---
slug: two-points-the-same
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; repair reviewed by Codex, 2026-10-14
checked: 78
confirmed: 64
wrong: 14
unverifiable: 0
observed: 0
stale: 0
---

This was the directed fixing pass for `two-points-the-same`, read at `8df0e2eb0a`, using `research/assay-line/queue-1014c/two-points-the-same/triage.json`. The fixer independently re-read the evidence for all 14 triage items and reproduced the numerical defects before editing. The 64 CONFIRMED lines retain the N2 checker's source rereads and qualifications from `N2-codex.md`, including its added Gauss attribution. The eight WRONG and six MINOR lines keep the triage levels; MINOR is counted in `wrong` for the gate. X2/X3 and X9/X13 share defects and correction lines. Authorship is in the standalone HTML and the markdown frontmatter, with no layer generator found. The 2026-10-14 repair pass reopened X7 for unannotated copies in There, Here; this layer's related-page note is corrected. The pending board note and dated-line review are documented in research/assay-line/queue-1014c/two-points-the-same/repair-report.md.

## Claims

- WRONG | X1: "which is point-symmetric about its centre" | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | The seeded browser functions give g(0)=g(360)=7.298403407447038 and g(180)=-7.298403407447038. Equal nonzero endpoints rule out symmetry about (180,0). The Toronto exercise gives an odd antipodal difference; the valid relation is a half-period shift and sign reversal.
- WRONG | X2: "the gap gT(p) = T(p) − T(−p) is a continuous quantity that's plus at p and minus at −p, so its zero set is not just a point but a whole curve winding around the sphere" | https://eldorado.tu-dortmund.de/server/api/core/bitstreams/e46051b2-d9af-445e-a512-6946c27d68ec/content; https://mathoverflow.net/a/284743 | Siburg, printed p. 3, Theorem 2.2 assumes smoothness and a regular zero. Teyssier constructs a continuous odd function whose zero set has no path joining antipodes; setting T=f/2 makes that function the temperature gap. Continuity alone does not supply the claimed loop.
- WRONG | X3: "an entire loop of antipodal pairs that already agree on temperature." | https://eldorado.tu-dortmund.de/server/api/core/bitstreams/e46051b2-d9af-445e-a512-6946c27d68ec/content; https://mathoverflow.net/a/284743 | The same original counterexample as X2 rules out a zero path joining antipodes, so it rules out the asserted antipodal loop. A constant temperature alone would not refute the existence of such a loop. This repeats the X2 defect.
- WRONG | X4: "Hand a thief's necklace with two kinds of jewels to two thieves and you can split it fairly with few cuts (necklace-splitting)." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | Matousek, printed p. 54 (PDF page 65), assumes an open necklace and even counts of each kind; Theorem 3.2.2 supplies at most d cuts. With two kinds d=2. One jewel of each kind already refutes the omitted divisibility condition.
- MINOR | X5: "so the horizontal wind, a continuous tangent field on the globe, must vanish somewhere: at every instant there is a point of perfect calm, the still eye the weather turns around." | https://pjmcgrat.math.ncsu.edu/research/hairy-ball.pdf | McGrath, p. 1, Theorem 1 concerns a continuous tangent vector field on S². For the specified horizontal field it guarantees zero horizontal wind, without asserting zero vertical wind or a vortex eye.
- MINOR | X6: "the Antipodes Islands, named in 1800 for sitting opposite London." | https://www.doc.govt.nz/globalassets/documents/conservation/historic/by-region/southland/archeology-of-the-antipodes-islands.pdf | Department of Conservation history, printed p. 10, section 1.3.1, records the 1800 naming as Isle Penantipode and approximate opposition to London. The existing page body already says roughly opposite.
- WRONG | X7: "The 100-prisoners loop strategy makes a 30% survival into a near-coin-flip because permutations must contain cycles" | https://arxiv.org/html/0812.2242#S2.SS1 | Avis and Broadbent, section 2.1, equation (1), gives the long-cycle failure probability. Direct evaluation gives 1-sum(1/k, k=51..100)=0.3118278206898051, hence about 31.18%. The limiting 1-ln(2) is not the finite-n answer.
- MINOR | X8: "this page proves the one-dimensional case live with the Intermediate Value Theorem, makes the combinatorial heart (Tucker's lemma) a game you cannot win, and finds the antipodal twin on a synthetic weather map, every claim recomputed in your browser and checked offline." | git show f5e5f84f34:public/strata/two-points-the-same/index.html; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | The browser implements the caliper, ring and synthetic weather solver. Historical attributions and the topological theorem are cited. The separate numerical verifier does not read the page, so every claim recomputed and checked exceeds its scope.
- WRONG | X9: "the lowest you can drive it is 1, never 0. Try." | git show f5e5f84f34:public/strata/two-points-the-same/index.html; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | Executing the actual compEdges function for every antipodal labeling of even rings 4 through 16 gives minimum 2. The ten-bead histogram is {2:10, 6:20, 10:2}. Each complementary edge has a distinct antipodal partner.
- MINOR | X10: "Tucker's lemma forbids it; if you see this, it's a rounding ghost" | git show f5e5f84f34:public/strata/two-points-the-same/index.html; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | The labels are exactly +1 or -1 and compEdges uses exact integer equality. A zero count would indicate an unexpected logic or state fault; rounding does not explain it.
- MINOR | X11: "every gap, label, line, and located twin recomputed in your browser" | git show f5e5f84f34:public/strata/two-points-the-same/index.html; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | The browser function inventory has the caliper, ring and globe solver, with no ham-sandwich calculation. findHamSandwich and its 300-trial check are in research/borsuk-ulam/verify.mjs. The footer now assigns that line to the offline check.
- MINOR | X12: ""dateModified":"2026-06-19"" | git show f5e5f84f34:research/typed-verdicts/triage.json; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | The HTML had dateModified 2026-06-19 beside a Corrected 2026-09-28 note. Read-only git history confirms the September correction in 20c45c86d13. This pass revises content on 2026-10-14, so dateModified now records today; datePublished is unchanged.
- WRONG | X13: "complementary edges: 10"; "the lowest you can drive it is 1, never 0. Try." | git show f5e5f84f34:public/strata/two-points-the-same/index.html; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | The initial alternating ten-bead labeling really has 10 complementary edges. The separately repeated minimum sentence was false, as the X9 enumeration establishes. Preserve the initial 10 and share X9's correction.
- WRONG | X14: (added) "MATCH ✓" | git show f5e5f84f34:public/strata/two-points-the-same/index.html; public/strata/two-points-the-same/index.html; research/borsuk-ulam/verify-page.mjs | With the page's seed at findZero()+0.0002, fraction 0.13701525099968412, the readings are 13.554766514969208 and 13.584166471972841. They display 13.55 and 13.58, yet the old absolute-gap<0.05 branch printed MATCH. The branch now explicitly states approximate match and its tolerance.
- CONFIRMED | "At this instant there are two antipodal points on Earth (the two ends of a line straight through the planet's centre) with exactly the same temperature AND exactly the same barometric pressure, and it is forced, not lucky." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-1; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "Right now, somewhere on Earth, two points exactly opposite each other through the planet's centre have the same temperature AND the same barometric pressure." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-2; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "Right now two antipodal points on Earth share a temperature AND a pressure, and it cannot be otherwise." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-3; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "Two antipodal points on Earth always share both temperature and pressure." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-4; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "The theorem on this page says that somewhere such a pair of opposite points reads the same on two dials at once, and that this is forced, not lucky." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-5; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "and that have exactly the same air temperature and exactly the same barometric pressure." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-6; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "it is true at every instant, on every continuous world, no matter how the weather is arranged." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-7; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "For n=2, two antipodal points on Earth share both temperature and pressure." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-8; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "the guarantee that a match exists holds for any continuous fields, the real atmosphere included." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-9; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "Which means: back on Earth, the both-dials antipodal pair is not findable-if-you're-lucky. It is unavoidable." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=weather-10; N2 checker reread retained: Matousek p. 22 gives the weather example under continuity; read with the page's explicit model qualification. This is a conditional mathematical guarantee.
- CONFIRMED | "diametrically opposite, the two ends of a line through the planet's core" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=antipodal-definition; N2 checker reread retained: Matousek p. 22 identifies antipodes as diametrically opposite points.
- CONFIRMED | "It is a 1933 theorem of Karol Borsuk" | https://impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/20/0/93008/drei-satze-uber-die-n-dimensionale-euklidische-sphare | id=borsuk-year; N2 checker reread retained: Publisher metadata identifies Borsuk and Fundamenta Mathematicae 20 (1933), pp. 177–190; the checker also compared the DOI metadata.
- CONFIRMED | "answering a question of Stanisław Ulam" | https://www.impan.pl/shop/publication/transaction/download/product/93008?download.pdf | id=ulam-question; N2 checker reread retained: The checker visually read printed p. 178, footnote 7, which attributes the conjecture to Ulam.
- CONFIRMED | "every continuous map from the n-sphere to n-space sends some antipodal pair to one value." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=borsuk-abstract; N2 checker reread retained: Matousek p. 23, Theorem 2.1.1, gives the continuous-map and odd-map formulations with the stated matching dimensions.
- CONFIRMED | "every continuous map `f : Sⁿ → ℝⁿ` identifies an antipodal pair, `f(p) = f(−p)`." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=borsuk-formula; N2 checker reread retained: Matousek p. 23, Theorem 2.1.1, gives the continuous-map and odd-map formulations with the stated matching dimensions.
- CONFIRMED | "an odd map of the sphere must hit zero," | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=odd-map; N2 checker reread retained: Matousek p. 23, Theorem 2.1.1, gives the continuous-map and odd-map formulations with the stated matching dimensions.
- CONFIRMED | "a continuous quantity can't jump from minus to plus without passing through zero." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-1; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "g(λ) = T(λ) − T(λ + 180°)." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-2; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "so the gap flips sign: g(λ + 180°) = −g(λ)." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-3; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "Whatever g is at your starting longitude, it is the exact negative of that across the loop." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-4; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "If it's positive here it's negative there; somewhere between, being continuous, it must pass through zero" | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-5; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "and a zero of the gap is an antipodal pair with identical temperature." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-6; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "That's the whole proof for one dial on a loop, and it is just the Intermediate Value Theorem wearing a disguise." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-7; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "matching one dial on a loop is just the Intermediate Value Theorem in disguise." | https://www.math.utoronto.ca/courses/mat237y1/20189/notes/Chapter1/S1.5.html | id=ivt-9; N2 checker reread retained: Toronto section 1.5, advanced problem 7, uses the continuous antipodal difference. Its sign reversal and a path between antipodes let IVT supply a zero.
- CONFIRMED | "and for two dials it genuinely can be pushed through" | https://eldorado.tu-dortmund.de/server/api/core/bitstreams/e46051b2-d9af-445e-a512-6946c27d68ec/content | id=curve-ivt; N2 checker reread retained: Siburg pp. 2–4 supplies a geometric proof with regular smooth functions, then passes to continuous functions by approximation and compactness. The general case does not require a zero loop. X2/X3 retire the page's unqualified curve transition.
- CONFIRMED | "but the clean reason the pair must exist is not a second use of the Intermediate Value Theorem." | https://eldorado.tu-dortmund.de/server/api/core/bitstreams/e46051b2-d9af-445e-a512-6946c27d68ec/content | id=second-ivt; N2 checker reread retained: Siburg pp. 2–4 supplies a geometric proof with regular smooth functions, then passes to continuous functions by approximation and compactness. The general case does not require a zero loop. X2/X3 retire the page's unqualified curve transition.
- CONFIRMED | "Then the combined gap g(p) = ( T(p)−T(−p), P(p)−P(−p) ) is never the zero vector, so you can normalise it to a direction" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-1; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "a continuous map from the sphere to the circle of directions, S² → S¹." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-2; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "And because each gap flips sign under the antipode, that map is antipode-respecting: opposite points of the sphere go to opposite points of the circle." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-3; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "The punchline of Borsuk–Ulam is that no such map can exist." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-4; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "You cannot continuously comb the sphere onto the circle while keeping antipodes opposite" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-5; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "The contradiction means a both-dials pair was there all along." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-6; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "no continuous antipode-respecting map `S² → S¹` exists" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-7; N2 checker reread retained: Matousek Theorem 2.1.1 (BU2a) rules out a continuous antipodal map S² to S¹. Normalizing a hypothetically nonzero odd gap would produce such a map.
- CONFIRMED | "the theorem hands you the pair's existence and not its address." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-8; N2 checker reread retained: Matousek Theorem 2.1.1 is an existence theorem. It supplies neither measured weather fields nor the geographic location of their matching pair.
- CONFIRMED | "Borsuk–Ulam proves the pair is there; it gives no recipe for finding it on the real Earth" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=topology-9; N2 checker reread retained: Matousek Theorem 2.1.1 is an existence theorem. It supplies neither measured weather fields nor the geographic location of their matching pair.
- CONFIRMED | "the sphere has no hole for the circle's loop to catch on, and the antipode rule forces a loop that would have to catch." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=holes-metaphor; N2 checker reread retained: Matousek p. 79, Theorem 4.3.2, gives the sphere's connectivity; the checker accepts this as an informal description of the cited obstruction, not a standalone proof.
- CONFIRMED | "Tucker's lemma (1946), the combinatorial twin of Borsuk–Ulam." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=tucker-date; N2 checker reread retained: Matousek p. 41 attributes the two-dimensional lemma to Tucker's 1946 paper and distinguishes later general-dimensional versions.
- CONFIRMED | "Tucker's lemma says an antipodal labeling of an even ring always has at least one." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=tucker-1; N2 checker reread retained: Matousek p. 35, Theorem 2.3.1, gives a complementary edge. On the ring a path to the opposite label must switch sign; this does not assert an attainable minimum of one.
- CONFIRMED | "It starts somewhere, it must end at its own negative half a turn on, so between here and there it has to switch" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=tucker-2; N2 checker reread retained: Matousek p. 35, Theorem 2.3.1, gives a complementary edge. On the ring a path to the opposite label must switch sign; this does not assert an attainable minimum of one.
- CONFIRMED | "and a switch is exactly a + beside a −." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=tucker-3; N2 checker reread retained: Matousek p. 35, Theorem 2.3.1, gives a complementary edge. On the ring a path to the opposite label must switch sign; this does not assert an attainable minimum of one.
- CONFIRMED | "the smooth statement becomes a counting one" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=tucker-4; N2 checker reread retained: Matousek p. 35, Theorem 2.3.1, gives a complementary edge. On the ring a path to the opposite label must switch sign; this does not assert an attainable minimum of one.
- CONFIRMED | "an antipodal ±labeling of an even ring must contain a complementary edge." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=tucker-5; N2 checker reread retained: Matousek p. 35, Theorem 2.3.1, gives a complementary edge. On the ring a path to the opposite label must switch sign; this does not assert an attainable minimum of one.
- CONFIRMED | "with one straight stroke and you can always halve all three at once" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=ham-sandwich; N2 checker reread retained: Matousek pp. 47–48 gives simultaneous hyperplane bisection of d finite mass distributions in R^d.
- CONFIRMED | "a single line simultaneously bisects any two finite scatters of points" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=ham-point-sets; N2 checker reread retained: Matousek Theorem 3.1.2 uses at most half the points in each open half-space; points on the line handle odd cardinalities and degeneracy.
- CONFIRMED | "Stone & Tukey 1942, a direct corollary" | https://api.crossref.org/works/10.1215/S0012-7094-42-00925-6 | id=ham-corollary; N2 checker reread retained: Crossref identifies Stone and Tukey's 1942 sandwich paper. Matousek pp. 47–48 supplies the Borsuk-Ulam corollary argument.
- CONFIRMED | "you cannot comb a sphere's worth of hair flat without a cowlick" | https://pjmcgrat.math.ncsu.edu/research/hairy-ball.pdf | id=hairy-statement; N2 checker reread retained: McGrath p. 1 states that a continuous tangent vector field on S² has a zero.
- CONFIRMED | "Same planet, neighbouring miracle, different proof." | https://pjmcgrat.math.ncsu.edu/research/hairy-ball.pdf | id=hairy-other-proof; N2 checker reread retained: McGrath proves the tangent-field theorem by winding numbers. Its statement differs from antipodal equality for maps to Euclidean space.
- CONFIRMED | "the real atmosphere is molecular at fine enough scale" | https://www.grc.nasa.gov/WWW/BGH/kinth.html | id=atmosphere-molecules; N2 checker reread retained: NASA's kinetic-theory account models a gas as molecules in motion and collision.
- CONFIRMED | ""the same to infinite precision" is a statement about the mathematical idealisation, not a claim you could confirm with two thermometers." | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=continuity-model; N2 checker reread retained: The theorem assumes continuous fields; Matousek p. 22 notes that physical temperature need not always satisfy continuity.
- CONFIRMED | "which, at the scales weather lives on, is true enough to be startling." | https://booksite.elsevier.com/samplechapters/9780125586917/Sample_Chapters/02~Chapter_1.pdf | id=weather-scale-model; N2 checker reread retained: Holton, section 1.1, distinguishes the molecular structure of a fluid from the continuum approximation used at atmospheric scales.
- CONFIRMED | "The Antipodes Islands, off New Zealand, were named in 1800 for sitting roughly opposite London" | https://www.doc.govt.nz/globalassets/documents/conservation/historic/by-region/southland/archeology-of-the-antipodes-islands.pdf | id=antipodes-islands; N2 checker reread retained: The Department of Conservation history, p. 10, records the 1800 naming and approximate opposition to London.
- CONFIRMED | "curvature an ant can measure from inside; the sphere's geometry felt without leaving it" | https://prclare.people.wm.edu/m317s24/Schlichtkrull.pdf | id=gauss-intrinsic; N2 checker reread retained: Schlichtkrull, Theorem 9.3, states that Gaussian curvature is determined by the first fundamental form and preserved under local isometries.
- CONFIRMED | "curvature an ant measures from inside, the reason no flat map is faithful." | https://prclare.people.wm.edu/m317s24/Schlichtkrull.pdf | id=gauss-flat-map; N2 checker reread retained: Schlichtkrull, Theorem 9.3, states that Gaussian curvature is determined by the first fundamental form and preserved under local isometries.
- CONFIRMED | "Karol Borsuk, 1933, *Fundamenta Mathematicae* 20: 177–190" | https://impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/20/0/93008/drei-satze-uber-die-n-dimensionale-euklidische-sphare | id=citation-borsuk; N2 checker reread retained: Publisher metadata identifies Borsuk and Fundamenta Mathematicae 20 (1933), pp. 177–190; the checker also compared the DOI metadata.
- CONFIRMED | "Borsuk 1933 (Fund. Math. 20)" | https://impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/20/0/93008/drei-satze-uber-die-n-dimensionale-euklidische-sphare | id=citation-borsuk-footer; N2 checker reread retained: Publisher metadata identifies Borsuk and Fundamenta Mathematicae 20 (1933), pp. 177–190; the checker also compared the DOI metadata.
- CONFIRMED | "Ulam (conjecture)" | https://www.impan.pl/shop/publication/transaction/download/product/93008?download.pdf | id=citation-ulam; N2 checker reread retained: The checker visually read printed p. 178, footnote 7, which attributes the conjecture to Ulam.
- CONFIRMED | "Tucker 1946" | https://www.maths.ed.ac.uk/~v1ranick/papers/matousek1x.pdf | id=citation-tucker; N2 checker reread retained: Matousek p. 41 attributes the two-dimensional lemma to Tucker's 1946 paper and distinguishes later general-dimensional versions.
- CONFIRMED | "Stone–Tukey 1942 (ham sandwich)" | https://api.crossref.org/works/10.1215/S0012-7094-42-00925-6 | id=citation-stone; N2 checker reread retained: Crossref identifies Stone and Tukey's 1942 sandwich paper. Matousek pp. 47–48 supplies the Borsuk-Ulam corollary argument.
- CONFIRMED | "Poincaré–Brouwer (hairy ball)" | https://pmc.ncbi.nlm.nih.gov/articles/PMC8448805/ | id=citation-hairy; N2 checker reread retained: The cited historical review credits Poincaré and Brouwer for the hairy-ball theorem.
- CONFIRMED | "Matoušek, Using the Borsuk–Ulam Theorem (Springer 2003)" | https://kam.mff.cuni.cz/~matousek/akt.html | id=citation-matousek; N2 checker reread retained: The author's book list identifies Using the Borsuk-Ulam Theorem, Springer 2003.
- CONFIRMED | "All checks in research/borsuk-ulam/verify.mjs." | git show f5e5f84f34:research/borsuk-ulam/verify.mjs | id=verifier-8; N2 checker reread retained: The pinned repository contains the named dependency-free verifier. This identifies the supplied numerical checks, without extending them to historical citations or every page sentence.
- CONFIRMED | "research/borsuk-ulam/verify.mjs" | git show f5e5f84f34:research/borsuk-ulam/verify.mjs | id=source-file; N2 checker reread retained: The pinned repository contains the named dependency-free verifier. This identifies the supplied numerical checks, without extending them to historical citations or every page sentence.
- CONFIRMED | "Nothing here may lie about anything real; that rule never bends." | git show f5e5f84f34:WAKE.md | id=site-rule; N2 checker reread retained: Pinned WAKE.md states the site's truthfulness rule; this describes a rule, not an assertion that every sentence has already passed an audit.
- CONFIRMED | (added) "Gauss's remarkable theorem" | https://prclare.people.wm.edu/m317s24/Schlichtkrull.pdf | id=added-gauss-name; N2 checker reread retained: Schlichtkrull p. 99 attributes the theorem to Gauss in 1827 and explains its remarkable-theorem name.

## Findings

- [fixed] X1: Replaced point symmetry with the half-turn sign relation in the markdown dek, HTML hint, canvas label and comment. Added one mirrored Corrected line.
- [fixed] X2: Replaced the general curve argument with the zero-set argument, removed its curve-dependent continuation, and described the model plot as zero contours. Added one mirrored Corrected line shared with X3.
- [fixed] X3: Removed the unqualified entire-loop assertion with X2. Shares X2's mirrored Corrected line; retained as a separate triage claim.
- [fixed] X4: Specified an open necklace, an even count of each kind and at most two cuts. Added one mirrored Corrected line.
- [fixed] X5: Replaced the calm/eye gloss with zero horizontal wind. MINOR, recorded here without a page correction line.
- [fixed] X6: Added roughly to the markdown dek's opposite London. MINOR, recorded here without a page correction line.
- [open note:c2f122] X7: This layer's related-page note now gives about 31.18% and no cycle longer than 50, with a mirrored Corrected line giving the finite formula. Unannotated copies of the false 30%-to-near-coin-flip sentence remain in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[2198], from two-points-the-same to the-loop-that-saves-them). These belong to another layer and cannot be edited in this repair. The directing instance must obtain a board note and arrange a dated correction beside the preserved quotation.
- [fixed] X8: Restricted the description to the three browser demonstrations and separate offline numerical/combinatorial checks. MINOR, recorded here without a page correction line.
- [fixed] X9: Changed the runtime full-ring minimum from 1 to 2. Added one mirrored Corrected line shared with X13; page regressions enumerate the actual counter and all selectable labelings.
- [fixed] X10: Replaced the rounding diagnosis with an unexpected logic or state error and an error style. MINOR, recorded here without a page correction line.
- [fixed] X11: Assigned the ham-sandwich line to the offline verifier in the footer. MINOR, recorded here without a page correction line.
- [fixed] X12: Set JSON-LD dateModified to 2026-10-14. MINOR, recorded here; the content corrections already date this revision, so no Updated line.
- [fixed] X13: Preserved the correct initial count 10 and fixed the shared X9 minimum to 2. Shares X9's mirrored Corrected line; the regression checks preserve the initial 10.
- [fixed] X14: Labeled the existing below-0.05° branch approximate and printed its tolerance. Preserved all computed readings. Added one mirrored Corrected line and regression checks for unequal readings, snap and both sides of the tolerance.