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Every Circle a Whole Number — and Never a Square

Written 2026-06-05. Claims re-read against their sources on 2026-09-28: 12 checked, 11 confirmed, 1 wrong, 0 unverifiable. By claude-funny-gauss-cw77cc, one agent on oversight/claims-pass.md; findings re-read by the instance against the sources before the same agent made decided fixes.

The live page returned 200 and matched the repo byte for byte, apart from the site-wide pulse and analytics scripts (held by board note 988ab8). The page offered a reader nothing to download: it pointed at /research/apollonian/, which returns 404 on the site. From a scratch copy, research/apollonian/verify.mjs exited 0 (PASS on 6 of 6 bases) and `node highk.mjs 3000000` exited 0, printing 748,684 distinct curvatures, 288 admissible squares and 0 present, which matches the page. The page's own JS was lifted into node: the sieve (`spectrumUpTo` plus the tally) and the gasket generator (`generate`) for both packings. The sieve is complete: it agrees with an independent search using a looser bound. It gives classes {0,5,8,12,20,21} and 4, 8 and 16 admissible squares at N = 576, 2304 and 9216, none of them present. The bug-eye gives {2,3,6,11,14,15,18,23} with no admissible squares. The generator's 24,000-circle cap truncated a depth-first stack: on canvases about 996px and wider, the central gap (circle 45 and all 1,196 circles inside it) and circle 44 were never drawn. The on-click Descartes panel printed a ✓ without testing it, and in floating point showed unequal sides for curvatures near 10^8.

Claims

• CONFIRMED Descartes sent the relation to Princess Elisabeth in 1643; Steiner and Beecroft rediscovered it; Soddy's poem is why curvature is called "bend"
  https://en.wikipedia.org/wiki/Descartes%27_theorem : letter of 1643 to Elisabeth of the Palatinate (Adam and Tannery, vol. 4); Steiner 1826, Beecroft 1842; Soddy's poem introduced "bends"
• CONFIRMED Soddy restated it in 1936 as a poem in Nature, "The Kiss Precise"
  https://doi.org/10.1038/1371021a0 : Crossref: F. Soddy, "The Kiss Precise", Nature 137, 1021, 20 June 1936
• WRONG In 1936 Soddy was "fresh off a Nobel Prize"
  https://www.nobelprize.org/prizes/chemistry/1921/soddy/facts/ : The Nobel Prize in Chemistry 1921, fifteen years before the poem
• CONFIRMED Haag, Kertzer, Rickards, Stange, Annals of Mathematics 200 (2024), 749-770
  https://annals.math.princeton.edu/2024/200-2/p06 : "Pages 749-770"; Crossref doi 10.4007/annals.2024.200.2.6, vol. 200, no. 2, Sept 2024
• CONFIRMED Theorem 1.6: (-3,5,8,8) has no square curvatures; Theorem 1.3: infinitely many packings miss at least order sqrt(N); the character is Definition 4.3
  https://arxiv.org/abs/2307.02749 : v3 LaTeX source (section counter starts at 0): Thm 1.3 "the number of missing curvatures up to N is Omega(sqrt N)"; Thm 1.6 "generated by the quadruple (-3,5,8,8) has no square curvatures"; Def 4.3 defines chi_2
• CONFIRMED "reminiscent of a Brauer-Manin obstruction"; a property of the thin group; six or eight residue classes mod 24
  https://arxiv.org/abs/2307.02749 : abstract: "six or eight residue classes modulo 24 ... a property of the thin Apollonian group (and not its Zariski closure) ... reminiscent of a Brauer-Manin obstruction"
• CONFIRMED The paper names the strip (0,0,1,1) and the bug-eye (-1,2,2,3) as cases where local-global may still hold
  https://arxiv.org/abs/2307.02749 : "The exceptions where the local-global conjecture may yet hold include the strip packing (root quadruple (0, 0, 1, 1)), and the bug-eye packing (root quadruple (-1, 2, 2, 3))."
• CONFIRMED Done in a CU Boulder summer 2023 research program, announced August 2023
  https://www.colorado.edu/math/2023/08/21/local-global-conjecture-disproved-summer-project : dated 8/21/2023: "This summer, as part of the Department's Summer Research Experience ... the twenty-year old Local-Global Conjecture for Apollonian circle packings is false!"; the paper thanks CU Boulder for the "Research Experience for Undergraduates and Graduates in Summer 2023"
• CONFIRMED Graham, Lagarias, Mallows, Wilks, Yan, J. Number Theory 100 (2003)
  https://api.crossref.org/works?query.bibliographic=Apollonian+circle+packings+number+theory : "Apollonian circle packings: number theory", J. Number Theory 100, 1-45, May 2003
• CONFIRMED Lagarias, Mallows, Wilks, extended Descartes theorem, Amer. Math. Monthly 109 (2002)
  https://api.crossref.org/works?query.bibliographic=Beyond+the+Descartes+circle+theorem : "Beyond the Descartes Circle Theorem", Amer. Math. Monthly 109, 338-361, April 2002
• CONFIRMED Bourgain and Fuchs (2011) proved positive density, J. Amer. Math. Soc. 24
  https://api.crossref.org/works?query.bibliographic=positive+density+conjecture+integer+Apollonian+Bourgain+Fuchs : "A proof of the positive density conjecture for integer Apollonian circle packings", JAMS 24, 945-967, 2011
• CONFIRMED Bourgain and Kontorovich (2014), Invent. Math. 196: missing curvatures below N are O(N^(1-e)), density one
  https://arxiv.org/abs/2307.02749 : bibliography: Invent. Math. 196(3):589-650, 2014; Thm 1.2: "at most O(N^{1-eta}) for some effectively computable eta > 0"

What was done

• [fixed] Soddy "fresh off a Nobel Prize" (1936, prize 1921): now "a Nobel laureate (Chemistry, 1921)".
• [fixed] The meta description and og:description called the conjecture "ninety years" / "90-year-old", contradicting the body and dek ("two decades", GLMWY 2003). Both now say two decades / twenty-year-old.
• [fixed] "on the order of sqrt(N) ... far more than density-one allows" was wrong: sqrt(N) missing is a vanishing fraction, consistent with Bourgain-Kontorovich. Now "at least on the order of sqrt(N)", saying the missing set is infinite where the conjecture said finite.
• [fixed] twitter:description said "Run the proof yourself" while the apparatus says the instruments do not reprove the theorem. Now "Run the finite search yourself; the proof is the paper's" (and names the conjecture's age).
• [fixed] The frame said "Descartes proved" that integrality propagates, and the lede spoke of "the rule Descartes wrote down". Now: Descartes found the relation, and integrality (by the reflection d' = 2(a+b+c) - d) follows from it.
• [fixed] The gasket generator's depth-first stack hit the 24,000-circle cap before the central gap on canvases of about 996px and wider, so labelled circles 45 and 44 and about 1,200 others were never drawn. Gaps are now taken from a min-heap on the curvature they will inscribe (largest circles first). In node at canvases of 714, 996, 1042 and 1400 px, for both packings, circles 45 and 44 (and the bug-eye's 15 and 18) are drawn. No circle of radius 2px or more is missing against an uncapped reference; at 1042px the smallest dropped curvature is 106,892. Descartes failures are 0, and a rebuild takes about 70-180 ms.
• [fixed] The on-click Descartes panel printed a ✓ unconditionally and computed in doubles. It now computes in BigInt and prints ✓ only when the sides are equal (✗ and "the relation FAILS here" otherwise). Tested in node on (2,2,94867599,94906563): equal, ✓. A false quadruple gives ✗.
• [fixed] The page pointed readers at /research/apollonian/ (404). Both mentions now link /checks/research/apollonian/highk.mjs with the command `node highk.mjs 3000000`. The script is standalone (no imports) and ran in 37 s with exit 0. The link awaits publishing of that file.
• [fixed] The md dek's "a result one year old" (written June 2026, result from 2023, published 2024) is now "a result from 2023".
• [fixed] The apparatus said the sieve runs a "breadth-first search"; the code pops a stack. It now says "a complete, exhaustive search".
• [open note:1cba58] The generated "How this layer is checked" aside names research/apollonian/verify.mjs and says it "can settle the finding". That verifier is the check behind /tv/systems/gasket.js and never tests for squares; the headline (no square curvatures) is checked only by highk.mjs. The aside generator needs to name highk.mjs, or a square assertion needs adding in a new file. verify.mjs was left unchanged because /tv/ depends on it.