---
slug: the-three-gap-theorem
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 and a Claude agent
checked: 133
confirmed: 89
wrong: 37
unverifiable: 0
observed: 6
stale: 1
---

This was the directed repair pass for the-three-gap-theorem at 41fdb2b77b, using queue-1014c's N1 source record, the Codex and Claude checker records, and all 40 triage decisions. The fixer re-read each item's evidence before editing, reran the mathematical verifier, and added page regressions. The record retains one line per triage item and every supported CONFIRMED claim from the checkers; overlapping shorthand is stated with the repaired scope rather than reconfirmed as false. MINOR findings count in wrong only for bookkeeping and receive no page correction. Two rejected findings remain CONFIRMED; the two shared placard findings remain WRONG under note 81816d. Five dated corrections are mirrored in HTML and the markdown body, and the JSON-LD date is updated under its explicit exception.

## Claims

- WRONG | X1: “each step replaces a long arc with two shorter ones drawn from the lengths already present.” | https://doi.org/10.1017/S1446788700031062; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | van Ravenstein p. 366 permits a new length. Re-evaluating the page at α=1/4 gives lengths {1/4,3/4} at N=2 and {1/4,1/2} at N=3.
- MINOR | X2: “it is the best fraction you could use to fake α.” | https://doi.org/10.1017/S1446788700031062 | van Ravenstein p. 368 defines approximation of the second kind. At the default N=13, p/q=3/8 while the smallest gap is 0.0557280900; the gap is the minimum scaled error, not the fraction.
- MINOR | X3: “the error of α's best rational approximation with denominator below N.” | https://doi.org/10.1017/S1446788700031062 | For α=√2−1 and N=4, 1/2 gives scaled error 0.1715728753 versus 0.2426406871 for 1/3; ordinary errors instead are 0.0857864376 and 0.0808802290. van Ravenstein p. 368 specifies the second kind.
- MINOR | X4: “the one whose denominators have to grow fastest before they catch it.” | https://arxiv.org/pdf/1904.10815 | Akiyama p. 4 gives q_(k+1)=a_(k+1)q_k+q_(k-1). All ones give the slowest denominator growth per convergent index. The AMS/Stony Brook account supports the extremal Hurwitz constant.
- WRONG | X5: “the proof that the count is never four lives in an offline verifier” | git show 37137afa5c:research/three-gap/verify.mjs | Re-read the complete verifier: the exhaustive loop covers reduced 0<p<q, 2≤q≤220, 1≤N≤q, then 4,000 seeded random cases. It contains no universal proof. Haynes and Marklof p. 1 and van Ravenstein supply the mathematical result.
- MINOR | X6: “exactly two lengths at every Fibonacci count” | git show 37137afa5c:research/three-gap/verify.mjs; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | The verifier tests k=4 through 30: 27 counts F4=3 through F30=832040. At N=1 the page returns one gap of one turn; the true golden-angle two-length statement needs Fibonacci N≥2.
- WRONG | X7: “the page proves the theorem for fractions, exhaustively” | git show 37137afa5c:research/three-gap/verify.mjs | Re-read the complete verifier: the exhaustive loop covers reduced 0<p<q, 2≤q≤220, 1≤N≤q, then 4,000 seeded random cases. It contains no universal proof. Haynes and Marklof p. 1 and van Ravenstein supply the mathematical result.
- MINOR | X8: “its gap ratio is bounded for all N while a near-rational angle's runs away.” | https://arxiv.org/pdf/1904.10815; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Akiyama Corollary 5 concerns badly approximable irrational turns. For the rational 2500001/10000000, N=5 has gaps in units {4,2499997,2500001}/10000000, giving ratio 625000.25. At N=10000000 all distinct marks are equally spaced.
- WRONG | X9: “Running it needs the repository around it.” | https://artwaste.land/checks/research/three-gap/verify.mjs | The verifier imports nothing and reads no files. The manifest reason is a 45-second timeout; scripts/check-placard.mjs derives a repository-dependency sentence from that read-only status. Both source checkers ran the published file alone, 9/9. The updated byte-identical local source also runs 9/9.
- WRONG | X10: “here that is exactly why its marks stay evenest at every count” | https://arxiv.org/pdf/1904.10815; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Akiyama gives uniform bounds, not a fixed-count optimizer. Re-evaluation at N=3 gives golden ratio of gaps φ≈1.6180339887; α=√2/4 instead gives (1+√2)/2≈1.2071067812.
- MINOR | X11: “exact, live arithmetic” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | The inline code uses Number, Math.sqrt(5) and a 1e-9 tolerance. The reverse relation originated in another layer and reversed here/there on this page.
- MINOR | X12: “with the longest always the sum of the other two.” | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | X13: “every value computed live and checked exactly.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html; git show 37137afa5c:research/three-gap/verify.mjs | The live code uses floating-point values and a 1e-9 grouping tolerance. The separate verifier reads no displayed values; its golden-proxy ratio comparison uses Number division and a 1e-9 tolerance.
- MINOR | X14: “The smallest gap is α's best fraction” | https://doi.org/10.1017/S1446788700031062 | van Ravenstein p. 368 defines approximation of the second kind. At the default N=13, p/q=3/8 while the smallest gap is 0.0557280900; the gap is the minimum scaled error, not the fraction.
- MINOR | X15: “the smallest gap IS the best fraction.” | https://doi.org/10.1017/S1446788700031062 | van Ravenstein p. 368 defines approximation of the second kind. At the default N=13, p/q=3/8 while the smallest gap is 0.0557280900; the gap is the minimum scaled error, not the fraction.
- MINOR | X16: “the longest equal to the sum of the other two” | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | X17: “the longest is always the sum of the other two.” | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | X18: “the longest is the sum of the other two.” | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | X19: “the longest equal to the sum of the other two.” | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- WRONG | X20: “the seeds fall into spokes, leaving the head wastefully gapped” | https://arxiv.org/pdf/1904.10815; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Akiyama Theorem 2 includes real quadratic irrationals such as √2−1 among angles producing Delone spiral sets. The old code assigned the same spokes caption to every non-golden preset.
- WRONG | X21: “‖qα‖ = Infinity = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | At N=1, structure() returns gaps=[1] but bestApprox() has no q<N candidate and leaves err=Infinity. The old readout asserted equality to the smallest gap despite there being no pair.
- CONFIRMED | K1: [fourth-colour] A fourth colour never appears. | public/strata/the-three-gap-theorem/index.html @37137afa5c | REJECTED with evidence: the palette really has three colours, and the independent length-count readout already has a >3 alarm. The prose never claims palette capacity itself is an independent falsification test. This matches the triage rejection.
- WRONG | K2: [old-lengths-reused] That is why no arc is ever left absurdly large, and why a fourth length can't accumulate: each step replaces a long arc with two shorter ones drawn from the lengths already present. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EA75E140919DEA9A55FEFD01EB2F677F/S1446788700031062a.pdf/the-three-gap-theorem-steinhaus-conjecture.pdf | van Ravenstein p. 366 permits a new length. Re-evaluating the page at α=1/4 gives lengths {1/4,3/4} at N=2 and {1/4,1/2} at N=3.
- MINOR | K3: [denominators-fastest] the one whose denominators have to grow fastest before they catch it. | https://arxiv.org/pdf/1904.10815 | Akiyama p. 4 gives q_(k+1)=a_(k+1)q_k+q_(k-1). All ones give the slowest denominator growth per convergent index. The AMS/Stony Brook account supports the extremal Hurwitz constant.
- WRONG | K4: [offline-exact] the proof that the count is never four lives in an offline verifier that works in exact integer arithmetic | research/three-gap/verify.mjs @37137afa5c | Re-read the complete verifier: the exhaustive loop covers reduced 0<p<q, 2≤q≤220, 1≤N≤q, then 4,000 seeded random cases. It contains no universal proof. Haynes and Marklof p. 1 and van Ravenstein supply the mathematical result.
- CONFIRMED | K5: [splitting-cases] The new point always lands in a current largest gap, which never grows -- 2,166/2,166. | recreated: curl https://artwaste.land/checks/research/three-gap/verify.mjs into an empty dir; node verify.mjs (2026-10-14, exit 0, 9/9) | REJECTED with evidence: splitting any gap cannot increase the maximum gap. That general invariant remains a valid implementation check; the verifier separately tests whether the host was a largest gap. The run again gives 2166/2166 for both and 9/9 overall, as triage anticipated.
- MINOR | K6: [golden-and-round-check] Golden-angle gap ratio bounded by φ² for all N; a round angle blows past 6·10⁵×. | recreated: curl https://artwaste.land/checks/research/three-gap/verify.mjs into an empty dir; node verify.mjs (2026-10-14, exit 0, 9/9) | The source builds 714 samples N=5,12,…,4996 for F41/F42. It converts exact integer lengths to Number for the ratio and compares with φ²+1e-9. The checker note's endpoint 4999 was itself inaccurate; triage correctly says 4996.
- WRONG | K7: [fractions-exhaustive] the page proves the theorem for fractions, exhaustively | research/three-gap/verify.mjs @37137afa5c | Re-read the complete verifier: the exhaustive loop covers reduced 0<p<q, 2≤q≤220, 1≤N≤q, then 4,000 seeded random cases. It contains no universal proof. Haynes and Marklof p. 1 and van Ravenstein supply the mathematical result.
- WRONG | K8: [repository-needed] Running it needs the repository around it. | https://artwaste.land/checks/research/three-gap/verify.mjs | The verifier imports nothing and reads no files. The manifest reason is a 45-second timeout; scripts/check-placard.mjs derives a repository-dependency sentence from that read-only status. Both source checkers ran the published file alone, 9/9. The updated byte-identical local source also runs 9/9.
- MINOR | K9: [nearby-exact-live] Both take an infinite process and pin it down with exact, live arithmetic instead of hand-waving: there, an endless walk around a circle yields only ever three gap… | git show 37137afa5c:src/content/strata/does-point-nine-equal-one.md | The inline code uses Number, Math.sqrt(5) and a 1e-9 tolerance. The reverse relation originated in another layer and reversed here/there on this page.
- MINOR | K10: [dek-theorem] Drop a mark at every equal step around a circle and the arcs between your marks come in at most three lengths (never a fourth) with the longest always the sum of the other two. | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | K11: [body-theorem] The three-gap (three-distance) theorem: place the points {kα} for k = 0…N−1 on the unit circle and the arcs between cyclically-consecutive points take at most three distinct lengths, the longest equal to the sum of the other two -- for every real α and every N. | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | K12: [description-2] No matter how many points, the gaps between them take at most three different lengths, and the longest is always the sum of the other two. | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | K13: [og-description-2] The gaps between them take at most three lengths, and the longest is the sum of the other two. | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- MINOR | K14: [structured-description-1] The three-gap (Steinhaus) theorem made playable: place points at the multiples of any number around a circle and the arcs between consecutive points take at most three distinct lengths, the longest equal to the sum of the other two. | https://arxiv.org/pdf/1904.10815 | Akiyama Theorem 4 allows gap classes with zero multiplicity. Re-evaluating α=1/4,N=4 gives one occurring length, 1/4. The page's main statement and sum-test guard already require three lengths.
- STALE | K15: [modification-metadata] "dateModified":"2026-06-21" | git log 37137afa5c -- public/strata/the-three-gap-theorem/index.html | The old JSON-LD dateModified was 2026-06-21. Read-only git history shows subsequent page edits, including 2026-10-01. This repair is dated 2026-10-14.
- WRONG | K16: [off-golden-spokes] the seeds fall into spokes, leaving the head wastefully gapped | public/strata/the-three-gap-theorem/index.html @37137afa5c | Akiyama Theorem 2 includes real quadratic irrationals such as √2−1 among angles producing Delone spiral sets. The old code assigned the same spokes caption to every non-golden preset.
- WRONG | K17: [readout-ring-n-1] 1 distinct gap length ≤ 3 ✓short 360.000° × 1closest pair: 1 steps apart → best fraction 1/1, ‖qα‖ = Infinity = the smallest gap | public/strata/the-three-gap-theorem/index.html @37137afa5c, recreated with a stub DOM (readouts.mjs) | At N=1, structure() returns gaps=[1] but bestApprox() has no q<N candidate and leaves err=Infinity. The old readout asserted equality to the smallest gap despite there being no pair.
- MINOR | K18: (added) "<title>The Three-Gap Theorem: Why Marks on a Circle Have 3 Gaps</title>" | public/strata/the-three-gap-theorem/index.html @37137afa5c | Four equally spaced quarter-turn marks give four gaps of one length. The theorem bounds distinct lengths, not the number of gaps.
- MINOR | K19: (added) the link "Vogel's rule" (line 143) goes to https://en.wikipedia.org/wiki/Phyllotaxis | https://en.wikipedia.org/wiki/Phyllotaxis | The linked Phyllotaxis article has no Vogel match. Re-fetched The Algorithmic Beauty of Plants, read printed pp. 99-100 (PDF pp. 111-112), and verified equation (4.1): angle n times the divergence angle and radius c√n.
- CONFIRMED | N2-1 (specimen-history): “THE THREE-GAP THEOREM · STEINHAUS'S CONJECTURE · CONJ. HUGO STEINHAUS · PROVED SÓS 1957 · SURÁNYI 1958 · ŚWIERCZKOWSKI 1959” | https://arxiv.org/pdf/1707.04094 | Entry 1, specimen-history: Haynes and Marklof, p. 1: "in 1957 by Sós", "in 1958 by Surányi", "in 1959 by Świerczkowski"; van Ravenstein p. 360 attributes the conjecture to Steinhaus.
- CONFIRMED | N2-2 (opening-three-lengths): “Wherever you stop” [...] “three steps in, three hundred” [...] “the gaps between your marks come in at most three lengths. Never a fourth.” | https://arxiv.org/pdf/1707.04094 | Entry 2, opening-three-lengths: The source states "for all α ∈ R" and "G(α, D) ≤ 3". This includes each finite stopping point.
- CONFIRMED | N2-3 (fractional-part-points): “You are dropping the points {α}, {2α}, {3α}, …” [...] “the fractional parts of the multiples of α” [...] “around the rim.” | https://arxiv.org/pdf/1707.04094 | Entry 3, fractional-part-points: The source abstract explicitly describes "the fractional parts of α, 2α, . . . , N α". The chosen excerpt alone omits this definition; the preceding source text supplies it.
- CONFIRMED | N2-4 (theorem-all-choices): “However many points you have placed, and whatever α you chose, the arcs come in no more than three distinct lengths” | https://arxiv.org/pdf/1707.04094 | Entry 4, theorem-all-choices: The source states "for all α ∈ R" and "any interval D"; this is the claimed arbitrary-step, arbitrary-count theorem.
- CONFIRMED | N2-5 (three-length-sum): “when there are exactly three, the longest is precisely the sum of the other two.” | https://arxiv.org/pdf/1904.10815 | Entry 5, three-length-sum: Akiyama Theorem 4 gives lengths ηk, ηk−1−sηk, ηk−1−(s−1)ηk; the third is the sum of the first two whenever all three occur.
- CONFIRMED | N2-6 (steinhaus-noticed): “Hugo Steinhaus noticed it” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EA75E140919DEA9A55FEFD01EB2F677F/S1446788700031062a.pdf/the-three-gap-theorem-steinhaus-conjecture.pdf | Entry 6, steinhaus-noticed: Van Ravenstein p. 360: "originally a conjecture of H. Steinhaus".
- CONFIRMED | N2-7 (three-proofs-1950s): “it took three separate proofs in the late 1950s to nail it down.” | https://arxiv.org/pdf/1707.04094 | Entry 7, three-proofs-1950s: The cited paper records three first proofs, in 1957, 1958 and 1959. The sentence is a summary of that independent-proof history.
- CONFIRMED | N2-8 (circle-draws): “Below, the circle draws itself.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 8, circle-draws: Pinned HTML lines 275-337 computes the positions and draws arcs and marks; local Node DOM/canvas stubs successfully execute the complete inline script.
- CONFIRMED | N2-9 (colours-by-length): “The arcs are coloured by length: short, medium, long.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 9, colours-by-length: Pinned HTML: "var COL = ['#ffc94a', '#59d3c4', '#f0608f'];" and "rx.strokeStyle = COL[Math.min(s.cls[i],2)];"; colours correspond to sorted length classes.
- CONFIRMED | N2-10 (fourth-colour): “A fourth colour never appears.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 10, fourth-colour: Pinned HTML defines three arc colours and caps the index at 2. The actual length-count readout is separate and can signal a count above three; the mathematical theorem supports the intended statement.
- CONFIRMED | N2-11 (circle-point-index): “{kα} for k = 0 … N−1” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 11, circle-point-index: Pinned HTML: "for (var k=0;k<N;k++) raw[k] = frac(k*a);". This includes the origin; translating all marks by α gives the introductory convention.
- CONFIRMED | N2-12 (new-point-largest): “First, the new point always falls into one of the widest gaps and cuts it in two” [...] “it never lands in a sliver.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EA75E140919DEA9A55FEFD01EB2F677F/S1446788700031062a.pdf/the-three-gap-theorem-steinhaus-conjecture.pdf | Entry 12, new-point-largest: Van Ravenstein Proposition 4.2: "the oldest of the largest gaps". This concerns a new distinct point; rational repeated marks are explicitly handled separately by the page.
- CONFIRMED | N2-13 (largest-only-shrinks): The largest gap cannot grow when a new point is inserted. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EA75E140919DEA9A55FEFD01EB2F677F/S1446788700031062a.pdf/the-three-gap-theorem-steinhaus-conjecture.pdf | The maximum can remain unchanged while another equally large gap remains. Clarified wording alongside X1/K2.
- CONFIRMED | N2-17 (gaps-visible-cf): The gaps make α's continued-fraction approximation errors visible. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EA75E140919DEA9A55FEFD01EB2F677F/S1446788700031062a.pdf/the-three-gap-theorem-steinhaus-conjecture.pdf | van Ravenstein Theorem 3.3 and section 4 relate gap lengths to convergent errors. Precision made explicit alongside X2/X3.
- CONFIRMED | N2-18 (fraction-readout): “The readout names the fraction.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 18, fraction-readout: Pinned HTML lines 319-320 print the numerator and denominator returned by bestApprox. Reproduced in evidence/readouts.json; the N=1 defect is assessed separately at entry 106.
- CONFIRMED | N2-19 (companion-link): “it ties this layer to The Most Irrational Number.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 19, companion-link: The anchor is exactly "href="/strata/most-irrational/"" with title "The Most Irrational Number".
- CONFIRMED | N2-20 (evenest-hardest): The golden ratio is badly approximable and makes Hurwitz's approximation bound sharp. | https://arxiv.org/pdf/1904.10815; https://www.math.stonybrook.edu/~tony/whatsnew/column/irrational-0799/irrational4.html | The source checkers supported only uniform resistance to clumping, not pointwise optimality. Revised under X4/K3 and X10; Akiyama pp. 2-4 and the AMS/Stony Brook Hurwitz account support this scope.
- CONFIRMED | N2-22 (golden-cf-all-ones): φ=(1+√5)/2 has an all-ones continued fraction and makes Hurwitz's approximation bound sharp. | https://www.math.stonybrook.edu/~tony/whatsnew/column/irrational-0799/irrational4.html | The AMS/Stony Brook source gives Fibonacci convergents and the sharp Hurwitz constant. Revised wording under X4/K3 specifies the intended approximation sense.
- CONFIRMED | N2-23 (golden-angle): “Its turn is the golden angle, ≈ 137.5°.” | https://www.algorithmicbotany.org/papers/abop/abop.pdf | Entry 23, golden-angle: The Algorithmic Beauty of Plants p. 99 gives "360◦ τ−2" and "137.5◦". Direct calculation gives 360/φ² = 137.507764... degrees.
- CONFIRMED | N2-24 (golden-phi-squared): “For the golden angle the ratio of the largest gap to the smallest stays bounded by φ² ≈ 2.618 forever” | https://arxiv.org/pdf/1904.10815; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 24, golden-phi-squared: The recorded excerpt gives only "B − 1 ≤ R ≤ B + 2". The exact φ² bound follows from Theorem 4 plus ηk−1/ηk=φ for the all-ones expansion: two-length ratios are φ, three-length ratios φ²; the reflected golden angle has the same gaps. N=1 has ratio 1.
- CONFIRMED | N2-25 (factor-three-every-count): “the marks are within a factor of three of perfectly even at every count, not just in some limit.” | https://arxiv.org/pdf/1904.10815; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 25, factor-three-every-count: The φ² bound is below 3. Since the average gap is 1/N and lies between minimum and maximum, each gap is between (1/N)/φ² and φ²/N. This derives the stated comparison to perfect spacing.
- CONFIRMED | N2-26 (near-quarter-ratio): “Nudge α a hair off a simple fraction” [...] “to 0.2500001, say” [...] “and the marks rush toward four spokes before crawling onward: the gap ratio runs past 10⁵.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 26, near-quarter-ratio: Recreated from the pinned page: "chosen worst ratio over N≤600: 6.25e+5×" for α=0.2500001. Exact rational arithmetic gives a large spike after the fifth point.
- CONFIRMED | N2-27 (pi-spike): “Even π misbehaves where it is well-approximated” [...] “its famous near-miss 355/113 shows up as a spike to ≈ 290×.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 27, pi-spike: Recreated: "chosen worst ratio over N≤600: 293.63×" for the π preset. The close return at denominator 113 corresponds to 355/113 for π.
- CONFIRMED | N2-28 (vogel-citation): “Vogel's rule (angle n·α, radius √n)” | https://www.algorithmicbotany.org/papers/abop/abop.pdf | Entry 28, vogel-citation: The Algorithmic Beauty of Plants p. 100, formula (4.1), gives angular position proportional to n and radius "r = c√n". The page chooses a scale c in its drawing.
- CONFIRMED | N2-29 (chart-and-seed-head): “the running gap ratio as you add points, and the same points grown into a seed head by Vogel's rule (angle n·α, radius √n).” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 29, chart-and-seed-head: Pinned HTML ratios() recomputes gaps as n increases to 600; drawFlower() uses "var r = c*Math.sqrt(n);" and angle 2π frac(n*a), for the first 380 seeds.
- CONFIRMED | N2-30 (browser-values): “Every value above is recomputed in your browser from α and N alone” [...] “nothing is stored.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 30, browser-values: Pinned HTML structure(), bestApprox(), ratios() and drawFlower() compute from the selected parameters, with no saved result table. This is a calculation claim, not an assertion of exact floating-point values.
- CONFIRMED | N2-31 (floating-point): “a browser uses floating-point” | https://tc39.es/ecma262/multipage/ecmascript-data-types-and-values.html#sec-ecmascript-language-types-number-type | Entry 31, floating-point: ECMA-262 section 6.1.6.1 specifies "double-precision floating point IEEE 754-2019 binary64 values" for Number. The inline page uses Number.
- CONFIRMED | N2-33 (integer-units): “take α = p/q and every gap becomes a whole number of 1/q units” [...] “no rounding, no tolerance” | git show 37137afa5c:research/three-gap/verify.mjs | Entry 33, integer-units: Pinned verifier positions() and gapList() use BigInt residues modulo den; differences are integer multiples of 1/den.
- CONFIRMED | N2-42 (irrational-extension): “that it holds for every real α (including the irrationals the sunflower actually uses) is the cited theorem of Sós, Surányi, and Świerczkowski, not something a finite computation can finish.” | https://arxiv.org/pdf/1707.04094 | Entry 42, irrational-extension: Haynes and Marklof explicitly state "for all α ∈ R" and name the three initial proofs. The sentence appropriately attributes the infinite theorem to mathematics outside the computation.
- CONFIRMED | N2-43 (cf-count-structure): “exactly how many short, medium, and long arcs there are at a given N, and precisely when there are two versus three, is given by α's continued fraction (the Steinhaus–Slater structure, N. B. Slater, 1967).” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EA75E140919DEA9A55FEFD01EB2F677F/S1446788700031062a.pdf/the-three-gap-theorem-steinhaus-conjecture.pdf | Entry 43, cf-count-structure: Van Ravenstein Theorem 3.3 and section 4 relate gap counts and transitions to the continued fraction; p. 368 describes Slater's derivation and reference [8] supplies the 1967 paper.
- CONFIRMED | N2-44 (slater-citation): “the Steinhaus–Slater structure, N. B. Slater, 1967” | https://doi.org/10.1017/S0305004100042195 | Entry 44, slater-citation: Cambridge identifies "Noel B. Slater", "October 1967", "pp. 1115 - 1123" and the supplied DOI. The online checker also returned "DOI MATCH". The 2008 date is online publication, not the original journal year.
- CONFIRMED | N2-45 (formula-cited): “We cite that formula rather than re-derive it” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 45, formula-cited: The page identifies "N. B. Slater, 1967" and does not present or claim to rederive the full continued-fraction formula. Its named citation corresponds to the verified paper in entry 44.
- CONFIRMED | N2-48 (scope-optimality): “It is not a claim that φ is uniquely optimal by every measure a botanist might use” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 48, scope-optimality: The quoted limitation is explicitly present in the page's scope paragraph. Entry 60 flags the incompatible pointwise superlative elsewhere.
- CONFIRMED | N2-49 (botanical-qualification): “the phyllotaxis literature is richer than one ratio, and real seed heads are shaped by more than geometry.” | https://www.nature.com/articles/nature02081 | Entry 49, botanical-qualification: The Nature abstract says "proteins involved in auxin transport regulate phyllotaxis". curl recovered the article text despite the online tool's unreadable response.
- CONFIRMED | N2-50 (footer-attribution): “CONJ. HUGO STEINHAUS · PROOFS V. T. SÓS (1957), J. SURÁNYI (1958), S. ŚWIERCZKOWSKI (1959)” | https://arxiv.org/pdf/1707.04094 | Entry 50, footer-attribution: Haynes and Marklof p. 1 gives the proof years 1957, 1958, 1959; van Ravenstein p. 360 supplies the Steinhaus attribution.
- CONFIRMED | N2-51 (footer-slater): “GAP STRUCTURE N. B. SLATER (1967)” | https://doi.org/10.1017/S0305004100042195 | Entry 51, footer-slater: Cambridge's journal record identifies "Noel B. Slater" and "October 1967". It discusses both the lengths and their numbers.
- CONFIRMED | N2-52 (footer-computation): Live floating-point values; finite rational gap checks in research/three-gap/verify.mjs. | public/strata/the-three-gap-theorem/index.html; research/three-gap/verify.mjs | The browser uses Number; the finite core rational checks use BigInt. The footer now explicitly scopes exactness, alongside X13.
- CONFIRMED | N2-53 (independent-verifier): “This layer's check works the answer out again from scratch, independently of this page.” | git show 37137afa5c:research/three-gap/verify.mjs | Entry 53, independent-verifier: The complete verifier has no file input and reconstructs residues and gaps. Running the only downloaded file in reader/ reproduces all nine checks.
- CONFIRMED | N2-55 (verifier-capability): “It can settle the finding; it cannot notice a figure that drifted on this page.” | git show 37137afa5c:research/three-gap/verify.mjs | Entry 55, verifier-capability: Pinned verifier calculates its own test cases and reads no page. It cannot detect a textual change to a displayed figure; the published wording states this limitation.
- CONFIRMED | N2-56 (published-source): “It is published to read.” | git show 37137afa5c:public/checks/manifest.json | Entry 56, published-source: The linked source download returned HTTP 200, 9,554 bytes, identical to both pinned copies of verify.mjs.
- CONFIRMED | N2-59 (nearby-hardest): The companion explores the golden ratio's extremal role in Hurwitz's approximation theorem. | https://www.math.stonybrook.edu/~tony/whatsnew/column/irrational-0799/irrational4.html | The AMS/Stony Brook source supports the normalized asymptotic claim. The local relation now states the φ² gap-ratio bound instead of pointwise optimality, under X10.
- CONFIRMED | N2-61 (nearby-sqrt-two): “√2's incommensurability, and the gaps that never settle into a fourth length.” | https://mathcs.clarku.edu/~djoyce/elements/bookX/propX9.html | Entry 61, nearby-sqrt-two: Euclid X.9 gives the square-ratio criterion for commensurability. The ratio 2 is not the square of a rational; the separate three-gap assertion is established by Haynes and Marklof, not by the Euclid excerpt.
- CONFIRMED | N2-62 (nearby-comma): “Quantities that won't divide evenly: the Pythagorean comma” | https://en.wikipedia.org/wiki/Pythagorean_comma | Entry 62, nearby-comma: The cited account describes the discrepancy of "12 justly tuned perfect fifths" and "seven octaves". Independent arithmetic gives (3/2)^12 / 2^7 = 531441/524288, not 1.
- CONFIRMED | N2-63 (irrational-never-closes): “the circle that an irrational turn can never close.” | https://arxiv.org/pdf/1904.10815 | Entry 63, irrational-never-closes: The source definition is nα modulo 1. Returning after nonzero q steps requires qα to be integral, hence α rational. The excerpt's rational-lines observation alone is not the irrational-direction proof; the short contrapositive supplies it.
- CONFIRMED | N2-64 (bellringers-steinhaus): “Steinhaus twice: the bell-ringers' Steinhaus–Johnson–Trotter order, and Steinhaus's three-gap conjecture about points on a circle.” | https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Johnson%E2%80%93Trotter_algorithm | Entry 64, bellringers-steinhaus: The cited algorithm history connects the procedure to "change ringing of church bells"; the three-gap Steinhaus attribution is independently supported by van Ravenstein.
- CONFIRMED | N2-69 (dek-sunflower): “The structural reason the sunflower packs uniformly at every count.” | https://www.scribd.com/document/816008523/A-better-way-to-construct-the-sunflower-head; https://arxiv.org/pdf/1904.10815; https://www.algorithmicbotany.org/papers/abop/abop.pdf#page=112 | Entry 69, dek-sunflower: Vogel p. 181 discusses uniform angular filling under refinement, and then separately supplies the square-root radial law. Taken as the displayed Vogel model with bounded gaps, this supports uniform filling; it does not prove equal nearest-neighbour distances or biological uniqueness. The fixer re-read Akiyama and ABOP for this geometric scope; the Vogel p. 181 reading is the source checkers' recorded evidence.
- CONFIRMED | N2-70 (plain-theorem): “Whatever the step size, the gaps between marks come in at most three lengths, and when there are three the longest is exactly the sum of the other two.” | https://arxiv.org/pdf/1904.10815 | Entry 70, plain-theorem: Akiyama Theorem 4 supplies the three lengths and their counts. The claim correctly includes "when there are three".
- CONFIRMED | N2-71 (plain-add-marks): “Add marks here and watch a fourth length never appear.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 71, plain-add-marks: The button increments N and recomputes the partition. The theorem bounds its distinct lengths; the readout is reproduced in the local harness.
- CONFIRMED | N2-73 (standalone-body): “An immersive layer; it renders as its own standalone page at the `href` above (source: `public/strata/the-three-gap-theorem/index.html`). This file exists so the layer appears on the ground with the others” [...] “its body is not rendered.” | git show 37137afa5c:src/content/strata/the-three-gap-theorem.md | Entry 73, standalone-body: The live response is the standalone pinned HTML, except injected analytics. The frontmatter identifies the same href; the explanatory Markdown body is not in the response.
- CONFIRMED | N2-75 (body-history): “Conjectured by Hugo Steinhaus; first proved independently by Vera T. Sós (1957), János Surányi (1958), and Stanisław Świerczkowski (1959);” | https://arxiv.org/pdf/1707.04094 | Entry 75, body-history: Haynes and Marklof p. 1 supplies the three proof dates and authors; van Ravenstein p. 360 attributes the conjecture to Steinhaus. Bibliographic references identify the same authors.
- CONFIRMED | N2-76 (body-slater): “the exact gap structure is the Steinhaus–Slater description (N. B. Slater, 1967).” | https://doi.org/10.1017/S0305004100042195 | Entry 76, body-slater: Cambridge's source record identifies Slater's 1967 article and its gap-count and ordering problems; van Ravenstein explains the continued-fraction derivation.
- CONFIRMED | N2-77 (body-two-instruments): Two live instruments recompute the circle and chart with floating-point arithmetic; the smallest gap for N≥2 distinct marks is the minimum scaled error, and the near-quarter example has a finite spike. | public/strata/the-three-gap-theorem/index.html; research/three-gap/verify.mjs | The original checker supported the instruments and computations but deferred the metric and scope defects to their specific entries. Those copies are repaired under X2/X3/X8/X13.
- CONFIRMED | N2-79 (body-sunflower): The golden angle keeps its largest-to-smallest gap ratio at most φ² for every N; the layer connects that uniform bound to a Vogel seed head. | https://arxiv.org/pdf/1904.10815; https://www.math.stonybrook.edu/~tony/whatsnew/column/irrational-0799/irrational4.html | Akiyama Theorems 2 and 4 support bounded geometric uniformity. The old pointwise superlative is repaired under X10; no equal-spacing or biological-uniqueness claim is made.
- CONFIRMED | N2-80 (body-hardest): “that layer shows the golden ratio is hardest to approximate” | https://www.math.stonybrook.edu/~tony/whatsnew/column/irrational-0799/irrational4.html | Entry 80, body-hardest: The AMS/Stony Brook explanation gives the golden mean's Fibonacci convergents and extremal Hurwitz constant. This supports the companion's standard approximation claim.
- CONFIRMED | N2-81 (description-1): “Place points at the multiples of any number around a circle.” | https://arxiv.org/pdf/1904.10815 | Entry 81, description-1: Theorem 4 uses "{nα (mod 1)" for the circle points; translating the starting index does not change any gap.
- CONFIRMED | N2-83 (description-3): “The three-gap (Steinhaus) theorem, made playable, every value computed live.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 83, description-3: The inline controls and calculations execute successfully in the local harness; the numerical values are generated from the selected α and N.
- CONFIRMED | N2-84 (description-4): “The exact reason the golden angle packs a sunflower evenly at every count.” | https://www.scribd.com/document/816008523/A-better-way-to-construct-the-sunflower-head; https://arxiv.org/pdf/1904.10815; https://www.algorithmicbotany.org/papers/abop/abop.pdf#page=112 | Entry 84, description-4: The cited Vogel argument concerns the geometric model. The source additionally needs the square-root radial law, supplied in the following paragraph and implemented on the page; uniform means bounded spacing, not exactly equal spacing. The fixer re-read Akiyama and ABOP for this geometric scope; the Vogel p. 181 reading is the source checkers' recorded evidence.
- CONFIRMED | N2-85 (og-description-1): “Place points at the multiples of any number around a circle.” | https://arxiv.org/pdf/1904.10815 | Entry 85, og-description-1: Theorem 4 defines the circle by multiples of α modulo 1, agreeing with the description.
- CONFIRMED | N2-87 (og-description-3): “The three-gap theorem, made playable.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 87, og-description-3: The two inline interactive instruments execute and respond to preset and N changes in the local harness.
- CONFIRMED | N2-88 (og-description-4): “Why the golden angle packs a sunflower evenly at every count.” | https://www.scribd.com/document/816008523/A-better-way-to-construct-the-sunflower-head; https://arxiv.org/pdf/1904.10815; https://www.algorithmicbotany.org/papers/abop/abop.pdf#page=112 | Entry 88, og-description-4: Vogel's model supports the qualified geometric uniformity explanation; Akiyama supplies a rigorous Delone result. No claim of botanical uniqueness is adopted. The fixer re-read Akiyama and ABOP for this geometric scope; the Vogel p. 181 reading is the source checkers' recorded evidence.
- CONFIRMED | N2-89 (twitter-description-1): “No matter how many points, the gaps take at most three lengths.” | https://arxiv.org/pdf/1904.10815 | Entry 89, twitter-description-1: Theorem 4 says "at most three lengths". The general real-α statement is also explicit in Haynes and Marklof.
- CONFIRMED | N2-90 (twitter-description-2): “The three-gap theorem, playable, every value live.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 90, twitter-description-2: The controls update the calculated partitions, readouts and plots; all sampled readouts are reproduced from the pinned script.
- CONFIRMED | N2-92 (structured-description-2): Values are computed live with floating-point arithmetic; an offline verifier checks a finite set of rational cases using exact integer gaps. | public/strata/the-three-gap-theorem/index.html; research/three-gap/verify.mjs | The revised JSON-LD preserves the supported claim and states the finite exact-check scope, alongside X5/X7/X13.
- CONFIRMED | N2-93 (structured-description-3): “The structural reason the golden angle packs a sunflower head uniformly at every seed count.” | https://www.scribd.com/document/816008523/A-better-way-to-construct-the-sunflower-head; https://arxiv.org/pdf/1904.10815; https://www.algorithmicbotany.org/papers/abop/abop.pdf#page=112 | Entry 93, structured-description-3: Vogel's model combines angular spacing with equal-area radial growth. This supports bounded uniformity in the model; the circle theorem alone does not specify radial packing. The fixer re-read Akiyama and ABOP for this geometric scope; the Vogel p. 181 reading is the source checkers' recorded evidence.
- CONFIRMED | N2-96 (social-card-size): “<meta property="og:image" content="https://artwaste.land/og/the-three-gap-theorem.png" /> <meta property="og:image:width" content="1200" /> <meta property="og:image:height" content="630" />” | git show 37137afa5c:research/three-gap/og-gen.mjs | Entry 96, social-card-size: Read the PNG IHDR from the pinned image: width 1200, height 630. The generator independently declares "body{width:1200px;height:630px;".
- CONFIRMED | N2-97 (seeds-never-align): “no two seeds ever line up” | https://arxiv.org/pdf/1904.10815 | Entry 97, seeds-never-align: For the exact irrational golden angle, radial alignment requires (n-m)α to be an integer or half-integer, impossible for distinct integer indices. The quoted source defines the model; its rational-only sentence by itself is insufficient, so this short deduction completes the check.
- CONFIRMED | N2-98 (head-even-every-count): “the head fills evenly at every count” | https://www.scribd.com/document/816008523/A-better-way-to-construct-the-sunflower-head; https://arxiv.org/pdf/1904.10815; https://www.algorithmicbotany.org/papers/abop/abop.pdf#page=112 | Entry 98, head-even-every-count: Vogel p. 181 uses uniform filling of the circle, followed by the radial equal-area model. Akiyama Theorem 2 confirms the geometric model is Delone. This supports bounded even filling, not identical inter-seed distances. The fixer re-read Akiyama and ABOP for this geometric scope; the Vogel p. 181 reading is the source checkers' recorded evidence.
- CONFIRMED | N2-100 (golden-reference): “(golden, in amber, stays ≤ φ²)” | https://arxiv.org/pdf/1904.10815; git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 100, golden-reference: Akiyama Theorem 4 plus the all-ones remainder ratios gives φ² as detailed in entry 24. The amber plot recomputes golden ratios through N=600.
- CONFIRMED | N2-101 (rational-collapse-tail): “the circle never fills further.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 101, rational-collapse-tail: For a reduced rational p/q, there are exactly q distinct residues. The pinned deduplication branch reports repeat landings; presets 1/4 and 1/7 recreate four and seven points.
- CONFIRMED | N2-102 (readout-ring-initial): “2 distinct gap lengths ≤ 3 ✓short 20.062° × 5medium 32.461° × 8closest pair: 8 steps apart → best fraction 3/8, ‖qα‖ = 0.05573 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 102, readout-ring-initial: At the initial golden α and N=13, reproduced exactly. Independent 55-digit Decimal arithmetic gives lengths 20.062° (5) and 32.461° (8), fraction 3/8, error 0.05573.
- CONFIRMED | N2-103 (readout-chart-initial): “chosen worst ratio over N≤600: 6.25e+5× (golden, in amber, stays ≤ φ²)” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 103, readout-chart-initial: Initial second instrument uses α=0.2500001; the output is reproduced exactly, with "6.25e+5×" over 2≤N≤600.
- CONFIRMED | N2-104 (readout-ring-circle-preset-1-4): “1 distinct gap length ≤ 3 ✓short 90.000° × 4α is rational here → only 4 distinct points (the 13 steps land on old marks); the circle never fills further.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 104, readout-ring-circle-preset-1-4: Reproduced at α=1/4, N=13: exactly four distinct points and equal 90° gaps. This also follows from integer residues modulo 4.
- CONFIRMED | N2-105 (readout-ring-circle-preset-1-7): “1 distinct gap length ≤ 3 ✓short 51.429° × 7α is rational here → only 7 distinct points (the 13 steps land on old marks); the circle never fills further.” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 105, readout-ring-circle-preset-1-7: Reproduced at α=1/7, N=13: seven points with equal gaps 360/7 = 51.428571... degrees, rounded to 51.429°.
- CONFIRMED | N2-107 (readout-ring-n-2): “2 distinct gap lengths ≤ 3 ✓short 137.508° × 1medium 222.492° × 1closest pair: 1 steps apart → best fraction 1/1, ‖qα‖ = 0.38197 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 107, readout-ring-n-2: Reproduced exactly after selecting the "0.618" preset, α=φ−1, then N=2. The source record's condition omits α; at the initial golden preset the fraction would instead be 0/1.
- CONFIRMED | N2-108 (readout-ring-n-14): “3 distinct gap lengths ≤ 3 ✓short 12.399° × 1medium 20.062° × 6long 32.461° × 7longest − (short + medium) = 0.0e+0 ≈ 0 ✓closest pair: 13 steps apart → best fraction 8/13, ‖qα‖ = 0.03444 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 108, readout-ring-n-14: Reproduced exactly at α=φ−1, N=14. The record omits the preset condition; the initial golden angle gives 5/13, whereas this preset gives 8/13.
- CONFIRMED | N2-109 (readout-ring-n-300): “3 distinct gap lengths ≤ 3 ✓short 0.691° × 67medium 1.118° × 156long 1.809° × 77longest − (short + medium) = 2.1e-14 ≈ 0 ✓closest pair: 233 steps apart → best fraction 144/233, ‖qα‖ = 0.00192 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 109, readout-ring-n-300: Reproduced exactly at α=φ−1, N=300. The record omits the preset condition; the golden-angle numerator would be 89, not 144.
- CONFIRMED | N2-110 (readout-chart-chart-preset-golden): “golden hugs φ² ≈ 2.618” [...] “worst ratio over N≤600: 2.618” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 110, readout-chart-chart-preset-golden: Reproduced "worst ratio over N≤600: 2.618"; mathematical bound and rounding independently agree with φ².
- CONFIRMED | N2-111 (readout-chart-sqrt2): “chosen worst ratio over N≤600: 3.41× (golden, in amber, stays ≤ φ²)” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 111, readout-chart-sqrt2: Reproduced "3.41×" for √2−1 through N=600; the exact gap-ratio bound for the all-twos tail is 2+√2 = 3.41421356....
- CONFIRMED | N2-112 (readout-chart-pi): “chosen worst ratio over N≤600: 293.63× (golden, in amber, stays ≤ φ²)” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 112, readout-chart-pi: Reproduced "293.63×" for the fractional part of π through N=600, matching the rounded prose spike.
- CONFIRMED | N2-113 (readout-chart-chart-preset-e): “chosen worst ratio over N≤600: 7.53× (golden, in amber, stays ≤ φ²)” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 113, readout-chart-chart-preset-e: Reproduced "7.53×" for the fractional part of e through N=600.
- CONFIRMED | N2-114 (readout-chart-chart-preset-1-4): “chosen worst ratio over N≤600: 6.25e+5× (golden, in amber, stays ≤ φ²)” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 114, readout-chart-chart-preset-1-4: Reproduced "6.25e+5×" for α=0.2500001 through N=600.
- CONFIRMED | N2-115 (readout-chart-chart-preset-1-7): “chosen worst ratio over N≤600: 3.57e+5× (golden, in amber, stays ≤ φ²)” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 115, readout-chart-chart-preset-1-7: Reproduced "3.57e+5×" for α=0.1428572 through N=600.
- CONFIRMED | N2-116 (added-circle-sqrt2): (added) “3 distinct gap lengths ≤ 3 ✓short 10.597° × 1medium 25.584° × 8long 36.182° × 4longest − (short + medium) = 0.0e+0 ≈ 0 ✓closest pair: 12 steps apart → best fraction 5/12, ‖qα‖ = 0.02944 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 116, added-circle-sqrt2: Omitted source-record circle readout: preset √2, N=13. Recreated by node readouts.mjs and independently checked with 55-digit Decimal arithmetic in independent.py. See evidence/independent-readouts.json.
- CONFIRMED | N2-117 (added-circle-pi): (added) “3 distinct gap lengths ≤ 3 ✓short 3.187° × 6medium 47.787° × 5long 50.973° × 2longest − (short + medium) = 0.0e+0 ≈ 0 ✓closest pair: 7 steps apart → best fraction 1/7, ‖qα‖ = 0.00885 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 117, added-circle-pi: Omitted source-record circle readout: preset π, N=13. Recreated by node readouts.mjs and independently checked with 55-digit Decimal arithmetic in independent.py. See evidence/independent-readouts.json.
- CONFIRMED | N2-118 (added-circle-e): (added) “3 distinct gap lengths ≤ 3 ✓short 10.070° × 6medium 35.604° × 2long 45.674° × 5longest − (short + medium) = 4.4e-16 ≈ 0 ✓closest pair: 7 steps apart → best fraction 5/7, ‖qα‖ = 0.02797 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 118, added-circle-e: Omitted source-record circle readout: preset e, N=13. Recreated by node readouts.mjs and independently checked with 55-digit Decimal arithmetic in independent.py. See evidence/independent-readouts.json.
- CONFIRMED | N2-119 (added-circle-phi-minus-one): (added) “2 distinct gap lengths ≤ 3 ✓short 20.062° × 5medium 32.461° × 8closest pair: 8 steps apart → best fraction 5/8, ‖qα‖ = 0.05573 = the smallest gap” | git show 37137afa5c:public/strata/the-three-gap-theorem/index.html | Entry 119, added-circle-phi-minus-one: Omitted source-record circle readout: preset 0.618, N=13. Recreated by node readouts.mjs and independently checked with 55-digit Decimal arithmetic in independent.py. See evidence/independent-readouts.json.
- CONFIRMED | N2-94 (publication-metadata): datePublished is 2026-06-21. | src/content/strata/the-three-gap-theorem.md; research/assay-line/queue-1014c/the-three-gap-theorem/N2-claude.md | The markdown date is 2026-06-21T12:00:00Z; the Claude checker confirmed the creation commit. This is distinct from K15, the stale modification date.
- OBSERVED | The exhaustive rational grid contains 2,160,746 cases with no core violations. | research/three-gap/verify.mjs; research/three-gap/facts.md | The rerun counts 29,450 one-length, 834,168 two-length and 1,297,128 three-length cases; each gap sum covers one full circle.
- OBSERVED | The seeded random stress run checks 4,000 rational cases. | research/three-gap/verify.mjs; research/three-gap/facts.md | No case has more than three lengths or violates the three-length sum rule.
- OBSERVED | The scaled-error identity passes 826,129 exact cases. | research/three-gap/verify.mjs; research/three-gap/facts.md | The test uses N≥2 and reduced denominators through 160.
- OBSERVED | The largest-host and non-increasing-maximum checks each pass 2166/2166. | research/three-gap/verify.mjs; research/three-gap/facts.md | Both assertions remain present; K5 is rejected, not silently removed.
- OBSERVED | The golden proxy has 714 passing sampled ratio checks and 27/27 passing Fibonacci counts. | research/three-gap/verify.mjs; research/three-gap/facts.md | Counts are N=5,12,…,4996 and Fibonacci F4=3 through F30=832040 respectively; the near-quarter sampled ratio is 625000.25 at N=5.
- OBSERVED | The layer verifier passes 9/9 after the repairs. | research/three-gap/verify.mjs; research/three-gap/facts.md | node research/three-gap/verify.mjs exits 0 on 2026-10-14. The published mirror is byte-identical.

## Findings

- [fixed] X1 (new-gap-length-denied): Fixed the splitting explanation in the HTML and check log; one dated correction covers X1 and K2.
- [fixed] X2 (gap-is-fraction): Named the minimum scaled error ‖qα‖=abs(qα-p), distinguished p/q, and repaired the dek, prose, markdown body, relation note, check log and verifier wording.
- [fixed] X3 (approximation-error-metric): Specified the scaled-error metric, 1≤q<N, and N≥2 distinct marks; the browser calculation already used that metric.
- [fixed] X4 (denominator-growth-metric): Replaced the fastest-denominator metaphor with bad approximability and the sharp Hurwitz bound; recorded as MINOR without a page correction.
- [fixed] X5 (finite-check-as-proof): Fixed both proof claims and their copies in the page, markdown, check log and verifier comments. One dated correction covers X5, X7, K4 and K7; the published verifier mirror matches the source.
- [fixed] X6 (fibonacci-one-point-exception): Qualified the theorem endpoint and stated the 27 tested counts in the page, markdown, check log and both verifier copies.
- [fixed] X7 (finite-check-as-proof): Fixed both proof claims and their copies in the page, markdown, check log and verifier comments. One dated correction covers X5, X7, K4 and K7; the published verifier mirror matches the source.
- [fixed] X8 (near-rational-runaway): Replaced the unbounded-growth implication with the finite N=5 spike and eventual rational equal spacing in the page, markdown and check log.
- [open note:81816d] X9 (repository-required): NOT FIXED: shared placard/manifest repair belongs to board note 81816d. Left the placard and shared files unchanged; retained WRONG.
- [fixed] X10 (golden-pointwise-optimum): Replaced the pointwise superlative with the all-N φ² gap-ratio bound in the dek, markdown body and source relation note; regenerated this page's Nearby copy. Added one dated correction.
- [fixed] X11 (live-arithmetic-exactness): Added a local outward relation naming the circle and repeating-decimal subjects explicitly, with floating-point live values and finite exact rational checks; regenerated only this page's Nearby block. Did not edit the other layer.
- [fixed] X12 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] X13 (all-values-exactly-checked): Separated live floating-point calculations from finite exact rational gap checks in the dek, metadata, footer, markdown, check log and verifier comments. MINOR only; no dated page line.
- [fixed] X14 (gap-is-fraction): Named the minimum scaled error ‖qα‖=abs(qα-p), distinguished p/q, and repaired the dek, prose, markdown body, relation note, check log and verifier wording.
- [fixed] X15 (gap-is-fraction): Named the minimum scaled error ‖qα‖=abs(qα-p), distinguished p/q, and repaired the dek, prose, markdown body, relation note, check log and verifier wording.
- [fixed] X16 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] X17 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] X18 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] X19 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] X20 (nongolden-spokes): Replaced that blanket caption with the chosen angle and the actual 380-seed model parameters; recorded the counterexample in the check log and one dated correction for X20 and K16.
- [fixed] X21 (single-point-closest-pair): Handled N<2 before bestApprox(): the readout now says there is no closest pair and the only gap is a full turn (360°). One dated correction covers X21 and K17; page regressions cover every ring preset.
- [fixed] K2 (new-gap-length-denied): Fixed the splitting explanation in the HTML and check log; one dated correction covers X1 and K2.
- [fixed] K3 (denominator-growth-metric): Replaced the fastest-denominator metaphor with bad approximability and the sharp Hurwitz bound; recorded as MINOR without a page correction.
- [fixed] K4 (finite-check-as-proof): Fixed both proof claims and their copies in the page, markdown, check log and verifier comments. One dated correction covers X5, X7, K4 and K7; the published verifier mirror matches the source.
- [fixed] K6 (all-n-ratio-check): Stated the rational proxy, sample count, exact endpoint and tolerance in the page, markdown, check log and verifier outputs. Tightened the near-quarter assertion from >100 to >600000, matching the displayed claim.
- [fixed] K7 (finite-check-as-proof): Fixed both proof claims and their copies in the page, markdown, check log and verifier comments. One dated correction covers X5, X7, K4 and K7; the published verifier mirror matches the source.
- [open note:81816d] K8 (repository-required): NOT FIXED: shared placard/manifest repair belongs to board note 81816d. Left the placard and shared files unchanged; retained WRONG.
- [fixed] K9 (live-arithmetic-exactness): Added a local outward relation naming the circle and repeating-decimal subjects explicitly, with floating-point live values and finite exact rational checks; regenerated only this page's Nearby block. Did not edit the other layer.
- [fixed] K10 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] K11 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] K12 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] K13 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [fixed] K14 (missing-three-length-condition): Added the three-distinct-length condition in all five abbreviated copies: dek, markdown body, HTML description, Open Graph description and JSON-LD description.
- [dated] K15 (modification-date): Set JSON-LD dateModified to 2026-10-14. No Updated line under the explicit JSON-LD exception; the five Corrected lines already date this revision.
- [fixed] K16 (nongolden-spokes): Replaced that blanket caption with the chosen angle and the actual 380-seed model parameters; recorded the counterexample in the check log and one dated correction for X20 and K16.
- [fixed] K17 (single-point-closest-pair): Handled N<2 before bestApprox(): the readout now says there is no closest pair and the only gap is a full turn (360°). One dated correction covers X21 and K17; page regressions cover every ring preset.
- [fixed] K18 (gap-count-title): Changed the standalone HTML title to The Three-Gap Theorem: At Most Three Gap Lengths. The shared historical scripts/seo-titles.tsv row was not edited under the task's shared-file prohibition.
- [fixed] K19 (vogel-link-target): Changed only the Vogel citation target to https://www.algorithmicbotany.org/papers/abop/abop.pdf#page=112; retained the supported formula.