---
slug: a-number-that-isnt-whole
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 recheck by Codex, 2026-10-14
checked: 88
confirmed: 56
wrong: 31
unverifiable: 0
observed: 0
stale: 1
---

This was the directed repair pass for the 33 queue-1014c triage items at repository read 0643fce469, with the 55 CONFIRMED claims from the checker re-read and retained. Sources were reopened, the live numerical functions and pinned coastline data were independently exercised, and the layer verifier was rerun. The 31 findings comprise 12 WRONG and 19 MINOR claims; MINOR is included in wrong for the gate arithmetic. X20 is rejected with evidence and recorded as CONFIRMED; X33 is dated under the explicit JSON-LD exception. Only this layer's source, its published verifier copy and its pass artifacts were changed. Image assets and historical logs were not regenerated or edited.

## Claims

- MINOR | measure with a finer ruler and the kilometres keep climbing, with no ceiling | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf; git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html | X1 (coast-no-ceiling): Empirical scaling has a finite range; the sibling names a data floor. Unlimited divergence belongs to the ideal model.
- MINOR | Length that diverges. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf; git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html | X2 (length-diverges): This summary inherits the unqualified physical-coastline claim. The finite-data limitation applies here too.
- WRONG | so to cover it at scale 3⁻⁷ you need exactly 4⁷ boxes | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf; git show 0643fce469:research/fractal-dimension/verify.mjs; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X3 (koch-exact-boxes): Theorem 9.3 establishes a limiting dimension. The old verifier only multiplied segments by four. Stage-eight Koch vertices occupy 19,343 cells of side 3^-7 at offsets (0.123,0.456), exceeding 4^7 = 16,384. Similarity pieces are not exact grid counts.
- WRONG | Switch the ruler between them and watch the slope tilt to each in turn. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X4 (switch-all): The live catalogue has circle, square, Koch, triangle, carpet and Mandelbrot. No Cantor builder or button exists; Cantor is enumerated offline.
- WRONG | the count machine and a one-line calculation agree to the decimal, which is the proof that the fractional dimension is the object's, not the ruler's. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json; https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | X5 (agreement-proof): Live Koch returns 1.282638369941022, displayed as 1.283, versus log4/log3 = 1.261859507142915. The cited theorem supplies exact dimension; finite numerical agreement is not its proof.
- WRONG | The same ruler that nailed the circle, the square, and all four self-similar sets returns about 1.1: barely more than a smooth curve. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X6 (mandel-live-four): There are three live self-similar choices, not four. Koch agrees only approximately; live Mandelbrot is 1.169504745285289.
- MINOR | For the Mandelbrot boundary it is a whole number you cannot | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | X7 (unmeasurable-integer): A finite estimate does not certify the limiting value. This experiment does not prove that estimation is impossible.
- MINOR | it exists only as a proof. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | X8 (only-proof): The theorem establishes the exact value. Numerical estimation and mathematical existence are not restricted to proofs.
- MINOR | length diverges · D ≈ 1.25 | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf; git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html | X9 (card-coast): The historical dimension may remain; divergence needs the ideal-model qualification.
- MINOR | Finer ruler, more kilometres, no ceiling. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf; git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html | X10 (card-no-ceiling): Finite coastline data cannot establish unlimited growth.
- WRONG | The slope is D−1, read off real coastlines. | git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X11 (card-slope): The linked axes are log L against log G, with L proportional to G^(1-D). Its signed slope is 1-D; the west-Britain rerun gives -0.3896210522523713.
- MINOR | Shishikura's 2, a dimension no box-count can reach. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | X12 (card-unreachable): A finite box-count does not certify the exact limit; the source gives no prohibition on a finite estimate returning 2.
- MINOR | Recomputed in your browser and offline (54/54 machine checks pass | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; git show 0643fce469:research/fractal-dimension/verify.mjs | X13 (browser-offline-checks): The 54 assertions are offline. Live Mandelbrot uses R=512 and 300 iterations; offline uses 256,512,1024 and 1000. Cantor is offline only.
- MINOR | checked by direct enumeration, the slopes coming out at the closed forms to nine decimals. | git show 0643fce469:research/fractal-dimension/verify.mjs | X14 (nine-decimals): Carpet enumeration stops at k=6, triangle at k=9 and Cantor at k=12. Their fitted slopes are checked within 1e-9; the old displayed k=7 carpet example was outside the enumerated range.
- MINOR | the small excess is the finite-resolution bias every real box-count carries, and it shrinks as the grid refines. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X15 (bias-shrinks): At unchanged Koch box scales, R=2187 gives 1.2669143543359074 and R=2916 gives 1.2700859466115848. The latter is farther from log4/log3; refinement need not improve every finite estimate.
- WRONG | these are reproduced from real data in the sibling layer and only quoted here. | git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X16 (sibling-reproduces): Pinned Natural Earth data gives divider dimensions 1.3896210522523713 and 1.4143909805829646 for west Britain and Norway; box estimates are 1.1688422873833653 and 1.2872843443382878. Neither reproduces the quoted historical pair 1.25,1.52.
- MINOR | Everything computable under that claim is recomputed above. | git show 0643fce469:research/fractal-dimension/verify.mjs | X17 (everything-recomputed): The suite enumerates specified levels, not every computable detail in the prose. The replacement names its actual calculations and ranges.
- MINOR | every computable claim recomputed live and re-checked in research/fractal-dimension/verify.mjs | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; git show 0643fce469:research/fractal-dimension/verify.mjs | X18 (footer-all-live): Cantor is absent from the live catalogue. The authored footer conflated six live choices with the separate 54-assertion offline suite.
- MINOR | Any small gap is finite-resolution bias, shrinking as the grid refines. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X19 (script-small-gap): The same R=2187 to 2916 counterexample disproves automatic improvement in this dynamic readout.
- CONFIRMED | dimension-2 crinkle lives below any finite render. This one only a proof can reach. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | X20 (script-below-render): REJECTED. In context the readout explicitly concerns the true, exact dimension. Finite images omit finer structure and finite counts alone do not certify a limit. Possible numerical estimation does not contradict this proof distinction. The quoted sentence is retained.
- MINOR | a coastline whose length only diverges | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf; git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html | X21 (dek-coast): The dek repeated the same unlimited physical-coastline implication; qualify the model.
- WRONG | One live instrument, a box-counting ruler, reads 1 off a smooth circle and 2 off a filled square (the integers, as it must), then log4/log3=1.26 off the Koch curve, log3/log2=1.58 and log8/log3=1.89 off the two Sierpinski sets, and even below one (log2/log3=0.63) off the Cantor set. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X22 (dek-instrument): The live catalogue omits Cantor, and Koch returns 1.283 rather than the theoretical 1.2619. The replacement separates live estimates, theoretical dimensions and offline Cantor.
- MINOR | a number no finite box-count can ever reach. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | X23 (dek-unreachable): Lack of exact certification is supported; universal numerical unreachability is not.
- MINOR | Four layers of this place each bump into the same wall: | research/fractal-dimension/README.md; git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html | X24 (meta-four): The README, body links and cards identify three member layers. The separate sound-shift relation does not make four member layers.
- WRONG | There a real coastline is measured with shrinking rulers and the kilometres climb without ceiling; the constant that survives is the log-log slope, D ≈ 1.25. | git show 0643fce469:public/strata/how-long-is-the-coast-of-britain/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X25 (relation-coast-observation): The modern divider value is about 1.390, not historical 1.25; the linked slope is 1-D, not D, and the finite data has a floor. All three parts are distinguished in the replacement.
- WRONG | the same instrument that nails every self-similar set is pointed at that boundary and returns ≈ 1.1 | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json | X26 (relation-instrument): The relationship note repeats absent live Cantor and exact Koch agreement. There are three live self-similar choices, with Koch approximate.
- WRONG | and is shown failing to climb to 2 at any resolution, because the dimension-2 crinkle lives below any finite render. | git show 0643fce469:research/fractal-dimension/verify.mjs | X27 (relation-resolution): The entire offline refinement experiment uses R=256,512,1024 at 1000 iterations. Independently rerun readings are 1.135,1.162,1.109. They establish three observations, not every resolution.
- WRONG | here a box-count holds across object after object, recovering the exact closed form for the self-similar sets (checked) | git show 0643fce469:research/fractal-dimension/verify.mjs | X28 (relation-closedforms): Offline Koch is 1.278451, with tolerance 0.03 against 1.261859507142915. Exact digit enumeration applies to Cantor, triangle and carpet, not this Koch raster estimate.
- WRONG | and lands on each closed form, the fraction proven to be the object's, not the ruler's. | git show 0643fce469:public/strata/a-number-that-isnt-whole/index.html; research/assay-line/queue-1014c/a-number-that-isnt-whole/recheck.json; https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | X29 (md-all-sets): The live instrument neither offers Cantor nor exactly recovers Koch. The theorem, under the open-set condition, establishes the exact dimensions.
- MINOR | Every computable claim is recomputed from scratch in `research/fractal-dimension/verify.mjs` (54/54 pass) | git show 0643fce469:research/fractal-dimension/verify.mjs | X30 (md-all-computable): All 54 offline checks pass. Their enumeration ranges are finite and do not include the formerly displayed seventh-level carpet example.
- MINOR | the real one is not a whole number. | https://annals.math.princeton.edu/articles/12909 | X31 (frame-noninteger): The page itself includes the integer example of Mandelbrot boundary dimension 2. Need not be whole is the supported framing.
- MINOR | there is a clean, honest line between the dimensions a counting machine can recover and the ones only a theorem can reach. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf; https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | X32 (dimension-versus-counting): Self-similar exact dimensions also rest on a theorem. The supported distinction is finite estimates versus proof, not exclusive classes of measurable and unmeasurable dimensions.
- STALE | "dateModified":"2026-06-21" | git log 0643fce469 -- public/strata/a-number-that-isnt-whole/index.html | X33 (modified-date): The June 21 creation date remained in dateModified despite substantive correction in 828996918ff on August 3. This pass sets dateModified to the assigned revision date 2026-10-14; the explicit metadata exception needs no Updated line.
- CONFIRMED | Spain and Portugal have reported their common border as 1,214 km and 987 km, and both are right. | https://archive.org/download/mandelbrot-the-fractal-geometry-of-nature_compress/mandelbrot-the-fractal-geometry-of-nature_compress_djvu.txt | N1:3 border-numbers. Mandelbrot, The Fractal Geometry of Nature, p. 33 records the pair 987 and 1214 km and explains ruler dependence. The page does not assign each number to a country.
- CONFIRMED | the leftover set (the points no circle ever covers) has area exactly zero | https://arxiv.org/pdf/math/0010298 | N1:4 gasket-zero. Theorem 4.2(2): zero Lebesgue area and dimension strictly between 1 and 2.
- CONFIRMED | its boundary is a curve, but one so crinkled that, by a theorem, it is as big as an area. | https://annals.math.princeton.edu/articles/12909 | N1:5 mandel-curve. The abstract establishes dimension 2; the comparison concerns dimension, not positive area.
- CONFIRMED | Area that vanishes but leaves something behind. | https://arxiv.org/pdf/math/0010298 | N1:7 area-vanishes. Theorem 4.2(2): zero Lebesgue area and dimension strictly between 1 and 2.
- CONFIRMED | A curve the size of a region. | https://annals.math.princeton.edu/articles/12909 | N1:8 curve-region. Dimension comparison, supported by the boundary theorem.
- CONFIRMED | the only answer that does not depend on your ruler is an exponent | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:9 only-exponent. Section 2.6(1), in the stated length/area scaling context.
- CONFIRMED | Lay a grid of boxes of side ε over a shape and count how many boxes it touches: N(ε). | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:10 grid-definition. Mesh-count definition, pp. 41–42.
- CONFIRMED | For a smooth thing the count grows like a clean power of 1/ε | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:11 smooth-power. Smooth-manifold box dimension, p. 48.
- CONFIRMED | a line needs N ∝ 1/ε boxes, a filled patch needs N ∝ (1/ε)² | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:12 line-patch-scaling. Line and plane scaling, pp. 41–48.
- CONFIRMED | Plot log N against log(1/ε) and read the slope. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:13 plot-slope. Finite slope estimates the limit, p. 42.
- CONFIRMED | a line that replaces its middle third with two sides of a bump, forever. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:17 koch-construction. Example 9.5 at ratio 1/3.
- CONFIRMED | It is built of four copies of itself, each shrunk by a third | https://arxiv.org/pdf/astro-ph/9803218 | N1:18 koch-copies. The Koch-construction passage gives four segments scaled by 1/3.
- CONFIRMED | log 4 / log 3 = 1.2619… | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:20 koch-value. Theorem 9.3; log4/log3 recomputes as 1.261859507142915.
- CONFIRMED | The number is not an artefact of a coarse grid: the construction proves the dimension is this irrational value, strictly between a line and a plane. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:22 koch-irrational. Theorem 9.3; irrationality also follows because 4^q cannot equal 3^p for positive integers.
- CONFIRMED | The Sierpinski triangle (three half-size copies) is log 3 / log 2 = 1.585 | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:23 triangle-value. Example 9.4; 1.584962500721156.
- CONFIRMED | the Sierpinski carpet (eight third-size copies) is log 8 / log 3 = 1.893 | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:24 carpet-value. Theorem 9.3 with eight third-size copies; 1.8927892607143724.
- CONFIRMED | Cut deeper, to the Cantor set (two third-size copies), and the dimension drops below one: log 2 / log 3 = 0.631, a thing more than a scatter of points and less than a line. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:25 cantor-value. Example 3.3; 0.6309297535714575.
- CONFIRMED | The coast of Britain's west side comes out near D ≈ 1.25 | https://gsp.humboldt.edu/OLM/courses/GSP_510/Articles/Mandelbrot1967.pdf | N1:28 britain-value. Historical west-Britain estimate, Mandelbrot p. 636 and Richardson quotation on p. 637; distinguished from modern measurements.
- CONFIRMED | real public-domain coastlines | https://www.naturalearthdata.com/about/terms-of-use/ | N1:30 coast-public-domain. Natural Earth terms place its map data in the public domain.
- CONFIRMED | a 1.25-dimensional thing has infinite 1-dimensional measure | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:31 dimension-length. Section 2.6(1): Hausdorff measure below the dimension is infinite.
- CONFIRMED | has area zero and dimension D ≈ 1.3057 (McMullen, 1998) | https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf; https://api.crossref.org/works/10.1353/ajm.1998.0031 | N1:32 gasket-dimension. McMullen abstract gives 1.305688; publication year checked against Crossref.
- CONFIRMED | positive dimension is the reason something survives a set of zero area. | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:33 positive-survival. Section 2.6(1): positive Hausdorff dimension suffices for nonemptiness here.
- CONFIRMED | In both, the integer measure you'd reach for is either infinite or zero, and the fraction in between is the only finite, ruler-independent answer. | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:34 only-finite. The critical-exponent interpretation, within the stated scaling problem and ideal coastline model.
- CONFIRMED | The true dimension of that boundary is exactly 2 | https://annals.math.princeton.edu/articles/12909 | N1:36 mandel-two. Abstract: boundary Hausdorff dimension 2.
- CONFIRMED | proved by Mitsuhiro Shishikura in 1991 | https://arxiv.org/abs/math/9201282 | N1:37 shishikura-1991. The arXiv submission history gives 12 April 1991.
- CONFIRMED | The crinkle that makes the dimension 2 lives at scales finer, and at iteration depths deeper, than any finite render reaches. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:39 all-finite-renders. Inference from the limit definition: a finite image omits finer scales.
- CONFIRMED | No box-count of any picture you can draw will ever certify it. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:40 no-certification. Finite counts alone do not prove an asymptotic limit, pp. 41–42.
- CONFIRMED | For the coastline, the gasket, the Koch curve, the dimension is a fraction you can measure. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:41 measurable-fractions. Measure here means numerical estimation, not proof of an exact limit.
- CONFIRMED | the scale-stable answer is always an exponent; that exponent need not be whole | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:44 always-exponent. Section 2.6(1), the critical Hausdorff exponent.
- CONFIRMED | area vanishes · D ≈ 1.306 | https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf; https://api.crossref.org/works/10.1353/ajm.1998.0031 | N1:48 card-gasket. McMullen abstract, 1.305688 rounded to 1.306.
- CONFIRMED | Circles swallow the disk; the dust they miss has zero area and positive dimension. | https://arxiv.org/pdf/math/0010298 | N1:49 card-dust. Theorem 4.2(2): zero Lebesgue area and dimension strictly between 1 and 2.
- CONFIRMED | the wall · D = 2 (theorem) | https://annals.math.princeton.edu/articles/12909 | N1:50 card-theorem. Boundary dimension theorem.
- CONFIRMED | The closed forms are not integers. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:54 forms-not-integers. All four log ratios recomputed by the offline verifier are nonintegers.
- CONFIRMED | Each self-similar set is N copies of itself scaled by 1/s, so D = log N / log s | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:55 self-similar-formula. Theorem 9.3 under the open-set condition for these four constructions.
- CONFIRMED | The Mandelbrot boundary has Hausdorff dimension exactly 2 | https://annals.math.princeton.edu/articles/12909 | N1:72 hausdorff-two. Boundary dimension theorem.
- CONFIRMED | M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Annals of Mathematics 147 (1998) 225–267; announced 1991 | https://annals.math.princeton.edu/articles/12909; https://arxiv.org/abs/math/9201282 | N1:73 reference-shishikura. Annals 147 (1998), issue 2, pp. 225–267; the arXiv record independently supplies 1991.
- CONFIRMED | Box-counting (Minkowski–Bouligand) dimension equals the self-similarity dimension for self-similar sets satisfying the open-set condition | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%209.pdf | N1:74 box-self-similarity. Theorem 9.3 identifies Hausdorff and box dimensions under the stated condition.
- CONFIRMED | Hutchinson 1981 | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:75 reference-hutchinson. Title page: Indiana University Mathematics Journal 30 (1981), 713–747.
- CONFIRMED | Falconer, Fractal Geometry | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf | N1:76 reference-falconer. Chapter footer identifies Falconer, Fractal Geometry, second edition.
- CONFIRMED | The Apollonian residual set has dimension ≈ 1.305688 | https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf; https://api.crossref.org/works/10.1353/ajm.1998.0031 | N1:78 apollonian-precision. McMullen abstract gives the numerical estimate 1.305688.
- CONFIRMED | C. McMullen, Hausdorff dimension and conformal dynamics III, 1998 | https://api.crossref.org/works/10.1353/ajm.1998.0031 | N1:79 reference-mcmullen. Crossref returned the title, author and print date August 1998; the author manuscript is dated 1997.
- CONFIRMED | bounds in D. Boyd 1973 | https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf | N1:80 reference-boyd. McMullen p. 2 gives the Boyd bounds; reference [5] is Mathematika 20 (1973), 170–174.
- CONFIRMED | B. Mandelbrot, How long is the coast of Britain?, Science 156 (1967), from L. F. Richardson's data | https://gsp.humboldt.edu/OLM/courses/GSP_510/Articles/Mandelbrot1967.pdf | N1:81 reference-mandelbrot. Paper header: Science 156 (1967), 636–638; Figure 1 and the text identify Richardson.
- CONFIRMED | Norway value, J. Feder, Fractals, 1988 | https://link.springer.com/book/10.1007/978-1-4899-2124-6 | N1:82 reference-feder. Publisher metadata confirms Feder, Fractals, 1988; the Norway attribution appears in astro-ph/9803218 Figure 8.
- CONFIRMED | a gasket whose leftover dust has zero area | https://arxiv.org/pdf/math/0010298 | N1:104 dek-gasket. Theorem 4.2(2): zero Lebesgue area and dimension strictly between 1 and 2.
- CONFIRMED | a Mandelbrot boundary that is a curve as big as a region. | https://annals.math.princeton.edu/articles/12909 | N1:105 dek-boundary. Boundary dimension theorem; no claim of positive area.
- CONFIRMED | while the truth, by Shishikura's 1991 theorem, is exactly 2 | https://annals.math.princeton.edu/articles/12909; https://arxiv.org/abs/math/9201282 | N1:108 dek-theorem. Annals confirms dimension 2; arXiv independently dates submission to 1991.
- CONFIRMED | the reason a 1.25-dimensional curve has infinite length is made explicit | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:113 relation-length. Conditional on Hausdorff dimension 1.25, the one-dimensional measure is infinite.
- CONFIRMED | residual set has area exactly zero | https://arxiv.org/pdf/math/0010298 | N1:114 relation-apollonian. Theorem 4.2(2): zero Lebesgue area and dimension strictly between 1 and 2.
- CONFIRMED | that set's dimension D ≈ 1.3057 (McMullen 1998) | https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf; https://api.crossref.org/works/10.1353/ajm.1998.0031 | N1:115 relation-apollonian-number. McMullen abstract, rounded to 1.3057; 1998 publication independently checked.
- CONFIRMED | positive dimension is the precise reason something survives a set of measure zero. | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:116 relation-survival. Positive Hausdorff dimension certifies a nonempty residual set here.
- CONFIRMED | At the corpus's wilder fractals (the coastline's 1.25, the gasket's 1.306) the dimension is still a number you can measure. | https://www.ma.imperial.ac.uk/~jswlamb/M345PA46/F03%20chap%203.pdf; https://gsp.humboldt.edu/OLM/courses/GSP_510/Articles/Mandelbrot1967.pdf; https://abel.math.harvard.edu/~ctm/papers/home/text/papers/dimIII/dimIII.pdf | N1:126 md-measured-values. Numerical estimation is supported; Mandelbrot supplies the historical coast value and McMullen the gasket estimate.
- CONFIRMED | Norway, more torn, runs to ≈ 1.52 | https://arxiv.org/pdf/astro-ph/9803218 | N1:127 norway-value. Figures 7–8 identify southern Norway and attribute 1.52 to Feder 1988; a historical finite-scale estimate.
- CONFIRMED | ≈1.52 Norway | https://arxiv.org/pdf/astro-ph/9803218 | N1:128 apparatus-norway. Same Figures 7–8 historical Norway estimate.
- CONFIRMED | the only scale-stable answer is an exponent (the dimension), and it need not be a whole number. | https://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf | N1:130 scale-stable-dimension. Section 2.6(1), read within the stated scaling question.

## Findings

- [fixed] X1 (coast-no-ceiling): Qualified the body coastline growth as finite-range evidence and unlimited growth as an ideal model. MINOR, record only.
- [fixed] X2 (length-diverges): Qualified the compressed divergence summary, including its generated nearby teaser and repeated summaries. MINOR, record only.
- [fixed] X3 (koch-exact-boxes): Replaced exact grid boxes with similarity pieces and the cited dimension theorem; corrected the verifier assertion, explanation and published mirror. Correction C1.
- [fixed] X4 (switch-all): Named Koch, triangle and carpet as the live fractal choices and Cantor as offline. Correction C2.
- [fixed] X5 (agreement-proof): Replaced decimal agreement as proof with approximate Koch agreement and the cited theorem. Corrected repeated exact-agreement claims, including the image-card HTML source. Correction C2.
- [fixed] X6 (mandel-live-four): Removed the claim that the live ruler nailed all four sets; named the implemented choices and used the reproduced live Mandelbrot value about 1.17. Correction C2.
- [fixed] X7 (unmeasurable-integer): Said this finite experiment does not certify the whole-number value. MINOR, record only.
- [fixed] X8 (only-proof): Attributed the exact value to Shishikura without claiming it exists only as proof. MINOR, record only.
- [fixed] X9 (card-coast): Qualified the card as ideal-model divergence with a historical D estimate. MINOR, record only.
- [fixed] X10 (card-no-ceiling): Qualified the card growth as finite-range and the lack of a ceiling as the model. MINOR, record only.
- [open note:875363] X11 (card-slope): This layer's signed coastline slope is corrected to 1-D with its axes named in Correction C3. The same sign error remains in the authored heading at public/strata/how-long-is-the-coast-of-britain/index.html:293, which says the slope is D - 1; its plotted axes and lines 296-298 give 1-D. That other layer is outside this repair's edit scope and needs a board note.
- [fixed] X12 (card-unreachable): Replaced the card impossibility claim with lack of certification from these finite counts. MINOR, record only.
- [fixed] X13 (browser-offline-checks): Separated live estimates from the 54 offline assertions and stated the distinct Mandelbrot parameters. MINOR, record only.
- [fixed] X14 (nine-decimals): Used the actually enumerated sixth-level carpet example, 8^6 cells at resolution 3^6, and named all enumeration ranges and the 1e-9 tolerance. MINOR, record only.
- [fixed] X15 (bias-shrinks): Qualified refinement error in the apparatus and research README; it depends on rasterisation and fitted scales and need not improve at each refinement. MINOR, record only.
- [fixed] X16 (sibling-reproduces): Distinguished historical coastline estimates from modern measurements in body, apparatus and research README; inserted divider values from the pinned data. Correction C4.
- [fixed] X17 (everything-recomputed): Replaced universal checking prose with the listed calculations and enumeration ranges. MINOR, record only.
- [fixed] X18 (footer-all-live): Replaced the universal live footer claim with six live choices and a separate offline suite. MINOR, record only.
- [open note:fd889d] X19 (script-small-gap): The dynamic refinement explanation is corrected. research/fractal-dimension/shot-koch.png, shot-tri.png and shot-carpet.png still display the unqualified shrinking-error sentence; OCR also confirms the triangle's bottom lines. README.md labels those captures as predating the 2026-10-14 correction and gives the qualification supported by the Koch refinement counterexample. Image changes are prohibited; the three retained copies need a board note. MINOR, record only.
- [fixed] X21 (dek-coast): Qualified coastline divergence in the dek, JSON-LD, Markdown summary and image-card source. The public OG bitmap shows only a truncated old dek fragment, not the complete disputed assertion; share card rebuilt at landing from the corrected dek. MINOR, record only.
- [fixed] X22 (dek-instrument): Separated actual live readings from theoretical dimensions and offline Cantor in the dek and metadata. Correction C2.
- [fixed] X23 (dek-unreachable): Replaced the dek and JSON-LD absolute impossibility wording with lack of exact certification. MINOR, record only.
- [fixed] X24 (meta-four): Changed Four to Three in meta description and JSON-LD. MINOR, record only.
- [fixed] X25 (relation-coast-observation): Corrected the coastline relationship: finite range, signed slope 1-D, historical 1.25 and modern divider 1.390. Corrections C3 and C4.
- [fixed] X26 (relation-instrument): Corrected the Mandelbrot relationship to name actual live coverage and approximate Koch agreement. Correction C2.
- [open note:e70da8] X27 (relation-resolution): Limited relationship, body, readout, apparatus, README and verifier explanations to the three tested resolutions and 1000 iterations. research/fractal-dimension/shot-mandel.png still displays the unsupported general refinement claim. README.md labels the capture as predating the 2026-10-14 correction and states the tested resolutions, iteration count and readings. Image changes are prohibited; the retained copy needs a board note. The rejected X20 sentence remains intact. WRONG, answered on the page by Correction C5.
- [fixed] X28 (relation-closedforms): Limited exact closed-form recovery to digit-defined sets and described Koch as approximate in the sound-shift relationship. Correction C2.
- [fixed] X29 (md-all-sets): Corrected the Markdown body to distinguish the three live fractals, offline Cantor, approximate Koch and exact theorem. Correction C2.
- [fixed] X30 (md-all-computable): Named the actual offline calculations and enumeration ranges in the Markdown body. MINOR, record only.
- [fixed] X31 (frame-noninteger): Changed the frame to need not be a whole number. MINOR, record only.
- [fixed] X32 (dimension-versus-counting): Distinguished finite estimates from exact theorems across repeated framing and verifier comments. MINOR, record only.
- [dated] X33 (modified-date): Set JSON-LD dateModified to 2026-10-14 for this substantive pass. The five Corrected lines date the revision; the user explicitly exempts this metadata update from an Updated line.

Correction C1 answers X3. C2 answers X4, X5, X6, X22, X26, X28 and X29. C3 answers X11 and the slope part of X25. C4 answers X16 and the historical-data part of X25. C5 answers X27. All five lines are mirrored at the end of the Markdown body. No public correction line was added for a MINOR item, and no Updated line was added for X33.

Repair recheck, 2026-10-14: The Mandelbrot, Koch, triangle and carpet screenshots retain the disputed sentences; their README references now carry the corrections and limits of the evidence. No image was rendered or changed. The coastline sibling heading is also left for its own layer. The three PENDING dispositions require real board note ids from the directing instance; no id has been invented. Counts remain 56 CONFIRMED + 12 WRONG + 19 MINOR + 1 STALE = 88. All five dated corrections were checked against the original wording, claims record and reproduced evidence and remain unchanged. See research/assay-line/queue-1014c/a-number-that-isnt-whole/repair-report.md for the complete review and note text. The public share card is rebuilt at landing from the corrected dek. Shared publication metadata is also rebuilt at landing. The checker's two UNVERIFIABLE causal statements about difficulty measuring Mandelbrot area were not triaged; this pass neither edits nor certifies them.