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How Big Is the Mandelbrot Set?

Written 2026-06-05. Claims re-read against their sources on 2026-09-28: 12 checked, 8 confirmed, 3 wrong, 1 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.

Live page https://artwaste.land/strata/how-big-is-the-mandelbrot-set/ returned 200 and matched the repo apart from the site-wide injected pulse and analytics snippet (held by board note 988ab8). Reader check before the fix: the page pointed readers at /research/mandelbrot-area/series_newton.py and /research/mandelbrot-area/, which are not served (404). The same files are published at /checks/research/mandelbrot-area/pixel.py and series_newton.py (200, identical to the repo), but the page never linked there. Downloaded into an empty directory, they need Python 3 with numpy, which the page did not say. `python3 series_newton.py 65536` exits 0 (9/9 exact rationals, 10/10 checkpoints), and `python3 pixel.py` exits 0 and prints 1.512807, 1.509043, 1.507401. The page's own code was lifted into node. Instrument 03, summed from the live b_coeffs.json (4,097 values), matches every published checkpoint up to 4,096 terms to within 5e-6. Instrument 01, run verbatim, did not do what the prose said: at the default 800 iterations the refine sequence read 1.51180, 1.51163, 1.51104, 1.51127, 1.51128, 1.51132 (flat, and never 1.509 or 1.507). At 10,000 iterations even the coarsest grid gave 1.50660, so the excess was the iteration cap, not the grid. research/mandelbrot-area/verify.mjs asserted nothing and exited 0 on a copy with two sentences falsified. After the fix, verify.mjs asserts 17 checks and passes on the fixed page at 1100px and 390px. It exits 1 on a copy with one prose number changed, and on a copy with the census cap-doubling and the 1.7274 reverted. `series_newton.py 262144` passes, including 240,000 → 1.7274, and exits 1 when that checkpoint is set back to 1.7275.

Claims

• CONFIRMED best estimate 1.5065918849 ± 2.8×10⁻⁹, Thorsten Förstemann, 2012, from roughly 88 trillion sample points
  http://www.mrob.com/pub/muency/areahistory.html : "Thorsten Forstemann used twin Radeon HD 5970 GPUs ... grid size of 2097152×2097152. The resulting area estimate is 1.5065918849(28)"; Bittner et al. (arXiv:1410.1212 p.2): "a resolution of almost 88 trillion pixels"
• CONFIRMED Bittner et al.'s 5,000,000 terms → 1.6829 (1.68288), Involve 10 (2017) 555–572, arXiv:1410.1212
  https://arxiv.org/pdf/1410.1212 : eq. (14) "A5,000,000≈ 1.68288"; Crossref 10.2140/involve.2017.10.555: Involve 10, 555-572, 2017
• CONFIRMED Chen–Kawahira–Li–Yuan 1,000,000 terms → 1.703927
  https://arxiv.org/pdf/1410.1212 : p.2: "they reported the upper bound A1,000,000 = 1.703927"
• CONFIRMED Jay Hill (1997) component areas 1.1780972450961724644 (cardioid), 0.1963495408493620774 (disk), 0.0565424181276813797 (period 3), total 1.506303622
  http://mrob.com/pub/math/jay-hill-2003-area-mandelbrot.html : Table 1 as quoted; dated August 18, 1997
• CONFIRMED in 1992 Keith Briggs wondered whether the area is exactly 3/2
  http://www.mrob.com/pub/muency/areahistory.html : "On 1992 Sep 18, Keith Briggs ... reported his estimate ... 1.499936. He conjectured that the area was exactly 3/2."
• CONFIRMED citations: Ewing & Schober, Numer. Math. 61 (1992) 59–72; Gronwall, Ann. of Math. 16 (1914) 72–76; Bielefeld, Fisher & von Haeseler, Adv. Appl. Math. 14 (1993) 25–38; Buff & Chéritat, Ann. of Math. 176 (2012)
  https://api.crossref.org/works/10.1007/BF01385497 : Crossref records for 10.1007/BF01385497, 10.2307/1968044, 10.1006/aama.1993.1002, 10.4007/annals.2012.176.2.1 match authors, volumes, pages and years
• CONFIRMED the b_m fall off only about as fast as 1/m, no faster
  https://arxiv.org/pdf/1410.1212 : p.2: "Bielefeld, Fisher, and Haeseler [3] proved that no constants ϵ and K exist so that |bm|<K/m^{1+ϵ} for all m"
• CONFIRMED Shishikura: ∂M has Hausdorff dimension 2 (Ann. of Math. 147 (1998) 225–267); whether ∂M has positive area is open
  https://arxiv.org/pdf/1410.1212 : p.1: "Shishikura [18] proved that M has fractal boundary of Hausdorff dimension 2. However, it is unknown whether the boundary has positive Lebesgue measure."
• WRONG Ewing–Schober's 240,000 terms → 1.7275
  https://arxiv.org/pdf/1410.1212 : p.2: "their calculation of A240,000≈ 1.7274"; Munafo's Area History agrees ("Using 240,000 terms, they got an estimate of 1.7274"); this repo's recomputation gives 1.727439, and the same run matches all 22 of Huddleston's published entries (72 to 115,232 terms) to under 5e-6
• WRONG "Jay Hill summed 430,809 of them (periods 1–16) to get a rigorous lower bound of 1.5063"
  http://mrob.com/pub/math/jay-hill-2003-area-mandelbrot.html : periods 1 to 16 hold 65,243 components; the 430,809 include the attached buds, recursed "stopping with bud periods greater than 256, or when the bud radii were smaller than 10-5"; the values are "believed to be correct to about 10-17", computed numerically and not proved, and Hill calls the total a "lower bound", never a rigorous one
• WRONG "I. Bittner, L. Cheong, D. Gates, H. Nguyen"
  https://api.crossref.org/works/10.2140/involve.2017.10.555 : authors: Daniel Bittner, Long Cheong, Dante Gates, Hieu Nguyen (Hieu D. Nguyen on the arXiv PDF)
• UNVERIFIABLE "Jungreis (1985) gave the first effective coefficient algorithm (and an early Gronwall bound around 1.7)"
  https://projecteuclid.org/euclid.dmj/1077304731 : the Duke paper was not reachable; Bittner et al. and the Integers 18 (2018) #A57 paper (read with pdftotext) say only that the coefficients "have been studied in depth, first by Jungreis"; no source found for the algorithm priority or for a bound near 1.7

What was done

• [fixed] Instrument 01 (the census) did not come down as the grid tightened: at a fixed cap the count sits near 1.511 however fine the grid, because the over-count is slow-escaping points the iteration cap has not caught. Also, only cell centres are tested, so it is not true that "every cell that merely touches" the set is counted. Refine now makes the grid 1.4 times finer AND doubles the cap (as pixel.py does), and the buttons pick the starting cap. The sequence is now 1.5118, 1.5092, 1.5081, 1.5070, 1.5072, 1.5067 at the default start (under 2 s per step in node). The prose, readouts, apparatus, meta description and md dek now say what it does. "double the grid" is gone (the code multiplies by 1.4). The claim that boundary cells "barely" shrink is replaced by the measured rate: halving the spacing leaves about 60% of the band's area, not 50%.
• [fixed] Ewing–Schober 240,000 → 1.7274 (was 1.7275) in the Instrument 03 readouts, the plotted checkpoint, the apparatus, the md body, research README, area_ladder.json and series_newton.py's checkpoint table. The README's excuse that the 6e-5 gap was a float64 ceiling is replaced by the evidence that 1.72750 was the misquote.
• [fixed] Hill's 1.5063 is now a "numerical lower bound" from 430,809 components (period up to 16 plus their buds followed out to period 256), not "rigorous" and not "(periods 1–16)". Fixed in Instrument 02, the apparatus, the code comment, the md body and the relates note.
• [fixed] The Bittner et al. first author's initial is now D. (and H. D. Nguyen).
• [fixed] The period-3 figure (0.056542) is Hill's total for three components, not "two bulbs"; the row, readout and comment now say two bulbs plus the cardioid of the copy near −1.75, and "all the rest" is periods ≥4. The unsupported claim that their boundaries are "transcendental" was removed (a hyperbolic component's boundary lies on a real algebraic curve).
• [fixed] Reader paths now link /checks/research/mandelbrot-area/series_newton.py and pixel.py (both 200) and say they need Python 3 with numpy.
• [fixed] "sits thousands of error-bars away" from 3/2 is now "about two million" ((1.5065918849 − 1.5)/2.8×10⁻⁹ ≈ 2.35 million).
• [fixed] The published dots the browser reaches (n ≤ 4,096) are from Scott Huddleston's 1990 table (via Munafo); they were credited to Ewing & Schober and Bittner et al. Huddleston is now credited in the readout, apparatus and sources.
• [fixed] Jungreis cut back to what the sources support: "was the first to study these coefficients".
• [fixed] research/mandelbrot-area/verify.mjs asserted nothing and could not fail. It now asserts 17 checks (cap doubling, grid step, descent, final count near 1.50659, the prose sequence against the live readouts, period-3 value, Instrument 03 against Huddleston at 72/128/1024/4096, the quoted 1.7274 and 1.6829, overflow, no JS errors) and exits 1 on any failure. series_newton.py's docstring no longer says its validation lives in verify.mjs; it now also checks 16384, 115232 and 240000.
• [open note:448509] The generated "How this layer is checked" aside still says the browser check was "Not yet put to that test" and "Last verified 5 June 2026"; verify.mjs now asserts and passes, so the aside should be regenerated. verify.mjs is not published under /checks/ (it needs Playwright and a local server, so it is not a from-an-empty-directory reader check).