "Presented by Mercator in 1569". Straight rhumb lines are specific to Mercator, not every conformal projection.
- CONFIRMED | which is why Mercator was a navigator's instrument in 1569, not a poster. | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: mercator-date: Read as claims about the explicitly named Mercator projection. Snyder p. 38 says "Conformal", "Loxodromes (rhumb lines) are straight lines", "Used for navigation", and "Presented by Mercator in 1569". Straight rhumb lines are specific to Mercator, not every conformal projection.
- CONFIRMED | Equal-area maps (Lambert, Gall–Peters, Mollweide) preserve area, so no country is inflated | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: equal-area-examples: Snyder p. 76 identifies Lambert (1772), Gall and Peters as cylindrical equal-area variants; p. 250 labels Mollweide "An equal-area projection of the world".
- CONFIRMED | A map that did both at once would be an isometry, and the ant has already forbidden it. | https://pubs.usgs.gov/pp/1395/report.pdf; https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: both-implies-isometry: Snyder p. 4: "No map can be both equal-area and conformal." Locally conformality gives two equal singular values and exact area preservation forces their product to one, hence a local isometry. Gauss section 12 and the unequal sphere/plane curvatures rule that out. A globe retains the curved metric.
- CONFIRMED | On the Mercator projection that still backs most web maps, | https://pro.arcgis.com/en/pro-app/3.5/help/mapping/properties/mercator.htm | N2 source reread, 2026-10-14: web-map: Esri: "The Web Mercator variant of the projection is the de facto standard for web maps and online services." Read the variant qualifications too: ellipsoidal Web Mercator is not exactly conformal. The page uses the spherical model.
- CONFIRMED | Greenland looks about the size of Africa. | https://equalmap.org/ | N2 source reread, 2026-10-14: greenland-looks: The cited page states "On Mercator, Greenland looks about the size of Africa." Independent Natural Earth polygon integration also produces comparable apparent areas, ratio about 1.08 before normalization to the page's cited true areas.
- CONFIRMED | Africa is 14 times larger | https://stat.gl/publ/en/GF/2025/pdf/Greenland%20in%20Figures%202025.pdf; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: africa-ratio: Africa source: "Area30,370,000 km2". Statistics Greenland: "Area 2,166,086 km2" and "Ice-free area 410,449 km2". Fixer recomputation of the cited areas gives 14.0206806193; the rounded 30.4 million and 2.17 million agree. The Greenland total includes the ice-covered part.
- CONFIRMED | 30.4 million square kilometres to Greenland's 2.17 million. | https://stat.gl/publ/en/GF/2025/pdf/Greenland%20in%20Figures%202025.pdf; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: rounded-areas: Africa source: "Area30,370,000 km2". Statistics Greenland: "Area 2,166,086 km2" and "Ice-free area 410,449 km2". Fixer recomputation of the cited areas gives 14.0206806193; the rounded 30.4 million and 2.17 million agree. The Greenland total includes the ice-covered part.
- CONFIRMED | Multiply it out and a body fourteen times smaller is made to look its equal. | https://equalmap.org/ | N2 source reread, 2026-10-14: looks-equal: The cited page states "On Mercator, Greenland looks about the size of Africa." Independent Natural Earth polygon integration also produces comparable apparent areas, ratio about 1.08 before normalization to the page's cited true areas.
- CONFIRMED | where Africa lives | https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: latitude-slide-2: Africa straddles the equator; the cited geographic extent runs from Cape Agulhas south of zero to Ras ben Sakka north of zero. The page explicitly uses a schematic equatorial position.
- CONFIRMED | It answers the question it was built for | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: navigation-intent: Read as claims about the explicitly named Mercator projection. Snyder p. 38 says "Conformal", "Loxodromes (rhumb lines) are straight lines", "Used for navigation", and "Presented by Mercator in 1569". Straight rhumb lines are specific to Mercator, not every conformal projection.
- CONFIRMED | perfectly, and it pays for that with area it never promised to keep. | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: navigation-perfect: Read as claims about the explicitly named Mercator projection. Snyder p. 38 says "Conformal", "Loxodromes (rhumb lines) are straight lines", "Used for navigation", and "Presented by Mercator in 1569". Straight rhumb lines are specific to Mercator, not every conformal projection.
- CONFIRMED | You may keep the angles, or you may keep the areas. The one map that keeps both is the globe, and the globe does not fold. | https://pubs.usgs.gov/pp/1395/report.pdf; https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: globe-angles-area-1: Snyder p. 4: "No map can be both equal-area and conformal." Locally conformality gives two equal singular values and exact area preservation forces their product to one, hence a local isometry. Gauss section 12 and the unequal sphere/plane curvatures rule that out. A globe retains the curved metric.
- CONFIRMED | this equality is the theorem. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: curvature-verification-2: Gauss sections 8 and 11 identify the extrinsic curvature product with a formula determined by the metric. The verifier demonstrates that equality on spheres, within numerical tolerance.
- CONFIRMED | Greenland 2,166,086 km², | https://stat.gl/publ/en/GF/2025/pdf/Greenland%20in%20Figures%202025.pdf; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: greenland-area: Africa source: "Area30,370,000 km2". Statistics Greenland: "Area 2,166,086 km2" and "Ice-free area 410,449 km2". Fixer recomputation of the cited areas gives 14.0206806193; the rounded 30.4 million and 2.17 million agree. The Greenland total includes the ice-covered part.
- CONFIRMED | Africa 30,370,000 km² (ratio ≈ 14.0). | https://stat.gl/publ/en/GF/2025/pdf/Greenland%20in%20Figures%202025.pdf; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: africa-area: Africa source: "Area30,370,000 km2". Statistics Greenland: "Area 2,166,086 km2" and "Ice-free area 410,449 km2". Fixer recomputation of the cited areas gives 14.0206806193; the rounded 30.4 million and 2.17 million agree. The Greenland total includes the ice-covered part.
- CONFIRMED | Greenland 60°–84° N, | https://www.marineregions.org/gazetteer.php?id=2260&p=details; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: greenland-band: The supplied Wikipedia excerpt says 59 and 83 degrees and does not demonstrate the claimed rounding. Marine Regions gives "59.7903°" and "83.6236°", which round to 60 and 84. Africa's 34°51'15" S and 37°21' N round to 35 S and 37 N.
- CONFIRMED | Africa 35° S–37° N | https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: africa-band: The cited geography text gives "Ras ben Sakka in Tunisia (37°21' N)" and "Cape Agulhas in South Africa (34°51'15" S)". Rounding gives the stated band.
- CONFIRMED | The Greenland figure of 2,166,086 km² includes the ice sheet. | https://stat.gl/publ/en/GF/2025/pdf/Greenland%20in%20Figures%202025.pdf; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: greenland-includes-ice: Africa source: "Area30,370,000 km2". Statistics Greenland: "Area 2,166,086 km2" and "Ice-free area 410,449 km2". Fixer recomputation of the cited areas gives 14.0206806193; the rounded 30.4 million and 2.17 million agree. The Greenland total includes the ice-covered part.
- CONFIRMED | The latitude bands are the bodies' true extents, rounded to the nearest degree. | https://www.marineregions.org/gazetteer.php?id=2260&p=details; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: rounded-latitude-extents: The supplied Wikipedia excerpt says 59 and 83 degrees and does not demonstrate the claimed rounding. Marine Regions gives "59.7903°" and "83.6236°", which round to 60 and 84. Africa's 34°51'15" S and 37°21' N round to 35 S and 37 N.
- CONFIRMED | the projection runs to infinity at the poles | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: pole-infinity: Snyder pp. 38 and 41 gives "Poles are at infinity" and the log-tangent, arcsinh(tan) and arctanh(sin) forms of y. Read the PDF text directly.
- CONFIRMED | Gauss, C. F., Disquisitiones generales circa superficies curvas (1827) | https://www.gutenberg.org/files/36856/36856-pdf.pdf | N2 source reread, 2026-10-14: cite-gauss: The title is exact. The translator calls it "the Paper of 1827", gives presentation on 8 October 1827, the 5 November 1827 abstract, and full-volume publication in 1828. A conventional (1827) theorem citation is supported.
- CONFIRMED | Girard, A. (1629) | https://fr.wikisource.org/wiki/Page:Joseph_Louis_de_Lagrange_-_%C5%92uvres,_Tome_7.djvu/339 | N2 source reread, 2026-10-14: cite-girard: Lagrange's historical account explicitly attributes the result to Albert Girard and his Invention nouvelle en Algèbre, "imprimé à Amsterdam en 1629". Read the full cited page.
- CONFIRMED | Tissot, N. A., Mémoire sur la représentation des surfaces (1881) | https://books.google.com/books/about/M%C3%A9moire_sur_la_repr%C3%A9sentation_des_surf.html?id=a5YOAAAAYAAJ | N2 source reread, 2026-10-14: cite-tissot: The catalogue identifies Auguste Tissot, the full title Mémoire sur la représentation des surfaces et les projections des cartes géographiques, and "Gauthier-Villars, 1881". The page uses a shortened title.
- CONFIRMED | Mercator, G. (1569); | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: cite-mercator: Read as claims about the explicitly named Mercator projection. Snyder p. 38 says "Conformal", "Loxodromes (rhumb lines) are straight lines", "Used for navigation", and "Presented by Mercator in 1569". Straight rhumb lines are specific to Mercator, not every conformal projection.
- CONFIRMED | Lambert, J. H. (1772) | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: cite-lambert: Snyder p. 76 identifies Lambert (1772), Gall and Peters as cylindrical equal-area variants; p. 250 labels Mollweide "An equal-area projection of the world".
- CONFIRMED | The Gauss–Bonnet theorem (Gauss, local; Bonnet, 1848). | https://mathshistory.st-andrews.ac.uk/Extras/Bonnet_publications/ | N2 source reread, 2026-10-14: cite-bonnet: MacTutor lists "Mémoire sur la théorie général des surfaces" in Journal de l’École Polytechnique 19 (1848), 1-133. Gauss section 20 supplies the geodesic-triangle predecessor; the page does not claim Bonnet proved the modern global formulation.
- CONFIRMED | Land areas after standard references (the Greenland and Africa figures as commonly cited; the verifier carries the exact numbers used). | git show f5e5f84f34:research/egregium/verify.mjs; https://stat.gl/publ/en/GF/2025/pdf/Greenland%20in%20Figures%202025.pdf; https://en.wikipedia.org/wiki/Africa | N2 source reread, 2026-10-14: cite-areas: The verifier carries `A_GREENLAND = 2_166_086, A_AFRICA = 30_370_000`, corroborated by Statistics Greenland and the cited Africa reference. The quantities are total surface areas including Greenland ice.
- CONFIRMED | Latin ēgregius = ex grege, "out of the flock" | https://www.oxfordlearnersdictionaries.com/us/definition/english/egregious | N2 source reread, 2026-10-14: latin-quoted: The source gloss is "standing out from the flock", not the page's shorter "out of the flock". This is a valid literal gloss of ex grege, not a quotation attributed to Oxford. The tool's verbatim-quote failure concerns how the record tagged it, not a false etymology.
- CONFIRMED | standing apart, distinguished, hence Gauss's "remarkable." | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: remarkable-quoted: Gauss section 12: "the remarkable" followed by "Theorem". The Latin gloss is corroborated by Oxford; the English word remarkable is present in the translation.
- CONFIRMED | English borrowed it in the 1530s still meaning eminent, then within a century turned it ironic, so that egregious now means conspicuously, remarkably bad. | https://www.etymonline.com/word/egregious | N2 source reread, 2026-10-14: english-date: curl returned a challenge page; opened the source through web. It says "1530s" and "Disapproving sense, now predominant, arose late 16c., originally ironic." Those dates support within a century.
- CONFIRMED | The word that praises and the word that damns are the same word, pulled in opposite directions by usage | https://www.oxfordlearnersdictionaries.com/us/definition/english/egregious | N2 source reread, 2026-10-14: word-same: Oxford: "originally meaning remarkably good"; "standing out from the flock"; "The current sense probably arose as an ironical use."
- CONFIRMED | Curvature is intrinsic; the map's lie is a theorem; the checks are in the open. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces; https://artwaste.land/checks/research/egregium/verify.mjs | N2 source reread, 2026-10-14: footer-1: Gauss sections 11 and 12 support the intrinsic-curvature/no-isometry argument; both published check URLs return the same runnable program. The correct theorem does not validate every drawing.
- CONFIRMED | from its angles (Girard, 1629) | https://fr.wikisource.org/wiki/Page:Joseph_Louis_de_Lagrange_-_%C5%92uvres,_Tome_7.djvu/339 | N2 source reread, 2026-10-14: girard-date: Lagrange's historical account explicitly attributes the result to Albert Girard and his Invention nouvelle en Algèbre, "imprimé à Amsterdam en 1629". Read the full cited page.
- CONFIRMED | K = −f″/f for the meridian profile f(u)=R·cos(u/R) | https://phys.uri.edu/gerhard/PHY510/wgex368.pdf | N2 source reread, 2026-10-14: curvature-formula: Opened the university PDF with curl -k. Its surface-of-revolution formula reduces to K=-f″/f when the meridian parameter is arc length, so f′²+g′²=1. For f=R cos(u/R), direct differentiation gives 1/R². Gauss section 11 establishes that the metric alone suffices. The supplied excerpt is extrinsic, so this reduction is needed.
- CONFIRMED | ∫K dA = 4π = 2π·χ for the sphere's Euler characteristic χ = 2. | https://math.uchicago.edu/~may/REU2015/REUPapers/Butt.pdf | N2 source reread, 2026-10-14: sphere-euler: Butt, example 4.7: "If S is a sphere, we know χ(S) = 2" and the displayed integral is 4π. The reader verifier independently evaluates K times 4πR² at two radii.
- CONFIRMED | An ant who never leaves a surface can still prove that surface is curved: by measuring a single triangle. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces; https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: dek-1: Gauss section 11 gives intrinsic curvature and section 12 gives invariance under isometry; Todhunter section 97 supplies the spherical triangle area/excess identity. For a nondegenerate great-circle triangle on a sphere this establishes the claimed measurement and no-flat-isometry conclusion.
- CONFIRMED | Gauss called the result the Theorema Egregium, the Remarkable Theorem (Latin egregium, 'out of the flock'; the same word English soured into egregious). | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces; https://www.oxfordlearnersdictionaries.com/us/definition/english/egregious | N2 source reread, 2026-10-14: dek-2: Oxford supports the Latin gloss and English pejoration. Gauss section 12 supplies the name translated as "remarkable". The dictionary excerpt alone does not establish what Gauss called the theorem.
- CONFIRMED | Its remarkable claim is that curvature is intrinsic: written into the distances on a surface, readable from inside, with no view from outside. | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: dek-3: Gauss section 11: "the general expression for the magnitude of any linear element is sufficient". Section 12 calls the resulting invariant theorem "remarkable". The 1827 date is supported by the dated abstract in the Gutenberg edition.
- CONFIRMED | And that is exactly why no flat map of the Earth can ever be faithful: the sphere's curvature is not zero, the plane's is, and curvature survives any distance-preserving copy. | https://pubs.usgs.gov/pp/1395/report.pdf; https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: dek-4: Snyder p. 4: "No map can be both equal-area and conformal." Locally conformality gives two equal singular values and exact area preservation forces their product to one, hence a local isometry. Gauss section 12 and the unequal sphere/plane curvatures rule that out. A globe retains the curved metric.
- CONFIRMED | You may keep angles, or you may keep areas; never both. | https://pubs.usgs.gov/pp/1395/report.pdf; https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: dek-5: Snyder p. 4: "No map can be both equal-area and conformal." Locally conformality gives two equal singular values and exact area preservation forces their product to one, hence a local isometry. Gauss section 12 and the unequal sphere/plane curvatures rule that out. A globe retains the curved metric.
- CONFIRMED | The cartographic sibling. There a coastline's length refuses to be a single number (a fractal dimension survives instead); here the whole flat map refuses to be faithful (a curvature invariant forbids it). Both are about measuring the round Earth honestly and naming what cannot be reduced | git show f5e5f84f34:src/content/strata/how-long-is-the-coast-of-britain.md | N2 source reread, 2026-10-14: related-how-long-is-the-coast-of-britain-1: The linked pinned layer explicitly describes changing divider lengths, recovered dimensions, a chosen base map, fit windows and named discrepancies. The relation is a fair account of its subject and stated modelling assumptions.
- CONFIRMED | and both reuse the discipline of stating the modelling assumption rather than hiding it. | git show f5e5f84f34:src/content/strata/how-long-is-the-coast-of-britain.md | N2 source reread, 2026-10-14: related-how-long-is-the-coast-of-britain-2: The linked pinned layer explicitly describes changing divider lengths, recovered dimensions, a chosen base map, fit windows and named discrepancies. The relation is a fair account of its subject and stated modelling assumptions.
- CONFIRMED | The geodesy sibling in the venue's spirit: a fact about the Earth's shape re-derived from first principles (there, Chimborazo as the farthest point from the centre, from the WGS84 constants; here, that no map of that shape can be flat-and-faithful, from its curvature). Both compute the surprising truth in front of the reader rather than asserting it. | git show f5e5f84f34:src/content/strata/the-farthest-point.md | N2 source reread, 2026-10-14: related-the-farthest-point-1: The linked layer states Chimborazo is farthest from the centre, describes the WGS84 constants, elevation and latitude inputs, and gives the independent verifier. This relation summarizes that declared method accurately.
- CONFIRMED | Two theorems about what geometry is and isn't intrinsic. 'You can't hear the shape of a drum' shows the spectrum does NOT determine the shape; the Theorema Egregium shows the metric DOES determine the curvature. Cousins in spectral and differential geometry | git show f5e5f84f34:src/content/strata/hearing-the-shape.md | N2 source reread, 2026-10-14: related-hearing-the-shape-1: The linked layer identifies the Gordon-Webb-Wolpert counterexamples: "Two non-congruent planar domains" with "identical Laplacian spectra". It contrasts a non-unique spectrum with Gauss's metric-determined curvature accurately.
- CONFIRMED | one a limit on what intrinsic data reveals, the other a guarantee of what it must. | git show f5e5f84f34:src/content/strata/hearing-the-shape.md | N2 source reread, 2026-10-14: related-hearing-the-shape-2: The linked layer identifies the Gordon-Webb-Wolpert counterexamples: "Two non-congruent planar domains" with "identical Laplacian spectra". It contrasts a non-unique spectrum with Gauss's metric-determined curvature accurately.
- CONFIRMED | the ant reads the curvature without leaving the surface | https://www.gutenberg.org/files/19770/19770-pdf.pdf; https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: live-3: Todhunter section 97 gives area = radian excess times R²; Gauss section 11 makes the resulting curvature intrinsic. The instrument evaluates that identity.
- CONFIRMED | angles preserved (conformal) | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: live-6: For the theoretical Mercator projection, Snyder p. 38 says "Conformal" and p. 44 gives h=k=sec φ. Its local angle preservation is mathematically correct.
- CONFIRMED | "about equal," from 1/14 the land | git show f5e5f84f34:research/egregium/verify.mjs | N2 source reread, 2026-10-14: live-9: Using the stated areas and band factors gives a true Greenland/Africa ratio 0.0713232 and model apparent ratio 0.789257, supporting this qualitative comparison.
- CONFIRMED | Equirectangular (keeps neither | git show f5e5f84f34:research/egregium/verify.mjs; https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: live-13: An equirectangular map with the equator as standard parallel has h=1, k=sec φ, so away from the equator it preserves neither area nor angle. The displayed nominal factors agree.
- CONFIRMED | ⓘ equirectangular preserves neither area nor angle away from the equator | git show f5e5f84f34:research/egregium/verify.mjs; https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: live-15: An equirectangular map with the equator as standard parallel has h=1, k=sec φ, so away from the equator it preserves neither area nor angle. The displayed nominal factors agree.
- CONFIRMED | areas preserved exactly | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: live-19: Snyder p. 77 gives h=cos φ and k=1/h, hence area factor exactly one for Lambert cylindrical equal-area.
- CONFIRMED | Gauss's *Theorema Egregium* (1827) proved that the Gaussian curvature of a surface is *intrinsic* | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces | N2 source reread, 2026-10-14: markdown-body-1: Gauss section 11: "the general expression for the magnitude of any linear element is sufficient". Section 12 calls the resulting invariant theorem "remarkable". The 1827 date is supported by the dated abstract in the Gutenberg edition.
- CONFIRMED | determined by distances measured within the surface alone, with no reference to how (or whether) it sits in a surrounding space. An ant confined to a sphere can prove its world is curved by measuring one triangle: the interior angles sum to more than 180°, and the excess divided by the triangle's area equals the curvature, `K = 1/R²`, exactly. Because curvature is preserved by any isometry (distance-preserving map), and the sphere's curvature is non-zero while the plane's is zero, **no flat map of the Earth can preserve all distances** | https://en.wikisource.org/wiki/General_Investigations_of_Curved_Surfaces; https://www.gutenberg.org/files/19770/19770-pdf.pdf | N2 source reread, 2026-10-14: markdown-body-2: Gauss section 11 gives intrinsic curvature and section 12 gives invariance under isometry; Todhunter section 97 supplies the spherical triangle area/excess identity. For a nondegenerate great-circle triangle on a sphere this establishes the claimed measurement and no-flat-isometry conclusion.
- CONFIRMED | ln tan(π/4+φ/2) = asinh(tan φ) = atanh(sin φ) | https://pubs.usgs.gov/pp/1395/report.pdf | N2 source reread, 2026-10-14: mercator-formulas: Snyder pp. 38 and 41 gives "Poles are at infinity" and the log-tangent, arcsinh(tan) and arctanh(sin) forms of y. Read the PDF text directly.
- CONFIRMED | Nothing here may lie about anything real; that rule never bends. | git show f5e5f84f34:README.md | N2 source reread, 2026-10-14: shared-help-3: README.md lists the rule "It must never lie about anything real." This is a declared rule, not an assertion that no page currently has an error.

## Findings

- [fixed] X1: Changed the radius label, explanatory caption and code comment to match the fixed angular triangle. Dated correction C1.
- [fixed] X2: Changed every grid crossing to selected grid crossings. Dated correction C2.
- [fixed] X3: Unified projection-coordinate units, adjusted the frame and graticule to longitude limits of plus/minus pi, and clarified the local indicatrix caption. Dated correction C3.
- [fixed] X4: Identified 12.7 as a uniform-band mean, not a point factor near 70 degrees. MINOR; recorded here with no page correction line.
- [fixed] X5: Included the last-to-first polygon edge and kept the full Greenland outline in view. Dated correction C4.
- [fixed] X6: The shared closure repair restores the promised true-size ratio in the drawn paths. Shares C4 with X5.
- [fixed] X7: Narrowed the apparatus, HTML and JSON-LD descriptions, and verifier header to the actual formula-check scope. MINOR; no page correction line.
- [fixed] X8: Replaced the precision/sample overclaim with 3,990 accepted triangles from 4,000 attempts, about 1.2e-12 radians observed discrepancy, and a 1e-9 threshold. MINOR; no page correction line.
- [fixed] X9: Removed by a little from the coastline qualification and corresponding verifier comment, preserving the band model and original build observation. MINOR; no page correction line.
- [fixed] X10: The quantitative-area promise is supported by the shared closure repair; schematic-coastline qualification retained. Shares C4 with X5.
- [fixed] X11: Narrowed the markdown dek to curvature, triangle, projection and band-area formulas. MINOR; no page correction line.
- [open note:4bc9c9] X12: Restored the source quotation in egregium's relates source and HTML note, with dated correction C5. The inaccurate quotation remains unannotated in notes[366] at public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1. That other layer needs a copy-correction annotation preserving its historical quotation.
- [open note:4bc9c9] X13: Attributed the dinosaur to the Datasaurus Dozen in egregium, with dated correction C6. The Anscombe attribution remains unannotated in the same two explorer payloads as X12; the other layer needs a copy-correction annotation preserving its historical quotation.
- [open note:4bc9c9] X14: Duplicate of X12. Egregium's quotation is corrected under C5, but the same unannotated explorer copies remain open. Shares X12's pending board note and required copy-correction annotation.
- [fixed] X15: Directed both the paragraph and live hint to Show true sizes for the equal-area comparison. MINOR; no page correction line.
- [fixed] X16: The repaired drawn ratio now supports the retained 14.0 and drawn-so readout. Shares C4 with X5.
- [fixed] X17: Narrowed the markdown body to the formula families actually checked. MINOR; no page correction line.
- [fixed] X18: Same projection repair as X3, retaining the now-supported canvas description. Shares C3 with X3.
- [dated] X19: Updated JSON-LD dateModified to 2026-10-14; preserved datePublished. No Updated line under the explicit metadata exception; C1-C6 date this revision.

## Checks

- Original formula verifier: 35/35 pass; 3,990 accepted triangles and about 1.2e-12 radians worst discrepancy.
- Original page through research/egregium/verify-page.mjs: 2 pass, 10 fail, reproducing the projection scales, polygon area and clipping defects at both tested widths.
- Corrected page: 12/12 pass. Independent screen-map derivatives agree with the marks; independent triangle-fan areas agree with cited areas and numerically integrated band means. This is a Node VM and canvas-command check, not browser rendering.
- Fixer claims gate, before repair: node scripts/check-claims-pass.mjs exited 1 for 39 other layers; egregium was not among its errors. The exported checkRecord function also validated egregium before the three PENDING dispositions were added. Current repair results are in research/assay-line/queue-1014c/egregium/repair-report.md.
- Corrected standalone formula verifier: 35/35 pass.
- Full command results and before/after locations: research/assay-line/queue-1014c/egregium/fix-report.md.