the assay office / record

Double the Square

Written 2026-06-20. Claims re-read against their sources on 2026-09-27: 8 checked, 4 confirmed, 4 wrong, 0 unverifiable. By claims-audit controls 2026-09-27 (audit agent + fix agent).

All five audit findings were re-read against the sources by the fix agent before editing. Verifier research/geometric-irrationality-root2/verify.mjs: ALL CHECKS PASS before and after (only a comment changed in it).

Claims

• WRONG the overlapping-squares descent is "the Greeks' geometric proof" (dek, og:description, <title> and scripts/seo-titles.tsv)
  https://www.cut-the-knot.org/proofs/sq_root.shtml : cut-the-knot: "Conway attributes the proof to the Princeton mathematician Stanley Tennenbaum ... in the early 1950s"; so does Conway & Shipman, Math. Intelligencer 35 (2013) 2-7, doi 10.1007/s00283-013-9373-9; the page's own Sources said Tennenbaum
• WRONG Conway popularized the proof, "e.g. Conway & Guy, The Book of Numbers, 1996"
  https://www.cut-the-knot.org/proofs/sq_root.shtml : cut-the-knot and divisbyzero.com name Conway's chapter "The Power of Mathematics" in Power (Blackwell & MacKay eds., CUP, doi 10.1017/CBO9780511541407.004); The Book of Numbers pp. 183-185 gives a paper-folding proof to the same effect. The book itself was not opened
• MINOR the diagonal is "the first length ever shown to be no fraction at all"
  https://en.wikipedia.org/wiki/Hippasus : tradition, not settled: the pentagon's diagonal has been argued to be the first (von Fritz 1945); counted as wrong for stating a contested priority as fact
• WRONG "Plato's slave boy does it in a single stroke"
  https://www.gutenberg.org/cache/epub/1643/pg1643.txt : after two wrong guesses, Socrates introduces "this line, reaching from corner to corner" and the boy assents
• CONFIRMED the doubling construction and slave-boy scene are Meno 82b-85b
  https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0178%3Atext%3DMeno%3Asection%3D84e : the "drawn from corner to corner" line sits at 84e, inside the range
• CONFIRMED Aristotle alludes to the odd-even reductio in Prior Analytics I.23
  https://classics.mit.edu/Aristotle/prior.1.i.html : Part 23: the diagonal is incommensurate because odd numbers would equal evens
• CONFIRMED the area identity a² = 2b² − (2b−a)² + 2(a−b)² holds for all integers
  research/geometric-irrationality-root2/verify.mjs : expanded by hand; the verifier's finite grid is also a proof (degree ≤ 2 per variable, 81 points per variable)
• CONFIRMED √2 = [1;2,2,2,...], Pell convergents 1/1, 3/2, 7/5, 17/12 with p² − 2q² = ±1
  research/geometric-irrationality-root2/verify.mjs : exact BigInt checks, 2000 partial quotients

What was done

• [fixed] "The Greeks' geometric proof" now "Tennenbaum's geometric proof" in the dek (src/content) and og:description; the <title> and its scripts/seo-titles.tsv line now read "Why Is √2 Irrational? The Overlapping-Squares Proof"; dated correction line added under Sources
• [fixed] The Book of Numbers citation replaced by Conway, "The Power of Mathematics" (2005) and Conway & Shipman (2013); the correction line says the Book of Numbers gives a paper-folding proof (per cut-the-knot; book not opened)
• [fixed] "the first length ever shown to be no fraction at all" now marked "by tradition" in the dek and the header frame; the correction line names the pentagon alternative
• [fixed] "does it in a single stroke" replaced: Socrates draws the diagonal for the boy, who then sees why it works; the Meno paragraph now says the boy agrees after Socrates draws the corner-to-corner line
• [declined] "the verifier's all-integers identity check is a finite box": the difference of the two sides is a polynomial of degree at most 2 in each variable, so vanishing on [−40,40]² (and C2's [−30,30]²) proves it for all integers; the claim stands. The verifier's comment, which said "symbolically", now gives that argument