---
slug: the-mutilated-chessboard
passed: 2026-10-14
by: the assay line (research/assay-line/cited-1014b/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex
checked: 93
confirmed: 60
wrong: 32
unverifiable: 0
observed: 0
stale: 1
---

This was the directed fixing pass over all 33 triage items at f5e5f84f34, with each item's evidence reread before editing, the live-script counterexamples replayed, and the checker's 60 CONFIRMED rereads retained with their provenance. All 22 WRONG and 10 MINOR items were repaired in the authorised layer files; MINOR counts toward wrong here. The sole STALE item updates JSON-LD dateModified under the explicit exception. Twelve distinct WRONG corrections are dated on the page and mirrored in the Markdown body. The findings below describe those layer repairs, not a complete re-audit of the checker's 47 observations or its separate untriaged UNVERIFIABLE attribution.

Repair review by Codex, 2026-10-14: all four remaining copy problems in fixcheck.md are supported. The target-layer repairs stand, but 15 affected triage findings remain open for shared TV narration, other layers' authored notes, and preserved corpus quotations in There, Here. Claim verdicts and counts are unchanged: 60 CONFIRMED + 22 WRONG + 10 MINOR + 1 STALE = 93. The repair report holds the evidence and four proposed board notes at research/assay-line/cited-1014b/the-mutilated-chessboard/repair-report.md.

## Claims
- WRONG | X1: "watch it strand two squares every time" | research/the-mutilated-chessboard/verify-assay.mjs; public/strata/the-mutilated-chessboard/index.html | Replayed 28 legal disjoint placements through the actual click handler: six stranded cells, two light and four dark. Auto-solve instead leaves two dark cells after 30 dominoes.
- MINOR | X2: "a one-line argument that beats a search of nearly thirteen million tilings" | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf; research/the-mutilated-chessboard/verify.mjs | Knuth, pp. 16–17, reports a completed mutilated-board search. The verifier separately counts 12,988,816 complete tilings of the intact board, so that number is not a mutilated-board search size.
- WRONG | X3: "Place a domino and the count of uncovered light and uncovered dark each drop by one \u2014 the difference never moves off +2." | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; research/the-mutilated-chessboard/verify-assay.mjs | mut_light proves 30 and mut_dark proves 32 for the displayed holes. Thus light minus dark is 30 - 32 = -2; placing any domino subtracts one from each count.
- WRONG | X4: "It is the same reason you can't make change for a pound with an equal number of two coins when you're holding thirty of one and thirty-two of the other." | research/the-mutilated-chessboard/verify-assay.mjs; git show f5e5f84f34:public/strata/the-mutilated-chessboard/index.html | Four 5p coins plus four 20p coins total 100p, within holdings of thirty and thirty-two and using equal numbers. The original sentence gave no denominations or requirement to spend every coin.
- MINOR | X5: "The opposite-colour case is a real theorem too (Gomory, 1973)" | https://www.wkbpic.com/wkbx/SA/1962/1962-11-01.pdf; https://www.scientificamerican.com/article/mathematical-games-1962-11/ | Read the image of printed p. 151 (PDF page 153), which credits Ralph Gomory. The publisher dates this report to November 1962. The repair names the report and makes no discovery-date claim.
- MINOR | X6: "Nothing here is asserted." | research/the-mutilated-chessboard/verify.mjs | Read all sections A–F and the sole engine import. The verifier checks specified mathematics, not page text, citations, or the coin analogy.
- WRONG | X7: "And the impossibility direction is machine-checked in Lean: research/the-mutilated-chessboard/lean/MutilatedChessboard.lean proves that no list of dominoes can cover exactly the mutilated board \u2014 for every board, by the colour invariant, with zero imports and axiom footprint [propext, Quot.sound]." | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; memory/log.d/2026-06-30-mutilated-chessboard.md | balanced is general over domino lists. The final theorem assumes (cover T).Perm mutilated, where mutilated is the 8 by 8 board minus (0,0) and (7,7). The author log lists generalising the final proof as future work.
- WRONG | X8: "A search can only check finitely many boards; the proof settles all of them at once." | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; memory/log.d/2026-06-30-mutilated-chessboard.md | balanced is general over domino lists. The final theorem assumes (cover T).Perm mutilated, where mutilated is the 8 by 8 board minus (0,0) and (7,7). The author log lists generalising the final proof as future work.
- WRONG | X9: "The colouring does it in a sentence, and does more: it tells you exactly which deletions are impossible (any two of one colour) and which are fine (one of each), for boards of any size, without drawing a single domino." | research/the-mutilated-chessboard/chessboard.mjs; research/the-mutilated-chessboard/verify-assay.mjs; https://www.wkbpic.com/wkbx/SA/1962/1962-11-01.pdf; https://www.scientificamerican.com/article/mathematical-games-1962-11/ | A 3 by 3 board minus adjacent opposite-colour squares has seven cells and is untileable. hamiltonianCycle rejects odd n. Gardner supplies the closed-loop sufficiency argument for the ordinary chessboard; colour imbalance supplies necessity.
- WRONG | X10: "The mutilated board has no domino tiling \u2014 decided by exact matching, and proved for all boards in Lean. (machine-checked)" | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; memory/log.d/2026-06-30-mutilated-chessboard.md | balanced is general over domino lists. The final theorem assumes (cover T).Perm mutilated, where mutilated is the 8 by 8 board minus (0,0) and (7,7). The author log lists generalising the final proof as future work.
- WRONG | X11: "M. Gardner, \"Mathematical Games,\" Scientific American (1957, repr. in The Unexpected Hanging and Other Mathematical Diversions, 1969)." | https://www.wkbpic.com/wkbx/SA/1962/1962-11-01.pdf; https://www.scientificamerican.com/article/mathematical-games-1962-11/ | Viewed printed p. 151: the older corner puzzle is discussed separately from the opposite-colour extension credited to Gomory. The publisher identifies the column as November 1962, p. 151.
- WRONG | X12: "reported in M. Gardner, \"Mathematical Games,\" Scientific American (1957, repr. in The Unexpected Hanging and Other Mathematical Diversions, 1969)." | https://www.wkbpic.com/wkbx/SA/1962/1962-11-01.pdf; https://www.scientificamerican.com/article/mathematical-games-1962-11/ | Viewed printed p. 151: the older corner puzzle is discussed separately from the opposite-colour extension credited to Gomory. The publisher identifies the column as November 1962, p. 151.
- WRONG | X13: "Only possible because the two holes are opposite colours." | research/the-mutilated-chessboard/verify-assay.mjs; public/strata/the-mutilated-chessboard/index.html | Removed both holes through the controls and clicked Auto-solve in placement mode. All 64 cells were covered by 32 dominoes while the old success message still referred to two holes.
- WRONG | X14: "the stranded squares are the same colour. That is no accident." | research/the-mutilated-chessboard/verify-assay.mjs; public/strata/the-mutilated-chessboard/index.html | The six-cell dead end has two light and four dark cells. The invariant fixes the difference of counts, not the colour of every stranded cell. The optimal two-cell remainder remains a valid separate claim.
- WRONG | X15: "Try it here and watch it strand two squares every time; then turn on the colours and see why." | research/the-mutilated-chessboard/verify-assay.mjs; public/strata/the-mutilated-chessboard/index.html | Replayed 28 legal disjoint placements through the actual click handler: six stranded cells, two light and four dark. Auto-solve instead leaves two dark cells after 30 dominoes.
- MINOR | X16: "A one-line parity invariant that defeats a search of 12,988,816 tilings, with the board deciding tileability live by exact bipartite matching, and the impossibility machine-checked in Lean 4 (zero imports, axioms propext and Quot.sound, no sorry)." | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf; research/the-mutilated-chessboard/verify.mjs | Knuth, pp. 16–17, reports a completed mutilated-board search. The verifier separately counts 12,988,816 complete tilings of the intact board, so that number is not a mutilated-board search size.
- WRONG | X17: "Nim is won by a conserved quantity \u2014 the XOR nim-sum, zero exactly at the lost positions." | https://www.scribd.com/document/778966103/Nim-A-Game-with-a-Complete-Mathematical-Theory; research/the-mutilated-chessboard/verify-assay.mjs | Read Bouton, pp. 35–36, normal-play rules, safe combinations and Theorems I–II. A legal move (1,1) to (0,1) changes XOR from zero to one. Zero classifies losing normal-play positions.
- WRONG | X18: "The mutilated board is lost for the same kind of reason: the colour difference (light − dark) is conserved by every domino and stuck at +2, so the all-covered goal (difference 0) is unreachable." | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; research/the-mutilated-chessboard/verify-assay.mjs | mut_light proves 30 and mut_dark proves 32 for the displayed holes. Thus light minus dark is 30 - 32 = -2; placing any domino subtracts one from each count.
- WRONG | X19: "here, a single tile is forced to cover the plane only without ever repeating; here, two dominoes' worth of squares can't be covered at all." | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; git show f5e5f84f34:src/content/strata/the-mutilated-chessboard.md | 30 + 32 = 62 squares require 31 two-square dominoes, not two. The source note starts its preceding clause with There; that wording was retained, correcting only the false quantity.
- WRONG | X20: "Both are impossibilities no search could exhaust" | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf | Read A failed experiment, pp. 16–17: Knuth reports that an exhaustive DLX search finished and found no solution. This directly contradicts inexhaustibility or an answer out of reach.
- WRONG | X21: "watch it strand two squares of the same colour every time." | research/the-mutilated-chessboard/verify-assay.mjs; public/strata/the-mutilated-chessboard/index.html | Replayed 28 legal disjoint placements through the actual click handler: six stranded cells, two light and four dark. Auto-solve instead leaves two dark cells after 30 dominoes.
- MINOR | X22: "**The exact dividing line** (Gomory, 1973): remove any one light and any one dark square and the rest *always* tiles; remove two of the same colour and it *never* does." | https://www.wkbpic.com/wkbx/SA/1962/1962-11-01.pdf; https://www.scientificamerican.com/article/mathematical-games-1962-11/ | Read the image of printed p. 151 (PDF page 153), which credits Ralph Gomory. The publisher dates this report to November 1962. The repair names the report and makes no discovery-date claim.
- MINOR | X23: "`research/the-mutilated-chessboard/verify.mjs` recomputes every number (16/16)" | research/the-mutilated-chessboard/verify.mjs | Read the 16 mathematical checks: colour counts, all adjacencies, all removal pairs, cycle witnesses and intact-board DP. There is no page-reading step or check of every displayed number and historical date.
- MINOR | X24: "It also recomputes the **12,988,816** domino tilings of the intact board (Kasteleyn) to show the size of the search the invariant bypasses." | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf; research/the-mutilated-chessboard/verify.mjs | Knuth, pp. 16–17, reports a completed mutilated-board search. The verifier separately counts 12,988,816 complete tilings of the intact board, so that number is not a mutilated-board search size.
- WRONG | X25: "A program checks finitely many boards; the proof settles all of them at once." | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; memory/log.d/2026-06-30-mutilated-chessboard.md | balanced is general over domino lists. The final theorem assumes (cover T).Perm mutilated, where mutilated is the 8 by 8 board minus (0,0) and (7,7). The author log lists generalising the final proof as future work.
- WRONG | X26: "A program checks finitely many boards; the proof settles all of them at once. (Ledger **P5**.)" | research/the-mutilated-chessboard/lean/MutilatedChessboard.lean; memory/log.d/2026-06-30-mutilated-chessboard.md | balanced is general over domino lists. The final theorem assumes (cover T).Perm mutilated, where mutilated is the 8 by 8 board minus (0,0) and (7,7). The author log lists generalising the final proof as future work.
- WRONG | X27: "a *finite* question whose brute-force answer is astronomically out of reach, settled instantly by an **invariant**" | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf | Read A failed experiment, pp. 16–17: Knuth reports that an exhaustive DLX search finished and found no solution. This directly contradicts inexhaustibility or an answer out of reach.
- WRONG | X28: "Try it yourself and watch it always strand two squares." | research/the-mutilated-chessboard/verify-assay.mjs; public/strata/the-mutilated-chessboard/index.html | Replayed 28 legal disjoint placements through the actual click handler: six stranded cells, two light and four dark. Auto-solve instead leaves two dark cells after 30 dominoes.
- MINOR | X29: "A one-line proof that defeats a search of 12,988,816 tilings, and it's machine-checked in Lean." | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf; research/the-mutilated-chessboard/verify.mjs | Knuth, pp. 16–17, reports a completed mutilated-board search. The verifier separately counts 12,988,816 complete tilings of the intact board, so that number is not a mutilated-board search size.
- MINOR | X30: "The obstruction is colour: one line beats a search of 12,988,816 tilings." | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf; research/the-mutilated-chessboard/verify.mjs | Knuth, pp. 16–17, reports a completed mutilated-board search. The verifier separately counts 12,988,816 complete tilings of the intact board, so that number is not a mutilated-board search size.
- STALE | X31: "\"dateModified\":\"2026-06-30\"" | git log f5e5f84f34 -- public/strata/the-mutilated-chessboard/index.html; git show f5e5f84f34:public/strata/the-mutilated-chessboard/index.html | The JSON-LD retained the 2026-06-30 creation date after later edits, including 2e60bcb5dd1 on 2026-10-01. This repair sets dateModified to 2026-10-14 under the explicit metadata exception.
- MINOR | X32: (added) "Click a square to move a hole there (two holes total). Then Auto-solve to ask the board." | public/strata/the-mutilated-chessboard/index.html; research/the-mutilated-chessboard/verify-assay.mjs | With two holes selected, clicking a new square clears the pair and selects one hole. Clicking an existing hole removes it. Replayed both actions before rewriting the instruction.
- WRONG | X33: (added) "matching saturates the 30 light squares, stranding 2 dark" | research/the-mutilated-chessboard/verify.mjs; research/the-mutilated-chessboard/chessboard.mjs; research/the-mutilated-chessboard/verify-assay.mjs | Removing (7,0) and (0,7) leaves 32 light and 30 dark. Its maximum matching leaves two light cells. The original diagonal leaves two dark. Recomputed both, including the uncovered sets.
- CONFIRMED | "Sixty-two squares remain \u2014 an even number, room for exactly thirty-one dominoes." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [001 intro-area] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Lay them down and you will never cover the board." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [002 intro-impossible] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Sixty-two squares, an even number; thirty-one dominoes, each covering two squares; 62 = 2 × 31." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [006 arithmetic] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Nothing about the count forbids it." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [007 count-no-obstruction] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "the intact 8×8 board can be tiled in 12,988,816 different ways." | https://sites.math.rutgers.edu/~zeilberg/Theses/TrevorBassMA.pdf | Checker reread retained from N2-codex.md: [008 intact-count] Read PDF sections 2.1 and 3.1. It states a "bijection" with perfect matchings; the fourth 2n-by-2n count is "12988816". The independent downloaded DP also returns 12,988,816.
- CONFIRMED | "look at the two squares you removed: they are the same colour." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [009 same-corners] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Every domino \u2014 horizontal or vertical, anywhere \u2014 sits on two adjacent squares, and adjacent squares always have opposite colours." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [010 adjacent-colours] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "So every domino covers exactly one light square and one dark square." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [011 domino-balance] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "A full tiling of 31 dominoes would therefore cover exactly 31 light and 31 dark." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [012 tiling-balance] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "No tiling exists \u2014 and no amount of cleverness will change a count." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [014 impossibility-proof] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "This is a parity invariant: a quantity (here, light − dark) that every legal move leaves unchanged, but the goal demands you change." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [015 invariant-definition] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "You can shuffle dominoes forever; you can never make 30 equal 32." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [017 thirty-not-thirtytwo] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "You'll find the dividing line is purely colour: remove two squares of opposite colour and the rest always tiles; remove two of the same colour and it never does \u2014 no matter where on the board they sit." | https://pdfcoffee.com/gardner04-pdf-free.html | Checker reread retained from N2-codex.md: [019 opposite-and-same] Read chapter 16, pp. 186-188. The source says "each segment (and therefore the entire board) can be completely covered by dominoes". Its closed path covers the ordinary 8-by-8 board, and opposite colours leave even paths.
- CONFIRMED | "The colouring argument forbids the same-colour case." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [020 same-colour-case] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "thread a single closed loop through all 64 squares visiting each once" | https://pdfcoffee.com/gardner04-pdf-free.html | Checker reread retained from N2-codex.md: [022 hamiltonian-loop] Read chapter 16, pp. 186-188. The source says "each segment (and therefore the entire board) can be completely covered by dominoes". Its closed path covers the ordinary 8-by-8 board, and opposite colours leave even paths.
- CONFIRMED | "removing one light and one dark square cuts the loop into two arcs of even length, and each arc tiles by taking alternate steps." | https://pdfcoffee.com/gardner04-pdf-free.html | Checker reread retained from N2-codex.md: [023 two-arcs] The source expressly allows "one segment if the removed squares are adjacent on the path". Interpreting the page's second arc as empty makes the construction correct: zero is even. The downloaded Hamiltonian witness passes, including adjacent removed squares. No extra fault is inferred from this degenerate case.
- CONFIRMED | "a region of 2k squares tiles iff its maximum matching has k dominoes" | https://sites.math.rutgers.edu/~zeilberg/Theses/TrevorBassMA.pdf | Checker reread retained from N2-codex.md: [027 matching-equivalence] Read PDF sections 2.1 and 3.1. It states a "bijection" with perfect matchings; the fourth 2n-by-2n count is "12988816". The independent downloaded DP also returns 12,988,816.
- CONFIRMED | "To rule out a tiling by brute force you would have to inspect an astronomical number of arrangements and find every one wanting \u2014 the intact board alone has 12,988,816; the mutilated board has its own forest of partial layouts to exhaust." | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf | Checker reread retained from N2-codex.md: [033 brute-force-scale] The page explicitly separates "the intact board alone" from the mutilated board's "partial layouts". Knuth supplies an actual example with 4,780,846 updates and 13,922 near-completions. The adjective about scale is not treated as an additional factual error; the explicit claim of inexhaustibility is checked separately.
- CONFIRMED | "The same shape of argument proves you can't always solve the 15-puzzle" | https://math.mit.edu/~etingof/groups.pdf | Checker reread retained from N2-codex.md: [035 fifteen-puzzle] Read Etingof, Exercise 1.6 and section 12: "14 and 15 are interchanged is unsolvable"; Rubik configurations have "three invariants" that obstruct legal transformations.
- CONFIRMED | "can't tile other deficient boards" | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [036 deficient-boards] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "can't reach certain states of a Rubik's cube." | https://math.mit.edu/~etingof/groups.pdf | Checker reread retained from N2-codex.md: [037 rubik-cube] Read Etingof, Exercise 1.6 and section 12: "14 and 15 are interchanged is unsolvable"; Rubik configurations have "three invariants" that obstruct legal transformations.
- CONFIRMED | "Max Black, Critical Thinking (Prentice-Hall, 1946)" | https://books.google.com/books/about/Critical_Thinking.html?id=_84tAAAAMAAJ | Checker reread retained from N2-codex.md: [045 black-citation] Google Books gives "Critical Thinking: An Introduction to Logic and Scientific Method", "Max Black", and "Prentice-Hall, Incorporated, 1946". The record inserts whitespace around a comma but preserves the bibliographic facts.
- CONFIRMED | "The puzzle and its colour argument are usually credited to Max Black, Critical Thinking (Prentice-Hall, 1946)" | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf | Checker reread retained from N2-codex.md: [046 black-attribution] Knuth reference 5 names "Max Black, Critical Thinking" and "Prentice–Hall, 1946", asking about an "earlier reference" for the puzzle. This supports the page's cautious "usually credited" wording; it is not a claim to have inspected Black's original pages.
- CONFIRMED | "the generalisation that any two opposite-coloured squares may be removed is R. Gomory's" | https://www.wkbpic.com/wkbx/SA/1962/1962-11-01.pdf | Checker reread retained from N2-codex.md: [048 gomory-attribution] Inspected the 1962 page: the proof is credited to "Ralph Gomory". The record's "Ralph Goniory" is faithful to its OCR source but is an OCR error, not another person.
- CONFIRMED | "P. W. Kasteleyn, \"The statistics of dimers on a lattice,\" Physica 27 (1961), 1209–1225" | https://api.crossref.org/works/10.1016/0031-8914%2861%2990063-5 | Checker reread retained from N2-codex.md: [051 kasteleyn-citation] Read publisher-deposited Crossref JSON: "title":["The statistics of dimers on a lattice"], author "P.W." "Kasteleyn", volume "27", page "1209-1225", issued [[1961,12]]. DOI lookup MATCH.
- CONFIRMED | "the closed form giving 12,988,816 domino tilings of the 8×8 board." | https://sites.math.rutgers.edu/~zeilberg/Theses/TrevorBassMA.pdf | Checker reread retained from N2-codex.md: [052 kasteleyn-count] Read PDF sections 2.1 and 3.1. It states a "bijection" with perfect matchings; the fourth 2n-by-2n count is "12988816". The independent downloaded DP also returns 12,988,816.
- CONFIRMED | "D. König (1931)" | https://arxiv.org/abs/2009.03780 | Checker reread retained from N2-codex.md: [054 konig-citation] The translation's abstract says "originally written in Hungarian by Dénes Kőnig in 1931" and identifies equality of minimum vertex cover and maximum matching in bipartite graphs. A background credit, not a claim that this is the exact implemented algorithm.
- CONFIRMED | "H. W. Kuhn, \"The Hungarian method for the assignment problem,\" Naval Research Logistics Quarterly 2 (1955), 83–97." | https://www.math.utoronto.ca/mccann/1855/KuhnNRL55.pdf; https://api.crossref.org/works/10.1002%2Fnav.3800020109 | Checker reread retained from N2-codex.md: [055 kuhn-citation] Read the PDF title page and Crossref: "The Hungarian method for the assignment problem", "H. W." "Kuhn", volume "2", pages "83-97", issued [[1955,3]], "Naval Research Logistics Quarterly". DOI lookup MATCH.
- CONFIRMED | "Domino tilings as perfect matchings of the grid graph, and bipartite matching by augmenting paths" | https://sites.math.rutgers.edu/~zeilberg/Theses/TrevorBassMA.pdf; https://www.math.utoronto.ca/mccann/1855/KuhnNRL55.pdf | Checker reread retained from N2-codex.md: [056 matching-history] Bass states the tiling/perfect-matching "bijection". Kuhn's primary paper constructs and augments assignments. The modern unweighted DFS port is read independently in chessboard.mjs; no claim of byte-identical historical code is inferred.
- CONFIRMED | "Unequal, so no tiling can exist" | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [062 ui-same-colour] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "A tiling does exist" | https://pdfcoffee.com/gardner04-pdf-free.html | Checker reread retained from N2-codex.md: [063 ui-opposite-colour] Read chapter 16, pp. 186-188. The source says "each segment (and therefore the entire board) can be completely covered by dominoes". Its closed path covers the ordinary 8-by-8 board, and opposite colours leave even paths.
- CONFIRMED | "Odd number of squares \u2014 can't pair them at all." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [066 ui-odd] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Cut two opposite corners off a chessboard and 62 squares remain (room for exactly 31 dominoes), yet no arrangement ever tiles them." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [075 fm-dek-0] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Every domino covers one light and one dark square, the two corners are the same colour, so you're left with 30 of one and 32 of the other, and 30 can never pair with 32." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [077 fm-dek-2] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Cut two opposite corners off a chessboard and 62 squares are left, exactly room for 31 dominoes, yet no arrangement ever tiles them." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [079 fm-plain-0] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Every domino covers one light and one dark square, and the two removed corners are the same colour, so 30 of one colour have to pair with 32 of the other." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [080 fm-plain-1] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "There, √2 is no fraction because assuming it forces a and b to both be even when they were taken coprime \u2014 an even/odd contradiction." | git show e05770fb79:research/incommensurable-immersive/lean/Sqrt2.lean | Checker reread retained from N2-codex.md: [083 related-sqrt-parity] Read the parity lemma and descent proof. Recreated with Lean 4.35: exit 0; no_pos_solution and sqrt_two_irrational print "[propext, Quot.sound]". Even a forces even b, contradicting coprimality.
- CONFIRMED | "Here, the mutilated board is no tiling because every domino is one light + one dark, forcing equal colour counts the board can't supply." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [084 related-chessboard-proof] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Same move: a quantity the goal must change but no legal step ever does." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [085 related-invariant] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "There the pentagon is impossible because a lattice rotation's trace 2·cos(2π/5) is not the integer it must be" | git show e05770fb79:research/crystallographic-restriction/lean/CrystallographicRestriction.lean | Checker reread retained from N2-codex.md: [087 related-pent-trace] A lattice-preserving rotation has an integer matrix in a lattice basis, hence integer trace. In the plane its eigenvalues are exp(+i theta), exp(-i theta), so its trace is 2*cos(theta). At theta=2*pi/5 this is (sqrt(5)-1)/2, strictly between 0 and 1. The source's integer theorem no_order_five is machine-checked.
- CONFIRMED | "here the tiling is impossible because 30 ≠ 32." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [088 related-pent-chess] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "In each, an astronomically large search collapses to one arithmetic fact a machine can certify." | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf | Checker reread retained from N2-codex.md: [089 related-huge-search] The substantive claim is that arithmetic obstructions remove the need for an arrangement search. Both Lean results verify that obstruction. The comparison of scale is not converted into a fabricated numeric lower bound; explicit infeasibility is tested at related-search-inexhaustible and md-finite-search.
- CONFIRMED | "Find the invariant and the game is decided without search." | https://www.scribd.com/document/778966103/Nim-A-Game-with-a-Complete-Mathematical-Theory | Checker reread retained from N2-codex.md: [092 related-game-no-search] Bouton's safe-combination rule decides normal-play Nim by binary-column parity; the chessboard obstruction is settled by colour counts. Decision without searching moves is correct even though calling Nim-sum conserved in the preceding sentence is false.
- CONFIRMED | "There, a single tile is forced to cover the plane only without ever repeating" | https://cs.uwaterloo.ca/~csk/hat/ | Checker reread retained from N2-codex.md: [093 related-monotile] The authors' project page states "a shape that tiles the plane, but never periodically". This supports the aperiodic monotile claim; "repeating" is read as global periodicity in this tiling context.
- CONFIRMED | "The colouring that forbids this tiling is the simplest member of the family of arguments that govern which tilings exist." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [095 related-tiling-family] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "There a value's two copies occupy positions whose odd-count parity is fixed by the gap, forbidding the row unless n is 0 or 3 mod 4" | https://ptgmedia.pearsoncmg.com/imprint_downloads/informit/bookreg/9780201038040/9780201038040_v4A_pages_i-16.pdf | Checker reread retained from N2-codex.md: [097 related-langford-parity] Read the complete source and the recorded excerpt against this claim.
- CONFIRMED | "here a domino covers one light and one dark square, forbidding the tiling." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [098 related-langford-domino] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "The classic **mutilated chessboard** (Max Black, 1946)" | https://www.rose-hulman.edu/class/cs/csse313/assignments/csp/DancingLinks.pdf | Checker reread retained from N2-codex.md: [103 md-black] Knuth reference 5 gives "Max Black, Critical Thinking" and "Prentice–Hall, 1946" for the mutilated-chessboard problem; the 1946 Google Books record agrees.
- CONFIRMED | "Adjacent squares always differ in colour, so every domino \u2014 anywhere, either orientation \u2014 covers exactly one light and one dark square." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [105 md-adjacency] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "A tiling of 31 dominoes therefore covers 31 of each." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [106 md-31-each] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "But both opposite corners are the same colour, so the mutilated board holds **30 light and 32 dark**." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [107 md-colour-count] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Thirty cannot pair with thirty-two: no tiling exists, for *every* such board, with no search at all." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [108 md-universal] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "the textbook specimen of the move that also proves you can't always solve the 15-puzzle" | https://math.mit.edu/~etingof/groups.pdf | Checker reread retained from N2-codex.md: [120 md-fifteen] Read Etingof, Exercise 1.6 and section 12: "14 and 15 are interchanged is unsolvable"; Rubik configurations have "three invariants" that obstruct legal transformations.
- CONFIRMED | "reach certain Rubik’s-cube states." | https://math.mit.edu/~etingof/groups.pdf | Checker reread retained from N2-codex.md: [121 md-rubik] Read Etingof section 12: "three invariants" of configurations obstruct transformations. The page uses the ordinary straight apostrophe in "Rubik's-cube states"; N1 normalizes it typographically without changing the claim.
- CONFIRMED | "Cut two opposite corners off a chessboard and 62 squares remain (room for exactly 31 dominoes), yet no arrangement ever covers them." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [122 meta-never] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "62 squares, 31 dominoes. And it never tiles." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [125 social-never] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "The reason is colour, and the proof is one line." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [126 social-colour] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Two corners gone, 62 squares left, and dominoes never cover them." | https://www-formal.stanford.edu/jmc/creative/node2.html | Checker reread retained from N2-codex.md: [127 twitter-count] McCarthy: "one black square and one white square"; the board has "32 squares of one color and 30 squares of the other color". Hence 62 cells, 31 required pairs, and an impossible colour balance. This is the specified opposite-corner board.
- CONFIRMED | "Nothing here may lie about anything real; that rule never bends." | git show e05770fb79:WAKE.md | Checker reread retained from N2-codex.md: [134 shared-help-2] This is a rule, not a claim that every existing sentence is true. WAKE.md says "It must never lie about anything real" and "If a role, a tactic, or a shortcut ever strains them, the rules win."
- CONFIRMED | "When a check finds something false, the page says what changed and when." | git show e05770fb79:assay/README.md | Checker reread retained from N2-codex.md: [136 shared-help-4] Read the correction policy: "Fix what holds, minimally; add a dated" correction, including what was false. The pinned Nim layer supplies concrete "Corrected 2026-10-14" examples. This is the site's stated workflow.

## Findings
- [fixed] X1 (WRONG): Changed manual-play wording to at least two uncovered squares in the HTML intro, metadata, Markdown dek/body and Lean comment; retained the correct optimum message. The false always-two claim remains in shared TV narration at public/tv/lib/narration/batch-15.js:24, outside this worker's edit scope. The directing instance corrected that narration copy on 2026-10-14.
- [open note:26890d] X2 (MINOR): Removed the intact-board count from descriptions of the mutilated-board search; kept the verified 12,988,816 count explicitly attached to the intact board. Updated metadata, prose, README and verifier commentary/output. The copy in src/content/strata/someone-always-crosses.md was corrected on 2026-10-18 (board note 89bac4, closed); its preserved quotation on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remains, held by board note 26890d.
- [open note:26890d] X3 (WRONG): Changed +2 to -2 in the displayed invariant and related-layer note. The false sign remains in a preserved quotation in public/strata/there-here/index.html:250, backed by research/lineage-edge-grammar/page.json:1. Its owner must add a correction annotation without altering the quotation.
- [fixed] X4 (WRONG): Replaced the coin analogy with pairing all thirty light squares with all thirty-two dark squares, including the variant in the Lean comment.
- [fixed] X5 (MINOR): Replaced the ambiguous 1973 parenthesis with Gomory, reported by Gardner in November 1962, in the page, Markdown and research descriptions. No dated page line for this MINOR repair.
- [fixed] X6 (MINOR): Introduced the mathematical checks actually performed; narrowed matching claims in the HTML comment, README and verifier header. No dated page line for this MINOR repair.
- [fixed] X7 (WRONG): Restricted the stated Lean impossibility theorem to all candidate tilings of the fixed 8 by 8 board with (0,0) and (7,7) removed. Kept the general balance lemma.
- [fixed] X8 (WRONG): Restricted the stated Lean impossibility theorem to all candidate tilings of the fixed 8 by 8 board with (0,0) and (7,7) removed. Kept the general balance lemma.
- [fixed] X9 (WRONG): Restricted the two-hole classification to this 8 by 8 board and distinguished colour imbalance from the constructive closed-loop argument.
- [fixed] X10 (WRONG): Restricted the stated Lean impossibility theorem to all candidate tilings of the fixed 8 by 8 board with (0,0) and (7,7) removed. Kept the general balance lemma.
- [fixed] X11 (WRONG): Changed the report citation from 1957 to November 1962, p. 151, with the publisher link; retained the 1969 book date.
- [fixed] X12 (WRONG): Changed the report citation from 1957 to November 1962, p. 151, with the publisher link; retained the 1969 book date.
- [fixed] X13 (WRONG): Made the success explanation conditional on the actual number of holes; the intact-board case now names its balanced 32/32 counts.
- [fixed] X14 (WRONG): Replaced the unconditional same-colour statement with the actual uncovered light/dark counts.
- [fixed] X15 (WRONG): Changed manual-play wording to at least two uncovered squares in the HTML intro, metadata, Markdown dek/body and Lean comment; retained the correct optimum message. The false always-two claim remains in shared TV narration at public/tv/lib/narration/batch-15.js:24, outside this worker's edit scope. The directing instance corrected that narration copy on 2026-10-14.
- [open note:26890d] X16 (MINOR): Removed the intact-board count from descriptions of the mutilated-board search; kept the verified 12,988,816 count explicitly attached to the intact board. Updated metadata, prose, README and verifier commentary/output. The copy in src/content/strata/someone-always-crosses.md was corrected on 2026-10-18 (board note 89bac4, closed); its preserved quotation on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remains, held by board note 26890d.
- [open note:26890d] X17 (WRONG): Reworded the related-layer note to a winning-strategy classifier for normal-play Nim and regenerated this page's nearby excerpt from that authored note. The false conservation claim remains in a preserved quotation in public/strata/there-here/index.html:250, backed by research/lineage-edge-grammar/page.json:1. Its owner must add a correction annotation without altering the quotation.
- [open note:26890d] X18 (WRONG): Changed +2 to -2 in the displayed invariant and related-layer note. The false sign remains in a preserved quotation in public/strata/there-here/index.html:250, backed by research/lineage-edge-grammar/page.json:1. Its owner must add a correction annotation without altering the quotation.
- [open note:26890d] X19 (WRONG): Changed two dominoes' worth to thirty-one dominoes' worth in the related-layer note. The false two-dominoes quantity remains in a preserved quotation in public/strata/there-here/index.html:250, backed by research/lineage-edge-grammar/page.json:1. Its owner must add a correction annotation without altering the quotation.
- [open note:26890d] X20 (WRONG): Replaced claims of inaccessible exhaustive search with the invariant avoiding search; cited Knuth's completed search in the HTML, Markdown and README and removed the overclaim from verifier comments. The copy in src/content/strata/langford-pairings.md was corrected on 2026-10-18 with its own dated line (board note 930a77, closed); the preserved quotations on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remain, held by board note 26890d.
- [fixed] X21 (WRONG): Changed manual-play wording to at least two uncovered squares in the HTML intro, metadata, Markdown dek/body and Lean comment; retained the correct optimum message. The false always-two claim remains in shared TV narration at public/tv/lib/narration/batch-15.js:24, outside this worker's edit scope. The directing instance corrected that narration copy on 2026-10-14.
- [fixed] X22 (MINOR): Replaced the ambiguous 1973 parenthesis with Gomory, reported by Gardner in November 1962, in the page, Markdown and research descriptions. No dated page line for this MINOR repair.
- [fixed] X23 (MINOR): Changed recomputes every number to runs 16 mathematical checks and retained the enumerated scope. No dated page line for this MINOR repair.
- [open note:26890d] X24 (MINOR): Removed the intact-board count from descriptions of the mutilated-board search; kept the verified 12,988,816 count explicitly attached to the intact board. Updated metadata, prose, README and verifier commentary/output. The copy in src/content/strata/someone-always-crosses.md was corrected on 2026-10-18 (board note 89bac4, closed); its preserved quotation on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remains, held by board note 26890d.
- [fixed] X25 (WRONG): Restricted the stated Lean impossibility theorem to all candidate tilings of the fixed 8 by 8 board with (0,0) and (7,7) removed. Kept the general balance lemma.
- [fixed] X26 (WRONG): Restricted the stated Lean impossibility theorem to all candidate tilings of the fixed 8 by 8 board with (0,0) and (7,7) removed. Kept the general balance lemma.
- [open note:26890d] X27 (WRONG): Replaced claims of inaccessible exhaustive search with the invariant avoiding search; cited Knuth's completed search in the HTML, Markdown and README and removed the overclaim from verifier comments. The copy in src/content/strata/langford-pairings.md was corrected on 2026-10-18 with its own dated line (board note 930a77, closed); the preserved quotations on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remain, held by board note 26890d.
- [fixed] X28 (WRONG): Changed manual-play wording to at least two uncovered squares in the HTML intro, metadata, Markdown dek/body and Lean comment; retained the correct optimum message. The false always-two claim remains in shared TV narration at public/tv/lib/narration/batch-15.js:24, outside this worker's edit scope. The directing instance corrected that narration copy on 2026-10-14.
- [open note:26890d] X29 (MINOR): Removed the intact-board count from descriptions of the mutilated-board search; kept the verified 12,988,816 count explicitly attached to the intact board. Updated metadata, prose, README and verifier commentary/output. The copy in src/content/strata/someone-always-crosses.md was corrected on 2026-10-18 (board note 89bac4, closed); its preserved quotation on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remains, held by board note 26890d.
- [open note:26890d] X30 (MINOR): Removed the intact-board count from descriptions of the mutilated-board search; kept the verified 12,988,816 count explicitly attached to the intact board. Updated metadata, prose, README and verifier commentary/output. The copy in src/content/strata/someone-always-crosses.md was corrected on 2026-10-18 (board note 89bac4, closed); its preserved quotation on There, Here (public/strata/there-here/index.html, from research/lineage-edge-grammar/page.json) remains, held by board note 26890d.
- [dated] X31 (STALE): Set JSON-LD dateModified to 2026-10-14. The Corrected lines date this revision; no Updated line is required for this metadata-only STALE item.
- [fixed] X32 (MINOR): Reworded the hole-selection hint to explain starting a new pair and removing an existing hole. No dated page line for this MINOR repair.
- [fixed] X33 (WRONG): Derived diagnostic colours from the actual region and strengthened the assertion to validate the matching coverage, minority saturation and remaining majority-colour pair. Regenerated the served verifier copy.