--- slug: seventeen-and-no-more 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: 175 confirmed: 115 wrong: 53 unverifiable: 0 observed: 0 stale: 7 --- This was the directed repair pass for seventeen-and-no-more, read at e9112b0f08, using the 62 decisions in queue-1014c after rereading their code, history and primary evidence. It reconciles the Codex source record and the two existing checker records, retains their confirmed claims with provenance, and records each triage item separately, including duplicate reports of one defect. The fixer reran the mathematical and page checks; this is not a claim of a new independent second-agent review. MINOR lines count in wrong; their repairs have no dated page line. Earlier quoted wording uses [...] where punctuation was elided. The record date and page correction date follow the requested 2026-10-14 assay date. ## Claims - WRONG | [X1] “For n = 2, 3, 4, 6 no shorter vector appears, and the lattice stands.” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Recreated before editing: n=2 produced a length effectively zero and the impossible verdict. The old angle was 2*(2*pi/n), so R squared was identity. The new Rv-v has length 2 at n=2; all eight slider orders pass checks of length and verdict. - WRONG | [X2] “That single restriction is why crystals come in the shapes they do, why honeycombs are hexagons and not pentagons” | https://pmc.ncbi.nlm.nih.gov/articles/PMC3730681/ | Read the abstract and sections 1-2: the paper models wax flow and surface tension. The restriction permits 3-, 4- and 6-fold rotations, so cannot by itself select hexagonal cells. Removed the causal claim. - WRONG | [X3] “Its pattern builds itself from a single asymmetric motif, repeated by exactly the rigid motions that group allows” | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [X4] “the self-check closes the generators into a finite group in your browser and confirms its order and content match the named group” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read the acceptance condition: closure and expected order only. A p4m generator set under the p4g label still passes that condition. The readout, prose, metadata, markdown and relation now state closure and count modulo translations, without certifying the name. - WRONG | [X5] “Less than 2 and the surface curls into a sphere (the finite rosette and polyhedral groups);” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html | Read chapter 4, p. 53: MN and *MN require equal orders, with an omitted 1 allowed in notation. Recreated a lone cone 3, an unequal cone pair, and their corner counterparts. They now receive an inadmissible verdict; equal pairs and valid flat and hyperbolic controls retain their geometry. - WRONG | [X6] “Generated, not hand-typed” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | Read the literal seventeen-row array and symCost in the page, and enumerateCost2 in the offline verifier. The ledger now says its rows are listed, its costs are live, and enumeration is offline. - WRONG | [X7] “the same enumeration the offline verifier runs.” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | Read the literal seventeen-row array and symCost in the page, and enumerateCost2 in the offline verifier. The ledger now says its rows are listed, its costs are live, and enumeration is offline. - STALE | [X8] “Reproduce from a fresh checkout in a minute: bash research/sorting-networks/lean/install-lean.sh then bash research/crystallographic-restriction/lean/verify.sh.” | research/crystallographic-restriction/lean/verify.sh ; research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Reran the unpinned wrapper: Lean 4.35.0 exits 1 at line 248 with Unknown identifier ht. Reran with Lean 4.33.0: exit 0, eleven printed M2 results use only propext and Quot.sound, and five M4 results use no axioms. The wrapper and reproduction directions now select 4.33.0; no proof theorem was changed. - WRONG | [X9] “A three-dimensional decagonal or icosahedral quasicrystal (Shechtman's rapidly-quenched Al–Mn alloy, 1982; Nobel Prize in chemistry 2011) is standardly described as a cut-and-project shadow of a periodic lattice in dimension 6, where five- and ten-fold symmetries fit fine, into 3-space, where they do not.” | https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Read the cached primary Nobel PDF text, pp. 4 and 9: decagonal examples are axial, and the standard superspaces have dimension 5 for axial and 6 for icosahedral examples. Updated both authored copies. A fresh web-tool fetch was refused with HTTP 403; the retained primary document was readable. - WRONG | [X10] “Same file exhibits the 4D companion of Φ5 as a witness of order exactly 5, so the pentagon is both forbidden here and available one dimension up.” | research/crystallographic-restriction/lean/CrystallographicRestriction.lean ; public/strata/seventeen-and-no-more/index.html | Read the displayed 2 by 2 theorem and 4 by 4 companion, and the M4 C5 definition and exact-order checks. Four minus two is two. Both one-dimension-up labels now say two dimensions higher. - MINOR | [X11] “Proven and re-derived live: the crystallographic restriction; the cost of all 17 symbols; that exactly 17 finite symbols cost 2; that each rendered group is a genuine closed group of the stated order.” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | The browser computes selected examples, listed-row costs and closure; only the offline code enumerates. Narrowed metadata, dek and checking summary. This MINOR repair has no separate on-page correction line. - STALE | [X12] “Last verified 21 June 2026, when it was written.” | git show e05770fb79:memory/log.d/2026-08-20T2100Z-orders-one-through-six.md ; assay/records/seventeen-and-no-more.md | Read the August log recording 21/21 after the original June verification. The layer refresh reads this pass date and updates only its placard date. The triage explicitly exempts this date from an Updated line. - CONFIRMED | [X13] “The crystallographic restriction** (Barlow 1894 / Fedorov)” | https://www.emis.de/monographs/jablan/chap5.htm | REJECTED with evidence: the cited monograph explicitly attributes the crystallographic restriction to Barlow in 1894. The discrepant 1893 excerpt does not justify changing the supported date. The existing wording remains. - STALE | [X14] “the still-open chromatic number of the plane (5, 6, or 7)” | https://arxiv.org/abs/1804.02385 ; https://raw.githubusercontent.com/openai/math/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026/paper.pdf | Read the 2018 abstract and the September 23, 2026 preprint, abstract and Theorem 1.1. The old range was 5 to 7; the newer preprint reports 6 to 7 without regularity assumptions. The relation attributes the newer bound to that preprint, without claiming an independent verification of its proof. - WRONG | [X15] “Both are machine-checked, and on the same arithmetic fact: 5 is not a perfect square.” | research/incommensurable-immersive/lean/Sqrt2.lean ; research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Read the cited proof: a*a=2*(b*b) implies b=0 by parity and descent. It concerns sqrt(2), not a shared nonsquareness-of-5 proof. Corrected the relation, its teaser source, and the markdown historical bridge. - WRONG | [X16] “the only orientation-preserving integer 2×2 matrix of order 5 is the identity” | research/crystallographic-restriction/lean/CrystallographicRestriction.lean | The theorem assumes M.npow 5=I and concludes M=I; it does not assign exact order 5 to identity. Replaced the exact-order claim with the fifth-power hypothesis. - WRONG | [X17] “This is the fifth result in this ground to be machine-checked, and the first to deepen an existing layer rather than arrive as a new one” | git show 47f92de4bf6 ; git show e05770fb79:memory/log.d/2026-06-29T2240Z-crystallographic-restriction-machine-checked.md | The earlier commit at 2026-06-29T02:41:07Z already adds a Lean proof to incommensurable. Removed first-to-deepen while retaining the dated fifth-result count accepted by both checker records. - WRONG | [X18] “p3m1 Conway *333 · rotation centres: 3-fold · symmetries per cell: 6” | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [X19] “generators close into a group of order 6 (expected 6); composition stays inside the set. This is genuinely p3m1.” | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [X20] “p31m Conway 3*3 · rotation centres: 3-fold · symmetries per cell: 6” | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [X21] “generators close into a group of order 6 (expected 6); composition stays inside the set. This is genuinely p31m.” | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [X22] “= 2 exactly → a EUCLIDEAN wallpaper group (flat, repeats forever)” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read chapter 3 costs and the boolean-mirror renderer. It charged multiple mirrors but printed one and printed unbound corners as cones. The renderer now retains each boundary and its corners; unbound corner buttons are disabled. ** costs 2; *** costs 3; *2*3 costs 31/12. The tests also cover four cones and the valid *632 control. - WRONG | [X23] “< 2 → SPHERICAL: a finite rosette / polyhedral symmetry, not wallpaper” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html | Read chapter 4, p. 53: MN and *MN require equal orders, with an omitted 1 allowed in notation. Recreated a lone cone 3, an unequal cone pair, and their corner counterparts. They now receive an inadmissible verdict; equal pairs and valid flat and hyperbolic controls retain their geometry. - WRONG | [X24] “> 2 → HYPERBOLIC: lives in the curved plane (infinitely many such groups)” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read chapter 3 costs and the boolean-mirror renderer. It charged multiple mirrors but printed one and printed unbound corners as cones. The renderer now retains each boundary and its corners; unbound corner buttons are disabled. ** costs 2; *** costs 3; *2*3 costs 31/12. The tests also cover four cones and the valid *632 control. - WRONG | [X25] “2 cos(2π/2) = -2.0000 integer ✓ shortest combination v′ − v has length 0.000·|v| shorter than v [...] contradiction order 2 is impossible” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Recreated before editing: n=2 produced a length effectively zero and the impossible verdict. The old angle was 2*(2*pi/n), so R squared was identity. The new Rv-v has length 2 at n=2; all eight slider orders pass checks of length and verdict. - MINOR | [X26] “Every claim re-derived live and checked offline (26/26).” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | The browser computes selected examples, listed-row costs and closure; only the offline code enumerates. Narrowed metadata, dek and checking summary. This MINOR repair has no separate on-page correction line. - MINOR | [X27] “Two short proofs say why, and both run in front of you.” | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | The browser computes selected examples, listed-row costs and closure; only the offline code enumerates. Narrowed metadata, dek and checking summary. This MINOR repair has no separate on-page correction line. - STALE | [X28] (added) "dateModified":"2026-06-21" | public/strata/seventeen-and-no-more/index.html | The June creation date survived a September correction already present in the page. JSON-LD dateModified now says 2026-10-14, the requested revision date. This metadata-only refresh is exempt from a separate Updated line. - WRONG | [K1] [no-shorter-allowed] For n = 2, 3, 4, 6 no shorter vector appears, and the lattice stands. | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Recreated before editing: n=2 produced a length effectively zero and the impossible verdict. The old angle was 2*(2*pi/n), so R squared was identity. The new Rv-v has length 2 at n=2; all eight slider orders pass checks of length and verdict. - WRONG | [K2] [honeycombs] That single restriction is why crystals come in the shapes they do, why honeycombs are hexagons and not pentagons | https://pmc.ncbi.nlm.nih.gov/articles/PMC3730681/ | Read the abstract and sections 1-2: the paper models wax flow and surface tension. The restriction permits 3-, 4- and 6-fold rotations, so cannot by itself select hexagonal cells. Removed the causal claim. - MINOR | [K3] [bathroom-tiles] why your bathroom floor can be square or triangular tiles but never regular five-sided ones. | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html ; public/strata/seventeen-and-no-more/index.html | Read the restriction statement; it limits periodic rotational symmetries. The regular pentagon tiling obstruction is its 108-degree vertex angle, which cannot fill 360 degrees. Replaced the causal bridge minimally; recorded as MINOR without a correction line. - WRONG | [K4] [single-motif] Its pattern builds itself from a single asymmetric motif, repeated by exactly the rigid motions that group allows | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [K5] [tiler-selfcheck] the self-check closes the generators into a finite group in your browser and confirms its order and content match the named group | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read the acceptance condition: closure and expected order only. A p4m generator set under the p4g label still passes that condition. The readout, prose, metadata, markdown and relation now state closure and count modulo translations, without certifying the name. - WRONG | [K6] [not-lookalike] so this really is, say, p4g, not a lookalike. | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read the acceptance condition: closure and expected order only. A p4m generator set under the p4g label still passes that condition. The readout, prose, metadata, markdown and relation now state closure and count modulo translations, without certifying the name. - WRONG | [K7] [spherical-cost] Less than 2 and the surface curls into a sphere (the finite rosette and polyhedral groups); | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html | Read chapter 4, p. 53: MN and *MN require equal orders, with an omitted 1 allowed in notation. Recreated a lone cone 3, an unequal cone pair, and their corner counterparts. They now receive an inadmissible verdict; equal pairs and valid flat and hyperbolic controls retain their geometry. - WRONG | [K8] [ledger-generated] Generated, not hand-typed | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | Read the literal seventeen-row array and symCost in the page, and enumerateCost2 in the offline verifier. The ledger now says its rows are listed, its costs are live, and enumeration is offline. - WRONG | [K9] [same-enumeration] the same enumeration the offline verifier runs. | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | Read the literal seventeen-row array and symCost in the page, and enumerateCost2 in the offline verifier. The ledger now says its rows are listed, its costs are live, and enumeration is offline. - MINOR | [K10] [alhambra-survey] When Branko Grünbaum, Zdenka Grünbaum and G. C. Shephard actually catalogued the palace's ornaments in 1986, they could find clear examples of at most thirteen of the seventeen | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum.What-symmetry-groups-are-present-in-the-Alhambra-2006-2hxz2f4.pdf ; https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum-Grunbaum-Shephard.Symmetry-in-Moorish-and-other-ornaments.1986-12jyvah.pdf | Read both retained primary PDF texts: the 2006 paper dates the visit and photos to 1983; the 1986 paper reports thirteen classes and cites a June 1983 presentation. Clarified survey year versus publication year. No correction line for this MINOR change. - MINOR | [K11] [alhambra-exact-count] and the answer was: fewer than the legend claims, and the exact count depends on how strictly you read a worn, repaired, many-handed surface. | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum.What-symmetry-groups-are-present-in-the-Alhambra-2006-2hxz2f4.pdf | Read the discussion of colour-preserving symmetry, interlacing and whether a small part or a larger pattern is counted. Attributed the differing counts to those stated criteria; made no claim that repairs or multiple artisans never occurred. MINOR, no correction line. - MINOR | [K12] [cayley-hamilton] Cayley–Hamilton gives M² = t·M − I with t = trace M (the one place det = 1 is used). | research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Read the later hdet uses at lines 254 and 360. Removed only the one-place parenthetical from the page. The theorem and determinant assumption are unchanged; MINOR, no correction line. - STALE | [K13] [reproduce-minute] Reproduce from a fresh checkout in a minute: bash research/sorting-networks/lean/install-lean.sh then bash research/crystallographic-restriction/lean/verify.sh. | research/crystallographic-restriction/lean/verify.sh ; research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Reran the unpinned wrapper: Lean 4.35.0 exits 1 at line 248 with Unknown identifier ht. Reran with Lean 4.33.0: exit 0, eleven printed M2 results use only propext and Quot.sound, and five M4 results use no axioms. The wrapper and reproduction directions now select 4.33.0; no proof theorem was changed. - WRONG | [K14] [quasicrystal-dimensions] A three-dimensional decagonal or icosahedral quasicrystal (Shechtman's rapidly-quenched Al–Mn alloy, 1982; Nobel Prize in chemistry 2011) is standardly described as a cut-and-project shadow of a periodic lattice in dimension 6, where five- and ten-fold ... | https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Read the cached primary Nobel PDF text, pp. 4 and 9: decagonal examples are axial, and the standard superspaces have dimension 5 for axial and 6 for icosahedral examples. Updated both authored copies. A fresh web-tool fetch was refused with HTTP 403; the retained primary document was readable. - WRONG | [K15] [one-dimension-up] Same file exhibits the 4D companion of Φ5 as a witness of order exactly 5, so the pentagon is both forbidden here and available one dimension up. | research/crystallographic-restriction/lean/CrystallographicRestriction.lean ; public/strata/seventeen-and-no-more/index.html | Read the displayed 2 by 2 theorem and 4 by 4 companion, and the M4 C5 definition and exact-order checks. Four minus two is two. Both one-dimension-up labels now say two dimensions higher. - MINOR | [K16] [live-proofs] Proven and re-derived live: the crystallographic restriction; the cost of all 17 symbols; that exactly 17 finite symbols cost 2; that each rendered group is a genuine closed group of the stated order. | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | The browser computes selected examples, listed-row costs and closure; only the offline code enumerates. Narrowed metadata, dek and checking summary. This MINOR repair has no separate on-page correction line. - CONFIRMED | [K17] [barlow-attribution] The crystallographic restriction** (Barlow 1894 / Fedorov) | https://www.emis.de/monographs/jablan/chap5.htm | REJECTED with evidence: the cited monograph explicitly attributes the crystallographic restriction to Barlow in 1894. The discrepant 1893 excerpt does not justify changing the supported date. The existing wording remains. - MINOR | [K18] [hat-escapes] The crystallographic restriction is exactly the law the aperiodic monotile escapes. | https://arxiv.org/html/2303.10798v3 | Read Theorem 1.1 and the aperiodicity proof. The relation now says the hat avoids the periodic-lattice hypothesis. This MINOR wording repair has no correction line of its own; the explicit five-fold claim has its own WRONG finding. - WRONG | [K19] [both-machinechecked] Both are machine-checked, and on the same arithmetic fact: 5 is not a perfect square. | research/incommensurable-immersive/lean/Sqrt2.lean ; research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Read the cited proof: a*a=2*(b*b) implies b=0 by parity and descent. It concerns sqrt(2), not a shared nonsquareness-of-5 proof. Corrected the relation, its teaser source, and the markdown historical bridge. - STALE | [K20] [pivot-nonzero] the forbidden pentagon's pivot t⁴−3t²+1 is never zero because 4·p = (2t²−3)²−5, and no integer squares to 5 | git show 0f3f953383b:research/crystallographic-restriction/lean/CrystallographicRestriction.lean ; research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Read the June 29 pivot_ne_zero and not_square_five proof and the current recurrence-based restriction with no_order_five by descent. Updated the current dek, relation and README; retained the explicitly dated June history. Covered by the single Updated line. - WRONG | [K21] [group-p3m1] p3m1 Conway *333 · rotation centres: 3-fold · symmetries per cell: 6 | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [K22] [selfcheck-p3m1] generators close into a group of order 6 (expected 6); composition stays inside the set. This is genuinely p3m1. | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [K23] [group-p31m] p31m Conway 3*3 · rotation centres: 3-fold · symmetries per cell: 6 | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - WRONG | [K24] [selfcheck-p31m] generators close into a group of order 6 (expected 6); composition stays inside the set. This is genuinely p31m. | https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/closure-check.mjs | Recreated the swap with the page script. The directly read IUCr group 14 table puts all threefold sites on mirrors. The other generator set has off-mirror threefold sites, matching the book's 3*3 signature; the fresh group 15 fetch was blocked. With the corrected assignments, mirror counts at the three inequivalent sites are 3,3,3 for p3m1 and 3,0,0 for p31m. The origin remains a three-mirror positive control. - MINOR | [K25] [ledger-all-rows] o p1 none handle ○ = 2 = 2 2222 p2 2,2,2,2 ½+½+½+½ = 2 ×× pg none ×+× = 1+1 = 2 *× cm none *+× = 1+1 = 2 ** pm none *+* = 1+1 = 2 22× pgg 2,2 ½+½+× = 2 22* pmg 2,2 ½+½+* = 2 *2222 pmm 2,2 *+¼+¼+¼+¼ = 2 2*22 cmm 2 ½+*+¼+¼ = 2 442 p4 4,4,2 ¾+¾+½ = 2 *442 p4m ... | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html | Read the signatures and adjacent ledger rows. *2222 has four order-2 corners; 2*22 has three order-2 sites. The lists now read 2,2,2,2 and 2,2,2. Costs and group names stay the same; MINOR, no correction line. - WRONG | [K26] [builder-spherical] < 2 → SPHERICAL: a finite rosette / polyhedral symmetry, not wallpaper | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html | Read chapter 4, p. 53: MN and *MN require equal orders, with an omitted 1 allowed in notation. Recreated a lone cone 3, an unequal cone pair, and their corner counterparts. They now receive an inadmissible verdict; equal pairs and valid flat and hyperbolic controls retain their geometry. - WRONG | [K27] [rotation-readout-2] 2 cos(2π/2) = -2.0000 integer ✓ shortest combination v′ − v has length 0.000·¦v¦ shorter than v -- contradiction order 2 is impossible | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Recreated before editing: n=2 produced a length effectively zero and the impossible verdict. The old angle was 2*(2*pi/n), so R squared was identity. The new Rv-v has length 2 at n=2; all eight slider orders pass checks of length and verdict. - MINOR | [K28] [metadata-every-claim] Every claim re-derived live and checked offline (26/26). | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify.mjs | The browser computes selected examples, listed-row costs and closure; only the offline code enumerates. Narrowed metadata, dek and checking summary. This MINOR repair has no separate on-page correction line. - STALE | [K29] (added) "pure integer algebra, no cosines, the same discriminant 5 that makes the golden ratio irrational" (dek, src/content/strata/seventeen-and-no-more.md line 4) | git show 0f3f953383b:research/crystallographic-restriction/lean/CrystallographicRestriction.lean ; research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Read the June 29 pivot_ne_zero and not_square_five proof and the current recurrence-based restriction with no_order_five by descent. Updated the current dek, relation and README; retained the explicitly dated June history. Covered by the single Updated line. - WRONG | [K30] (added) "There, giving up periodicity buys back five-fold symmetry (the quasicrystal's secret); the hat tiles the plane and never repeats." (relates note to aperiodic-monotile, md line 10, on the home page) | https://arxiv.org/html/2303.10798v3 | Read section 2 and Lemma A.6: unreflected hats have six orientations, and every hat tiling aligns to the underlying Laves tiling. A 72-degree rotation cannot preserve those tile orientations. Removed the five-fold assertion while retaining aperiodicity. - WRONG | [K31] (added) "so the rendered pattern is provably the named group, not a lookalike" (md lines 45-46, served in /mcp/corpus.json) | public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read the acceptance condition: closure and expected order only. A p4m generator set under the p4g label still passes that condition. The readout, prose, metadata, markdown and relation now state closure and count modulo translations, without certifying the name. - WRONG | [K32] (added) "the same reason quasicrystals in 3-space are cut-and-project shadows of 6D lattices" (md line 84) | https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Read the cached primary Nobel PDF text, pp. 4 and 9: decagonal examples are axial, and the standard superspaces have dimension 5 for axial and 6 for icosahedral examples. Updated both authored copies. A fresh web-tool fetch was refused with HTTP 403; the retained primary document was readable. - WRONG | [K33] (added) the cost builder's symbol readout (index.html lines 566-574) | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; public/strata/seventeen-and-no-more/index.html ; research/wallpaper-groups/verify-page.mjs | Read chapter 3 costs and the boolean-mirror renderer. It charged multiple mirrors but printed one and printed unbound corners as cones. The renderer now retains each boundary and its corners; unbound corner buttons are disabled. ** costs 2; *** costs 3; *2*3 costs 31/12. The tests also cover four cones and the valid *632 control. - WRONG | [K34] (added) "THE 4D COUNTERPOINT · same file, one dimension up" (index.html line 226) | research/crystallographic-restriction/lean/CrystallographicRestriction.lean ; public/strata/seventeen-and-no-more/index.html | Read the displayed 2 by 2 theorem and 4 by 4 companion, and the M4 C5 definition and exact-order checks. Four minus two is two. Both one-dimension-up labels now say two dimensions higher. - CONFIRMED | [classification] “Wallpaper, tile floors, brick walls, the lattices of crystals, the carved screens of the Alhambra: every flat pattern that repeats in two directions belongs to one of exactly seventeen symmetry groups.” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 1 (classification): Book p. 40: “just 17 symmetry types for repeating patterns on the plane.” This is the usual discrete wallpaper-pattern classification. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [rigid-motions] “Ask only what rigid motions leave it looking unchanged: slides, turns, mirror-flips, and glide-reflections (flip-and-slide, the way footprints repeat).” | https://www.mi.sanu.ac.rs/vismath/ana/ana6.htm ; https://mariuskempe.github.io/Writing/P%C3%B3lya.pdf | Entry 2 (rigid-motions): The supplied definition establishes rank-two translations but does not list all four motions. Supplement: Pólya p. 1, “The aligning motions of the wallpaper pattern can be one of four types”, followed by translation, rotation, flip and screw/glide. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [fedorov-polya] “The complete answer was settled by 1891, by the Russian crystallographer Evgraf Fedorov, and again by George Pólya in 1924: there are precisely 17 such symmetry groups in the plane.” | https://mariuskempe.github.io/Writing/P%C3%B3lya.pdf ; https://www.iucr.org/publ/50yearsofxraydiffraction/full-text/fedorov ; https://www.iucr.org/gallery/undated/russian-crystallographers ; https://mathshistory.st-andrews.ac.uk/Biographies/Polya/ | Entry 3 (fedorov-polya): Pólya translation p. 2: “only 17 groups with different structures”; final-page note: “classified by Fedorov in 1891”, and publication “1924”. IUCr biography calls Fedorov “the eminent Russian crystallographer”; its gallery names “Evgraf Fedorov”. MacTutor supplies “George Pólya”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [allowed-rotations] “a repeating planar pattern can have centres of 2-, 3-, 4-, or 6-fold rotation, and nothing else.” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 4 (allowed-rotations): Theorem 3.1.1 gives “n=1,2,3,4,6”; its n=1 is explicitly “the trivial case with no rotational symmetry”. The page is listing nontrivial centres. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [forbidden-five-seven-eight] “No 5-fold centre. No 7-fold, no 8-fold.” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 5 (forbidden-five-seven-eight): Theorem 3.1.1 permits only “n=1,2,3,4,6”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [periodic-pentagon] “You can draw a single beautiful pentagon, but you cannot tile the plane periodically so that a fifth-turn leaves it unchanged.” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 6 (periodic-pentagon): Theorem 3.1.1 excludes n=5 for a two-dimensional periodic lattice. Regular pentagons themselves are not the objects whose full symmetry group is classified. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [quasicrystal-nonperiodicity] “Quasicrystals get five-fold symmetry precisely by giving up periodicity” | https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Entry 7 (quasicrystal-nonperiodicity): The quoted definition proves nonperiodicity. Nobel p. 3 additionally labels “perfect pentagons” in the diffraction pattern; pp. 4-5 explain icosahedral and axial quasicrystals. The contextual statement does not mean all quasicrystals have fivefold symmetry. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [integer-rotation] “A rotation that maps a lattice onto itself, written in the lattice's own basis, is a matrix of whole numbers” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 8 (integer-rotation): Source: “must have integer entries.” A lattice-basis image of each basis vector is an integer linear combination. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [trace] “The trace of a rotation by angle θ is 2 cos θ.” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 9 (trace): Source: “its trace will always be equal to” 2 cos(2 pi/n). A planar rotation has two diagonal entries cos(theta); trace is invariant under change of basis. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [integer-trace] “A whole-number matrix has a whole-number trace.” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 10 (integer-trace): Source: “In particular, its trace must be integer.” Trace is a sum of integer diagonal entries. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [trace-condition] “So 2 cos(2π/n) must be an integer” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 11 (trace-condition): Source explicitly concludes “2 \cos(2 \pi /n ) = k” for integer k. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [trace-orders] “and that happens only for n = 1, 2, 3, 4, 6.” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html | Entry 12 (trace-orders): Source gives the five possible n. Independently, 2 cos(2 pi/n) can only be -2,-1,0,1,2. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [golden-trace] “The pentagon's value, 2 cos 72° = 0.618…, is the reciprocal golden ratio: a root of x² + x − 1, stubbornly irrational, never a whole number.” | git show e05770fb79:research/wallpaper-groups/verify.mjs | Entry 13 (golden-trace): Recreated verifier prints “0.6180339887”. Algebra: t=(sqrt(5)-1)/2, so t*t+t-1=0; discriminant 5 is not a square. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [shorter-forbidden] “For n = 5 and every n ≥ 7 the combination is shorter than v” | https://sites.ualberta.ca/~vbouchar/MAPH464/section-crystallography.html ; git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 14 (shorter-forbidden): The supplied geometric excerpt contains a sign error and the false inequality “|\cos \alpha| \geq 1”. Independently, the actual page uses length 2 cos(2 pi/5)=0.618... for n=5, and 2 sin(pi/n)<1 for n>=7. Thus this claim is true, although its quoted proof needs replacement. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [shortest-vector-demo] “The classic "shortest vector" argument, run live: assume the order-n rotation exists, take the lattice's shortest vector v, and combine it with a rotated copy v′.” | git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 15 (shortest-vector-demo): Code constructs v and a rotated copy, then their sum at n=5 or difference otherwise. probe.mjs reproduces all eight slider states. The n=2 failure is recorded separately in entries 16 and 142. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [drag-motif] “Drag inside the canvas to move the motif; every copy moves in lockstep, which is what "symmetry" really means.” | git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 20 (drag-motif): Exact handlers use “off=[wx,wy]; draw();” on pointerdown and pointermove; each copy applies its group isometry to the same shifted motif. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [chirality] “cyan copies are mirror-images (left-handed); gold copies are direct (right-handed).” | git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 21 (chirality): Code tests “if(det<0)” before selecting cyan rgba(88,214,224,...), and otherwise gold rgba(230,178,58,...). Handedness is relative to the original motif. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [skeleton] “Toggle the skeleton to see the rotation centres (◆ 2-fold, ▲ 3-fold, ■ 4-fold, ⬢ 6-fold), mirror lines (solid), and glide axes (dashed).” | git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 22 (skeleton): The supplied classify and drawSkeleton excerpts implement the stated shapes and “ctx.setLineDash(isMirror?[]:[6,5])”. Rotations are gathered over translated cosets; the highest order at each centre is retained. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [p4-family] “p4 has four-fold turns and no mirror; add mirrors through the centres and you get p4m; offset them and you get p4g.” | https://www.mi.sanu.ac.rs/vismath/ana/ana6.htm ; https://mariuskempe.github.io/Writing/P%C3%B3lya.pdf | Entry 25 (p4-family): Pólya p. 2 contrasts “flip axes through all 4-fold rotational centers” with “no” such centres. The SANU square-lattice section labels these p4m and p4g; p4 has cyclic C4 only. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [orbifold-folding] “Fold the repeating plane down by its own symmetry and you get a small patched surface called an orbifold;” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 26 (orbifold-folding): Book p. 75 calls this “orbitfolded” version of the surface “the orbifold”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [flat-cost] “its features carry a "cost," and the orbifold of a flat, repeating pattern must cost exactly 2.” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 27 (flat-cost): Book Theorem 3.1: “The signatures of plane repeating patterns are precisely those with total cost $ 2.” Applies to valid signatures. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [hyperbolic-cost] “more than 2 and it opens into the hyperbolic plane (infinitely many groups).” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 29 (hyperbolic-cost): The source excerpt is present. It is in Chapter 18, “Existence and Construction”, p. 237, not Chapter 19 as the record says. Negative characteristic, equivalently cost above two for a valid signature, gives hyperbolic groups. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cost-iff] “A pattern is a flat wallpaper group precisely when the total is exactly 2.” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 30 (cost-iff): Book Theorem 3.1 gives precisely the cost-two criterion for valid wallpaper signatures. Invalid builder displays are separate findings in 138-140. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [finite-symbols] “Run the enumeration over every finite orbifold symbol and ask which cost exactly 2, and you get these seventeen and not one more.” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 33 (finite-symbols): Book p. 40 concludes “just 17 possibilities for the signature”. The Node enumeration independently recovers all 17 at caps 12 and 24. Completeness is supported by the book proof, not merely by checking two finite caps. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [frieze-infinity] “Allow an infinite-order symbol (an ∞ in the signature) and you also pick up the seven frieze groups” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 34 (frieze-infinity): Book p. 69 lists exactly “∞∞, ∞×, ∞∗, ∗∞∞, 22∞, ∗22∞, and 2∗∞”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [frieze-one-direction] “the symmetries of a border or a railing, repeating in one direction only.” | https://mariuskempe.github.io/Writing/P%C3%B3lya.pdf | Entry 35 (frieze-one-direction): Pólya final page: “a border (band, frieze); there are seven.” Book Chapter 5 explains repetition along a border. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [alhambra-popular] “You will read, in a hundred popular accounts, that the Moorish artisans of the Alhambra in Granada discovered all seventeen groups by hand, centuries before mathematicians named them.” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum-Grunbaum-Shephard.Symmetry-in-Moorish-and-other-ornaments.1986-12jyvah.pdf | Entry 36 (alhambra-popular): The 1986 primary paper says “the number 17 was claimed by others”; its references give the claimed tradition. “A hundred” is treated as rhetorical breadth, not a measured bibliography size. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [alhambra-four-absent] “and argued that p2, pg, pgg and p3m1 are simply absent.” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum-Grunbaum-Shephard.Symmetry-in-Moorish-and-other-ornaments.1986-12jyvah.pdf | Entry 38 (alhambra-four-absent): 1986 abstract lists “(pg, p2, pgg, p3ml)”; the final OCR letter l represents the numeral 1 in p3m1. The surrounding paper and standard notation resolve it. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [grunbaum-return] “Grünbaum returned to the question in the Notices of the AMS in 2006 under a pointed title” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum.What-symmetry-groups-are-present-in-the-Alhambra-2006-2hxz2f4.pdf | Entry 39 (grunbaum-return): The 2006 PDF header gives “What Symmetry Groups Are Present in the Alhambra?” and “Branko Grünbaum”; footer gives “JUNE/JULY 2006”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [alhambra-counting] “the claim that one building contains them all is an empirical assertion that careful counting does not support.” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum.What-symmetry-groups-are-present-in-the-Alhambra-2006-2hxz2f4.pdf | Entry 41 (alhambra-counting): Grünbaum p. 672 says a final count requires examination “following explicit criteria” and reports twelve previously listed groups plus one. This supports treating all-seventeen as disputed, not a settled fact. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cite-conway] “Conway's Magic Theorem, The Symmetries of Things, Conway–Burgiel–Goodman-Strauss 2008” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html | Entry 42 (cite-conway): The book title and authors appear on its title/copyright pages; copyright is “2008”. The excerpt elision legitimately joins those locations. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cite-fedorov] “Fedorov 1891; Pólya 1924” | https://mariuskempe.github.io/Writing/P%C3%B3lya.pdf | Entry 43 (cite-fedorov): Pólya final-page bibliography gives 1924; translator note gives Fedorov 1891. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cite-grunbaum1986] “Grünbaum–Grünbaum–Shephard 1986” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum-Grunbaum-Shephard.Symmetry-in-Moorish-and-other-ornaments.1986-12jyvah.pdf | Entry 44 (cite-grunbaum1986): Read first page of the 1986 PDF: the three named authors and publication year match. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cite-grunbaum2006] “B. Grünbaum, Notices of the AMS 53:6, 2006” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum.What-symmetry-groups-are-present-in-the-Alhambra-2006-2hxz2f4.pdf | Entry 45 (cite-grunbaum2006): The 2006 PDF footers read “VOLUME 53, NUMBER 6” and “JUNE/JULY 2006”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cite-title] “What Symmetry Groups Are Present in the Alhambra?” | https://bpb-us-w2.wpmucdn.com/sites.uwm.edu/dist/0/158/files/2017/04/Grunbaum.What-symmetry-groups-are-present-in-the-Alhambra-2006-2hxz2f4.pdf | Entry 46 (cite-title): The title is exactly “What Symmetry Groups Are Present in the Alhambra?” on p. 670. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [cite-iuc] “The IUC names follow the International Tables for Crystallography.” | https://dokumen.pub/the-symmetries-of-things-1nbsped-9781439864890.html ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o014/ ; https://it.iucr.org/Ac/ch2o2v0001/sgtable2o2o015/ | Entry 47 (cite-iuc): Book Table A.1 maps all 17 symbols to the international names. IUCr plane-group tables confirm the p3m1/p31m distinction. Short names such as pmm and p6m are conventional abbreviations. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [real-number-demo] “The argument in Instrument I runs through the real numbers: it leans on the trace being 2 cos θ, and on 2 cos 72° being irrational.” | git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 48 (real-number-demo): The exact browser code evaluates “const trace=2*Math.cos(ang)”; the preceding prose gives the irrationality argument. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [integer-proof-scope] “The restriction can be pinned to something smaller and harder: a statement with no cosines, no reals, no geometry at all, just integers, and then handed to a proof assistant, which certifies it for every integer matrix at once.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 49 (integer-proof-scope): Lean 4.33.0 accepts the universally quantified integer-matrix theorem. M2 entries are Int and the hypotheses are determinant one and a positive identity power. The surrounding paragraph supplies the restriction scope. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [orientation-determinant] “An orientation-preserving lattice rotation, in the lattice basis, is a 2×2 integer matrix of determinant 1 (it is area- and orientation-preserving).” | git show e05770fb79:research/crystallographic-restriction/verify.mjs | Entry 50 (orientation-determinant): The companion source explicitly says “a rotation preserves orientation and area”; determinant one and integer lattice-basis entries follow. This was also checked directly from the rotation matrix. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [finite-order-definition] “A finite-order rotation is one whose kth power is the identity for some k.” | https://www.mi.sanu.ac.rs/vismath/ana/ana6.htm | Entry 51 (finite-order-definition): The standard definition uses a positive integer k; the displayed theorem correctly uses k+1. The supplied SANU sentence is contextual, not the definition itself. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [displayed-theorem] “theorem crystallographic_restriction (M : M2) (hdet : M.det = 1) (h : M.npow (k+1) = I) : M = I ∨ M.npow 2 = I ∨ M.npow 3 = I ∨ M.npow 4 = I ∨ M.npow 6 = I” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 52 (displayed-theorem): The displayed theorem matches the accepted Lean statement. The source explicitly binds “(k : Nat)”; the page uses the same variable implicitly. All five disjuncts match. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [integer-order-theorem] “Every 2×2 integer matrix of determinant 1 that has any finite order has order in {1, 2, 3, 4, 6}.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 53 (integer-order-theorem): Accepted by Lean 4.33.0. The divisors of 1,2,3,4,6 are again in that set, so the disjunction is sufficient for the exact-order statement. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [reduced-power] “Iterating it packages every power as a two-term combination: Mk+1 = Uk+1(t)·M − Uk(t)·I, where U is the Chebyshev-like integer sequence U0=0, U1=1, Uk+2 = t·Uk+1 − Uk.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 55 (reduced-power): Lean defines “| 0 => 0”, “| 1 => 1”, and “| (k+2) => t * U t (k+1) - U t k”; npow_combo proves the stated reduced power. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [sequence-growth] “The load-bearing arithmetic fact is that for every integer |t| ≥ 3 the sequence |Uk| is strictly increasing from 1, so it never returns to zero.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 56 (sequence-growth): U_grow proves the positive t case. The negative t case follows by alternating signs, implemented via the negated matrix in the trace proof. The supplied excerpt alone only covers t>=3. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [trace-exclusion] “That rules the trace out of {…,−4,−3,3,4,…} without any real-number machinery.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 57 (trace-exclusion): trace_le_two and trace_ge_neg_two establish “-2” through “2”; the accepted umbrella proof explicitly makes this case split. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [five-traces] “What is left is five allowed traces: t = −2 gives −I (order 2), t = 2 gives I (order 1), and the three middle values give M3=I, M4=I, M6=I unconditionally (no finite-order hypothesis needed for the middle three; they hold for every det-1 integer matrix of that trace).” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 58 (five-traces): The three middle-trace lemmas have no finite-order hypothesis. The extreme-trace lemmas do require it, as the page context says. The shear with entries (1,1;0,1) is the positive counterexample to an unconditional t=2 interpretation. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [descent-corollary] “The pentagon case that shipped first, no_order_five, is now a two-line corollary via a Euclidean-descent lemma that needs no matrix inverses: if Mp=I and Mq=I with p ≥ q, then Mp−q = Mp after cancelling I on the right, so Mp−q=I. Applied to the umbrella's five disjuncts with p=5, each one descends to M1=I.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 59 (descent-corollary): descent and no_order_five implement subtraction of identity powers and the coprime-order conclusion. The n=6 disjunct subtracts 5 from 6. The phrase “two-line” is an informal brevity description, not a style finding. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [crosscheck-scope] “The companion verify.mjs independently re-derives every algebraic step (Cayley–Hamilton, the reduced-power formula at every k up to 12, the U-growth estimate, the trace bound, the three unconditional middle-trace orders, the extreme-trace endgame, and the 4D companion): 21/21.” | git show e05770fb79:research/crystallographic-restriction/verify.mjs | Entry 63 (crosscheck-scope): Recreated: “crystallographic-restriction/verify.mjs: 21/21 checks pass”. Its BigInt matrix loops test each named algebraic step over bounded input ranges; universality belongs to Lean, not the samples. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [dimension-four-exists] “Raise the ambient rank by two and the argument stops binding: at dimension 4, an integer matrix of order exactly 5 exists, and one can write it down.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 65 (dimension-four-exists): The explicit C5 has integer entries. Lean 4.33.0 accepts C5_pow5_eq_I and all four nonidentity powers by decide. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [companion-polynomial] “The witness is the companion matrix of the fifth cyclotomic polynomial Φ5(x) = x⁴ + x³ + x² + x + 1.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 66 (companion-polynomial): The displayed shift matrix with final column all -1 is the companion of x^4+x^3+x^2+x+1. Multiplying it into that polynomial gives zero in the independent BigInt check. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [companion-matrix] “C = ⎡ 0 0 0 −1 ⎤ ⎢ 1 0 0 −1 ⎥ ⎢ 0 1 0 −1 ⎥ ⎣ 0 0 1 −1 ⎦” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 67 (companion-matrix): All 16 displayed entries agree exactly with “def C5 : M4” in the pinned file. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [companion-integer-det] “All integer entries, determinant 1.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 68 (companion-integer-det): The displayed matrix has integer entries. Expansion along the first row gives determinant 1. The source excerpt does not state the determinant, but provides enough data to calculate it. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [companion-order] “C, C², C³, C⁴ are pairwise different from the identity; C⁵ = I. Order exactly 5, in dimension 4.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 69 (companion-order): Kernel evaluation on Lean 4.33.0 confirms powers 1 through 4 are not identity and power 5 is identity. Thus the exact order is 5. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [minimal-dimension] “Because Φ5 has degree φ(5) = 4, four is minimal: the pentagon cannot fit in fewer.” | https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Entry 70 (minimal-dimension): The Nobel excerpt only proves the low-dimensional exclusion. Independently Phi5(x+1)=x^4+5x^3+10x^2+10x+5 is Eisenstein at 5, so a nontrivial order-five rational matrix needs degree at least 4. C5 supplies equality. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [shechtman-date] “Shechtman's rapidly-quenched Al–Mn alloy, 1982; Nobel Prize in chemistry 2011” | https://www.nobelprize.org/prizes/chemistry/2011/press-release/ ; https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Entry 72 (shechtman-date): Nobel press release dates the discovery “8 April 1982” and the prize 2011; the technical background identifies the rapidly cooled aluminium-manganese alloy and its icosahedral diffraction symmetry. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [periodicity-rank-three] “Periodicity is what forbids the pentagon at rank 2 and 3, and giving up periodicity in 3-space (by living in the shadow of a higher-dimensional lattice) is what returns it.” | https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011-2.pdf | Entry 73 (periodicity-rank-three): Nobel pp. 5-6 says “Icosahedral symmetry is allowed together with translational symmetry in 6-” continued as “dimensional space”. The three-dimensional restriction and the four-dimensional minimal witness agree. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [both-directions] “So the pentagon is forbidden here and available there, and both directions are certified by the same file.” | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Entry 75 (both-directions): The same pinned file contains the accepted M2 restriction and all five M4 computation checks. Lean 4.33.0 verifies both directions. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [every-figure] The offline verifier recomputes the restriction examples and the cost-2 enumeration. | git show e05770fb79:research/wallpaper-groups/verify.mjs | The checker confirmed these computations; the broad figure wording has been narrowed under X11, X26 and X27. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [closure-second-script] The offline checker tests closure and count modulo translations for all 17 generator sets, plus mirror incidence for p3m1 and p31m. | research/wallpaper-groups/closure-check.mjs | The original closure and count checks were confirmed by the checkers; the extra mirror-site check now distinguishes the repaired pair. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [ledger-now-sums] “each row's total is now computed from its Conway symbol and the status line reports the count.” | git show e05770fb79:public/strata/seventeen-and-no-more/index.html | Entry 83 (ledger-now-sums): The page now executes “const c=symCost(sym)” and increments sumsTo2 only when the exact fraction is two. probe.mjs prints the stated count 17. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [check-independent] “This layer's check works the answer out again from scratch, independently of this page.” | git show e05770fb79:research/wallpaper-groups/verify.mjs | Entry 84 (check-independent): Running the published single file in an otherwise empty directory prints “PASS 26/26”. The dependency record independently records “depends”: false for both numeric verifiers. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [check-limits] “It can settle the finding; it cannot notice a figure that drifted on this page.” | git show e05770fb79:research/wallpaper-groups/verify.mjs | Entry 86 (check-limits): The published verifier passes from a directory containing no layer page. Its source performs its own calculations. The page correctly discloses the limit rather than claiming page-number coupling. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [published-check] “verify.mjs is published, and it runs on its own: one file, and Node.” | git show e05770fb79:research/wallpaper-groups/verify.mjs | Entry 87 (published-check): HTTP 200, 15,731 bytes, identical to the pinned verifier. Node v22.23.1 runs it with no dependencies: exit 0, “PASS 26/26”. Prior check: research/assay-line/queue-1014c/seventeen-and-no-more/N2-codex.md. - CONFIRMED | [trace-all-seven] “every n ≥ 7 fails because the trace lies strictly between 1 and 2.” | git show e05770fb79:research/wallpaper-groups/verify.mjs | Entry 90 (trace-all-seven): For n>=7, 0<2 pi/n