---
slug: one-straight-cut
passed: 2026-10-18
by: the assay line (research/assay-line/claude-1018b/), Claude fixer (claude-opus-5-5) directed by claude-assaying-boyle-1014; source record by a Claude agent (claude-sonnet-5-5), checked by a Claude agent (claude-opus-5-5)
checked: 135
confirmed: 102
wrong: 19             # 3 WRONG + 16 MINOR
unverifiable: 3
observed: 11
stale: 0
---

This was a directed repair pass over the 19 triage items in research/assay-line/claude-1018b/one-straight-cut/triage.json, read at 03d50827e7 (the checker read the layer at 4fcb19ad5e). The checker raised nineteen MINOR lines and nothing WRONG outright. The fixer re-read the two quotation sources itself (the Internet Archive full text of Gardner's New Mathematical Diversions, and p. 188 of Demaine and O'Rourke's survey from the SLMath PDF) and raised the three quotation lines to WRONG under the line's quotation rule: each printed non-contiguous source words as one continuous run. Those omissions are now marked with ellipses (markdown, HTML, the engine.js header comment, and the verify-claims.mjs assertion that held the unmarked Gardner form), with two dated Corrected lines on the page and in the markdown; JSON-LD dateModified moved to 2026-10-18. The sixteen other MINOR lines (overstatements about intractability, the page's own controls and print gate, two citation slips, loose wording) were listed by the tier B triage and left unchanged. The layer's verify-claims.mjs passed 29/29 after the edits. The record carries forward the checker's 135 claim lines from research/assay-line/claude-1018b/one-straight-cut/N2-claude.md, including its CONFIRMED and OBSERVED lines, without claiming a fresh audit of them; the three UNVERIFIABLE lines (which method printed patterns use, what nearly every implementation uses, printer rescaling) remain unsourced.

## Claims
- CONFIRMED | Every pattern of straight-line cuts can be made by folding flat and one complete straight cut. | https://erikdemaine.org/foldcut/ | [theorem-every-pattern] Demaine: "every pattern (plane graph) of straight-line cuts can be made by folding and one complete straight cut"
- CONFIRMED | Any collection of straight-sided shapes at all, however awkward | https://erikdemaine.org/foldcut/ | [theorem-collection] Demaine lists nonconvex, disjoint, nested, adjoining polygons, segments and points; JCDCG98 p.2 says the same
- CONFIRMED | A five-pointed star, the letter G, a swan, a hundred disjoint triangles, the outline of your own signature rendered in straight segments | https://erikdemaine.org/foldcut/ | [theorem-examples] each example is a straight-line cut pattern, so the theorem covers it; the swan is the survey Fig. 14 (p.188); offered as what the theorem allows, not as published examples
- CONFIRMED | Martin Gardner first posed the question in Scientific American in June 1960 | https://library.slmath.org/books/Book52/files/12dem.pdf | [gardner-month] survey ref [Gardner 1960] "Scientific American (June 1960)"; en.wikipedia List_of_Martin_Gardner_Mathematical_Games_columns: "1960 Jun || Recreations involving folding and cutting sheets of paper"
- CONFIRMED | and is the first to pose the general question | https://erikdemaine.org/foldcut/ | [gardner-first-to-pose] also JCDCG98 p.2: "Gardner [9] was the first to state cutting out complex polygons as an open problem"
- CONFIRMED | Inspired by Loe, Martin Gardner writes about it in Scientific American (June) | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [gardner-sciam-wp] Wikipedia wikitext as quoted; month from the survey reference
- WRONG | remarking that "more complicated designs present formidable problems," | https://archive.org/download/new-mathematical-diversions/New%20Mathematical%20Diversions_djvu.txt | [gardner-quote] the 1995 MAA reprint, ch. 5 "Paper Cutting", p.65, reads "but more complicated designs--the swastika for instance--present formidable problems" (dashes shown as double hyphens): the page drops the parenthetical with no ellipsis, as Wikipedia does; the same quote is in the 1960 row of the HTML (added line below). The record said no reprint was reachable; this full text is reachable [re-read by the fixer: raised to WRONG under the quotation rule, K1]
- CONFIRMED | it was answered at the end of the 1990s, twice, by two different methods | https://erikdemaine.org/foldcut/ | [answered-twice] FUN98 (1998), JCDCG98 (Dec 1998), SODA99 (Jan 1999)
- CONFIRMED | There are two known ways to solve the problem. | https://erikdemaine.org/foldcut/ | [two-ways] verbatim on Demaine's page
- WRONG | There are two solutions to the problem. The first (partial) solution … is based on a structure called the straight skeleton … This solution applies to a large class of instances, which we do not describe in detail here. The second (complete) solution … is based on disk packing. | https://library.slmath.org/books/Book52/files/12dem.pdf | [survey-quote] p.188 has "...which we do not describe in detail here. See Figure 14 for two examples. The second (complete) solution [Bern et al. 2002] is based on disk packing to make the problem more local": the page joins "here." to "The second" with no ellipsis for the dropped sentence, and closes "based on disk packing." with a period mid-sentence. Same text in public/strata/one-straight-cut/engine.js header (lines 17-19) [re-read by the fixer: raised to WRONG under the quotation rule, K2]
- CONFIRMED | Their own survey six years later calls that solution "(partial)", applying to "a large class of instances," against the disk-packing "(complete)" one | https://library.slmath.org/books/Book52/files/12dem.pdf | [survey-partial-md] all three fragments verbatim on p.188 (pdftotext)
- CONFIRMED | Their own survey six years later calls that solution | https://library.slmath.org/books/Book52/files/12dem.pdf | [survey-year] MSRI vol. 52, 2005; DDL 1999; six years
- CONFIRMED | The disk-packing method of Bern, Demaine, Eppstein and Hayes is complete. | https://library.slmath.org/books/Book52/files/12dem.pdf | [disk-complete] verbatim p.188
- CONFIRMED | is complete. It packs the sheet with disks, decomposes the gaps into triangles and quadrilaterals, and folds each with a known molecule. | https://erikdemaine.org/foldcut/ | [disk-complete-html] Demaine: disks, triangles and quadrangles, molecules
- CONFIRMED | The straight-skeleton method of Demaine, Demaine and Lubiw | https://erikdemaine.org/foldcut/ | [skeleton-attribution] verbatim on Demaine's page
- UNVERIFIABLE | is the one that produces the patterns people actually print, and the one this page implements. | none | [skeleton-is-what-people-print] no source for which method printed patterns come from; Demaine's examples and JOrigami are straight-skeleton, nothing counts the rest
- UNVERIFIABLE | The construction this page uses is the one nearly every implementation uses, and it is not a promise. | none | [every-implementation] no survey of implementations found; same as the record
- CONFIRMED | Wikipedia's article says "The first proof of the fold-and-cut theorem, solving the problem, was published in 1999 by Erik Demaine, Martin Demaine, and Anna Lubiw and was solved using straight skeleton method." | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [wikipedia-quote] verbatim in the wikitext; article unchanged since revision 1305053178 (2025-08-09), so the same on 2026-08-15
- CONFIRMED | Wikipedia's article on the theorem says, in full: | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [wikipedia-in-full] the sentence is whole, followed only by two refs
- CONFIRMED | Demaine's own page for the problem says of it, still, that "the full paper (which includes all proof details) is still in progress." | https://erikdemaine.org/foldcut/ | [demaine-in-progress] verbatim on Demaine's page, fetched 2026-10-19 (page last updated March 3, 2025)
- CONFIRMED | The full paper (which includes all proof details) is still in progress. | https://erikdemaine.org/foldcut/ | [demaine-in-progress-html] verbatim
- CONFIRMED | the construction on this page is a partial solution to it whose complete proof has never been published. | https://erikdemaine.org/foldcut/ | [never-published] supported by Demaine's later statement; but note JCDCG98 itself (p.2) says "we prove that any collection of straight edges can be cut along" and states Theorem 1/Corollary 1 (p.5), falling "just short" only of Conjecture 1 (arbitrary side assignments)
- MINOR | Every applet that hands you a straight-skeleton crease pattern is handing you an artifact whose correctness, for your particular shape, nobody has established. | https://erikdemaine.org/papers/JCDCG98/paper.pdf | [nobody-established] overstated: the published straight-skeleton paper states Theorem 1 (p.5) for any cut graph whose side assignment is constant or induces no circular corridors, and Corollary 1: "Given any cut graph, there exists a flat folding ... such that mathematically cutting along the line removes precisely the (folded) cut graph"; the survey says the method "applies to a large class of instances". For shapes in that class correctness is a published theorem (with some proof details unpublished, per Demaine), not something "nobody has established"
- CONFIRMED | it says the skeleton creases work together with "a subset of the perpendicular creases (and a few more auxiliary creases)," | https://erikdemaine.org/foldcut/ | [perp-subset-quote] verbatim on Demaine's page
- MINOR | and choosing that subset is part of the proof that has not been written down. | https://erikdemaine.org/papers/JCDCG98/paper.pdf | [subset-unwritten] not supported: the 1998 paper writes down which perpendiculars are folded, sec. 4.2 p.9: "We will only fold along real perpendiculars", a real perpendicular being "one that is incident to a skeleton vertex", and discusses infinite perpendiculars ("The answer is no"). Demaine's page says the difficult part is the proof that they fold, not that the choice is unwritten
- CONFIRMED | Whether real paper can reach that folded position without passing through itself is NP-hard (Bern and Hayes, 1996) | https://library.slmath.org/books/Book52/files/12dem.pdf | [nphard-bern-hayes-md] survey 3.3.2: deciding a flat folded state is NP-hard; with 3.3.3 (any folded state is reachable by a continuous motion) the reachability reading holds in general. Loose in applying a worst-case result to "that" instance; see the MINOR lines below
- CONFIRMED | Bern and Hayes proved in 1996 that deciding whether a crease pattern has a flat folded state at all is intractable in general | https://library.slmath.org/books/Book52/files/12dem.pdf | [nphard-bern-hayes-html] survey 3.3.2 verbatim; the page says "in general"
- CONFIRMED | testing whether each vertex folds flat on its own is easy and fast, and the hard part is finding an order for the layers that no two sheets of paper have to swap through. | https://library.slmath.org/books/Book52/files/12dem.pdf | [local-easy] survey 3.3.1-3.3.2: "local foldability can be tested in linear time"; "computing a valid overlap order" is the hard part
- MINOR | Which way each of those goes is settled by the layer order, and finding the layer order is the intractable step. | https://erikdemaine.org/papers/JCDCG98/paper.pdf | [layer-order-decides] right for crease patterns in general (survey 3.3.2), but for these patterns the method's own paper says (p.2) "we present an algorithm that computes the creases and the actual flat origami that lines up precisely the given plane graph", and the abstract "We prove that the crease pattern is flat foldable by demonstrating a family of folded states"; NP-hardness of the general problem does not make this step intractable for this class
- MINOR | you are running a search no algorithm runs efficiently. | https://erikdemaine.org/papers/JCDCG98/paper.pdf | [no-algorithm-efficiently] stronger than the source: NP-hard means no polynomial algorithm unless P = NP, for crease patterns in general (survey 3.3.2); for straight-skeleton patterns the authors give an algorithm that computes the folded state (JCDCG98 p.2, abstract)
- MINOR | you have just performed a search no algorithm does efficiently. | https://erikdemaine.org/papers/JCDCG98/paper.pdf | [no-algorithm-does] same as the line above (HTML "The half your hands decide")
- CONFIRMED | The earliest known statement of the problem: Wakoku Chiyekurabe (Mathematical Contests) by Kan Chu Sen, a Japanese puzzle book. | https://erikdemaine.org/foldcut/ | [sen-1721] Demaine "first published reference ... of which we are aware"; Wikipedia "earliest known description"
- CONFIRMED | One problem asks the reader to fold a rectangle flat and make one complete straight cut to produce the crest called sangaibisi, "three folded rhombics". | https://erikdemaine.org/foldcut/ | [sen-sangaibisi] verbatim on Demaine's page and JCDCG98 p.1
- CONFIRMED | A solution is given, as a sequence of simple folds. | https://erikdemaine.org/foldcut/ | [sen-solution] verbatim on Demaine's page
- CONFIRMED | Fold-and-cut appears as a parlour trick in The Girl's Own Book by Lydia Maria Child. | https://erikdemaine.org/foldcut/ | [girls-own-book] Demaine: 1833, "early magic tricks"; Demaine spells "Lydia Marie", the standard name is Lydia Maria
- CONFIRMED | "National Standards and Emblems" by Kate Putnam Osgood, Harper's New Monthly Magazine vol. 47 no. 278, pp. 171–181 | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [osgood-citation] Wikipedia's citation names Osgood; the issue prints no bylines (no name near the article in the OCR), so the attribution rests on Wikipedia alone; pages 171-181 also in the survey reference
- CONFIRMED | tells the story that in 1777 Betsy Ross argued for a five-pointed star over a six-pointed one | https://archive.org/download/sim_harpers-magazine_1873-07_47_278/sim_harpers-magazine_1873-07_47_278_djvu.txt | [osgood-primary] the excerpt is verbatim in the OCR (lines 3062-3081 after whitespace normalisation): the tool's NOT FOUND is a tool miss, not a record error
- CONFIRMED | "she showed them how such a star could be made, by folding a sheet of paper and producing the pattern by a single cut." | https://archive.org/download/sim_harpers-magazine_1873-07_47_278/sim_harpers-magazine_1873-07_47_278_djvu.txt | [osgood-quote] matches the primary scan word for word apart from the OCR line-break hyphen ("sin- gle")
- CONFIRMED | The truth of the story is unclear | https://erikdemaine.org/foldcut/ | [osgood-truth-unclear] verbatim on Demaine's page
- OBSERVED | the Library of Congress scan was unreachable when this page was written, so that sentence is quoted from the transcription in Wikipedia's citation of the article rather than from the page itself. | not kept: memory/log.d/2026-08-15T1000Z-one-straight-cut.md, research/one-straight-cut/README.md and the commit messages 0f809aa4ec1..9e3b8600601 record no LoC fetch | [osgood-loc-unreachable] the log only says the 1873 quote was not read from the primary; on 2026-10-19 the LoC URL (survey and Wikipedia link) returned HTTP 522, consistent with the claim. The quote does match the Internet Archive scan, so the hedge could now be retired
- CONFIRMED | Will Blyth, Paper Magic (C. Arthur Pearson, London), one of several magicians publishing fold-and-cut examples in this period. | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [blyth-1920] Wikipedia text and reference
- CONFIRMED | Will Blyth, Paper Magic (C. Arthur Pearson, London) | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [blyth-ref] Wikipedia reference
- CONFIRMED | Harry Houdini, Houdini's Paper Magic (E. P. Dutton, pp. 176–177), gives a method for the five-pointed star. | https://erikdemaine.org/foldcut/ | [houdini-1922] Demaine: 1922, E. P. Dutton & Company, pp. 176-177, five-pointed star
- CONFIRMED | The book appears to have been ghostwritten by another magician, Walter Gibson. | https://erikdemaine.org/foldcut/ | [houdini-ghost] Demaine, citing David Lister; the page keeps the hedge "appears"
- CONFIRMED | Gerald Loe, Paper Capers (Magic, Inc.), works the idea hard: chains of stars, and any letter of the alphabet. | https://erikdemaine.org/foldcut/ | [loe-1955] Demaine; Gardner's own text (1995 reprint p.65) says the book explains "any desired letter of the alphabet ... a circular chain of stars" and names the publisher "Ireland Magic Company of Chicago"
- CONFIRMED | works the idea hard: chains of stars, and any letter of the alphabet. | https://erikdemaine.org/foldcut/ | [loe-letters] Demaine; Gardner reprint p.65 agrees
- CONFIRMED | Inspired by Loe, Martin Gardner writes about it in Scientific American (June) | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [inspired-by-loe] Wikipedia text
- CONFIRMED | Erik Demaine, Martin Demaine and Anna Lubiw answer it with the straight skeleton (JCDCG'98, LNCS 1763, pp. 104–117; SODA'99, pp. 891–892). | https://library.slmath.org/books/Book52/files/12dem.pdf | [ddl-1998-99] survey references [Demaine et al. 2000c] and [1999b]; see the page-range MINOR at cite-ddl98
- CONFIRMED | Marshall Bern, Erik Demaine, David Eppstein and Barry Hayes give the second solution, by disk packing (FUN'98, then OSME 2001, published 2002). | https://library.slmath.org/books/Book52/files/12dem.pdf | [disk-1998-2002] survey reference [Bern et al. 2002]; FUN98 from Demaine's page
- CONFIRMED | This is the one the same survey calls complete, and it is why the theorem is a theorem. | https://library.slmath.org/books/Book52/files/12dem.pdf | [disk-this-is-why] survey p.188 "second (complete) solution"
- CONFIRMED | This is the solution Wikipedia calls the first proof and the authors' own survey calls partial. | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [wiki-calls-first-proof] Wikipedia wikitext
- CONFIRMED | Take the shape and shrink it, keeping every edge parallel to where it started and moving each one inward at the same speed. | https://erikdemaine.org/foldcut/ | [shrink-description] Demaine's page
- CONFIRMED | a roof whose every surface slopes at forty-five degrees | https://en.wikipedia.org/wiki/Straight_skeleton | [roof-45] "constant slope"; with height = arrival time and unit speed the slope is 1, i.e. 45 degrees
- CONFIRMED | The rest are perpendiculars: from each corner of the skeleton, drop a line straight at the edge whose front got there, and let it carry on through the outline and reflect off every skeleton crease it meets. | https://erikdemaine.org/foldcut/ | [perp-description] Demaine: "shoot a ray perpendicular to each reachable cut edge, and the ray bounces (reflects) through any skeleton edges"; JCDCG98 sec. 4.2 defines it the same way
- MINOR | Erik D. Demaine, Martin L. Demaine, Anna Lubiw. "Folding and Cutting Paper." Revised Papers from the Japan Conference on Discrete and Computational Geometry (JCDCG'98), Lecture Notes in Computer Science vol. 1763, Tokyo, December 1998, pp. 104–117. | https://api.crossref.org/works/10.1007/978-3-540-46515-7_9 | [cite-ddl98] Crossref (Springer) gives pages 104-118 and Demaine's paper page says "The paper is 15 pages"; the page says pp. 104-117, following the survey reference. Same range in the 1998-99 timeline row
- CONFIRMED | Erik D. Demaine, Martin L. Demaine, Anna Lubiw. "Folding and one straight cut suffice." Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'99), pp. 891–892. | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [cite-ddl99] Wikipedia reference and survey [Demaine et al. 1999b]
- CONFIRMED | Marshall Bern, Erik D. Demaine, David Eppstein, Barry Hayes. "A Disk-Packing Algorithm for an Origami Magic Trick." Proceedings of the 3rd International Meeting of Origami Science, Math, and Education (OSME 2001). Earlier version: Proceedings of the International Conference on Fun with Algorithms (FUN'98). | https://erikdemaine.org/foldcut/ | [cite-bdeh] Demaine's page; survey reference gives Origami3, 2002
- MINOR | Erik D. Demaine and Joseph O'Rourke. "Folding and Unfolding in Computational Geometry." Combinatorial and Computational Geometry, MSRI Publications vol. 52 (2005), pp. 167–211. | https://library.slmath.org/books/Book52/contents.html | [cite-survey] the printed title is "A Survey of Folding and Unfolding in Computational Geometry" (title page, contents page, Crossref 10.1017/9781009701259.013); the page drops "A Survey of"
- MINOR | Erik D. Demaine and Joseph O'Rourke, "Folding and Unfolding in Computational Geometry", Combinatorial and Computational Geometry, MSRI Publications vol. 52 (2005), p. 188 | https://library.slmath.org/books/Book52/files/12dem.pdf | [cite-survey-md] same title slip in the attribution under the quote (and in engine.js header)
- CONFIRMED | Marshall Bern and Barry Hayes. "The complexity of flat origami." Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms (1996), pp. 175–183. | https://library.slmath.org/books/Book52/files/12dem.pdf | [cite-bern-hayes] survey reference verbatim
- CONFIRMED | Flat foldability of a crease pattern is NP-hard; local foldability is linear-time. | https://library.slmath.org/books/Book52/files/12dem.pdf | [cite-bern-hayes-content] survey 3.3.1 and 3.3.2
- CONFIRMED | Oswin Aichholzer, Franz Aurenhammer, David Alberts, Bernd Gärtner. "A novel type of skeleton for polygons." Journal of Universal Computer Science 1(12), 1995. The straight skeleton itself. | https://www.jucs.org/jucs_1_12/a_novel_type_of/Aichholzer_O.html | [cite-aichholzer] JUCS page: vol. 1 (1995) issue 12, authors Aichholzer, Aurenhammer, Alberts, Gaertner in that order on the article
- CONFIRMED | Kan Chu Sen. Wakoku Chiyekurabe, 1721. Scans of the relevant pages are linked from Erik Demaine's fold-and-cut page. | https://erikdemaine.org/foldcut/ | [cite-sen] Demaine links sen_book.html
- CONFIRMED | "National Standards and Emblems." Harper's New Monthly Magazine vol. 47, no. 278, July 1873. | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [cite-harpers] IA record "1873-07: Vol 47 Iss 278"; survey "(July 1873), 171-181"
- CONFIRMED | Harry Houdini. Paper Magic. E. P. Dutton & Company, 1922, pp. 176–177. | https://erikdemaine.org/foldcut/ | [cite-houdini] Demaine
- CONFIRMED | Gerald Loe. Paper Capers. Magic, Inc., Chicago, 1955. | https://erikdemaine.org/foldcut/ | [cite-loe] Demaine "Magic, Inc., Chicago"; Gardner's bibliography says "Chicago: Ireland Magic Co., 1955", the firm's earlier name; not counted as an error
- CONFIRMED | Martin Gardner. "Paper cutting," chapter 5 of New Mathematical Diversions (revised edition), Mathematical Association of America, 1995. Originally in Scientific American, 1960. | https://library.slmath.org/books/Book52/files/12dem.pdf | [cite-gardner] survey reference; the 1995 MAA reprint text has "CHAPTER FIVE Paper Cutting"
- CONFIRMED | The CGAL Project. CGAL User and Reference Manual, 5.6. Straight_skeleton_2, used here only as an independent check, never in the page. | https://doc.cgal.org/5.6/Straight_skeleton_2/index.html | [cite-cgal] manual title; the oracle includes only CGAL headers and this machine has libcgal-dev 5.6 (CGAL_VERSION 5.6)
- CONFIRMED | an unrelated implementation by other people using exact predicates | https://doc.cgal.org/5.6/Straight_skeleton_2/index.html | [cgal-exact-predicates] manual sentence verbatim; oracle/skel.cpp line 24 uses Exact_predicates_inexact_constructions_kernel; authors Cacciola, Loriot, Rouxel-Labbe
- CONFIRMED | a separate implementation by other people using exact predicates | https://doc.cgal.org/5.6/Straight_skeleton_2/index.html | [cgal-exact-predicates-html] as above
- CONFIRMED | 629 assertions from node verify-skeleton.mjs --random 200, over fixed cases, 200 pseudo-random polygons, 100 pseudo-random sheets with holes, and the fully degenerate five-pointed star whose every edge touches one inscribed circle. | research/one-straight-cut/verify-skeleton.mjs | [data-sum-629] recreated 2026-10-19 in a scratch copy at 4fcb19ad5e: "verify-skeleton: 629 passed, 0 failed"
- MINOR | 629 assertions, including 200 pseudo-random polygons, 100 pseudo-random sheets with holes, and the fully degenerate pentagram whose every edge touches one inscribed circle | research/one-straight-cut/verify-skeleton.mjs | [data-sum-629-md] the count holds, but "every edge touches one inscribed circle" is not so for the ten-edge outline the engine is given: with R = 1 the five lines lie at 0.309 from the centre (the inscribed circle's radius) while every outline edge runs from a tip (1.0) to an inner vertex (0.382), so no edge touches the circle; only the edges' extended lines do (they touch it at the inner pentagon's side midpoints, inside the star). The degeneracy claimed (the front vanishes at one instant) is real. Same wording in the HTML "How this was checked" and verify-skeleton.mjs line 114
- CONFIRMED | a separate implementation by other people using exact predicates, and the two are required to agree | research/one-straight-cut/verify-skeleton.mjs | [data-cgal-header] header lines 4-6
- CONFIRMED | carry 69 assertions of their own | research/one-straight-cut/verify-foldcut.mjs | [data-69] recreated 2026-10-19: "verify-foldcut: 69 passed, 0 failed" (scratch copy, and again from the 3 served files)
- MINOR | an undamaged pattern must certify, a crease rotated by half a degree must be refused, a deleted perpendicular must be refused, and folding along the outline instead of across it must be refused. | research/one-straight-cut/verify-foldcut.mjs | [data-four-controls] "four negative controls" lists "an undamaged pattern must certify", which is the positive baseline; the file itself says "Three ways to break a working pattern" (line 83)
- MINOR | The construction and its certificate carry 69 more, four of them negative controls | research/one-straight-cut/verify-foldcut.mjs | [data-69-md] same: three negative controls and one baseline
- CONFIRMED | A further 33 assertions | research/one-straight-cut/verify-page.mjs | [data-33] recreated 2026-10-19 from research/one-straight-cut/ (working-tree files identical to 4fcb19ad5e): "verify-page: 33 passed, 0 failed"
- CONFIRMED | the slider is moved through the actual DOM, and the certificate is required to turn red and to name which check failed. | research/one-straight-cut/verify-page.mjs | [data-slider-dom] verify-page.mjs lines 101-108
- CONFIRMED | The engine on this page is the same file the verifiers import. Not a copy of it. | research/one-straight-cut/skeleton.mjs | [data-same-file] skeleton.mjs, foldcut.mjs, arrangement.mjs re-export engine.js; index.html imports ./engine.js
- CONFIRMED | The comparison against CGAL is itself given something to reject: a 0.1 mm displaced arc and a 0.01 mm arrival-time error, each of which a different code path has to notice. | research/one-straight-cut/verify-skeleton.mjs | [data-cgal-selftest] verify-skeleton.mjs lines 127-146: bent[0].b.y += 0.1 and slow[0].tb += 0.01, separate checks
- CONFIRMED | the two are required to agree segment for segment and arrival time for arrival time | research/one-straight-cut/verify-skeleton.mjs | [data-cgal-agree] header and README
- CONFIRMED | Over 300 pseudo-random polygons of 3 to 14 vertices, this implementation certified 297. | research/one-straight-cut/census.mjs | [data-census-297] recreated 2026-10-19: "certified : 297 / refused : 3"
- CONFIRMED | Over 300 random polygons the certificate passes 297 | research/one-straight-cut/census.mjs | [data-census-md] as above
- CONFIRMED | Each demanded a single perpendicular crease that reflects its way across the sheet 18,475, 7,686 and 22,907 times before it terminates. | research/one-straight-cut/census.mjs | [data-census-reflections] recreated: "# 73 terminates after 18475", "#167 ... 7686", "#178 ... 22907"; at the 4096 cap 3, 3 and 7 perpendiculars were truncated, so "a single" means the longest one
- CONFIRMED | For a typical shape that number is 6; nine in ten stay under 204. | research/one-straight-cut/census.mjs | [data-census-typical] recreated: median 6, 90th 204; counted directly, 270 of 300 maxima are below 204 (271 at or below), so "nine in ten stay under 204" is exact
- CONFIRMED | (node census.mjs reproduces all of it from a fixed seed.) | research/one-straight-cut/census.mjs | [data-census-reproduces] two runs on 2026-10-19 printed the same distribution (median 6, 90th 204, 99th 4097)
- CONFIRMED | why a perpendicular can occasionally bounce twenty thousand times | research/one-straight-cut/census.mjs | [data-bounce-twenty-thousand] max 22907
- CONFIRMED | Working code, the CGAL cross-check, the verifiers, the census and printable sheets are in `research/one-straight-cut/`. | research/one-straight-cut/README.md | [data-all-in-dir] ls-tree at 4fcb19ad5e
- CONFIRMED | Then it asks four questions of the result, and the fourth is the one that matters. | public/strata/one-straight-cut/index.html | [data-four-questions] writeCertificate() four rows
- CONFIRMED | it walks from region to region across the crease pattern, composing one reflection per crease, until every region has an isometry saying where it lands. | public/strata/one-straight-cut/engine.js | [data-fold-walk] engine.js foldAndCertify doc comment
- CONFIRMED | The fourth check never looks at your shape. | public/strata/one-straight-cut/engine.js | [data-blade-never-looks] bladePreimage uses the folded line and the faces' isometries, then compares with cut edges
- CONFIRMED | The straight skeleton is computed here, in your browser, by simulating the shrinking front. | public/strata/one-straight-cut/index.html | [data-in-browser] index.html imports engine.js; engine.js header "No imports, no dependencies, no network"
- CONFIRMED | used here only as an independent check, never in the page | public/strata/one-straight-cut/engine.js | [data-no-cgal-in-page] no CGAL in engine.js or index.html
- CONFIRMED | The pattern is drawn at true millimetres | public/strata/one-straight-cut/index.html | [data-true-mm] buildSheet sets mm units via AW.printable.sheet
- CONFIRMED | So the sheet carries a printed ruler and four registration marks. | public/_kit/printable.js | [data-ruler-marks] index.html line 549 uses AW.printable.sheet (fiducials + ruler)
- MINOR | The page hands you no crease pattern it has not checked, and it will refuse rather than print a shape it cannot certify. | public/strata/one-straight-cut/index.html | [data-refuse-print] the print handler (lines 590-596) prints an uncertified pattern after confirm("This pattern did NOT certify. ... Print it anyway?"); the page warns and asks, it does not refuse
- CONFIRMED | One A4 or US Letter page. | public/strata/one-straight-cut/index.html | [data-one-page] 150 mm square plus kit frame fits A4 and Letter
- CONFIRMED | Rotate one crease and watch what happens: the certificate goes red, and the blue line, which is computed only from the folded state, walks off your shape and starts cutting through the middle of it. | research/one-straight-cut/verify-page.mjs | [data-sabotage-slider] verify-page asserts the certificate turns red and the blade leaves the outline; "through the middle" is description, not asserted
- CONFIRMED | Choose a shape below and the page will work out the creases | public/strata/one-straight-cut/index.html | [data-presets-certify] PRESETS; verify-page asserts every preset certifies (33 passed)
- OBSERVED | Two real defects were found this way and are worth naming, because both are the same kind of mistake. | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [obs-two-defects] log and README list four defects; the page names two of them, defect 2 "CGAL found it"
- OBSERVED | The skeleton engine silently dropped an entire arc whenever a shape sat off-centre in a much larger sheet | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [obs-off-centre-arc] defect 2 verbatim
- OBSERVED | its split-event guard excluded candidates by vertex adjacency when what must be excluded is edge adjacency, and once two shrinking fronts merge those two things stop being the same. | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [obs-vertex-edge-adjacency] defect 2
- OBSERVED | the construction accepted an edge as owning a skeleton corner whenever the corner sat at the right distance from that edge's line, when what it must also require is that the perpendicular point into that corner's own wedge. | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [obs-owner-test] defect 3; README adds "Emitting it anyway put two non-parallel creases through one point"
- OBSERVED | Both passed every test written before them. | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [obs-both-passed] log and README
- OBSERVED | this page did get it backwards first | recreated: deleting one perpendicular from the L-shape pattern (verify-foldcut control b) gives foldAndCertify ok=false, "fold is inconsistent: two routes to the same face differ by 2.60e+2" | [obs-loose-end] log defect 1 records the mistake; the certificate does refuse a loose end
- OBSERVED | retrieved 15 August 2026 | research/one-straight-cut/README.md | [obs-retrieved] README "retrieved 2026-08-15"; the sentence is still on the page (last updated March 3, 2025)
- OBSERVED | Over 300 random polygons the certificate passes 297; the three it refuses each demand one perpendicular crease that reflects 18,475, 7,686 and 22,907 times before terminating. | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [obs-297-log] and recreated by census.mjs
- CONFIRMED | You can get any shape with straight sides this way, and any number of them at once, however awkward. | https://erikdemaine.org/foldcut/ | [dek-any-shape] Demaine theorem; "any number of them at once" = multiple disjoint polygons
- CONFIRMED | Martin Gardner was the first to pose it as a problem, in 1960 | https://library.slmath.org/books/Book52/files/12dem.pdf | [gardner-first-pose-html] survey p.188 verbatim
- CONFIRMED | which several tools will draw for you | https://en.wikipedia.org/wiki/Fold-and-cut_theorem | [several-tools] Wikipedia names JOrigami; Demaine's page says Lang's TreeMaker also solves the problem when the faces are convex
- CONFIRMED | the time at which the shrinking front arrives at a point is the height of a roof over the shape | https://en.wikipedia.org/wiki/Straight_skeleton | [roof-height-time] Wikipedia straight skeleton, Applications
- CONFIRMED | The skeleton gives most of the creases. | https://erikdemaine.org/foldcut/ | [skeleton-most-creases] Demaine verbatim
- CONFIRMED | Every point at height zero, which is to say every point of your outline, ends up on one line; everything inside ends up on one side of it and everything outside on the other. | https://library.slmath.org/books/Book52/files/12dem.pdf | [fold-to-a-line] survey p.188 states the line condition; JCDCG98 sec. 3 the above/below side assignment
- CONFIRMED | a loose end cannot fold flat: go once around it and you have composed a single reflection instead of the identity. | https://library.slmath.org/books/Book52/files/12dem.pdf | [loose-end-reflection] sec. 3.3.1: a single-vertex pattern folds flat "precisely if the alternate angles around the vertex sum to 180", which needs an even number of creases; one crease ending at a point fails it (the record had no source)
- CONFIRMED | Whether paper can reach that position without passing through itself is a known hard problem, left to your hands. | https://library.slmath.org/books/Book52/files/12dem.pdf | [plain-hard-problem] survey 3.3.2, general case
- CONFIRMED | So this page marks the ridge and gutter folds, which follow from the geometry, and deliberately leaves the perpendicular creases unmarked. | public/strata/one-straight-cut/index.html | [perp-unmarked] buildSheet draws perps dotted with no sense
- CONFIRMED | so the ridge and gutter folds are marked and the perpendicular creases deliberately are not. | public/strata/one-straight-cut/index.html | [perp-unmarked-md] skeleton dashed by sense(s)
- UNVERIFIABLE | Nearly every driver defaults to fit to page or shrink oversized pages, lands somewhere around 94 to 96 per cent, and never says so. | none | [printers-rescale] no outside source; the only statement is public/_kit/printable.js, the page's own claim
- MINOR | creases have to reach the edge of the sheet, and no printer prints to the edge of the paper. | https://en.wikipedia.org/wiki/Gutenprint | [no-printer-edge] overstated: borderless printing exists ("borderless or bordered" for Canon printers; "Borderless selection, added in version 5.2.9"); most office printers and browser print paths do leave a margin, so the advice stands
- CONFIRMED | And the sheet in your hands is bounded by something none of them mention, which is how small a crease your fingers can actually make. | https://erikdemaine.org/foldcut/ | [nobody-mentions-crease-size] "none of them" are the theorem, the method and the certificate; neither Demaine's page nor the survey sec. 3.2.3 states a minimum crease size (both do discuss creases, the positive control)
- CONFIRMED | Both are about what paper can do that other tools cannot, and both turn on folding as an exact operation. | src/content/strata/what-paper-can-do.md | [related-wpcd-1] what-paper-can-do dek
- CONFIRMED | That page proves a fold outruns compass and straightedge | src/content/strata/what-paper-can-do.md | [related-wpcd-2] what-paper-can-do plain
- CONFIRMED | Both manufacture a precision instrument on the reader's own printer, carry the same registration marks and printed ruler, and refuse to claim accuracy until the reader has measured what their printer actually did. | public/strata/the-paper-computer/index.html | [related-tpc] both use public/_kit/printable.js; the-paper-computer prints only when calibrated
- CONFIRMED | Both are about verification that only appears to verify. | src/content/strata/the-check-that-cannot-fail.md | [related-tccf] the-check-that-cannot-fail dek
- CONFIRMED | Both put a negative control in the reader's hands rather than in the apparatus: there a planted witness the same unmodified search must find, here a slider that damages one crease and makes the certificate say so. | public/strata/twenty-empty-squares/index.html | [related-tes] that page has reader buttons "insert probe", "remove probe", "run browser probe" on the same browser search, so the control is in the reader's hands
- OBSERVED | Last verified 15 August 2026, when it was written. | memory/log.d/2026-08-15T1000Z-one-straight-cut.md | [placard-verified] log dated 2026-08-15 records all five runs
- CONFIRMED | verify-foldcut.mjs is published, and it runs from 3 files this site serves (all 3, and the command). | https://artwaste.land/checks/ | [placard-three-files] recreated 2026-10-19 in an empty directory from the 3 served files and the /checks/ recipe: exit 0, "verify-foldcut: 69 passed, 0 failed"
- CONFIRMED | It is that this one will not hand you a pattern it has not checked. | public/strata/one-straight-cut/index.html | [md-not-hand-unchecked] every pattern is certified before the print button; an uncertified one is labelled REFUSED and needs a confirm (see data-refuse-print)
- CONFIRMED | It takes the blade's line in the folded sheet, pulls it back through each region's own isometry, and draws where the scissors actually travel across the unfolded paper, without ever consulting your shape. | public/strata/one-straight-cut/engine.js | [md-blade-preimage] engine.js bladePreimage
- CONFIRMED | requires your whole outline to land on a single line with nothing else on it | public/strata/one-straight-cut/index.html | [md-outline-one-line] writeCertificate rows 2 and 3
- MINOR | A pattern that folds beautifully and cuts out the wrong thing fails there rather than passing quietly. | research/one-straight-cut/verify-foldcut.mjs | [md-wrong-shape-fails] true by construction of bladePreimage, but no control shows the fourth check catching it: run on the L-shape, the rotated crease fails check 1 (consistency 2.52e+2), the missing perpendicular fails check 1 with the blade at 0, and the "folding ALONG the outline" control, the one meant for "folds correctly and cuts out nothing", is refused only because "no cut edges found on the sheet"
- WRONG | (added) He gets close enough to see the shape of it: "more complicated designs present formidable problems." (HTML, 1960 row) | https://archive.org/download/new-mathematical-diversions/New%20Mathematical%20Diversions_djvu.txt | same unmarked omission as gardner-quote ("designs--the swastika for instance--present") [re-read by the fixer: raised to WRONG under the quotation rule, K17]
- MINOR | (added) 33 further assertions drive the live page in a real browser (markdown body) | research/one-straight-cut/verify-page.mjs | lines 17-34 serve public/ from 127.0.0.1 and drive that local copy, not the live site; the HTML says "this page itself", which is accurate
- OBSERVED | (added) so the check was rebuilt to ask where the blade actually goes (relates note to the-check-that-cannot-fail) | not kept: the first commit 0f809aa4ec1 already ships bladePreimage ("pulls the blade's line back through each face's isometry"); the log describes no earlier fold-only check | an observation from before the first commit, not kept
- MINOR | (added) seoTitle: "Cut Any Shape With One Straight Cut" | https://erikdemaine.org/foldcut/ | the theorem is for straight-line cuts ("every pattern (plane graph) of straight-line cuts"); the dek and plain say "straight sides" but the title drops it

## Findings
- [fixed] gardner-quote-unmarked-omission (K1, raised from MINOR to WRONG): the markdown summary quoted Gardner as "more complicated designs present formidable problems"; his chapter "Paper Cutting" (New Mathematical Diversions, 1995, Internet Archive full text) has "the swastika for instance" between "designs" and "present". The omission is now marked with an ellipsis, no word of Gardner's changed, with a dated Corrected line on the page and in the markdown; verify-claims.mjs, which held the unmarked form as verbatim, now holds the marked form.
- [fixed] gardner-quote-unmarked-omission (K17, the HTML 1960 timeline row, raised from MINOR to WRONG): the same quotation, ellipsis added; covered by the same Corrected line.
- [fixed] survey-quote-unmarked-omission (K2, raised from MINOR to WRONG): the blockquote under "The proof that is still in progress" and the engine.js header joined "detail here." to "The second (complete) solution" over the dropped sentence "See Figure 14 for two examples." and stopped "based on disk packing." where the survey (p. 188, re-read from the SLMath PDF) goes on "to make the problem more local". Both omissions are now marked with ellipses, with a dated Corrected line on the page and in the markdown.
- [declined] nobody-established-overstated (K3, MINOR): Tier B MINOR, decided from the checker's note: JCDCG98 states Theorem 1 and Corollary 1 for a large class of cut graphs, so "nobody has established" is stronger than the record, though Demaine's page does say the full proof details are unpublished. An overstatement, not in a sentence being fixed; listed.
- [declined] subset-unwritten-unsupported (K4, MINOR): Tier B MINOR, from the checker's note: JCDCG98 sec. 4.2 writes down that only real perpendiculars are folded; the unpublished part per Demaine is the proof details. Loose wording about which part is unwritten; listed.
- [declined] intractability-applied-to-class (K5, MINOR): Tier B MINOR, from the checker's note: the NP-hardness of flat foldability (Bern and Hayes) is a general result; for straight-skeleton patterns JCDCG98 gives an algorithm for the folded state. Overstated scope, the page says "in general" a sentence earlier; listed.
- [declined] intractability-applied-to-class (K6, MINOR): Tier B MINOR, same overstatement as K5 ("no algorithm runs efficiently" for this class); listed.
- [declined] intractability-applied-to-class (K7, MINOR): Tier B MINOR, the HTML copy of K6 ("The half your hands decide"); listed.
- [declined] cite-ddl98-page-range (K8, MINOR): Tier B MINOR, from the checker's note: Crossref gives pp. 104-118 where the page, following the survey's reference list, says 104-117. A citation precision slip; listed.
- [declined] cite-survey-title (K9, MINOR): Tier B MINOR: the survey's printed title begins "A Survey of" (I saw it on the PDF's title page while re-reading K2); the page drops those words. Metadata slip in the Sources list, not in a sentence being fixed; listed.
- [declined] cite-survey-title (K10, MINOR): Tier B MINOR, the same title slip as K9 in the attribution line under the blockquote and the engine.js header. The attribution is a separate line from the quotation being fixed for K2, so the tier B rule lists it; listed.
- [declined] pentagram-edges-vs-lines (K11, MINOR): Tier B MINOR, from the checker's note: the edges' supporting lines are tangent to one circle (the degeneracy is real), but the outline's edges themselves do not touch it. Loose wording about a test case; listed.
- [declined] four-negative-controls (K12, MINOR): Tier B MINOR, from the checker's note: verify-foldcut.mjs line 83 says "Three ways to break a working pattern"; the fourth listed item is the positive baseline. An overstatement of the page's own checking; listed.
- [declined] four-negative-controls (K13, MINOR): Tier B MINOR, the markdown copy of K12; listed.
- [declined] refuse-print-overstated (K14, MINOR): Tier B MINOR: I read the print handler (index.html lines 590-596); an uncertified pattern prints after confirm("This pattern did NOT certify ... Print it anyway?"). The page refuses by default and lets the reader override, so "will refuse" overstates the gate. Kept at the checker's level; listed.
- [declined] no-printer-edge (K15, MINOR): Tier B MINOR, from the checker's note: borderless printing exists, though most office and browser print paths leave a margin, so the advice stands. Overstatement; listed.
- [declined] fourth-check-uncontrolled (K16, MINOR): Tier B MINOR, from the checker's note: true by construction of bladePreimage, but no control isolates the fourth check. An overstatement of the page's own demonstrated checking; listed.
- [declined] live-page-is-local-copy (K18, MINOR): Tier B MINOR, from the checker's note: verify-page.mjs lines 17-34 serve public/ locally and drive that copy, so "the live page" in the markdown is loose; the HTML's "this page itself" is accurate. Listed.
- [declined] seo-title-drops-straight (K19, MINOR): Tier B MINOR, from the checker's note: the seoTitle "Cut Any Shape" drops the straight-sided qualifier the dek and plain carry. Metadata; listed.