---
slug: what-the-bridges-knew
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
checked: 98
confirmed: 68
wrong: 28
unverifiable: 0
observed: 0
stale: 2
---

This was the directed fixer pass for what-the-bridges-knew, read at f5e5f84f34, against research/assay-line/queue-1014c/what-the-bridges-knew/triage.json. Each triage item's evidence was re-read before editing. The 67 CONFIRMED claims from the Codex N2 check are retained as that checker's source readings, not presented as 67 new independent source fetches in this fixer pass. All 29 triage items are recorded individually, followed by those confirmed claims and two corrections to verifier documentation found during this pass. MINOR lines count in wrong. The two STALE items use the triage's date-field exceptions. X8 remains pending on the generated placard; this record does not claim a completed placard rebuild.

## Claims
- MINOR | [X1; body-9] "The proof is exact, the verifier runs in the page, and every figure here is independently reproduced offline in research/what-the-bridges-knew/ (35/35 checks pass)." | git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | The offline executable defines seven graph cases including HOUSE, K4, K5 and FIG8; it has no envelope instance. Narrowed the page's lede and source-list wording and the layer README to the actual coverage. The fresh run still reports 35 passed, 0 failed.
- MINOR | [X2; body-18] "Euler's first move" ... "and the one that founded graph theory" ... "was to throw the map away." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | Re-read E53 sections 4-7 and Hopkins and Wilson's authorized reprint p. 410: Euler uses sequences of region letters; the dot-and-line diagram is a modern presentation. The shared paragraph now says 'In modern notation'.
- MINOR | [X3; body-21] "The town becomes a picture with four dots (one per landmass) and seven lines (one per bridge), and the question becomes one about that picture alone." | https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | The four-dot graph is mathematically correct, but the surrounding historical attribution blurred the distinction from Euler's letter-sequence method. Added the modern-notation qualification, also mirrored in the dek under X21.
- WRONG | [X4; body-31] "The house (square, roof, one diagonal) has two odd vertices" ... "drawable, starting at one corner of the diagonal and ending at the other." | git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs; git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html | Independently counted the authored house edges ab, bc, cd, da, ae, be, ac: a=4, b=3, c=3, d=2, e=2. The diagonal joins a and c; the trail endpoints are b and c. Corrected the prose and verifier comments, strengthened the existing odd-vertex assertion to require b and c, and refreshed the layer's published verifier from the source. One correction line covers this error.
- CONFIRMED | [X5; body-43] "The Krämerbrücke and the Grüne Brücke survived the war but were demolished in the early 1970s when a concrete overpass (the Estakada) was driven through the same crossings." | https://de.wikipedia.org/wiki/K%C3%B6nigsberger_Pregelbr%C3%BCcken; https://www.rudnikov.com/koenigsberg/sem-mostov-kjonigsberga-zadachu-jejler/; https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg | REJECTED with evidence. The German article says the Green and old Kramer structures did not survive WWII; the local-history report at rudnikov.com explicitly says both survived and were demolished in 1972. The English bridge article distinguishes two wartime losses from two later highway replacements. These sources conflict; the proposed destruction correction is not established. 'Survived' does not assert undamaged survival. The same author's book is not an independent witness. Retained the sentence as directed, with no claim that this resolves the archival disagreement.
- MINOR | [X6; body-49] "Every figure on this page is recomputed by the script below, live, the moment the page loads." | git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | The live graph keyed modern is labelled '+1 bridge' and has eight edges; the separate five-site modern model is checked offline. Replaced the universal live-coverage sentence with 'The graph tests below are recomputed live, the moment the page loads.'
- WRONG | [X7; body-52] "The phrase did not catch; we now call it topology upstream and graph theory downstream, and the bridges are remembered as the founding stone of both." | https://prussia.online/Data/Book/le/leonhard-euler/Leonhard%20Euler.%20Life%2C%20Work%20and%20Legacy%20%282007%29%2C%20OCR.pdf | Re-read Hopkins and Wilson's authorized reprint p. 413: Leibniz's term circulated among eighteenth-century followers. Replaced the categorical non-adoption claim with 'The names changed'; shares one correction line with X28.
- STALE | [X8; last-verified] "Last verified 14 June 2026, when it was written." | git show f5e5f84f34:research/self-check/verifier-runs.json; git show f5e5f84f34:memory/log.d/2026-06-14T0230Z-what-the-bridges-knew.md | The original June 14 verification date was overtaken by the committed run at 2026-08-01T13:02:09.145Z, exit 0, 35 passes. This fixer also ran the offline verifier on 2026-10-14 and recorded 35 passes. The generated placard still says June 14: triage directs this date through the record and the landing placard rebuild, and the user forbids the writing form of scripts/check-placard.mjs. NOT FIXED on the page pending that rebuild. No Updated or Corrected line under the explicit placard exception.
- WRONG | [X9; epigraph-is-title] "The Latin sentence quoted as the epigraph is the paper's own title" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf | Downloaded and visually read original E53 PDF page 1, printed p. 128. The title sits above section 1; the epigraph begins that section. Corrected the explanatory source note only. The Latin passage and explicitly authorial gloss remain unchanged.
- MINOR | [X10; wikipedia-source-description] "For the bridge-name correspondences and the post-WWII status of the surviving bridges." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; https://de.wikipedia.org/wiki/K%C3%B6nigsberger_Pregelbr%C3%BCcken | Re-read the English article's Present state section and the German per-bridge article. The English text supplies the restricted postwar model, not the bridge-name crosswalk. Narrowed its description and added a separate German citation for names and endpoints.
- WRONG | [X11; body-72] "a Krämerbrücke Altstadt ↔ Kneiphof" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Euler's printed plate and the geographic crosswalk assign c to Kramer, while the page uses a. The endpoints are correct. Chose the triage's attribution fix: the table, header and walker now explicitly call a-g this page's labels. No graph or bridge name was changed.
- WRONG | [X12; body-73] "b Schmiedebrücke Altstadt ↔ Kneiphof" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Euler's printed plate and the geographic crosswalk assign d to Schmiede, while the page uses b. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- WRONG | [X13; body-74] "c Grünebrücke Vorstadt ↔ Kneiphof" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Euler's printed plate and the geographic crosswalk assign a to Green, while the page uses c. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- WRONG | [X14; body-75] "d Köttelbrücke Vorstadt ↔ Kneiphof" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Euler's printed plate and the geographic crosswalk assign b to Kottel, while the page uses d. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- WRONG | [X15; body-77] "f Holzbrücke Altstadt ↔ Lomse" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Euler's printed plate and the geographic crosswalk assign g to Wood, while the page uses f. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- WRONG | [X16; body-78] "g Hohe Brücke Vorstadt ↔ Lomse" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Euler's printed plate and the geographic crosswalk assign f to High, while the page uses g. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- WRONG | [X17; euler-lettering-caption] "The seven bridges of Königsberg, after Euler's lettering" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.mathouriste.eu/Koenigsberg/koenigsberg.html | Visually read E53 PDF page 2, plate VIII, Figure 1: a/b join A-B, c/d join A-C, f joins B-D, and g joins C-D. Read the geographic name crosswalk separately; handwritten names on the scan were not treated as Euler's printed labels. Six page assignments differ. Corrected the attribution across the table and walker; one dated line covers X11-X17.
- WRONG | [X18; jubilee-bridge] "The modern bridge count in Kaliningrad is contested between sources and the city has built new spans (most prominently the Jubilee Bridge, 2005, connecting Kant Island to the south bank)" | https://de.wikipedia.org/wiki/K%C3%B6nigsberger_Pregelbr%C3%BCcken; https://www.klgd.ru/city/tourism/Walking_in_Kaliningrad_The_map_of_the_city.pdf | Re-read the German Kaiserbrucke section and municipal guide PDF page 3. The successor Jubilee Bridge was built in 2005 at the Imperial Bridge site, between Oktyabrskaya and Epronovskaya streets, joining Lomse/Oktyabrsky Island to the south bank. Replaced Kant Island with Lomse (now Oktyabrsky Island).
- WRONG | [X19; all-e-numbers] "the same ambiguity that haunts every E-numbered paper." | https://scholarlycommons.pacific.edu/euler-works/3/ | Re-read the Euler Archive E3 record: written 1727, published 1727, and Acta Eruditorum volume 1727. This counterexample refutes the universal three-way date-ambiguity claim. Removed that clause while preserving E53's documented dates.
- MINOR | [X20; colophon-recompute] "every figure recomputed in your browser." | git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | The colophon repeated X6's universal browser coverage claim. Changed it to 'the graph tests run in your browser'; the five-edge restricted modern model remains an offline case.
- MINOR | [X21; metadata-2] "He answered it, and in answering it invented graph theory: throw the map away, draw four dots for the four landmasses and seven lines for the seven bridges, and ask the question only of the picture that is left." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | Re-read E53 sections 4-7 and Hopkins and Wilson p. 410. Added 'In modern notation' before the dek's dot-and-line construction, matching X2-X3.
- WRONG | [X22; hamiltonian-complexity] "For Eulerian trails Euler gave a one-line decision procedure in 1736; for Hamiltonian cycles there is still no general-purpose one, and the problem is NP-complete." | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/Karp.pdf | Read Karp's retrospective introduction, printed pp. 219-220, which distinguishes polynomial-time solvability and NP-completeness. A finite enumeration of all vertex permutations is a terminating general decision procedure. Qualified the relation note to say no polynomial-time algorithm is known for general Hamiltonian cycles, preserving NP-completeness. Repair review confirmed unannotated copies remain in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1; those copies are outside this layer's edit scope and remain open pending a board note.
- WRONG | [X23; gallery-caption-5] "Closed envelope (square + roof + both diagonals + roof-side, no bottom). Two odd vertices" ... "the classic schoolchildren's drawable." | git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html | The authored envelope edge array contains c-d and the coordinates put c and d at the two bottom corners. Counted eight edges and degrees a=4, b=4, c=3, d=3, e=2. Replaced the shared caption with square + roof + both diagonals. One dated line covers X23 and X26.
- WRONG | [X24; generated-2] "Euler's theorem: a graph has an Eulerian trail iff exactly 0 or 2 vertices have odd degree." | git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html | A connected triangle has zero odd vertices and a circuit; two disjoint triangles also have zero odd vertices but no Eulerian trail. The code rejects the disconnected pair and allows a triangle plus an isolated vertex. Added the condition that vertices incident to edges lie in one component to the aria-label, synopsis and README.
- WRONG | [X25; generated-12] "every one strands at most six of seven bridges." | git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html | Independently enumerated all 4*7! edge-order attempts from the authored seven edges. Crossed-edge histogram: 0:10080, 1:5520, 2:2448, 3:1104, 4:496, 5:256, 6:256. None crosses seven; half cross zero and leave all seven. Replaced 'strands at most six' with 'crosses at most six'.
- WRONG | [X26; panel-envelope] "Closed envelope (square + roof + both diagonals + roof-side, no bottom). a:4 b:4 c:3 d:3 e:2 edges 8 Σ degree 16 = 2·8 odd vertices 2 (c,d) connected? yes verdict trail possible" | git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html | The parity panel reuses X23's envelope caption and graph. Its c-d bottom edge is present; the corrected shared caption now accurately describes that same eight-edge graph. Shares X23's dated line.
- STALE | [X27; date-modified] ""dateModified":"2026-06-14"" | git show f5e5f84f34:memory/log.d/2026-07-26T2200Z-the-score-audit.md | Re-read the July 26 audit log recording this layer's 32-to-35 check-count edit after the June date. Set JSON-LD dateModified to 2026-10-14 for this revision. No Updated line under the explicit metadata-date exception; the Corrected lines date the revision.
- WRONG | [X28; geometry-name-usage] "The phrase did not catch" | https://prussia.online/Data/Book/le/leonhard-euler/Leonhard%20Euler.%20Life%2C%20Work%20and%20Legacy%20%282007%29%2C%20OCR.pdf | This is X7's same sentence. Hopkins and Wilson p. 413 documents eighteenth-century reception of the term. Replaced the non-adoption clause; shares X7's dated line.
- MINOR | [X29; added-hopkins-link] (added) "The truth about Königsberg." | https://old.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem; https://api.crossref.org/works/10.1080/07468342.2004.11922073 | The MAA article returned by the web index identifies a different author and title. Crossref identifies the Hopkins/Wilson paper, CMJ 35(3), 198-207, May 2004, DOI 10.1080/07468342.2004.11922073. Replaced the bibliography href and displayed link with that DOI.
- CONFIRMED | "Königsberg, Prussia, 1735." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [body-0] E53 section 2 names "Königsberg in Prussia", "seven bridges", and regions A, B, C, D. The supplied excerpt does not date the event; the Archive adds "Written Date 1735". Four regions and seven crossings agree with the plate.
- CONFIRMED | "Four landmasses, joined by seven bridges." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [body-1] E53 section 2 names "Königsberg in Prussia", "seven bridges", and regions A, B, C, D. The supplied excerpt does not date the event; the Archive adds "Written Date 1735". Four regions and seven crossings agree with the plate.
- CONFIRMED | "No one could do it, but no one could prove it impossible either." | https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-3] The supplied E53 section 2 excerpt only reports doubts. Euler's 13 March 1736 letter to Marinoni, reproduced by Hopkins and Wilson on printed p. 414, says he was informed nobody had demonstrated possibility or impossibility. This supplies the missing support for the pre-solution state.
- CONFIRMED | "He answered it, and in answering it invented graph theory." | https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [body-5] Archive: "the earliest paper in both topology and graph theory" and the problem is "unsolvable". Read as the usual founding-paper attribution, the claim is supported; it does not establish that Euler drew the modern node-and-edge picture.
- CONFIRMED | "Praeter illam Geometriae partem, quae circa quantitates versatur, … alterius partis etiamnum admodum ignotae primus mentionem fecit Leibnitzius, quam Geometriam situs vocavit." | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf | N2 evidence retained: [latin-epigraph] Opened the tool-unreadable PDF and visually read printed p. 128. The words and omission match section 1. The scan has "vocauit" and the page normalizes it to "vocavit". This is conventional orthographic normalization, not a substantive misquotation. The record accurately preserves the scan spelling.
- CONFIRMED | "Besides that part of geometry that deals with quantities, … Leibniz was the first to mention another part, still quite unknown, which he called the geometry of position." | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [english-epigraph-gloss] Behrend and the page translate the same section 1 sentence. The page explicitly says "the in-text gloss is my own". Its wording therefore need not match Behrend verbatim; the automated quotation mismatch is not a finding.
- CONFIRMED | "Euler 1741, §1, p. 128" | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [epigraph-citation] The scanned page is numbered "128" and the passage begins "§. 1."; the Archive records "Published Date 1741".
- CONFIRMED | "The Pregel river ran through Königsberg dividing it into four separable pieces of ground: the north bank (Altstadt), the south bank (Vorstadt), and two islands in midstream" ... "Kneiphof in the middle and Lomse upstream to the east." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; https://users.manchester.edu/Facstaff/SSNaragon/Kant/Helps/Glossary.htm | N2 evidence retained: [body-11] The supplied source names "Kneiphof and Lomse" and "Altstadt and Vorstadt". The historical map and bridge directions corroborate north and south banks and Lomse to the east. These are four connected land regions, not four separate municipalities.
- CONFIRMED | "Five of the seven touched Kneiphof" ... "making it the busy hub." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-13] E53 section 9 gives five bridges into A and three into each of B, C and D, then concludes "such a crossing of the seven Königsberg bridges cannot be achieved". Direct recount gives 5+3+3+3=14=2*7. The north/south/island names were cross-checked separately.
- CONFIRMED | "The other two crossed to Lomse, one from each bank." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-14] E53 sections 4-6 give the remaining links B-D and C-D. With A=Kneiphof and D=Lomse these are one bridge to Lomse from each bank.
- CONFIRMED | "The geometry of the river, the lengths of the bridges, the layout of the streets: none of it matters." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-19] E53 sections 1 and 4-7 deliberately discard metric information and keep region-to-region incidences. It says "only position is considered". The broad description of graph theory is explanatory framing, not a claim that the whole discipline reduces to vertex degrees.
- CONFIRMED | "All that matters is which landmass touches which bridge, and how many bridges meet at each landmass." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-20] E53 sections 1 and 4-7 deliberately discard metric information and keep region-to-region incidences. It says "only position is considered". The broad description of graph theory is explanatory framing, not a claim that the whole discipline reduces to vertex degrees.
- CONFIRMED | "The art of letting the answer be small was Euler's, and graph theory is just the discipline of asking the question only of the picture that is left." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-22] E53 sections 1 and 4-7 deliberately discard metric information and keep region-to-region incidences. It says "only position is considered". The broad description of graph theory is explanatory framing, not a claim that the whole discipline reduces to vertex degrees.
- CONFIRMED | "Every time you walk into it across a bridge, you must walk out of it across a different bridge." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-24] E53 sections 11-12 pair entry and exit at an intermediate region: "one writing of it will record both the entry into A and the exit". An open trail has one unmatched incidence at each endpoint; a closed trail has no odd endpoints.
- CONFIRMED | "The bridges at this landmass come in matched pairs" ... "one for arriving, one for leaving" ... "so the number of bridges meeting that landmass must be even." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-25] E53 sections 11-12 pair entry and exit at an intermediate region: "one writing of it will record both the entry into A and the exit". An open trail has one unmatched incidence at each endpoint; a closed trail has no odd endpoints.
- CONFIRMED | "The only landmasses that can dodge this rule are the two endpoints of the walk: at the start you only leave, and at the end you only arrive, each of those costs you one unmatched bridge." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-26] E53 sections 11-12 pair entry and exit at an intermediate region: "one writing of it will record both the entry into A and the exit". An open trail has one unmatched incidence at each endpoint; a closed trail has no odd endpoints.
- CONFIRMED | "So at most two landmasses can have an odd number of bridges." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-27] E53 section 20 states the 0-or-2 criterion. For finite undirected graphs the vertices incident to edges must lie in one component. The page's main synopsis includes connectivity and its code tests it. Pairing arrivals and departures proves necessity; Hierholzer supplies sufficiency.
- CONFIRMED | "If three or more do, the walk cannot exist." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-28] E53 section 20 states the 0-or-2 criterion. For finite undirected graphs the vertices incident to edges must lie in one component. The page's main synopsis includes connectivity and its code tests it. Pairing arrivals and departures proves necessity; Hierholzer supplies sufficiency.
- CONFIRMED | "The same one-line theorem decides every shape in the classic schoolbook puzzle, draw this in one stroke without lifting your pen." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-30] E53 section 20 states the 0-or-2 criterion. For finite undirected graphs the vertices incident to edges must lie in one component. The page's main synopsis includes connectivity and its code tests it. Pairing arrivals and departures proves necessity; Hierholzer supplies sufficiency.
- CONFIRMED | "The arithmetic the proof depends on is the same arithmetic that decides whether your morning route through any city" ... "or any subway map, any maze, any network" ... "can be done without doubling back." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-48] The statement applies to the undirected edge-once traversal problem just explained, not an arbitrary itinerary or directed/weighted routing problem. E53 section 20 gives the rule for "each bridge once only". Read in that stated scope, it is supported.
- CONFIRMED | "Königsberg's bridge counts are 5, 3, 3, 3." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-29] E53 section 9 gives five bridges into A and three into each of B, C and D, then concludes "such a crossing of the seven Königsberg bridges cannot be achieved". Direct recount gives 5+3+3+3=14=2*7. The north/south/island names were cross-checked separately.
- CONFIRMED | "K₄" ... "the square with both diagonals" ... "has four." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-32] Recounting K4 gives four vertices each of degree 3. E53 section 20 rules out a trail with more than two odd vertices. The inline and independent algorithms both return no walk.
- CONFIRMED | "K₅, where everyone is joined to everyone, has all even degrees" ... "drawable as a closed loop." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-33] K5 has five vertices, each adjacent to the other four. The verifier prints a valid ten-edge closed circuit and all degrees are 4, so the theorem applies.
- CONFIRMED | "The proof Euler wrote in 1736 still arbitrates the playground." | https://scholarlycommons.pacific.edu/euler-works/53/; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [body-34] The supplied Archive excerpt says "Written Date 1735" and cannot by itself support 1736. Supplement: Hopkins and Wilson p. 411 explicitly say Euler wrote up the solution in 1736, after presenting it in 1735. This common paper-year usage is supported, though the record should disclose the conflicting catalog date.
- CONFIRMED | "If you add up the degrees of every vertex" ... "the bridge-counts of every landmass" ... "you count every bridge twice (once for each end)." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-36] E53 section 16 says "each bridge is counted twice". Summing incidences counts two ends per edge, so the sum is 2*E and is even, including for loops counted twice.
- CONFIRMED | "The sum of the degrees is always twice the number of bridges: an even number, no matter what graph you draw." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-37] E53 section 16 says "each bridge is counted twice". Summing incidences counts two ends per edge, so the sum is 2*E and is even, including for loops counted twice.
- CONFIRMED | "So the number of odd-degree vertices is itself always even." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [body-38] E53 section 17 says odd bridge counts are "necessarily even" in number and expressly excludes one, three and five. Reducing the degree sum modulo 2 proves it. Hopkins and Wilson identify this as the handshaking lemma.
- CONFIRMED | "This is the handshake lemma" ... "at a party, the number of people who have shaken an odd number of hands is even" ... "and it is why Euler's "0 or 2" odd vertices is not a coincidence." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [body-39] E53 section 17 says odd bridge counts are "necessarily even" in number and expressly excludes one, three and five. Reducing the degree sum modulo 2 proves it. Hopkins and Wilson identify this as the handshaking lemma.
- CONFIRMED | "The next case after zero is two; the next after two is four." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [body-40] E53 section 17 says odd bridge counts are "necessarily even" in number and expressly excludes one, three and five. Reducing the degree sum modulo 2 proves it. Hopkins and Wilson identify this as the handshaking lemma.
- CONFIRMED | "1 is impossible. 3 is impossible." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [odd-counts-one-three] E53 section 17 says odd bridge counts are "necessarily even" in number and expressly excludes one, three and five. Reducing the degree sum modulo 2 proves it. Hopkins and Wilson identify this as the handshaking lemma.
- CONFIRMED | "Königsberg has changed." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; https://www.klgd.ru/city/tourism/Walking_in_Kaliningrad_The_map_of_the_city.pdf | N2 evidence retained: [body-41] The supplied current-state account says two bridges were destroyed and two replaced by a highway. The municipal map also documents replacement structures. The town has changed.
- CONFIRMED | "Three of the seven still stand at their old sites: the Honigbrücke, the Holzbrücke, and the Hohe Brücke (the last rebuilt 1938)." | https://de.wikipedia.org/wiki/K%C3%B6nigsberger_Pregelbr%C3%BCcken; https://www.klgd.ru/city/tourism/Walking_in_Kaliningrad_The_map_of_the_city.pdf | N2 evidence retained: [body-44] The supplied German excerpt dates the nearby High Bridge replacement to 1937, not 1938. A municipal tourist map says "The old High Bridge was demolished in 1938" and a new one was built several metres away. Thus 1938 has official support, while dates differ between sources. "Old sites" is read as the same river crossings, not identical foundations; the replacement caveat is explicit.
- CONFIRMED | "Counting the five connections that remain at Euler's seven sites" ... "three surviving bridges plus two flyover crossings at the Krämer and Grüne positions" ... "the modern degree sequence is 2, 2, 3, 3: two odd vertices, the others even." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-45] The supplied source explicitly limits the five-edge graph to the original seven sites and gives "degree 2" for two nodes and "degree 3" for two. Recreated degrees N=2,K=3,L=3,S=2; trail K→L→S→K→N→L. This is the stated restricted model, not an inventory of all present city bridges.
- CONFIRMED | "The walk Euler proved impossible became possible" ... "not as a circuit, but as a trail, beginning at one island and ending at the other." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [body-46] The supplied source explicitly limits the five-edge graph to the original seven sites and gives "degree 2" for two nodes and "degree 3" for two. Recreated degrees N=2,K=3,L=3,S=2; trail K→L→S→K→N→L. This is the stated restricted model, not an inventory of all present city bridges.
- CONFIRMED | "Euler called this geometriam situs" ... "the geometry of position" ... "and the new field he had cracked open behind his single problem." | https://math.dartmouth.edu/~euler/docs/originals/E053.pdf; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [body-51] E53 section 1 uses the term "geometria situs" and attributes its antecedent to Leibniz. The Latin title uses the inflected form "geometriam situs". Both mean geometry of position.
- CONFIRMED | "The 28-year-old who walked into the St Petersburg Academy with a small piece of arithmetic about a Prussian town walked out having defined a whole new kind of mathematics" ... "the kind that asks of an object only what cannot be straightened out of it." | https://mathshistory.st-andrews.ac.uk/Biographies/Euler/; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [body-53] Euler was born "15 April 1707" and presented the bridge solution on 26 August 1735, aged 28. The Archive calls it the earliest graph-theory/topology paper. The walk-in/walk-out wording is narrative framing of that presentation, not an independently documented literal itinerary.
- CONFIRMED | "Euler, L. (1741) "Solutio problematis ad geometriam situs pertinentis." Commentarii academiae scientiarum Petropolitanae 8: 128–140. (Written 1735; presented to the St Petersburg Academy in 1735; printed 1741. Eneström E53.)" | https://scholarlycommons.pacific.edu/euler-works/53/; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [euler-reference] The Archive gives the exact title, "Published Date 1741", "Written Date 1735", and volume 8 pp. 128-140. The scan/translation heading supplies nominal volume year 1736. Hopkins and Wilson p. 411 supplies the 1735 Academy presentation; the Archive excerpt alone omits presentation and nominal year.
- CONFIRMED | "presented to the St Petersburg Academy in 1735" | https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [euler-presented-1735] Hopkins and Wilson p. 411 dates the presentation "26 August 1735". This confirms the year in the source record and page.
- CONFIRMED | "Behrend, M. (n.d.), translating the full text of E53." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [behrend-reference] The linked translation presents all 21 numbered sections. The site belongs to Michael Behrend; the Euler Archive Spotlight also identifies his translation. No publication date is supplied, consistent with n.d.
- CONFIRMED | "Biggs, N. L., Lloyd, E. K. & Wilson, R. J. (1976) Graph Theory 1736–1936. Oxford: Clarendon Press. (Reprinted 1986, 1998.)" | https://books.google.com/books/about/Graph_Theory_1736_1936.html?id=XqYTk0sXmpoC; https://www.maths.ox.ac.uk/research/faculty-books/authors-u-z/wilson-robin | N2 evidence retained: [biggs-reference] Google Books names Biggs, Lloyd and Wilson, Clarendon Press, and "First published in 1976". Robin Wilson's Oxford faculty bibliography gives "2nd Revision, 1998", "Revised 1986" and "1st Edition 1976". The stated publication/reissue years are supported.
- CONFIRMED | "Reprinted 1986, 1998." | https://books.google.com/books/about/Graph_Theory_1736_1936.html?id=XqYTk0sXmpoC; https://www.maths.ox.ac.uk/research/faculty-books/authors-u-z/wilson-robin | N2 evidence retained: [biggs-reprints] Google Books names Biggs, Lloyd and Wilson, Clarendon Press, and "First published in 1976". Robin Wilson's Oxford faculty bibliography gives "2nd Revision, 1998", "Revised 1986" and "1st Edition 1976". The stated publication/reissue years are supported.
- CONFIRMED | "The standard scholarly translation of E53 and the surrounding history; carries Euler's diagrams." | https://books.google.com/books?id=XqYTk0sXmpoC&jscmd=SearchWithinVolume2&q=K%C3%B6nigsberg; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [biggs-translation-diagrams] The supplied marketing excerpt is insufficient by itself. Hopkins and Wilson pp. 412/419 identify this book as containing the full English translation. Google Books SearchWithinVolume returns the translated passage on p. 3 referring to "Fig . [ 1.2 ]" and the chapter's diagram references; this corroborates the diagram/translation claim. An image request for p. 4 returned "image not available", not evidence that the figure is absent. Positive control: the same volume search returned the expected p. 3 bridge passage.
- CONFIRMED | "Hopkins, B. & Wilson, R. J. (2004) "The truth about Königsberg." The College Mathematics Journal 35(3): 198–207." | https://api.crossref.org/works/10.1080/07468342.2004.11922073 | N2 evidence retained: [hopkins-reference] Fetched Crossref metadata: Brian Hopkins, Robin J. Wilson, "The Truth about Königsberg", 2004, volume 35 issue 3, "198-207". The printed bibliographic citation is correct. Its separate hyperlink identity is an added claim below.
- CONFIRMED | "A careful re-reading of what Euler actually proved." | https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [hopkins-source-description] Located and read the 2007 authorized reprint, pp. 409-420, whose first-page footnote identifies the original CMJ 35(3), 198-207 article. It distinguishes the original argument from later graph pictures and the unproved sufficiency claim. The description is accurate. The record's blocked fetches were real access limitations, not evidence that the article lacked this content.
- CONFIRMED | "Wikipedia: Seven Bridges of Königsberg." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg | N2 evidence retained: [wikipedia-reference] Fetched article title "Seven Bridges of Königsberg" matches the source-list title.
- CONFIRMED | "e Honigbrücke Kneiphof ↔ Lomse" | https://de.wikipedia.org/wiki/K%C3%B6nigsberger_Pregelbr%C3%BCcken; https://math.dartmouth.edu/~euler/docs/originals/E053.pdf | N2 evidence retained: [body-76] The German source says the Honey Bridge "verbindet die Lomse mit dem östlichen Kneiphof". Euler's plate marks the inter-island crossing e. Both the endpoints and e label are correct.
- CONFIRMED | "The bridge layout is the canonical reconstruction matching Euler's own figure (E53, plate VIII) and the standard reading in Biggs–Lloyd–Wilson 1976" | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://math.dartmouth.edu/~euler/docs/originals/E053.pdf | N2 evidence retained: [canonical-layout] The unlabeled multigraph matches E53 plate VIII and sections 4-6: two A-B edges, two A-C, one A-D, one B-D, one C-D. The named bank labels are checked separately. Letter-name assignment errors are recorded in the six table rows and caption, without calling the underlying graph incorrect.
- CONFIRMED | "The dates: written 1735, volume 8 of the Commentarii nominally 1736, actually printed 1741" | https://scholarlycommons.pacific.edu/euler-works/53/; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [publication-dates] The Archive gives the exact title, "Published Date 1741", "Written Date 1735", and volume 8 pp. 128-140. The scan/translation heading supplies nominal volume year 1736. Hopkins and Wilson p. 411 supplies the 1735 Academy presentation; the Archive excerpt alone omits presentation and nominal year.
- CONFIRMED | "Four of them odd. No walk." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [four-odd-no-walk] E53 section 9 gives five bridges into A and three into each of B, C and D, then concludes "such a crossing of the seven Königsberg bridges cannot be achieved". Direct recount gives 5+3+3+3=14=2*7. The north/south/island names were cross-checked separately.
- CONFIRMED | "Not drawable." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [k4-not-drawable] Recounting K4 gives four vertices each of degree 3. E53 section 20 rules out a trail with more than two odd vertices. The inline and independent algorithms both return no walk.
- CONFIRMED | "In 1735 the people of Königsberg wanted to know if they could walk through town crossing each of their seven bridges exactly once." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [metadata-0] E53 section 2 names "Königsberg in Prussia", "seven bridges", and regions A, B, C, D. The supplied excerpt does not date the event; the Archive adds "Written Date 1735". Four regions and seven crossings agree with the plate.
- CONFIRMED | "Whenever a walk passes through a landmass in the middle, it must arrive on one bridge and leave on another, so the bridges at any non-endpoint vertex come in matched pairs and that vertex's degree must be even." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [metadata-3] E53 sections 11-12 pair entry and exit at an intermediate region: "one writing of it will record both the entry into A and the exit". An open trail has one unmatched incidence at each endpoint; a closed trail has no odd endpoints.
- CONFIRMED | "Only the start and the end of the walk can have odd degree, so at most two vertices may." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [metadata-4] E53 section 20 states the 0-or-2 criterion. For finite undirected graphs the vertices incident to edges must lie in one component. The page's main synopsis includes connectivity and its code tests it. Pairing arrivals and departures proves necessity; Hierholzer supplies sufficiency.
- CONFIRMED | "Königsberg has four, all odd: 5, 3, 3, 3." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [metadata-5] E53 section 9 gives five bridges into A and three into each of B, C and D, then concludes "such a crossing of the seven Königsberg bridges cannot be achieved". Direct recount gives 5+3+3+3=14=2*7. The north/south/island names were cross-checked separately.
- CONFIRMED | "Then it turns to the town as it is now (two of the seven gone in the 1944 bombing, two more replaced by a flyover in the early 1970s) and reads off the modern graph (5 edges at the 7 original sites, degrees 2, 2, 3, 3): the walk Euler proved impossible became possible in his town's later life, as a trail from one island to the other." | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [metadata-9] The supplied source explicitly limits the five-edge graph to the original seven sites and gives "degree 2" for two nodes and "degree 3" for two. Recreated degrees N=2,K=3,L=3,S=2; trail K→L→S→K→N→L. This is the stated restricted model, not an inventory of all present city bridges.
- CONFIRMED | "The Seven Bridges of Königsberg: nobody could walk the town crossing each bridge exactly once, and in answering why not, Euler invented graph theory." | https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [metadata-11] Archive: "the earliest paper in both topology and graph theory" and the problem is "unsolvable". Read as the usual founding-paper attribution, the claim is supported; it does not establish that Euler drew the modern node-and-edge picture.
- CONFIRMED | "Euler, *Solutio problematis ad geometriam situs pertinentis* (E53, written 1735; *Commentarii academiae scientiarum Petropolitanae* **8**: 128–140, printed 1741)." | https://scholarlycommons.pacific.edu/euler-works/53/; https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf | N2 evidence retained: [synopsis-euler-reference] The Archive gives the exact title, "Published Date 1741", "Written Date 1735", and volume 8 pp. 128-140. The scan/translation heading supplies nominal volume year 1736. Hopkins and Wilson p. 411 supplies the 1735 Academy presentation; the Archive excerpt alone omits presentation and nominal year.
- CONFIRMED | "The verified core. Four landmasses (N=Altstadt, K=Kneiphof, L=Lomse, S=Vorstadt) joined by seven bridges, with the canonical degree sequence (5, 3, 3, 3)" ... "all four odd." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html | N2 evidence retained: [synopsis-graph] E53 section 9 gives five bridges into A and three into each of B, C and D, then concludes "such a crossing of the seven Königsberg bridges cannot be achieved". Direct recount gives 5+3+3+3=14=2*7. The north/south/island names were cross-checked separately.
- CONFIRMED | "an Eulerian walk exists iff all vertices incident to an edge lie in one connected component and 0 or 2 vertices have odd degree" | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [connected-parity-theorem] E53 section 20 states the 0-or-2 criterion. For finite undirected graphs the vertices incident to edges must lie in one component. The page's main synopsis includes connectivity and its code tests it. Pairing arrivals and departures proves necessity; Hierholzer supplies sufficiency. Fixer clarification under X24 explicitly excludes isolated vertices from the connectivity requirement.
- CONFIRMED | "five connections at the seven original sites after the 1944 bombing and the 1970s flyover, degrees 2, 2, 3, 3" | https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg; git show f5e5f84f34:research/what-the-bridges-knew/verify.mjs | N2 evidence retained: [modern-synopsis] The supplied source explicitly limits the five-edge graph to the original seven sites and gives "degree 2" for two nodes and "degree 3" for two. Recreated degrees N=2,K=3,L=3,S=2; trail K→L→S→K→N→L. This is the stated restricted model, not an inventory of all present city bridges.
- CONFIRMED | "In 1735 the people of Königsberg wanted to know whether they could walk through the town crossing each of seven bridges exactly once." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [html-description-0] E53 section 2 supplies the seven-bridge walking question and four regions. The Archive dates the written work 1735; these statements match that historical problem.
- CONFIRMED | "A 28-year-old Leonhard Euler proved that they could not, and in proving it invented graph theory." | https://mathshistory.st-andrews.ac.uk/Biographies/Euler/; https://pyrkov-professor.ru/Portals/0/Mediateka/18%20vek/bradley_leonhard_euler.pdf; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [html-description-1] The proof was presented on 26 August 1735, after Euler's 28th birthday on 15 April. The Archive confirms the impossibility result and the founding-paper attribution. This establishes age at proof, unlike the first-receipt claim.
- CONFIRMED | "Königsberg, 1735: four landmasses, seven bridges, and a question that wouldn't go away." | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [html-og-description-0] E53 section 2 supplies the seven-bridge walking question and four regions. The Archive dates the written work 1735; these statements match that historical problem.
- CONFIRMED | "Could you walk every bridge exactly once?" | https://www.cantab.net/users/michael.behrend/repubs/maze_maths/pages/euler_en.html; https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [html-og-description-1] E53 section 2 supplies the seven-bridge walking question and four regions. The Archive dates the written work 1735; these statements match that historical problem.
- CONFIRMED | "Euler proved you could not, and in proving it invented graph theory." | https://scholarlycommons.pacific.edu/euler-works/53/ | N2 evidence retained: [html-og-description-2] Archive: "the earliest paper in both topology and graph theory" and the problem is "unsolvable". Read as the usual founding-paper attribution, the claim is supported; it does not establish that Euler drew the modern node-and-edge picture.
- CONFIRMED | "Nothing here may lie about anything real; that rule never bends." | git show f5e5f84f34:WAKE.md | N2 evidence retained: [shared-help-1-0] WAKE rule 2 is "It must never lie about anything real." This is a stated editorial rule, not evidence that the corpus is free of factual errors.
- CONFIRMED | (added) "Königsberg's seven bridges, Euler's 1736 proof that they cannot all be walked once, recomputed live in the browser" | https://scholarlycommons.pacific.edu/euler-works/53/; git show f5e5f84f34:public/strata/what-the-bridges-knew/index.html | N2 evidence retained: [added-jsonld-description] The JSON-LD description is not separately represented in the source record. E53 is conventionally labeled by nominal volume year 1736; the impossible walk and live recomputation were independently checked. This is not a claim that publication occurred in 1736.
- MINOR | [D1; verifier documentation] The offline brute-force comments say 7! * 2^7 * 4, about 2.6 million tries. | research/what-the-bridges-knew/verify.mjs | The code has one edge-permutation loop per starting vertex; crossing orientation follows the current vertex. Independent enumeration gives 4*7!=20,160. Corrected the two comments in the authored and published verifier; the executable search was already correct.
- MINOR | [D2; verifier documentation] The README says adding N-K takes both N and K from degree 3 to 4. | research/what-the-bridges-knew/README.md; research/what-the-bridges-knew/verify.mjs | The original degrees are N=3 and K=5. Adding N-K gives N=4 and K=6. Corrected the documentation to 3 to 4 and 5 to 6 respectively. The executable graph and parity verdict were already correct.

## Findings
- [fixed] [X1] The offline executable defines seven graph cases including HOUSE, K4, K5 and FIG8; it has no envelope instance. Narrowed the page's lede and source-list wording and the layer README to the actual coverage. The fresh run still reports 35 passed, 0 failed.
- [fixed] [X2] Re-read E53 sections 4-7 and Hopkins and Wilson's authorized reprint p. 410: Euler uses sequences of region letters; the dot-and-line diagram is a modern presentation. The shared paragraph now says 'In modern notation'.
- [fixed] [X3] The four-dot graph is mathematically correct, but the surrounding historical attribution blurred the distinction from Euler's letter-sequence method. Added the modern-notation qualification, also mirrored in the dek under X21.
- [fixed] [X4] Independently counted the authored house edges ab, bc, cd, da, ae, be, ac: a=4, b=3, c=3, d=2, e=2. The diagonal joins a and c; the trail endpoints are b and c. Corrected the prose and verifier comments, strengthened the existing odd-vertex assertion to require b and c, and refreshed the layer's published verifier from the source. One correction line covers this error.
- [fixed] [X6] The live graph keyed modern is labelled '+1 bridge' and has eight edges; the separate five-site modern model is checked offline. Replaced the universal live-coverage sentence with 'The graph tests below are recomputed live, the moment the page loads.'
- [fixed] [X7] Re-read Hopkins and Wilson's authorized reprint p. 413: Leibniz's term circulated among eighteenth-century followers. Replaced the categorical non-adoption claim with 'The names changed'; shares one correction line with X28.
- [dated] [X8] The original June 14 verification date was overtaken by the committed run at 2026-08-01T13:02:09.145Z, exit 0, 35 passes. This fixer also ran the offline verifier on 2026-10-14 and recorded 35 passes. The generated placard still says June 14: triage directs this date through the record and the landing placard rebuild, and the user forbids the writing form of scripts/check-placard.mjs. NOT FIXED on the page pending that rebuild. No Updated or Corrected line under the explicit placard exception.
- [fixed] [X9] Downloaded and visually read original E53 PDF page 1, printed p. 128. The title sits above section 1; the epigraph begins that section. Corrected the explanatory source note only. The Latin passage and explicitly authorial gloss remain unchanged.
- [fixed] [X10] Re-read the English article's Present state section and the German per-bridge article. The English text supplies the restricted postwar model, not the bridge-name crosswalk. Narrowed its description and added a separate German citation for names and endpoints.
- [fixed] [X11] Euler's printed plate and the geographic crosswalk assign c to Kramer, while the page uses a. The endpoints are correct. Chose the triage's attribution fix: the table, header and walker now explicitly call a-g this page's labels. No graph or bridge name was changed.
- [fixed] [X12] Euler's printed plate and the geographic crosswalk assign d to Schmiede, while the page uses b. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- [fixed] [X13] Euler's printed plate and the geographic crosswalk assign a to Green, while the page uses c. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- [fixed] [X14] Euler's printed plate and the geographic crosswalk assign b to Kottel, while the page uses d. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- [fixed] [X15] Euler's printed plate and the geographic crosswalk assign g to Wood, while the page uses f. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- [fixed] [X16] Euler's printed plate and the geographic crosswalk assign f to High, while the page uses g. The endpoints are correct. The shared attribution fix identifies the existing labels as the page's own.
- [fixed] [X17] Visually read E53 PDF page 2, plate VIII, Figure 1: a/b join A-B, c/d join A-C, f joins B-D, and g joins C-D. Read the geographic name crosswalk separately; handwritten names on the scan were not treated as Euler's printed labels. Six page assignments differ. Corrected the attribution across the table and walker; one dated line covers X11-X17.
- [fixed] [X18] Re-read the German Kaiserbrucke section and municipal guide PDF page 3. The successor Jubilee Bridge was built in 2005 at the Imperial Bridge site, between Oktyabrskaya and Epronovskaya streets, joining Lomse/Oktyabrsky Island to the south bank. Replaced Kant Island with Lomse (now Oktyabrsky Island).
- [fixed] [X19] Re-read the Euler Archive E3 record: written 1727, published 1727, and Acta Eruditorum volume 1727. This counterexample refutes the universal three-way date-ambiguity claim. Removed that clause while preserving E53's documented dates.
- [fixed] [X20] The colophon repeated X6's universal browser coverage claim. Changed it to 'the graph tests run in your browser'; the five-edge restricted modern model remains an offline case.
- [fixed] [X21] Re-read E53 sections 4-7 and Hopkins and Wilson p. 410. Added 'In modern notation' before the dek's dot-and-line construction, matching X2-X3.
- [open note:790e36] [X22] The source relation and its dated correction are fixed, but public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still quote the false general-decision-procedure claim without an annotation. Exhaustive search is a terminating general procedure; no polynomial-time algorithm is known for the general problem. These copies are outside this layer's edit scope. The directing instance must arrange an annotation through research/lineage-edge-grammar/copy-corrections.mjs, regenerate those copies while preserving the quotation and corpus statistics, and replace PENDING with a board-issued note id.
- [fixed] [X23] The authored envelope edge array contains c-d and the coordinates put c and d at the two bottom corners. Counted eight edges and degrees a=4, b=4, c=3, d=3, e=2. Replaced the shared caption with square + roof + both diagonals. One dated line covers X23 and X26.
- [fixed] [X24] A connected triangle has zero odd vertices and a circuit; two disjoint triangles also have zero odd vertices but no Eulerian trail. The code rejects the disconnected pair and allows a triangle plus an isolated vertex. Added the condition that vertices incident to edges lie in one component to the aria-label, synopsis and README.
- [fixed] [X25] Independently enumerated all 4*7! edge-order attempts from the authored seven edges. Crossed-edge histogram: 0:10080, 1:5520, 2:2448, 3:1104, 4:496, 5:256, 6:256. None crosses seven; half cross zero and leave all seven. Replaced 'strands at most six' with 'crosses at most six'.
- [fixed] [X26] The parity panel reuses X23's envelope caption and graph. Its c-d bottom edge is present; the corrected shared caption now accurately describes that same eight-edge graph. Shares X23's dated line.
- [dated] [X27] Re-read the July 26 audit log recording this layer's 32-to-35 check-count edit after the June date. Set JSON-LD dateModified to 2026-10-14 for this revision. No Updated line under the explicit metadata-date exception; the Corrected lines date the revision.
- [fixed] [X28] This is X7's same sentence. Hopkins and Wilson p. 413 documents eighteenth-century reception of the term. Replaced the non-adoption clause; shares X7's dated line.
- [fixed] [X29] The MAA article returned by the web index identifies a different author and title. Crossref identifies the Hopkins/Wilson paper, CMJ 35(3), 198-207, May 2004, DOI 10.1080/07468342.2004.11922073. Replaced the bibliography href and displayed link with that DOI.
- [fixed] [D1] The code has one edge-permutation loop per starting vertex; crossing orientation follows the current vertex. Independent enumeration gives 4*7!=20,160. Corrected the two comments in the authored and published verifier; the executable search was already correct.
- [fixed] [D2] The original degrees are N=3 and K=5. Adding N-K gives N=4 and K=6. Corrected the documentation to 3 to 4 and 5 to 6 respectively. The executable graph and parity verdict were already correct.


## Fixer verification (before repair)

- node research/what-the-bridges-knew/verify.mjs: exit 0, 35 passed, 0 failed on 2026-10-14, including the strengthened house-endpoint assertion.
- The published verifier was refreshed byte-for-byte from the authored layer verifier; no shared manifest was edited.
- Independent edge counting reproduced the house, envelope and eight-edge +1 degree sequences. Exhaustive enumeration reproduced the 20,160-attempt histogram above.
- The source's connectivity implementation accepted one triangle and one triangle plus an isolated vertex, and rejected two disjoint triangles.
- Browser verifier: 32 passed, 0 failed at 1100x1600 and 390x844, using a scratch copy with the installed Playwright import and screenshot capture disabled. The authored browser verifier was not changed.
- Targeted revision checks passed for all ten HTML/markdown correction pairs, their placement, metadata, custom labels, generated wording, envelope and house readouts, connectivity controls, and unchanged epigraph/placard.
- node scripts/check-claims-pass.mjs: exit 1 for 40 other layer records; no failure for this layer. Direct checkRecord validation of this record returned no errors, checked 98, confirmed 68, wrong 28 (18 WRONG plus 10 MINOR), stale 2, open 0. Full details are in research/assay-line/queue-1014c/what-the-bridges-knew/fix-report.md.

## Scope left to landing

The generated June 14 Last verified placard is not hand-edited. Triage X8 directs the new record into the landing placard rebuild; the writing form of scripts/check-placard.mjs is forbidden in this worker's scope. JSON-LD dateModified was updated directly in the authored standalone HTML. Shared check-publishing manifests and site-wide generated artifacts are left for landing. Untriaged N2 uncertainties remain outside this work list and are not promoted to CONFIRMED here.

## Repair review (2026-10-14)

The Codex repair worker confirmed the fix-checker's X22 finding: the two outside copies still lack a correction annotation. X22 remains WRONG and its finding is now open board notes. All ten dated correction notices match their markdown copies, state the old claim and supported correction, and name the assay line and date. No notice required a wording change; the epigraph and its gloss remain unchanged.

Fresh verification: node research/what-the-bridges-knew/verify.mjs passed 35/35. The claims gate exited 1 for this record's required PENDING placeholder and 12 other layers; direct checkRecord validation found only that placeholder error. Counts still reconcile: 98 checked = 68 CONFIRMED + 18 WRONG + 10 MINOR + 2 STALE; wrong is 28 including MINOR. The em-dash gate, run with explicit base 9306f663bb5bb7f447014082bd7f342ea22e982d to avoid its default git fetch, exited 1 for four occurrences in dunning-kruger and the-only-other-pair, with none in this layer. The outside-layer failures were left alone. Full evidence, exact dated lines and the board-note request are in research/assay-line/queue-1014c/what-the-bridges-knew/repair-report.md.