---
slug: every-two-cards
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 and a Claude agent
checked: 159
confirmed: 124
wrong: 34
unverifiable: 0
observed: 0
stale: 1
---

This pass implements the directing triage for every-two-cards at 67088387e0, after the Codex fixer independently reread the evidence for all 35 triage items. It records those items separately, carries forward each remaining checker-confirmed source entry once with its checker attribution, and records one additional MINOR verifier-label correction. The triage takes precedence over earlier checker confirmations of the same wording. The wrong count includes MINOR claims; the sole STALE item is the JSON-LD date. Literal \u2014 and \n encode punctuation and line breaks in original claim quotations.

## Claims

- WRONG | X1 (html-deck-is-plane): It is a theorem: the deck is a finite projective plane. | https://research.ibm.com/blog/ponder-this-may-2024 | IBM distinguishes the 55-card retail deck from a complete 57-card order-7 plane. The retained cards keep their pairwise intersection, but the retail subset is not itself the full plane.
- WRONG | X2 (html-one-defining-rule): A projective plane is a world of points and lines with one defining rule: | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- MINOR | X3 (html-law-equals-definition): The game's only law and the plane's defining axiom are the same sentence. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- MINOR | X4 (html-counters-naming): The first counter is named for the human eye; the second, for the limit of the proven. | git show 67088387e0:public/strata/every-two-cards/index.html | The scorebar order is found, misses, best time, then pairs dealt with a shared-symbol count other than one. The named mathematical counter is fourth, not second.
- MINOR | X5 (html-every-number-live): Every number on this page is recomputed live by the same arithmetic | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck counts the built cards and checks only the displayed matrix; deal checks the dealt pair, and brcRun performs the selected arithmetic test. Retail 55, historical dates, and the recorded 51/51 are literal text, not browser recomputations.
- WRONG | X6 (citation-bruck-ryser-chowla): Bruck–Ryser–Chowla (1949) | https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/combinatorial-problems/CAEFF0578570DF77C7595FFACBEF55A1; https://people.math.harvard.edu/~elkies/M155.15/notes.html | The Cambridge record identifies Chowla and Ryser, Combinatorial Problems, 1950, pp. 93-99, and cites Bruck and Ryser, 1949. Elkies distinguishes the 1949 plane result from the 1950 generalization. The Crossref URL failed in this reread; the publisher record and original 1949 paper independently settle the date.
- WRONG | X7 (md-one-rule): A finite projective plane is a world of points and lines obeying one rule: any two lines meet in exactly one point. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- MINOR | X8 (md-defining-axiom): The game's whole law and the plane's defining axiom are the same sentence \u2014 so building a projective plane builds a perfect Dobble deck for free. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- WRONG | X9 (md-whole-deck-matrix): Then watch the whole deck get built mod q and the all-pairs shared-count matrix turn entirely to 1s | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses m=Math.min(deck.length, 10) for both matrix loops; diagonal cells contain dots. The complete deck is constructed and the offline verifier covers all pairs, but the visible matrix covers at most ten cards.
- MINOR | X10 (md-prime-power-exactly-as-above): A plane of order n exists for every prime power \u2014 built exactly as above. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534; git show 67088387e0:memory/log.d/2026-06-21T1845Z-every-two-cards.md | The page recipe uses integer arithmetic modulo a prime. Bruck and Ryser, p. 88, describe prime-power constructions using Galois fields; the author log also distinguishes GF(p^k). A composite prime power needs its finite-field operations, rather than ordinary arithmetic modulo that number.
- MINOR | X11 (md-every-number-live): Every number here is recomputed in the browser by the same arithmetic | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck counts the built cards and checks only the displayed matrix; deal checks the dealt pair, and brcRun performs the selected arithmetic test. Retail 55, historical dates, and the recorded 51/51 are literal text, not browser recomputations.
- MINOR | X12 (dek-defining-axiom): Call each card a line and each symbol a point, and the game's only rule becomes the defining axiom of a finite projective plane: any two lines meet in exactly one point. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- WRONG | X13 (dek-whole-deck-matrix): watch the whole deck get built mod q and the all-pairs shared-count matrix turn entirely to 1s | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses m=Math.min(deck.length, 10) for both matrix loops; diagonal cells contain dots. The complete deck is constructed and the offline verifier covers all pairs, but the visible matrix covers at most ten cards.
- WRONG | X14 (meta-deck-is-plane): The card game Dobble / Spot It! works because the deck is a finite projective plane: any two cards share exactly one symbol. | https://research.ibm.com/blog/ponder-this-may-2024 | IBM distinguishes the 55-card retail deck from a complete 57-card order-7 plane. The retained cards keep their pairwise intersection, but the retail subset is not itself the full plane.
- WRONG | X15 (related-two-card-games): Two card games, both built on a finite combinatorial design where structure is forced, not chosen. | git show 67088387e0:src/content/strata/perfect-codes.md | The pinned perfect-codes frontmatter describes an error-correcting code, bit flips, Hamming parity checks and three coding-theory instruments. Calling that linked page another card game misidentifies it.
- WRONG | X16 (related-fano-hamming): The order-2 plane (the seven-point Fano plane this page builds first) and the Hamming(7,4) code are two faces of the same object: seven points, seven lines, the same incidence written once as a parity-check matrix and once as a deck. | git show 67088387e0:public/strata/every-two-cards/index.html | The selector marks order 3 selected; initialization calls rebuild, which reads that selection and constructs 13 cards. Order 2, the seven-card Fano plane, is an available option. The mathematical Fano/Hamming relationship is retained.
- WRONG | X17 (built-full-deck-5): showing the full deck for order 5 \u2014 31 cards, each a list of its symbol indices | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses show=Math.min(deck.length, 13). Order 5 constructs 31 cards and shows 13 plus a +18 tile; order 7 constructs 57 and shows 13 plus a +44 tile. The caption must distinguish displayed and total counts.
- WRONG | X18 (built-full-deck-7): showing the full deck for order 7 \u2014 57 cards, each a list of its symbol indices | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses show=Math.min(deck.length, 13). Order 5 constructs 31 cards and shows 13 plus a +18 tile; order 7 constructs 57 and shows 13 plus a +44 tile. The caption must distinguish displayed and total counts.
- WRONG | X19 (related-fano-built-first): the seven-point Fano plane this page builds first | git show 67088387e0:public/strata/every-two-cards/index.html | The selector marks order 3 selected; initialization calls rebuild, which reads that selection and constructs 13 cards. Order 2, the seven-card Fano plane, is an available option. The mathematical Fano/Hamming relationship is retained.
- WRONG | K1 (html-one-defining-rule): A projective plane is a world of points and lines with one defining rule: | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- MINOR | K2 (html-primepower-list): 2,3,4,5,7,8,9,11,13…yesevery prime power (dash) built from a finite field, exactly as above | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | The page recipe uses integer arithmetic modulo a prime. Bruck and Ryser, p. 88, describe prime-power constructions using Galois fields; the author log also distinguishes GF(p^k). A composite prime power needs its finite-field operations, rather than ordinary arithmetic modulo that number.
- MINOR | K3 (html-counters-naming): The first counter is named for the human eye; the second, for the limit of the proven. | git show 67088387e0:public/strata/every-two-cards/index.html | The scorebar order is found, misses, best time, then pairs dealt with a shared-symbol count other than one. The named mathematical counter is fourth, not second.
- MINOR | K4 (html-every-number-live): Every number on this page is recomputed live by the same arithmetic | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck counts the built cards and checks only the displayed matrix; deal checks the dealt pair, and brcRun performs the selected arithmetic test. Retail 55, historical dates, and the recorded 51/51 are literal text, not browser recomputations.
- WRONG | K5 (citation-bruck-ryser-chowla): Bruck–Ryser–Chowla (1949) | https://api.crossref.org/works/10.4153/CJM-1950-009-8; https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/combinatorial-problems/CAEFF0578570DF77C7595FFACBEF55A1; https://people.math.harvard.edu/~elkies/M155.15/notes.html | The Cambridge record identifies Chowla and Ryser, Combinatorial Problems, 1950, pp. 93-99, and cites Bruck and Ryser, 1949. Elkies distinguishes the 1949 plane result from the 1950 generalization. The Crossref URL failed in this reread; the publisher record and original 1949 paper independently settle the date.
- WRONG | K6 (md-one-rule): A finite projective plane is a world of points and lines obeying one rule: any two lines meet in exactly one point. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Lam, Thiel and Swiercz, p. 1117, require both pairwise incidence axioms and regular point/line counts. Pairwise line intersection alone is insufficient: {a,b}, {a,c}, {a,d} meet once pairwise, but no line contains b and c.
- WRONG | K7 (md-whole-deck-matrix): Then watch the whole deck get built mod q and the all-pairs shared-count matrix turn entirely to 1s | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses m=Math.min(deck.length, 10) for both matrix loops; diagonal cells contain dots. The complete deck is constructed and the offline verifier covers all pairs, but the visible matrix covers at most ten cards.
- MINOR | K8 (md-prime-power-exactly-as-above): A plane of order n exists for every prime power (dash) built exactly as above. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | The page recipe uses integer arithmetic modulo a prime. Bruck and Ryser, p. 88, describe prime-power constructions using Galois fields; the author log also distinguishes GF(p^k). A composite prime power needs its finite-field operations, rather than ordinary arithmetic modulo that number.
- MINOR | K9 (md-every-number-live): Every number here is recomputed in the browser by the same arithmetic | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck counts the built cards and checks only the displayed matrix; deal checks the dealt pair, and brcRun performs the selected arithmetic test. Retail 55, historical dates, and the recorded 51/51 are literal text, not browser recomputations.
- WRONG | K10 (dek-whole-deck-matrix): watch the whole deck get built mod q and the all-pairs shared-count matrix turn entirely to 1s | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses m=Math.min(deck.length, 10) for both matrix loops; diagonal cells contain dots. The complete deck is constructed and the offline verifier covers all pairs, but the visible matrix covers at most ten cards.
- STALE | K11 (metadata-modified): "dateModified":"2026-06-21" | git log 67088387e0 -- public/strata/every-two-cards/index.html | The pinned JSON-LD says 2026-06-21 although git history includes description changes on 2026-08-16 and image-alt changes on 2026-10-01. This pass changes the content on 2026-10-14, so dateModified is refreshed to that date.
- WRONG | K12 (related-two-card-games): Two card games, both built on a finite combinatorial design where structure is forced, not chosen. | git show 67088387e0:src/content/strata/perfect-codes.md | The pinned perfect-codes frontmatter describes an error-correcting code, bit flips, Hamming parity checks and three coding-theory instruments. Calling that linked page another card game misidentifies it.
- CONFIRMED | K13 (nearby-capset): Each page makes finite geometry from arithmetic modulo a prime and checks the whole constructed object. | git show 67088387e0:public/strata/every-two-cards/index.html; research/dobble-projective-plane/verify.mjs; git show 67088387e0:public/strata/cap-set-polynomial/index.html | REJECTED with evidence: the nearby sentence does not say all checks run live. verify.mjs exhausts all card pairs and all symbol pairs at orders 2,3,5,7; cap-set-polynomial ambientAudit exhausts ambient triples and supportData checks all selected-point pairs. The limited visible matrix does not refute the broader check claim.
- WRONG | K14 (built-full-deck-5): showing the full deck for order 5 (dash) 31 cards, each a list of its symbol indices | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses show=Math.min(deck.length, 13). Order 5 constructs 31 cards and shows 13 plus a +18 tile; order 7 constructs 57 and shows 13 plus a +44 tile. The caption must distinguish displayed and total counts.
- WRONG | K15 (built-full-deck-7): showing the full deck for order 7 (dash) 57 cards, each a list of its symbol indices | git show 67088387e0:public/strata/every-two-cards/index.html | renderDeck uses show=Math.min(deck.length, 13). Order 5 constructs 31 cards and shows 13 plus a +18 tile; order 7 constructs 57 and shows 13 plus a +44 tile. The caption must distinguish displayed and total counts.
- WRONG | K16 (related-fano-built-first): the seven-point Fano plane this page builds first | git show 67088387e0:public/strata/every-two-cards/index.html | The selector marks order 3 selected; initialization calls rebuild, which reads that selection and constructs 13 cards. Order 2, the seven-card Fano plane, is an available option. The mathematical Fano/Hamming relationship is retained.
- CONFIRMED | title: Every Two Cards Share a Symbol | https://www.asmodee.co.uk/pages/games/dobble | Codex checker, N2-codex.md: Publisher: "any two cards in the deck will always have one and only one matching symbol." This is exactly the pairwise game rule, including after cards are removed.
- CONFIRMED | html-game-name: In the card game Dobble (Spot It!), | https://www.zygomatic-games.com/fr/contact/ | Codex checker, N2-codex.md: The publisher FAQ says "Dobble et Spot it! sont le même jeu, commercialisé sous des noms différents selon les pays." Opened with curl despite the checker labelling the page UNREADABLE.
- CONFIRMED | html-match: In the card game Dobble (Spot It!), any two cards you ever lay down share exactly one picture \u2014 never zero, never two. | https://www.asmodee.co.uk/pages/games/dobble | Codex checker, N2-codex.md: Publisher: "any two cards in the deck will always have one and only one matching symbol." This is exactly the pairwise game rule, including after cards are removed.
- CONFIRMED | html-no-hand-design: That is not careful design or luck. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM gives a mathematical construction and explicitly identifies a full Dobble-like deck with a projective plane: "A Dobble-like deck corresponding to a parameter NNN is equivalent to a **projective plane** of order N." Its next sentence identifies points with symbols and lines with cards. The claim about avoiding hand-tuning follows from that construction.
- CONFIRMED | html-translation: Think of each card as a line and each symbol as a point. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM gives a mathematical construction and explicitly identifies a full Dobble-like deck with a projective plane: "A Dobble-like deck corresponding to a parameter NNN is equivalent to a **projective plane** of order N." Its next sentence identifies points with symbols and lines with cards. The claim about avoiding hand-tuning follows from that construction.
- CONFIRMED | html-axiom: any two lines meet in exactly one point. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "any two distinct lines intersect at exactly one point". Under card=line and symbol=point this is precisely the asserted shared-symbol property. Distinct cards/lines are understood.
- CONFIRMED | html-game-law: Translate it back: any two cards share exactly one symbol. | https://www.asmodee.co.uk/pages/games/dobble | Codex checker, N2-codex.md: Publisher: "any two cards in the deck will always have one and only one matching symbol." This is exactly the pairwise game rule, including after cards are removed.
- CONFIRMED | html-construction-gives-deck: So if we can build a projective plane, we get a perfect Dobble deck for free \u2014 and we can, out of arithmetic over a finite field. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies defines points as "nonzero vectors modulo scaling" and lines as "2-dimensional subspaces" of a three-dimensional vector space. The dual-coordinate passage gives each line a nonzero coefficient triple. Over a prime field the operations are precisely arithmetic modulo q. PG(2,q) is the textbook notation, also explicitly mentioned by Elkies.
- CONFIRMED | html-coordinates: Every symbol is a direction in a tiny 3-D space over the numbers {0,1,…,q−1} with arithmetic done mod 3 \u2014 a triple [a,b,c], where scaling the whole triple by a nonzero number gives the same symbol (these are homogeneous coordinates). | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies: coordinates are "not all zero" and identify proportional vectors, with a multiplier in the field's nonzero elements. The page says "direction", so the zero triple is excluded; its selected q=3 is prime. The code updates this number when q changes.
- CONFIRMED | html-line-vector: Every card is also such a triple [L], read as a recipe. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies defines points as "nonzero vectors modulo scaling" and lines as "2-dimensional subspaces" of a three-dimensional vector space. The dual-coordinate passage gives each line a nonzero coefficient triple. Over a prime field the operations are precisely arithmetic modulo q. PG(2,q) is the textbook notation, also explicitly mentioned by Elkies.
- CONFIRMED | html-incidence: A symbol [x] sits on a card [L] exactly when (L₀·x₀ + L₁·x₁ + L₂·x₂) mod q = 0. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies' dual-coordinate passage says "the hyperplane x* contains the point x iff x* x = 0" and expands the usual inner product. In dimension three this is exactly the displayed dot-product incidence condition; two independent equations leave one projective direction.
- CONFIRMED | html-entire-incidence-rule: That dot-product being zero is the entire incidence rule. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies' dual-coordinate passage says "the hyperplane x* contains the point x iff x* x = 0" and expands the usual inner product. In dimension three this is exactly the displayed dot-product incidence condition; two independent equations leave one projective direction.
- CONFIRMED | html-common-solution: Now the magic: two different cards [L] and [M] share the symbols [x] with both L·x=0 and M·x=0 \u2014 and two independent linear equations in this projective space have exactly one common solution direction. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies: "any two distinct 2-dim. spaces W,W' intersect in a 1-dim. space"; the following dimension formula gives 2+2-3=1. Nonzero vectors in that intersection are one direction, hence one symbol.
- CONFIRMED | html-forced-symbol: Exactly one shared symbol, forced by linear algebra. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies: "any two distinct 2-dim. spaces W,W' intersect in a 1-dim. space"; the following dimension formula gives 2+2-3=1. Nonzero vectors in that intersection are one direction, hence one symbol.
- CONFIRMED | html-self-dual: Swap the roles of point and line and the same sentence holds \u2014 the plane is self-dual, so any two symbols also share exactly one card. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies: "the projective plane obtained from V* is the combinatorial dual of the one obtained from V." The constructed PG(2,q) is self-dual. The universal claim here concerns this field plane, not all abstract projective planes.
- CONFIRMED | html-plane-counts: For a plane of order n the arithmetic forces the counts exactly: n²+n+1 symbols, the same number of cards, and n+1 symbols on each card. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Elkies derives "(q^3-1) / (q-1) = q^2+q+1" and "(q^2-1)/(q-1) = q+1" for field planes. For the claim about every plane, Lam et al., p. 1117, explicitly define n^2+n+1 points and lines and n+1 incidences at each. Substitution gives 7,13,31,57 and 3,4,6,8.
- CONFIRMED | html-retail-order7: Retail Dobble shows 8 symbols per card, so its order is 7: a full deck of 7²+7+1 = 57 cards drawn from 57 symbols. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM: "each card contains eight symbols"; "A standard Dobble deck consists of 55 cards and 57 symbols"; and "it is possible to have a deck of 57 cards and 57 symbols." Arithmetic: 8-1=7 and 7^2+7+1=57. The publisher independently gives 55 cards, 57 symbols, and the one-match rule.
- CONFIRMED | html-production: The box ships 55 \u2014 two cards dropped, a manufacturing choice, not a mathematical one | https://forum.trictrac.net/t/dobble-et-la-geometrie-finie/95005 | Codex checker, N2-codex.md: The forum post of 29 August 2011 reproduces an email attributed to the publisher's project manager: "Ce sont des raisons pratiques de production qui nous ont amenés à imprimer 55 cartes et non 57." It also says every box contains the same 55 cards. This is a reported publisher email, not an independently archived publisher statement; no specific sheet-size explanation is claimed.
- CONFIRMED | html-removal-preserves-match: removing cards can never create a second match | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Deleting some cards leaves the symbol sets on every retained pair unchanged. Therefore their intersection stays exactly one. This follows directly from Lam's line-intersection axiom; the verifier also checks a 55-card subset.
- CONFIRMED | html-order6: 6noBruck–Ryser (1949): 6 ≡ 2 (mod 4) and 6 is not a sum of two squares ⇒ impossible | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | Codex checker, N2-codex.md: Opened the 1949 PDF and read Theorem 1, p. 88. It excludes orders congruent to 1 or 2 modulo 4 whose square-free part has a prime factor 3 modulo 4. Thus 6=2*3, 14=2*7, and 21=3*7 are excluded. Direct enumeration of squares gives no representation for these orders. Text extraction produces OCR slips in congruence symbols, not a source discrepancy.
- CONFIRMED | html-order10: 10noBruck–Ryser is silent (10 = 1²+3²); eliminated only by exhaustive computer search \u2014 Lam, Thiel & Swiercz, 1989, thousands of CPU-hours | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam, Thiel and Swiercz (1989), p. 1117: the computer search "when taken together with previous results, implies that a plane of order 10 does not exist." The same introduction says this shows the Bruck-Ryser necessary condition is not sufficient. 10=1^2+3^2. The paper describes the exhaustive remaining cases, not a theorem from BRC.
- CONFIRMED | html-cpu-hours: thousands of CPU-hours | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1119: "CRAY-1A and took an estimated 2,000 hours of computing"; the preceding paragraph additionally reports "800 days of CPU time on a VAX-11/780." Both the thousands-of-hours and supercomputer claims are supported.
- CONFIRMED | html-order12: 12OPENnobody on Earth knows. Not built, not forbidden. The smallest undecided order. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://arxiv.org/html/2510.19804v1#S4 | Codex checker, N2-codex.md: Elkies: "the first open case is n=12". Also checked the later primary paper Alexeev and Mixon (2025), section 4: "Order 12 is the smallest for which it is unknown whether a projective plane exists." The statements about making no claim are read as making no existence/nonexistence claim.
- CONFIRMED | html-open-predates-game: A children's matching game sits on top of a question \u2014 does a projective plane of order 12 exist? \u2014 that has been open since before the game was invented. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://api.mountainscholar.org/server/api/core/bitstreams/5e6d4e7f-8c45-4a48-81e9-a60d1bf03fd4/content; https://cdn.svc.asmodee.net/production-asmodeees/uploads/2023/06/Dobble123_ES_Rules.pdf | Codex checker, N2-codex.md: The source-record excerpt establishes open status but not historical precedence. Supplement: Cooper's 2015 dissertation, p. 67, describes order-12 investigations "over the period 1973-2009". Publisher rules trace the precursor to "En 1976, Jacques Cottereau" and modern Dobble to 2008/2009. Thus this question predates even the precursor. "Outlived" is read in its contextual sense of being older, not a claim that the still-sold game has ceased.
- CONFIRMED | html-supercomputer: it took a supercomputer. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1119: "CRAY-1A and took an estimated 2,000 hours of computing"; the preceding paragraph additionally reports "800 days of CPU time on a VAX-11/780." Both the thousands-of-hours and supercomputer claims are supported.
- CONFIRMED | html-order12-open: Order 12 is genuinely open; we make no claim about it. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://arxiv.org/html/2510.19804v1#S4 | Codex checker, N2-codex.md: Elkies: "the first open case is n=12". Also checked the later primary paper Alexeev and Mixon (2025), section 4: "Order 12 is the smallest for which it is unknown whether a projective plane exists." The statements about making no claim are read as making no existence/nonexistence claim.
- CONFIRMED | html-pg-notation: Projective plane construction: PG(2,q) over the field F_q. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies defines points as "nonzero vectors modulo scaling" and lines as "2-dimensional subspaces" of a three-dimensional vector space. The dual-coordinate passage gives each line a nonzero coefficient triple. Over a prime field the operations are precisely arithmetic modulo q. PG(2,q) is the textbook notation, also explicitly mentioned by Elkies.
- CONFIRMED | html-folklore: The Dobble↔projective-plane identification is folklore in recreational mathematics | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM gives a mathematical construction and explicitly identifies a full Dobble-like deck with a projective plane: "A Dobble-like deck corresponding to a parameter NNN is equivalent to a **projective plane** of order N." Its next sentence identifies points with symbols and lines with cards. The claim about avoiding hand-tuning follows from that construction.
- CONFIRMED | citation-lam-footnote: Lam–Thiel–Swiercz (1989) | https://geodesic.mathdoc.fr/articles/10.4153/CJM-1989-049-4/ | Codex checker, N2-codex.md: Mathdoc: "Lam, C. W. H. ; Thiel, L. ; Swiercz, S."; "Tome 41 (1989) no. 6, pp. 1117-1123"; DOI 10.4153/CJM-1989-049-4. Opened the corresponding primary PDF too.
- CONFIRMED | html-trademark: Spot It!/Dobble is a trademark of its publisher | https://cdn.svc.asmodee.net/production-unboxnownl/uploads/2022/03/dobble_demokit_nl.pdf | Codex checker, N2-codex.md: Opened the publisher PDF. It repeatedly prints "Spot it!, Dobble, and Dobble characters are trademarks of Asmodee Group". This supports the page's ownership wording.
- CONFIRMED | md-game: In the matching game Dobble (sold as Spot It!), every two cards share exactly one picture. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM: "each card contains eight symbols"; "A standard Dobble deck consists of 55 cards and 57 symbols"; and "it is possible to have a deck of 57 cards and 57 symbols." Arithmetic: 8-1=7 and 7^2+7+1=57. The publisher independently gives 55 cards, 57 symbols, and the one-match rule.
- CONFIRMED | md-match: Lay down any pair from the deck and there it is \u2014 one common symbol, never zero, never two. | https://www.asmodee.co.uk/pages/games/dobble | Codex checker, N2-codex.md: Publisher: "any two cards in the deck will always have one and only one matching symbol." This is exactly the pairwise game rule, including after cards are removed.
- CONFIRMED | md-no-hand-check: They did not need to. It is a theorem, and the deck is its proof. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM gives a mathematical construction and explicitly identifies a full Dobble-like deck with a projective plane: "A Dobble-like deck corresponding to a parameter NNN is equivalent to a **projective plane** of order N." Its next sentence identifies points with symbols and lines with cards. The claim about avoiding hand-tuning follows from that construction.
- CONFIRMED | md-card-point: Call each card a line and each symbol a point. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM gives a mathematical construction and explicitly identifies a full Dobble-like deck with a projective plane: "A Dobble-like deck corresponding to a parameter NNN is equivalent to a **projective plane** of order N." Its next sentence identifies points with symbols and lines with cards. The claim about avoiding hand-tuning follows from that construction.
- CONFIRMED | md-translated-law: Read it back through the translation and it says: any two cards share exactly one symbol. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "any two distinct lines intersect at exactly one point". Under card=line and symbol=point this is precisely the asserted shared-symbol property. Distinct cards/lines are understood.
- CONFIRMED | md-mod-prime-construction: Pick a prime *q*. Every symbol is a\ndirection `[a,b,c]` in a tiny three-dimensional space over the numbers\n`{0,…,q−1}` taken mod *q*; every card is such a triple `[L]` read as a recipe; a\nsymbol sits on a card exactly when `L₀x₀ + L₁x₁ + L₂x₂ ≡ 0 (mod q)`. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies defines points as "nonzero vectors modulo scaling" and lines as "2-dimensional subspaces" of a three-dimensional vector space. The dual-coordinate passage gives each line a nonzero coefficient triple. Over a prime field the operations are precisely arithmetic modulo q. PG(2,q) is the textbook notation, also explicitly mentioned by Elkies.
- CONFIRMED | md-two-equations: Two different cards share the symbols that solve both of their equations at once \u2014 and two independent linear equations in this projective space have exactly one common solution direction. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies: "any two distinct 2-dim. spaces W,W' intersect in a 1-dim. space"; the following dimension formula gives 2+2-3=1. Nonzero vectors in that intersection are one direction, hence one symbol.
- CONFIRMED | md-no-hand-tuning: One shared symbol, forced by linear algebra, with no hand-tuning anywhere. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies: "any two distinct 2-dim. spaces W,W' intersect in a 1-dim. space"; the following dimension formula gives 2+2-3=1. Nonzero vectors in that intersection are one direction, hence one symbol.
- CONFIRMED | md-retail-counts: retail Dobble shows 8 symbols per card: that means order 7, a plane of 57 cards drawn from 57 symbols, of which the box ships 55 | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM: "each card contains eight symbols"; "A standard Dobble deck consists of 55 cards and 57 symbols"; and "it is possible to have a deck of 57 cards and 57 symbols." Arithmetic: 8-1=7 and 7^2+7+1=57. The publisher independently gives 55 cards, 57 symbols, and the one-match rule.
- CONFIRMED | md-printing-choice: two dropped \u2014 a printing choice, not a mathematical one | https://forum.trictrac.net/t/dobble-et-la-geometrie-finie/95005 | Codex checker, N2-codex.md: The forum post of 29 August 2011 reproduces an email attributed to the publisher's project manager: "Ce sont des raisons pratiques de production qui nous ont amenés à imprimer 55 cartes et non 57." It also says every box contains the same 55 cards. This is a reported publisher email, not an independently archived publisher statement; no specific sheet-size explanation is claimed.
- CONFIRMED | md-six: But order 6 is impossible (Bruck–Ryser, 1949: 6 ≡ 2 mod 4 and is not a sum of two squares) | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | Codex checker, N2-codex.md: Opened the 1949 PDF and read Theorem 1, p. 88. It excludes orders congruent to 1 or 2 modulo 4 whose square-free part has a prime factor 3 modulo 4. Thus 6=2*3, 14=2*7, and 21=3*7 are excluded. Direct enumeration of squares gives no representation for these orders. Text extraction produces OCR slips in congruence symbols, not a source discrepancy.
- CONFIRMED | md-ten: order 10 is also impossible, but Bruck–Ryser is silent on it and its nonexistence fell only to an exhaustive computer search (Lam, Thiel & Swiercz, 1989) | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam, Thiel and Swiercz (1989), p. 1117: the computer search "when taken together with previous results, implies that a plane of order 10 does not exist." The same introduction says this shows the Bruck-Ryser necessary condition is not sufficient. 10=1^2+3^2. The paper describes the exhaustive remaining cases, not a theorem from BRC.
- CONFIRMED | md-twelve: and order 12 is open \u2014 nobody on Earth knows whether it exists. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://arxiv.org/html/2510.19804v1#S4 | Codex checker, N2-codex.md: Elkies: "the first open case is n=12". Also checked the later primary paper Alexeev and Mixon (2025), section 4: "Order 12 is the smallest for which it is unknown whether a projective plane exists." The statements about making no claim are read as making no existence/nonexistence claim.
- CONFIRMED | md-question-outlived-game: A children's matching game sits on top of a question that has outlived the game. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://api.mountainscholar.org/server/api/core/bitstreams/5e6d4e7f-8c45-4a48-81e9-a60d1bf03fd4/content; https://cdn.svc.asmodee.net/production-asmodeees/uploads/2023/06/Dobble123_ES_Rules.pdf | Codex checker, N2-codex.md: The source-record excerpt establishes open status but not historical precedence. Supplement: Cooper's 2015 dissertation, p. 67, describes order-12 investigations "over the period 1973-2009". Publisher rules trace the precursor to "En 1976, Jacques Cottereau" and modern Dobble to 2008/2009. Thus this question predates even the precursor. "Outlived" is read in its contextual sense of being older, not a claim that the still-sold game has ceased.
- CONFIRMED | md-supercomputer: it took a supercomputer | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1119: "CRAY-1A and took an estimated 2,000 hours of computing"; the preceding paragraph additionally reports "800 days of CPU time on a VAX-11/780." Both the thousands-of-hours and supercomputer claims are supported.
- CONFIRMED | md-order12-no-one-can: and order 12 we make no claim about, because no one can. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://arxiv.org/html/2510.19804v1#S4 | Codex checker, N2-codex.md: Elkies: "the first open case is n=12". Also checked the later primary paper Alexeev and Mixon (2025), section 4: "Order 12 is the smallest for which it is unknown whether a projective plane exists." The statements about making no claim are read as making no existence/nonexistence claim.
- CONFIRMED | dek-dobble-law: In the matching game Dobble (Spot It!), any two cards you ever lay down share exactly one picture (never none, never two), and that is not careful hand-tuning, it is a theorem. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM: "each card contains eight symbols"; "A standard Dobble deck consists of 55 cards and 57 symbols"; and "it is possible to have a deck of 57 cards and 57 symbols." Arithmetic: 8-1=7 and 7^2+7+1=57. The publisher independently gives 55 cards, 57 symbols, and the one-match rule.
- CONFIRMED | dek-arithmetic: So a perfect Dobble deck can be built out of nothing but arithmetic: every symbol a direction [a,b,c] in a tiny space over the numbers mod a prime q, every card a recipe L, a symbol on a card exactly when L·x ≡ 0; two cards then share the symbols solving two linear equations at once, of which a projective plane has exactly one. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies' dual-coordinate passage says "the hyperplane x* contains the point x iff x* x = 0" and expands the usual inner product. In dimension three this is exactly the displayed dot-product incidence condition; two independent equations leave one projective direction.
- CONFIRMED | dek-retail: retail Dobble has 8 symbols per card: order 7, a 57-card plane shipping 55. | https://research.ibm.com/blog/ponder-this-may-2024 | Codex checker, N2-codex.md: IBM: "each card contains eight symbols"; "A standard Dobble deck consists of 55 cards and 57 symbols"; and "it is possible to have a deck of 57 cards and 57 symbols." Arithmetic: 8-1=7 and 7^2+7+1=57. The publisher independently gives 55 cards, 57 symbols, and the one-match rule.
- CONFIRMED | dek-prime-powers: planes exist for every prime-power order | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | dek-order6: order 6 is forbidden by Bruck–Ryser (1949) | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | Codex checker, N2-codex.md: Opened the 1949 PDF and read Theorem 1, p. 88. It excludes orders congruent to 1 or 2 modulo 4 whose square-free part has a prime factor 3 modulo 4. Thus 6=2*3, 14=2*7, and 21=3*7 are excluded. Direct enumeration of squares gives no representation for these orders. Text extraction produces OCR slips in congruence symbols, not a source discrepancy.
- CONFIRMED | dek-order10: order 10 fell only to an exhaustive computer search (Lam–Thiel–Swiercz, 1989) | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam, Thiel and Swiercz (1989), p. 1117: the computer search "when taken together with previous results, implies that a plane of order 10 does not exist." The same introduction says this shows the Bruck-Ryser necessary condition is not sufficient. 10=1^2+3^2. The paper describes the exhaustive remaining cases, not a theorem from BRC.
- CONFIRMED | dek-order12: order 12 is OPEN: a children's card game sitting on top of a question nobody on Earth has answered. | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://arxiv.org/html/2510.19804v1#S4 | Codex checker, N2-codex.md: Elkies: "the first open case is n=12". Also checked the later primary paper Alexeev and Mixon (2025), section 4: "Order 12 is the smallest for which it is unknown whether a projective plane exists." The statements about making no claim are read as making no existence/nonexistence claim.
- CONFIRMED | related-perfect-code: There the codewords are spheres packed so tightly into the cube of all messages that a flipped bit always snaps back home \u2014 a perfect Hamming code. | https://math.mit.edu/~goemans/18310S15/Hamming-code-notes.pdf | Codex checker, N2-codex.md: MIT Hamming notes, section 3: "Every binary Hamming code is a perfect code" and the space is partitioned into decoding spheres. A one-bit error stays in the unique radius-one sphere of its codeword. The page's sphere language describes these correction neighborhoods.
- CONFIRMED | related-plane-meeting: Here the cards are the lines of a projective plane packed so that any two meet in exactly one symbol. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "any two distinct lines intersect at exactly one point". Under card=line and symbol=point this is precisely the asserted shared-symbol property. Distinct cards/lines are understood.
- CONFIRMED | related-parity: The sliding-tile puzzle is governed by one parity bit no move can flip, so half its arrangements can never be reached | https://personal.math.ubc.ca/~cass/courses/m308-02b/projects/grant/fifteen.html | Codex checker, N2-codex.md: The UBC account says "only half the possible arrangements are solvable" because of the parity class. Read the linked puzzle code as well: it uses permutation parity combined with the gap's checkerboard color, so a legal move preserves the combined bit.
- CONFIRMED | related-shuffle-law: Dobble is governed by an incidence rule no shuffle can break, so every pair of cards must share exactly one symbol. | https://www.asmodee.co.uk/pages/games/dobble | Codex checker, N2-codex.md: Publisher: "any two cards in the deck will always have one and only one matching symbol." This is exactly the pairwise game rule, including after cards are removed.
- CONFIRMED | related-ramsey: Ramsey's theorem says you cannot arrange six people to avoid a monochromatic triangle, however hard you try | https://math.mit.edu/research/highschool/primes/materials/2013/conf/1-3-Gandhi-Golowich.pdf | Codex checker, N2-codex.md: The cited MIT talk, slide 2, states: "Given 6 people, any 2 of whom are either friends or enemies" and guarantees three all mutual friends or all mutual enemies. This supports R(3,3)=6 in the intended two-color relation model.
- CONFIRMED | related-no-escape-meeting: a projective plane says you cannot lay down two of its cards without exactly one shared symbol, however you shuffle. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "any two distinct lines intersect at exactly one point". Under card=line and symbol=point this is precisely the asserted shared-symbol property. Distinct cards/lines are understood.
- CONFIRMED | citation-bruck-html: Bruck–Ryser (1949) | https://doi.org/10.4153/cjm-1949-009-2 | Codex checker, N2-codex.md: The DOI resolves to The Nonexistence of Certain Finite Projective Planes by "R. H. Bruck" and "H. J. Ryser", published "01 February 1949 , pp. 88 - 93". The online upload date in 2018 is not the paper's publication year.
- CONFIRMED | citation-bruck-md: Bruck–Ryser, 1949 | https://doi.org/10.4153/cjm-1949-009-2 | Codex checker, N2-codex.md: The DOI resolves to The Nonexistence of Certain Finite Projective Planes by "R. H. Bruck" and "H. J. Ryser", published "01 February 1949 , pp. 88 - 93". The online upload date in 2018 is not the paper's publication year.
- CONFIRMED | citation-lam-md: Lam, Thiel & Swiercz, 1989 | https://geodesic.mathdoc.fr/articles/10.4153/CJM-1989-049-4/ | Codex checker, N2-codex.md: Mathdoc: "Lam, C. W. H. ; Thiel, L. ; Swiercz, S."; "Tome 41 (1989) no. 6, pp. 1117-1123"; DOI 10.4153/CJM-1989-049-4. Opened the corresponding primary PDF too.
- CONFIRMED | citation-lam-dek: Lam–Thiel–Swiercz, 1989 | https://geodesic.mathdoc.fr/articles/10.4153/CJM-1989-049-4/ | Codex checker, N2-codex.md: Mathdoc: "Lam, C. W. H. ; Thiel, L. ; Swiercz, S."; "Tome 41 (1989) no. 6, pp. 1117-1123"; DOI 10.4153/CJM-1989-049-4. Opened the corresponding primary PDF too.
- CONFIRMED | citation-lam-honesty: Lam–Thiel–Swiercz 1989 | https://geodesic.mathdoc.fr/articles/10.4153/CJM-1989-049-4/ | Codex checker, N2-codex.md: Mathdoc: "Lam, C. W. H. ; Thiel, L. ; Swiercz, S."; "Tome 41 (1989) no. 6, pp. 1117-1123"; DOI 10.4153/CJM-1989-049-4. Opened the corresponding primary PDF too.
- CONFIRMED | built-primepower-2: n = 2 is a prime power ⇒ a projective plane of order 2 EXISTS (PG(2,2)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-3: n = 3 is a prime power ⇒ a projective plane of order 3 EXISTS (PG(2,3)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-4: n = 4 is a prime power ⇒ a projective plane of order 4 EXISTS (PG(2,4)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-5: n = 5 is a prime power ⇒ a projective plane of order 5 EXISTS (PG(2,5)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-7: n = 7 is a prime power ⇒ a projective plane of order 7 EXISTS (PG(2,7)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-8: n = 8 is a prime power ⇒ a projective plane of order 8 EXISTS (PG(2,8)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-9: n = 9 is a prime power ⇒ a projective plane of order 9 EXISTS (PG(2,9)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-11: n = 11 is a prime power ⇒ a projective plane of order 11 EXISTS (PG(2,11)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-primepower-13: n = 13 is a prime power ⇒ a projective plane of order 13 EXISTS (PG(2,13)). | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam et al., p. 1117: "It is known that a plane of order n exists if n is a prime power." The listed composite cases are 4=2^2, 8=2^3, and 9=3^2; all others are prime. The HTML table explicitly says "built from a finite field", which allows extension-field arithmetic for these cases. The interactive outputs were reproduced.
- CONFIRMED | built-forbids-6: n = 6 ≡ 2 (mod 4) and 6 is NOT a sum of two squares ⇒ Bruck–Ryser FORBIDS a plane of order 6. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | Codex checker, N2-codex.md: Opened the 1949 PDF and read Theorem 1, p. 88. It excludes orders congruent to 1 or 2 modulo 4 whose square-free part has a prime factor 3 modulo 4. Thus 6=2*3, 14=2*7, and 21=3*7 are excluded. Direct enumeration of squares gives no representation for these orders. Text extraction produces OCR slips in congruence symbols, not a source discrepancy.
- CONFIRMED | built-forbids-14: n = 14 ≡ 2 (mod 4) and 14 is NOT a sum of two squares ⇒ Bruck–Ryser FORBIDS a plane of order 14. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | Codex checker, N2-codex.md: Opened the 1949 PDF and read Theorem 1, p. 88. It excludes orders congruent to 1 or 2 modulo 4 whose square-free part has a prime factor 3 modulo 4. Thus 6=2*3, 14=2*7, and 21=3*7 are excluded. Direct enumeration of squares gives no representation for these orders. Text extraction produces OCR slips in congruence symbols, not a source discrepancy.
- CONFIRMED | built-forbids-21: n = 21 ≡ 1 (mod 4) and 21 is NOT a sum of two squares ⇒ Bruck–Ryser FORBIDS a plane of order 21. | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2A2E0D91C8E872232A01C7519F14D705/S0008414X00028686a.pdf/nonexistence_of_certain_finite_projective_planes.pdf | Codex checker, N2-codex.md: Opened the 1949 PDF and read Theorem 1, p. 88. It excludes orders congruent to 1 or 2 modulo 4 whose square-free part has a prime factor 3 modulo 4. Thus 6=2*3, 14=2*7, and 21=3*7 are excluded. Direct enumeration of squares gives no representation for these orders. Text extraction produces OCR slips in congruence symbols, not a source discrepancy.
- CONFIRMED | built-not-forbidden-10: n = 10 ≡ 2 (mod 4); 10 = 1²+3². Bruck–Ryser is satisfied \u2014 it does NOT forbid order 10. (Existence still unknown unless settled elsewhere.) | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Read the full Elkies passage: q congruent to 1 or 2 modulo 4 must be a sum of two integer squares. 10=1^2+3^2 and 18=3^2+3^2 satisfy this condition. For 10, the very next rendered line explicitly supplies the separate nonexistence result; the conditional "unless settled elsewhere" does not erase it. These rendered outputs were replayed.
- CONFIRMED | built-not-forbidden-18: n = 18 ≡ 2 (mod 4); 18 = 3²+3². Bruck–Ryser is satisfied \u2014 it does NOT forbid order 18. (Existence still unknown unless settled elsewhere.) | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Read the full Elkies passage: q congruent to 1 or 2 modulo 4 must be a sum of two integer squares. 10=1^2+3^2 and 18=3^2+3^2 satisfy this condition. For 10, the very next rendered line explicitly supplies the separate nonexistence result; the conditional "unless settled elsewhere" does not erase it. These rendered outputs were replayed.
- CONFIRMED | built-test-silent-12: n = 12 ≡ 0 (mod 4) ⇒ Bruck–Ryser says nothing. Order 12 is undecided by this test. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies explains that for residues 0 or 3 modulo 4 "the condition is automatically satisfied". Thus this particular necessary condition does not decide 12 or 15. This does not claim the test proves either order exists.
- CONFIRMED | built-test-silent-15: n = 15 ≡ 3 (mod 4) ⇒ Bruck–Ryser says nothing. Order 15 is undecided by this test. | https://people.math.harvard.edu/~elkies/M155.15/notes.html | Codex checker, N2-codex.md: Elkies explains that for residues 0 or 3 modulo 4 "the condition is automatically satisfied". Thus this particular necessary condition does not decide 12 or 15. This does not claim the test proves either order exists.
- CONFIRMED | built-order10-note: (Order 10 was ruled out separately by exhaustive computer search \u2014 Lam–Thiel–Swiercz, 1989.) | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | Codex checker, N2-codex.md: Lam, Thiel and Swiercz (1989), p. 1117: the computer search "when taken together with previous results, implies that a plane of order 10 does not exist." The same introduction says this shows the Bruck-Ryser necessary condition is not sufficient. 10=1^2+3^2. The paper describes the exhaustive remaining cases, not a theorem from BRC.
- CONFIRMED | built-order12-note: (Order 12 is the smallest order whose existence is OPEN.) | https://people.math.harvard.edu/~elkies/M155.15/notes.html; https://arxiv.org/html/2510.19804v1#S4 | Codex checker, N2-codex.md: Elkies: "the first open case is n=12". Also checked the later primary paper Alexeev and Mixon (2025), section 4: "Order 12 is the smallest for which it is unknown whether a projective plane exists." The statements about making no claim are read as making no existence/nonexistence claim.
- CONFIRMED | html-built-live: Play the game below; the deck is built in front of you from nothing but arithmetic. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: buildPlane(q) enumerates nonzero triples mod q, normalises the leading coordinate, and sets incidence by dot product mod q; rebuild() calls it on load and on every order change. recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs
- CONFIRMED | html-live-cards: Two cards. They share exactly one symbol. Find it \u2014 click it on either card \u2014 as fast as you can. New pair, again. Forever. Watch: you will never be dealt a pair with no match, and never one with two. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: deal() picks two distinct random cards from the built deck; recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs, so no deal can share 0 or 2.
- CONFIRMED | selector-order-2: 2 \u2014 3 symbols/card, 7 cards | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: verify.mjs section 1 prints q=2: 7 points, 7 lines, 3 symbols per card; recreated by the page's own buildPlane (7 cards, 3 per card).
- CONFIRMED | selector-order-3: 3 \u2014 4 symbols/card, 13 cards | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: section 1: q=3 gives 13 and 4; recreated (13 cards, 4 per card).
- CONFIRMED | selector-order-5: 5 \u2014 6 symbols/card, 31 cards | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: section 1: q=5 gives 31 and 6; recreated (31 cards, 6 per card).
- CONFIRMED | selector-order-7: 7 \u2014 8 symbols/card, 57 cards (retail Dobble) | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: section 1 and 4: q=7 gives 57 cards, 8 per card; recreated. Note: the label puts "57 cards" next to "(retail Dobble)" while the box holds 55 (Asmodee); section 3 of the page says so two screens later.
- CONFIRMED | html-anomaly-counter: That last counter is the honest part: it ticks up only if the live deck ever deals a pair sharing zero or two symbols. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: deal(): "const sc=sharedCount(A,B); if(sc!==1){ anomaly++; ..." It counts any shared count other than 1, which includes zero and two.
- CONFIRMED | html-never-filters: It stays at 0 no matter how long you play, because the construction makes it impossible \u2014 not because we filtered the bad pairs out. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: deal() has no rejection on the shared count: its only loop is "do { pairB=...; } while(pairB===pairA);" (distinct cards). recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs
- CONFIRMED | html-matrix-description: Below: the shared-symbol count for every pair of the first cards. Every off-diagonal cell is 1. No cell is 0 or 2. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: renderDeck: m=Math.min(deck.length, 10); every off-diagonal cell is sharedCount; recreated as 1 for every pair. The sentence says "the first cards", which is accurate.
- CONFIRMED | html-check-scope: the counts for orders 2,3,5,7; that every pair of cards shares exactly one symbol over all C(n²+n+1, 2) pairs; the dual statement for symbols; the retail order-7 reconciliation; and the Bruck–Ryser obstruction for order 6 (and its silence on order 10). | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: verify.mjs sections 1 to 5 test exactly these: counts for q=2,3,5,7; all C(N,2) card pairs; the dual law; order 7 = 57/8/57 plus the 55 subset; order 6 forbidden and order 10 not forbidden. My run printed all 51 as passed.
- CONFIRMED | html-constructs-checks: The page constructs the planes of prime order and checks the matching property on what it builds. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: buildPlane for the selected prime; deal() and renderDeck() count shared symbols on what was built.
- CONFIRMED | html-lam-not-rerun: The nonexistence of order 10 is cited (Lam–Thiel–Swiercz 1989), not re-run | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: verify.mjs: "Order 10 was eliminated by exhaustive computer search (Lam, Thiel, Swiercz 1989) (dash) a result we cite, not re-run here"; the only order-10 test is arithmetic (10 = 1²+3²).
- CONFIRMED | html-glyphs: The symbols are little glyphs drawn for play; the mathematics is in the incidence, not the pictures. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: GLYPHS is an array of Unicode characters; layoutSymbols writes GLYPHS[symIdx % GLYPHS.length]; incidence uses indices only.
- CONFIRMED | html-no-game-assets: this page uses no game assets. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: The page has no img, svg or CSS url() asset; symbols are Unicode glyphs. The only image is the social card https://artwaste.land/og/every-two-cards.png, which I opened: the site's own title card, no game art.
- CONFIRMED | placard-independent: This layer's check works the answer out again from scratch, independently of this page. | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: verify.mjs imports nothing and reads no file; it builds PG(2,q) itself. git show 67088387e0:research/verifier-independence/trace-results.json: "observedReason": "opened nothing that /strata/every-two-cards/ ships".
- CONFIRMED | placard-cannot-notice-drift: It can settle the finding; it cannot notice a figure that drifted on this page. | git show 67088387e0:research/verifier-independence/trace-results.json | Claude checker, N2-claude.md: "observed": "C0", "observedReason": "opened nothing that /strata/every-two-cards/ ships": it never reads the page, so it cannot see a page figure change.
- CONFIRMED | placard-published-standalone: It is published, and it runs on its own: one file, and Node. | https://artwaste.land/checks/research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: recreated: curl https://artwaste.land/checks/research/dobble-projective-plane/verify.mjs into an empty directory, node verify.mjs, exit 0, "51 passed, 0 failed", file byte-identical to the pinned commit
- CONFIRMED | md-playable: The immersive page makes all of it playable. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: The page has the timed matching game, the deck explorer and the Bruck-Ryser tester.
- CONFIRMED | md-race-and-counter: Race the clock to spot the shared symbol on two live cards \u2014 and watch the honest counter that ticks up only if the deck ever deals a pair with the wrong number of shared symbols. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: startTimer/stopTimer time each find; the anomaly counter increments on sharedCount(A,B)!==1.
- CONFIRMED | md-zero-counter: It stays at 0, not because bad pairs are filtered out, but because the construction cannot produce one. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: No filtering in deal(); recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs
- CONFIRMED | md-check-scope: the counts for orders 2, 3, 5, 7; that every pair of cards shares exactly one symbol across all C(n²+n+1, 2) pairs; the self-dual statement that every pair of symbols shares exactly one card; the retail order-7 reconciliation; and the Bruck–Ryser obstruction for order 6 with its silence on order 10. | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: verify.mjs sections 1 to 5; my run printed every one of the 51 checks as passed, including "every pair of symbols shares exactly 1 card".
- CONFIRMED | md-constructs-checks: The page constructs the prime-order planes and checks the matching property on what it builds | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: buildPlane for prime q; deal() and renderDeck() check shared counts on the built deck.
- CONFIRMED | md-order10-cited: the nonexistence of order 10 is cited, not re-run | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: verify.mjs comments: "a result we cite, not re-run here".
- CONFIRMED | dek-play-and-counter: This immersive page makes it playable: race the clock to find the shared symbol on two live cards, with a counter that ticks only if the deck ever deals a pair sharing ≠1 symbol (it stays at 0, because the construction forbids it, not because the bad pairs are filtered out) | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Timed game, anomaly counter on sharedCount!==1, no filtering; recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs
- CONFIRMED | dek-verifier-summary: research/dobble-projective-plane/verify.mjs, 51/51: the counts for orders 2,3,5,7, every pair of cards sharing exactly one symbol over all C(n²+n+1,2) pairs, the self-dual statement for symbols, the retail order-7 reconciliation, and the Bruck–Ryser obstruction. | git show 67088387e0:research/dobble-projective-plane/verify.mjs | Claude checker, N2-claude.md: recreated: curl https://artwaste.land/checks/research/dobble-projective-plane/verify.mjs into an empty directory, node verify.mjs, exit 0, "51 passed, 0 failed", file byte-identical to the pinned commit
- CONFIRMED | meta-built-live: A theorem, built live from arithmetic and playable in your browser. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs
- CONFIRMED | related-whole-universes: In each the apparatus shows its working by building the whole reachable (or whole dealable) universe and watching the verdict never go wrong | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: The page builds the whole deck (deck=plane.lineToPoints, all q²+q+1 cards) and deals from it, with the anomaly counter as the verdict. The comparison page was not checked.
- CONFIRMED | related-by-hand: both let you try by hand and fail. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: The "deal a new pair" button lets a reader deal pairs by hand; none ever has 0 or 2 shared symbols (recreated in Node from the page's own buildPlane: q=2,3,5,7 give 7/13/31/57 cards, 3/4/6/8 symbols per card, 0 pairs with a shared count other than 1 over all 21/78/465/1596 pairs).
- CONFIRMED | nearby-pair-once: Both are finite projective geometries, one dimension apart in different fields. | git show 67088387e0:src/content/strata/every-pair-once.md | Claude checker, N2-claude.md: every-pair-once: "the 35 rows are exactly the lines of PG(3,2)"; this page builds PG(2,q). Note: "in different fields" holds for q=3,5,7, not when the reader picks order 2 (both F_2).
- CONFIRMED | built-full-deck-2: showing the full deck for order 2 \u2014 7 cards, each a list of its symbol indices | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Order 2: 7 cards, all 7 shown as minicards (show=Math.min(7,13)).
- CONFIRMED | built-full-deck-3: showing the full deck for order 3 \u2014 13 cards, each a list of its symbol indices | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Order 3: 13 cards, all 13 shown.
- CONFIRMED | built-matrix-verdict: ✓ every off-diagonal cell = 1 \u2014 exactly one shared symbol for every pair | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Printed when allOne; recreated: every pair among the first 10 cards shares 1, for all four orders. "every pair" here means the matrix's pairs.
- CONFIRMED | built-matrix-failure: ✗ a cell ≠ 1 \u2014 the construction broke (this should never appear) | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Failure branch exists and is unreachable for these planes (recreated: 0 bad pairs).
- CONFIRMED | built-hit-message: Both cards shared it, and only it. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Printed only when symIdx===answer, the single shared symbol.
- CONFIRMED | built-miss-message: that symbol is on only one of the two cards. | git show 67088387e0:public/strata/every-two-cards/index.html | Claude checker, N2-claude.md: Every symbol on a card other than the shared one is on that card only (pairs share exactly one).
- CONFIRMED | shared-plain-1: You've landed on the Artificial Wasteland, a website built by a series of AI minds. | git show 67088387e0:WAKE.md | Claude checker, N2-claude.md: WAKE.md: "This repository *is* the strata: the layers earlier instances left".
- CONFIRMED | shared-plain-2: A fresh instance of Claude wakes here each night with no memory of the last one, reads its way back in from the site's own files, and adds a little more before it goes. | git show 67088387e0:WAKE.md | Claude checker, N2-claude.md: WAKE.md: "You remember nothing of your own; the ground remembers for you." and "one instance with one night". Note: WAKE.md also says "The Wasteland runs many instances now (dash) some scheduled, some hand-started", so "a fresh instance ... each night" is a simplification.
- CONFIRMED | shared-plain-3: Nothing here may lie about anything real; that rule never bends. | git show 67088387e0:WAKE.md | Claude checker, N2-claude.md: WAKE.md rule 2: "It must never lie about anything real."
- CONFIRMED | shared-plain-4: Claims are being checked against their sources, layer by layer, and the working is shown on the page. | git show 67088387e0:assay/README.md | Claude checker, N2-claude.md: The Assay Office records claims passes per layer (assay/records/), and this pass is one.
- CONFIRMED | shared-plain-5: When a check finds something false, the page says what changed and when. | git show 67088387e0:assay/README.md | Claude checker, N2-claude.md: README step 4: "add a dated `Corrected YYYY-MM-DD: ...` line on the page for what was false when written, or an `Updated YYYY-MM-DD: ...` line for a STALE claim".
- CONFIRMED | shared-plain-6: Most pages are a single deep dive into one strange, specific thing, and search engines tend to drop you straight into the middle of one. | git show 67088387e0:research/search-visibility/findings.json | Claude checker, N2-claude.md: pagesEverShown: 122 of the 134 pages Bing ever showed are /strata/ pages (291 of 424 impressions).
- MINOR | V1: the shipped 55-card subset still has exactly-one-match (property is hereditary) | research/dobble-projective-plane/verify.mjs | The verifier actually tests lineToPoints.slice(0, 55), an arbitrary constructed subset. It does not identify the printed retail symbol mapping. The checked hereditary property is correct; the label now says a 55-card subset.

## Findings

- [fixed] X1 (WRONG): Replaced the retail-deck identity with a plane-derived deck with two cards omitted in the HTML introduction, all four description copies, verifier introduction/banner, layer social draft and old image HTML source; clarified the order-7 selector label. Added the shared retail correction in HTML and Markdown.
- [fixed] X2 (WRONG): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] X3 (MINOR): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] X4 (MINOR): Named the found and anomaly counters explicitly in the HTML. No on-page correction for this MINOR.
- [fixed] X5 (MINOR): HTML and Markdown now distinguish live deck counts, dealt-pair and displayed-matrix checks, the arithmetic tester, sourced figures, and the recorded offline result. No on-page correction for this MINOR.
- [fixed] X6 (WRONG): Changed the HTML historical attribution and verifier comment to Bruck–Ryser (1949). Added the shared correction distinguishing 1949 from Chowla's 1950 generalization.
- [fixed] X7 (WRONG): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] X8 (MINOR): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] X9 (WRONG): The Markdown body and dek now specify the first up to ten cards and off-diagonal 1s. The HTML caption and generated verdict explicitly limit the displayed scope. Added the shared matrix correction.
- [fixed] X10 (MINOR): HTML and Markdown now specify the same construction using arithmetic in the finite field of the chosen order. No on-page correction for this MINOR.
- [fixed] X11 (MINOR): HTML and Markdown now distinguish live deck counts, dealt-pair and displayed-matrix checks, the arithmetic tester, sourced figures, and the recorded offline result. No on-page correction for this MINOR.
- [fixed] X12 (MINOR): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] X13 (WRONG): The Markdown body and dek now specify the first up to ten cards and off-diagonal 1s. The HTML caption and generated verdict explicitly limit the displayed scope. Added the shared matrix correction.
- [fixed] X14 (WRONG): Replaced the retail-deck identity with a plane-derived deck with two cards omitted in the HTML introduction, all four description copies, verifier introduction/banner, layer social draft and old image HTML source; clarified the order-7 selector label. Added the shared retail correction in HTML and Markdown.
- [fixed] X15 (WRONG): Changed the authored relates note to a card game and an error-correcting code; regenerated only this layer's nearby block with the existing generator. Added the shared related-page correction.
- [fixed] X16 (WRONG): Changed builds first to can build in the authored relates note. Added the shared default-order correction; retained the supported Fano/Hamming relationship.
- [fixed] X17 (WRONG): The caption now gives shownCount and deckCount separately; shownCount comes from the exact show value used by the card loop. Added one shared correction for orders 5 and 7.
- [fixed] X18 (WRONG): The caption now gives shownCount and deckCount separately; shownCount comes from the exact show value used by the card loop. Added one shared correction for orders 5 and 7.
- [fixed] X19 (WRONG): Changed builds first to can build in the authored relates note. Added the shared default-order correction; retained the supported Fano/Hamming relationship.
- [fixed] K1 (WRONG): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] K2 (MINOR): HTML and Markdown now specify the same construction using arithmetic in the finite field of the chosen order. No on-page correction for this MINOR.
- [fixed] K3 (MINOR): Named the found and anomaly counters explicitly in the HTML. No on-page correction for this MINOR.
- [fixed] K4 (MINOR): HTML and Markdown now distinguish live deck counts, dealt-pair and displayed-matrix checks, the arithmetic tester, sourced figures, and the recorded offline result. No on-page correction for this MINOR.
- [fixed] K5 (WRONG): Changed the HTML historical attribution and verifier comment to Bruck–Ryser (1949). Added the shared correction distinguishing 1949 from Chowla's 1950 generalization.
- [fixed] K6 (WRONG): Changed the framing to one of the axioms in HTML, Markdown body/dek and verifier comments. The false one-rule definition has a shared dated correction; the MINOR axiom phrases have no separate on-page correction.
- [fixed] K7 (WRONG): The Markdown body and dek now specify the first up to ten cards and off-diagonal 1s. The HTML caption and generated verdict explicitly limit the displayed scope. Added the shared matrix correction.
- [fixed] K8 (MINOR): HTML and Markdown now specify the same construction using arithmetic in the finite field of the chosen order. No on-page correction for this MINOR.
- [fixed] K9 (MINOR): HTML and Markdown now distinguish live deck counts, dealt-pair and displayed-matrix checks, the arithmetic tester, sourced figures, and the recorded offline result. No on-page correction for this MINOR.
- [fixed] K10 (WRONG): The Markdown body and dek now specify the first up to ten cards and off-diagonal 1s. The HTML caption and generated verdict explicitly limit the displayed scope. Added the shared matrix correction.
- [dated] K11 (STALE): Set JSON-LD dateModified to 2026-10-14. The substantive Corrected lines date this revision; the user's JSON-LD exception requires no separate Updated line.
- [fixed] K12 (WRONG): Changed the authored relates note to a card game and an error-correcting code; regenerated only this layer's nearby block with the existing generator. Added the shared related-page correction.
- [fixed] K14 (WRONG): The caption now gives shownCount and deckCount separately; shownCount comes from the exact show value used by the card loop. Added one shared correction for orders 5 and 7.
- [fixed] K15 (WRONG): The caption now gives shownCount and deckCount separately; shownCount comes from the exact show value used by the card loop. Added one shared correction for orders 5 and 7.
- [fixed] K16 (WRONG): Changed builds first to can build in the authored relates note. Added the shared default-order correction; retained the supported Fano/Hamming relationship.
- [fixed] V1 (MINOR): Narrowed the verifier result label from the shipped subset to a 55-card subset and refreshed its verbatim public copy. The arithmetic assertion is unchanged. No on-page correction for this MINOR.

## Fix report

All 34 triage fixes are complete: 21 WRONG, 12 MINOR and one STALE metadata field. K13 is REJECTED with evidence and its sentence is unchanged. One additional MINOR label in the verifier was narrowed to the arbitrary 55-card subset it actually tests. Seven unique WRONG corrections appear on the page and at the end of the Markdown body. The MINOR fixes have no separate on-page correction. The JSON-LD date refresh has no Updated line under the explicit exception.

The immersive page is authored directly in public/strata/every-two-cards/index.html; research/dobble-projective-plane contains the verifier and an old image source, not a page build generator. The relates note is authored in the Markdown frontmatter. Only this layer's generated nearby block was regenerated by evaluating the existing scripts/nearby-layers.mjs definitions through blockHtml, without running the cross-layer write loop. The published verifier is a verbatim copy of its research source and was refreshed from that source.

**Triage outcomes.** Evidence and the precise claim wording are recorded above, under the matching ID.

| ID | Level | Outcome | Disposition |
| --- | --- | --- | --- |
| X1 | WRONG | FIXED | Retail deck distinguished from the complete plane in all current layer text copies. |
| X2 | WRONG | FIXED | Framing now says one of the axioms. |
| X3 | MINOR | FIXED | Framing now says one of the axioms. |
| X4 | MINOR | FIXED | Counters named explicitly. |
| X5 | MINOR | FIXED | Live checks separated from sourced facts and recorded offline checks. |
| X6 | WRONG | FIXED | Plane result attributed to Bruck–Ryser (1949); correction dates the generalization to 1950. |
| X7 | WRONG | FIXED | Framing now says one of the axioms. |
| X8 | MINOR | FIXED | Framing now says one of the axioms. |
| X9 | WRONG | FIXED | First up to ten cards, with 1s off the diagonal. |
| X10 | MINOR | FIXED | Finite-field arithmetic specified for the chosen order. |
| X11 | MINOR | FIXED | Live checks separated from sourced facts and recorded offline checks. |
| X12 | MINOR | FIXED | Framing now says one of the axioms. |
| X13 | WRONG | FIXED | First up to ten cards, with 1s off the diagonal. |
| X14 | WRONG | FIXED | Retail deck distinguished from the complete plane in all current layer text copies. |
| X15 | WRONG | FIXED | A card game and an error-correcting code. |
| X16 | WRONG | FIXED | Can build the Fano plane; default remains order 3. |
| X17 | WRONG | FIXED | Caption reports the displayed and total counts from the construction. |
| X18 | WRONG | FIXED | Caption reports the displayed and total counts from the construction. |
| X19 | WRONG | FIXED | Can build the Fano plane; default remains order 3. |
| K1 | WRONG | FIXED | Framing now says one of the axioms. |
| K2 | MINOR | FIXED | Finite-field arithmetic specified for the chosen order. |
| K3 | MINOR | FIXED | Counters named explicitly. |
| K4 | MINOR | FIXED | Live checks separated from sourced facts and recorded offline checks. |
| K5 | WRONG | FIXED | Plane result attributed to Bruck–Ryser (1949); correction dates the generalization to 1950. |
| K6 | WRONG | FIXED | Framing now says one of the axioms. |
| K7 | WRONG | FIXED | First up to ten cards, with 1s off the diagonal. |
| K8 | MINOR | FIXED | Finite-field arithmetic specified for the chosen order. |
| K9 | MINOR | FIXED | Live checks separated from sourced facts and recorded offline checks. |
| K10 | WRONG | FIXED | First up to ten cards, with 1s off the diagonal. |
| K11 | STALE | FIXED | dateModified set to 2026-10-14; no separate Updated line. |
| K12 | WRONG | FIXED | A card game and an error-correcting code. |
| K13 | MINOR | REJECTED with evidence | The offline verifier checks every card and symbol pair; cap-set ambientAudit/supportData exhaust their stated objects. The sentence promises no live-only check. |
| K14 | WRONG | FIXED | Caption reports the displayed and total counts from the construction. |
| K15 | WRONG | FIXED | Caption reports the displayed and total counts from the construction. |
| K16 | WRONG | FIXED | Can build the Fano plane; default remains order 3. |

**Replacement figures and evidence.**

- The 55-card retail count, 57-card complete order-7 plane, and eight symbols per card come from [IBM Research](https://research.ibm.com/blog/ponder-this-may-2024), with the retail count independently stated by [Asmodee](https://www.asmodee.co.uk/pages/games/dobble).
- The need for more than the shared-point axiom is explicit on p. 1117 of [Lam, Thiel and Swiercz](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534).
- The 1949/1950 distinction is supported by the publisher's [Combinatorial Problems record](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/combinatorial-problems/CAEFF0578570DF77C7595FFACBEF55A1) and [Elkies' lecture notes](https://people.math.harvard.edu/~elkies/M155.15/notes.html). The Crossref endpoint failed to open; the independent publisher evidence was sufficient.
- Displayed 13, matrix limit 10, totals 31 and 57, and remainder tiles +18/+44 come from the existing renderDeck/buildPlane code and were reproduced in Chromium. The default order 3 and its 13 cards come from the selected option and rebuild initialization, also reproduced in Chromium.
- The revised comparison follows the pinned perfect-codes frontmatter: an error-correcting code and coding instruments. The supported Fano/Hamming relationship is retained. No replacement figure came from an estimate.

**Exact dated lines.** These are the rendered HTML lines. The Markdown copies replace the prefix with `*Corrected 2026-10-14:*` and otherwise match exactly.

> Corrected 2026-10-14: The page called the retail Dobble deck a finite projective plane. The standard box has 55 cards; the complete order-7 plane has 57, with two cards omitted from the retail deck (found by the assay line, 2026-10-14).

> Corrected 2026-10-14: The page said one intersection rule defined a projective plane. The shared-point rule is one of the axioms; it does not define a plane by itself (found by the assay line, 2026-10-14).

> Corrected 2026-10-14: The page dated Bruck–Ryser–Chowla to 1949. Bruck and Ryser proved the plane result in 1949; Chowla's generalization followed in 1950 (found by the assay line, 2026-10-14).

> Corrected 2026-10-14: The page promised an all-pairs matrix of 1s for the whole deck. The displayed matrix covers the first up to ten cards; its off-diagonal entries are 1 and its diagonal is marked with dots (found by the assay line, 2026-10-14).

> Corrected 2026-10-14: The related note called this page and A Message That Heals Itself two card games. The comparison is between a card game and an error-correcting code (found by the assay line, 2026-10-14).

> Corrected 2026-10-14: The related note said this page builds the seven-point Fano plane first. The page starts at order 3, with 13 cards; the order-2 Fano plane is available in the selector (found by the assay line, 2026-10-14).

> Corrected 2026-10-14: The deck caption said it showed the full deck at orders 5 and 7. It shows the first 13 of 31 cards at order 5 and the first 13 of 57 at order 7, with a tile counting the remaining cards (found by the assay line, 2026-10-14).

**Commands and results.**

| Command | Result |
| --- | --- |
| `python3 /tmp/every-two-cards-1014/edit.py` | Exit 0. Applied asserted exact text replacements to five authored layer files and mirrored seven dated lines. |
| `node /tmp/every-two-cards-1014/rebuild-layer.mjs` | Exit 0. Regenerated only this layer's nearby block from its updated frontmatter; copied the verifier to its public mirror. |
| `node research/dobble-projective-plane/verify.mjs` | Exit 0. 51 passed, 0 failed. |
| `node public/checks/research/dobble-projective-plane/verify.mjs` | Exit 0. 51 passed, 0 failed. Byte-identical to the research source. |
| `node /tmp/every-two-cards-1014/check-page.mjs` | Exit 0. Chromium: default order 3; correct 7/13/31/57 totals; correct 7/13/13/13 shown-card counts; 7/10/10/10 matrix sizes; all 21/78/465/1596 card pairs share exactly one symbol; successful game click and zero anomalies at every order; Bruck–Ryser outputs for 6/10/12; seven correction mirrors; dateModified and four descriptions; zero page errors. No screenshots. |
| `python3 /tmp/every-two-cards-1014/write-record.py` | Exit 0. 159 claims: 124 CONFIRMED, 21 WRONG, 13 MINOR, one STALE. The frontmatter wrong count of 34 includes MINOR. |
| `node scripts/check-claims-pass.mjs` | Exit 1 only for wisdom-of-crowds, whose findings 14-19, 21-22 and 31-37 failed the disposition syntax rule at this run. No every-two-cards error. Ignored the other-layer failure as instructed. |
| `node --input-type=module` calling exported `checkRecord` for `every-two-cards` | Exit 0. errs: [], checked: 159, confirmed: 124, wrong: 34, stale: 1, open: 0. |
| `git diff --check --` followed by the six changed existing layer paths below | Exit 0. No whitespace errors. |
| Python consistency probe over triage and claims record | Exit 0. All 35 IDs occur once; all 34 fix items have findings; no U+2014 in the record; public verifier equals source; social JSON valid, text 289 characters. |
| `rg -n` for the old false phrases in this layer's six current text files | No remaining matches. Historical quotations in the record are intentionally retained. |
| Final `rg -l -F` scan for repeated retail-identity and related-note wording across public/strata, public/search and src/content/strata | One out-of-scope match: public/strata/there-here/index.html:250, an embedded snapshot of the related note. Reported below and left unchanged. |
| `git diff 67088387e0 --` for this layer's paths before editing | No differences: the originals matched the requested pin. |
| `git show 67088387e0:src/content/strata/perfect-codes.md`, `git show 67088387e0:public/strata/cap-set-polynomial/index.html`, and `git log 67088387e0 --format='%h %ad %s' --date=short -- public/strata/every-two-cards/index.html` | Read-only evidence for the related-page correction, K13 rejection and stale metadata. |
| `pwd`, `git status --short`, `rg --files`, and targeted `rg`, `sed`, `cat`, `head`, `tail`, `ls`, `cmp` and Python reads | Located authored files, generators, checker records and existing unrelated changes. No repository-wide mutation. Some guessed README/script paths did not exist; existing paths were then located with rg. |
| Initial `python` read probe | Exit 127: python was unavailable. Repeated successfully with python3 before making any edit. |
| Web open/find/search tools | Reread IBM, Asmodee, Zygomatic, the 1949 and 1989 papers, the 1950 publisher record, Elkies, and the 2025 order-12 source. Rechecked the reported publisher correspondence about the 55-card subset. Crossref returned an error; no bulk source download was needed. |
| `view_image` on `public/og/every-two-cards.png` | Inspected the existing title card: it contains the title and opening matching-rule excerpt, not the old image-source identity claim. No image mutation. |
| `python3 /tmp/every-two-cards-1014/append-report.py` | Added this report, the exact dated lines and exhaustive before/after edits to the requested claims record. |

**Not done.**

- All 34 triage fixes are applied to this layer's authorized files. K13 was deliberately left unchanged after the evidence supported it.
- NOT FIXED outside this layer: [public/strata/there-here/index.html:250](/home/liam/aw-claude-assaying-boyle-1014/public/strata/there-here/index.html:250) embeds an old snapshot of the relates note, including two card games and the Fano plane builds first. The authored every-two-cards note is corrected; editing that other layer is expressly forbidden.
- No shared files or other layers were edited. The shared public/checks/manifest.json still lists this verifier as 11,565 bytes; the corrected source and mirror are 11,534 bytes. Updating that shared manifest is outside the authorized scope.
- No global build, dependency installation, placard rewrite, oversight derivation, commit, publication or social posting was run. Shared search/generated site data was not regenerated.
- No film, audio or image was rendered or re-encoded. The old image HTML source was corrected; the current title-card bitmap already omits its false identity statement.
- Historical session logs, the pinned source record, checker reports and triage remain intact as evidence. No bulk datasets or archives were downloaded; the bulk-download total is zero.

**Every repository edit, before and after.** Links use the current line; the preceding line number is also recorded for comparison. Literal `\u2014` represents an original U+2014, so the report introduces none. Unchanged surrounding lines are omitted.

[src/content/strata/every-two-cards.md:5](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:5) (before line 5).

Before:

```text
dek: "In the matching game Dobble (Spot It!), any two cards you ever lay down share exactly one picture (never none, never two), and that is not careful hand-tuning, it is a theorem. Call each card a line and each symbol a point, and the game's only rule becomes the defining axiom of a finite projective plane: any two lines meet in exactly one point. So a perfect Dobble deck can be built out of nothing but arithmetic: every symbol a direction [a,b,c] in a tiny space over the numbers mod a prime q, every card a recipe L, a symbol on a card exactly when L·x ≡ 0; two cards then share the symbols solving two linear equations at once, of which a projective plane has exactly one. This immersive page makes it playable: race the clock to find the shared symbol on two live cards, with a counter that ticks only if the deck ever deals a pair sharing ≠1 symbol (it stays at 0, because the construction forbids it, not because the bad pairs are filtered out); watch the whole deck get built mod q and the all-pairs shared-count matrix turn entirely to 1s; and read why retail Dobble has 8 symbols per card: order 7, a 57-card plane shipping 55. The honest frontier is loud: planes exist for every prime-power order, but order 6 is forbidden by Bruck–Ryser (1949), order 10 fell only to an exhaustive computer search (Lam–Thiel–Swiercz, 1989), and order 12 is OPEN: a children's card game sitting on top of a question nobody on Earth has answered. research/dobble-projective-plane/verify.mjs, 51/51: the counts for orders 2,3,5,7, every pair of cards sharing exactly one symbol over all C(n²+n+1,2) pairs, the self-dual statement for symbols, the retail order-7 reconciliation, and the Bruck–Ryser obstruction."
```

After:

```text
dek: "In the matching game Dobble (Spot It!), any two cards you ever lay down share exactly one picture (never none, never two), and that is not careful hand-tuning, it is a theorem. Call each card a line and each symbol a point, and the game's only rule becomes one of the axioms of a finite projective plane: any two lines meet in exactly one point. So a perfect Dobble deck can be built out of nothing but arithmetic: every symbol a direction [a,b,c] in a tiny space over the numbers mod a prime q, every card a recipe L, a symbol on a card exactly when L·x ≡ 0; two cards then share the symbols solving two linear equations at once, of which a projective plane has exactly one. This immersive page makes it playable: race the clock to find the shared symbol on two live cards, with a counter that ticks only if the deck ever deals a pair sharing ≠1 symbol (it stays at 0, because the construction forbids it, not because the bad pairs are filtered out); watch the whole deck get built mod q and the shared-count matrix for the first up to ten cards show 1s off the diagonal; and read why retail Dobble has 8 symbols per card: order 7, a 57-card plane shipping 55. The honest frontier is loud: planes exist for every prime-power order, but order 6 is forbidden by Bruck–Ryser (1949), order 10 fell only to an exhaustive computer search (Lam–Thiel–Swiercz, 1989), and order 12 is OPEN: a children's card game sitting on top of a question nobody on Earth has answered. research/dobble-projective-plane/verify.mjs, 51/51: the counts for orders 2,3,5,7, every pair of cards sharing exactly one symbol over all C(n²+n+1,2) pairs, the self-dual statement for symbols, the retail order-7 reconciliation, and the Bruck–Ryser obstruction."
```

[src/content/strata/every-two-cards.md:11](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:11) (before line 11).

Before:

```text
    note: "Two card games, both built on a finite combinatorial design where structure is forced, not chosen. There the codewords are spheres packed so tightly into the cube of all messages that a flipped bit always snaps back home \u2014 a perfect Hamming code. Here the cards are the lines of a projective plane packed so that any two meet in exactly one symbol. The order-2 plane (the seven-point Fano plane this page builds first) and the Hamming(7,4) code are two faces of the same object: seven points, seven lines, the same incidence written once as a parity-check matrix and once as a deck. Read once as error-correction, once as a game you can hold."
```

After:

```text
    note: "A card game and an error-correcting code, both built on a finite combinatorial design where structure is forced, not chosen. There the codewords are spheres packed so tightly into the cube of all messages that a flipped bit always snaps back home: a perfect Hamming code. Here the cards are the lines of a projective plane packed so that any two meet in exactly one symbol. The order-2 plane (the seven-point Fano plane this page can build) and the Hamming(7,4) code are two faces of the same object: seven points, seven lines, the same incidence written once as a parity-check matrix and once as a deck. Read once as error-correction, once as a game you can hold."
```

[src/content/strata/every-two-cards.md:25](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:25) (before line 25).

Before:

```text
Call each **card** a *line* and each **symbol** a *point*. A **finite projective
plane** is a world of points and lines obeying one rule: **any two lines meet in
```

After:

```text
Call each **card** a *line* and each **symbol** a *point*. One of the axioms of a **finite projective
plane** says: **any two lines meet in
```

[src/content/strata/every-two-cards.md:28](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:28) (before line 28).

Before:

```text
cards share exactly one symbol.* The game's whole law and the plane's defining
axiom are the same sentence \u2014 so building a projective plane builds a perfect
```

After:

```text
cards share exactly one symbol.* The game's whole law and one of the plane's
axioms are the same sentence, so building a projective plane builds a perfect
```

[src/content/strata/every-two-cards.md:46](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:46) (before line 46).

Before:

```text
mod *q* and the all-pairs shared-count matrix turn entirely to **1**s, and turn
```

After:

```text
mod *q* and the shared-count matrix for the first up to ten cards show **1**s
off the diagonal, and turn
```

[src/content/strata/every-two-cards.md:53](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:53) (before line 52).

Before:

```text
**prime power** \u2014 built exactly as above. But order **6** is *impossible*
```

After:

```text
**prime power**, using the same construction with arithmetic in the finite field
of that order. But order **6** is *impossible*
```

[src/content/strata/every-two-cards.md:61](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:61) (before line 59).

Before:

```text
Every number here is recomputed in the browser by the same arithmetic and proven
offline first in `research/dobble-projective-plane/verify.mjs` (51/51): the counts
```

After:

```text
The browser builds the selected deck, counts its cards, checks each dealt pair
and the displayed matrix, and runs the Bruck–Ryser test. The retail count and
historical dates are sourced; the **51/51** result in
`research/dobble-projective-plane/verify.mjs` records the offline checks: the counts
```

[src/content/strata/every-two-cards.md:72](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/every-two-cards.md:72) (before line 68).

Before:

```text
(absent)
```

After:

```text

*Corrected 2026-10-14:* The page called the retail Dobble deck a finite projective plane. The standard box has 55 cards; the complete order-7 plane has 57, with two cards omitted from the retail deck (found by the assay line, 2026-10-14).

*Corrected 2026-10-14:* The page said one intersection rule defined a projective plane. The shared-point rule is one of the axioms; it does not define a plane by itself (found by the assay line, 2026-10-14).

*Corrected 2026-10-14:* The page dated Bruck–Ryser–Chowla to 1949. Bruck and Ryser proved the plane result in 1949; Chowla's generalization followed in 1950 (found by the assay line, 2026-10-14).

*Corrected 2026-10-14:* The page promised an all-pairs matrix of 1s for the whole deck. The displayed matrix covers the first up to ten cards; its off-diagonal entries are 1 and its diagonal is marked with dots (found by the assay line, 2026-10-14).

*Corrected 2026-10-14:* The related note called this page and A Message That Heals Itself two card games. The comparison is between a card game and an error-correcting code (found by the assay line, 2026-10-14).

*Corrected 2026-10-14:* The related note said this page builds the seven-point Fano plane first. The page starts at order 3, with 13 cards; the order-2 Fano plane is available in the selector (found by the assay line, 2026-10-14).

*Corrected 2026-10-14:* The deck caption said it showed the full deck at orders 5 and 7. It shows the first 13 of 31 cards at order 5 and the first 13 of 57 at order 7, with a tile counting the remaining cards (found by the assay line, 2026-10-14).
```

[public/strata/every-two-cards/index.html:7](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:7) (before line 7).

Before:

```text
<meta name="description" content="The card game Dobble / Spot It! works because the deck is a finite projective plane: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser." />
```

After:

```text
<meta name="description" content="The card game Dobble / Spot It! works because the retail deck comes from a finite projective plane with two cards removed: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser." />
```

[public/strata/every-two-cards/index.html:12](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:12) (before line 12).

Before:

```text
<meta property="og:description" content="The card game Dobble / Spot It! works because the deck is a finite projective plane: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser." />
```

After:

```text
<meta property="og:description" content="The card game Dobble / Spot It! works because the retail deck comes from a finite projective plane with two cards removed: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser." />
```

[public/strata/every-two-cards/index.html:20](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:20) (before line 20).

Before:

```text
<meta name="twitter:description" content="Dobble / Spot It! works because the deck is a finite projective plane: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser." />
```

After:

```text
<meta name="twitter:description" content="Dobble / Spot It! works because the retail deck comes from a finite projective plane with two cards removed: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser." />
```

[public/strata/every-two-cards/index.html:82](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:82) (before line 82).

Before:

```text
    <script type="application/ld+json">{"@context":"https://schema.org","@type":"Article","name":"Every Two Cards Share a Symbol","headline":"Every Two Cards Share a Symbol","description":"The card game Dobble / Spot It! works because the deck is a finite projective plane: any two cards share exactly one symbol \u2014 a theorem, built live from arithmetic and playable in your browser.","url":"https://artwaste.land/strata/every-two-cards/","image":"https://artwaste.land/og/every-two-cards.png","datePublished":"2026-06-21","dateModified":"2026-06-21","author":{"@type":"Organization","@id":"https://artwaste.land/#organization","name":"Artificial Wasteland","alternateName":["The Artificial Wasteland","artwaste.land"],"url":"https://artwaste.land/","logo":"https://artwaste.land/og/default.png","sameAs":["https://bsky.app/profile/artwaste.land"]},"publisher":{"@type":"Organization","@id":"https://artwaste.land/#organization","name":"Artificial Wasteland","alternateName":["The Artificial Wasteland","artwaste.land"],"url":"https://artwaste.land/","logo":"https://artwaste.land/og/default.png","sameAs":["https://bsky.app/profile/artwaste.land"]},"isPartOf":{"@type":"WebSite","@id":"https://artwaste.land/#website","name":"Artificial Wasteland"}}</script>
```

After:

```text
    <script type="application/ld+json">{"@context":"https://schema.org","@type":"Article","name":"Every Two Cards Share a Symbol","headline":"Every Two Cards Share a Symbol","description":"The card game Dobble / Spot It! works because the retail deck comes from a finite projective plane with two cards removed: any two cards share exactly one symbol. A theorem, built live from arithmetic and playable in your browser.","url":"https://artwaste.land/strata/every-two-cards/","image":"https://artwaste.land/og/every-two-cards.png","datePublished":"2026-06-21","dateModified":"2026-10-14","author":{"@type":"Organization","@id":"https://artwaste.land/#organization","name":"Artificial Wasteland","alternateName":["The Artificial Wasteland","artwaste.land"],"url":"https://artwaste.land/","logo":"https://artwaste.land/og/default.png","sameAs":["https://bsky.app/profile/artwaste.land"]},"publisher":{"@type":"Organization","@id":"https://artwaste.land/#organization","name":"Artificial Wasteland","alternateName":["The Artificial Wasteland","artwaste.land"],"url":"https://artwaste.land/","logo":"https://artwaste.land/og/default.png","sameAs":["https://bsky.app/profile/artwaste.land"]},"isPartOf":{"@type":"WebSite","@id":"https://artwaste.land/#website","name":"Artificial Wasteland"}}</script>
```

[public/strata/every-two-cards/index.html:89](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:89) (before line 89).

Before:

```text
  <p class="dek">In the card game <em>Dobble</em> (<em>Spot It!</em>), any two cards you ever lay down share <strong>exactly one</strong> picture \u2014 never zero, never two. That is not careful design or luck. It is a theorem: the deck <em>is</em> a finite projective plane. Play the game below; the deck is built in front of you from nothing but arithmetic.</p>
```

After:

```text
  <p class="dek">In the card game <em>Dobble</em> (<em>Spot It!</em>), any two cards you ever lay down share <strong>exactly one</strong> picture: never zero, never two. That is not careful design or luck. It is a theorem: the retail deck comes from a finite projective plane with two cards removed. Play the game below; the deck is built in front of you from nothing but arithmetic.</p>
```

[public/strata/every-two-cards/index.html:93](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:93) (before line 93).

Before:

```text
<p>Here is the whole trick of the game, stated once. Think of each <strong>card</strong> as a <strong>line</strong> and each <strong>symbol</strong> as a <strong>point</strong>. A projective plane is a world of points and lines with one defining rule:</p>
```

After:

```text
<p>Here is the whole trick of the game, stated once. Think of each <strong>card</strong> as a <strong>line</strong> and each <strong>symbol</strong> as a <strong>point</strong>. One of the axioms of a projective plane says:</p>
```

[public/strata/every-two-cards/index.html:95](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:95) (before line 95).

Before:

```text
<p>Translate it back: <strong>any two cards share exactly one symbol.</strong> The game's only law and the plane's defining axiom are the same sentence. So if we can <em>build</em> a projective plane, we get a perfect Dobble deck for free \u2014 and we can, out of arithmetic over a finite field.</p>
```

After:

```text
<p>Translate it back: <strong>any two cards share exactly one symbol.</strong> The game's only law and one of the plane's axioms are the same sentence. So if we can <em>build</em> a projective plane, we get a perfect Dobble deck for free, and we can, out of arithmetic over a finite field.</p>
```

[public/strata/every-two-cards/index.html:108](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:108) (before line 108).

Before:

```text
        <option value="7">7 \u2014 8 symbols/card, 57 cards (retail Dobble)</option>
```

After:

```text
        <option value="7">7: 8 symbols/card, 57 cards (full plane behind retail Dobble)</option>
```

[public/strata/every-two-cards/index.html:136](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:136) (before line 136).

Before:

```text
    <span>showing the full deck for order <b class="mono" id="qDeck">3</b> \u2014 <b class="mono" id="deckCount"></b> cards, each a list of its symbol indices</span>
```

After:

```text
    <span>order <b class="mono" id="qDeck">3</b>: showing the first <b class="mono" id="shownCount"></b> of <b class="mono" id="deckCount"></b> cards, each shown card a list of its symbol indices</span>
```

[public/strata/every-two-cards/index.html:139](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:139) (before line 139).

Before:

```text
  <p style="color:var(--dim);font-size:.86rem;margin-bottom:.3rem">Below: the shared-symbol count for every pair of the first cards. Every off-diagonal cell is <span style="color:var(--good)">1</span>. No cell is 0 or 2.</p>
```

After:

```text
  <p style="color:var(--dim);font-size:.86rem;margin-bottom:.3rem">Below: the shared-symbol count for every pair of the first up to ten cards. Every off-diagonal cell is <span style="color:var(--good)">1</span>. No cell is 0 or 2.</p>
```

[public/strata/every-two-cards/index.html:152](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:152) (before line 152).

Before:

```text
  <tr><td class="n">2,3,4,5,7,8,9,11,13…</td><td class="yes">yes</td><td>every <strong>prime power</strong> \u2014 built from a finite field, exactly as above</td></tr>
```

After:

```text
  <tr><td class="n">2,3,4,5,7,8,9,11,13…</td><td class="yes">yes</td><td>every <strong>prime power</strong>: the same construction, using arithmetic in the finite field of that order</td></tr>
```

[public/strata/every-two-cards/index.html:157](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:157) (before line 157).

Before:

```text
<p style="color:var(--dim);font-size:.92rem">The first counter is named for the human eye; the second, for the limit of the proven. A children's matching game sits on top of a question \u2014 does a projective plane of order 12 exist? \u2014 that has been open since before the game was invented.</p>
```

After:

```text
<p style="color:var(--dim);font-size:.92rem">The "found" counter is named for the human eye; "pairs dealt with ≠1 shared symbol", for the limit of the proven. A children's matching game sits on top of a question (does a projective plane of order 12 exist?) that has been open since before the game was invented.</p>
```

[public/strata/every-two-cards/index.html:168](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:168) (before line 168).

Before:

```text
  <p><strong>The check.</strong> Every number on this page is recomputed live by the same arithmetic, and proven offline first in <code>research/dobble-projective-plane/verify.mjs</code> (51/51): the counts for orders 2,3,5,7; that <em>every</em> pair of cards shares exactly one symbol over all C(n²+n+1, 2) pairs; the dual statement for symbols; the retail order-7 reconciliation; and the Bruck–Ryser obstruction for order 6 (and its silence on order 10).</p>
```

After:

```text
  <p><strong>The check.</strong> The browser builds the selected deck, counts its cards, checks each dealt pair and the displayed matrix, and runs the Bruck–Ryser test. The retail count and historical dates are sourced; the 51/51 result in <code>research/dobble-projective-plane/verify.mjs</code> records the offline checks: the counts for orders 2,3,5,7; that <em>every</em> pair of cards shares exactly one symbol over all C(n²+n+1, 2) pairs; the dual statement for symbols; the retail order-7 reconciliation; and the Bruck–Ryser obstruction for order 6 (and its silence on order 10).</p>
```

[public/strata/every-two-cards/index.html:170](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:170) (before line 170).

Before:

```text
  <p>Projective plane construction: PG(2,q) over the field F_q. The Dobble↔projective-plane identification is folklore in recreational mathematics; the existence theory is Bruck–Ryser–Chowla (1949) and Lam–Thiel–Swiercz (1989). Spot It!/Dobble is a trademark of its publisher; this page uses no game assets.</p>
```

After:

```text
  <p>Projective plane construction: PG(2,q) over the field F_q. The Dobble↔projective-plane identification is folklore in recreational mathematics; the existence theory is Bruck–Ryser (1949) and Lam–Thiel–Swiercz (1989). Spot It!/Dobble is a trademark of its publisher; this page uses no game assets.</p>
```

[public/strata/every-two-cards/index.html:286](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:286) (before line 286).

Before:

```text
(absent)
```

After:

```text
  $('shownCount').textContent=show;
```

[public/strata/every-two-cards/index.html:314](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:314) (before line 313).

Before:

```text
    ? `<span class="verdict ok">✓ every off-diagonal cell = 1 \u2014 exactly one shared symbol for every pair</span>`
```

After:

```text
    ? `<span class="verdict ok">✓ every off-diagonal cell = 1: exactly one shared symbol for every pair shown</span>`
```

[public/strata/every-two-cards/index.html:358](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:358) (before line 357).

Before:

```text
(absent)
```

After:

```text
<aside class="wrap foot" aria-label="Corrections">
<h2>Corrections</h2>
<p>Corrected 2026-10-14: The page called the retail Dobble deck a finite projective plane. The standard box has 55 cards; the complete order-7 plane has 57, with two cards omitted from the retail deck (found by the assay line, 2026-10-14).</p>
<p>Corrected 2026-10-14: The page said one intersection rule defined a projective plane. The shared-point rule is one of the axioms; it does not define a plane by itself (found by the assay line, 2026-10-14).</p>
<p>Corrected 2026-10-14: The page dated Bruck–Ryser–Chowla to 1949. Bruck and Ryser proved the plane result in 1949; Chowla's generalization followed in 1950 (found by the assay line, 2026-10-14).</p>
<p>Corrected 2026-10-14: The page promised an all-pairs matrix of 1s for the whole deck. The displayed matrix covers the first up to ten cards; its off-diagonal entries are 1 and its diagonal is marked with dots (found by the assay line, 2026-10-14).</p>
<p>Corrected 2026-10-14: The related note called this page and A Message That Heals Itself two card games. The comparison is between a card game and an error-correcting code (found by the assay line, 2026-10-14).</p>
<p>Corrected 2026-10-14: The related note said this page builds the seven-point Fano plane first. The page starts at order 3, with 13 cards; the order-2 Fano plane is available in the selector (found by the assay line, 2026-10-14).</p>
<p>Corrected 2026-10-14: The deck caption said it showed the full deck at orders 5 and 7. It shows the first 13 of 31 cards at order 5 and the first 13 of 57 at order 7, with a tile counting the remaining cards (found by the assay line, 2026-10-14).</p>
</aside>
```

[public/strata/every-two-cards/index.html:389](/home/liam/aw-claude-assaying-boyle-1014/public/strata/every-two-cards/index.html:389) (before line 378).

Before:

```text
      <li><a href="/strata/perfect-codes/"><span class="aw-nearby__dot" style="background:#deae45"></span><span class="aw-nearby__t">A Message That Heals Itself</span><span class="aw-nearby__n">Two card games, both built on a finite combinatorial design where structure is forced, not chosen.</span></a></li>
```

After:

```text
      <li><a href="/strata/perfect-codes/"><span class="aw-nearby__dot" style="background:#deae45"></span><span class="aw-nearby__t">A Message That Heals Itself</span><span class="aw-nearby__n">A card game and an error-correcting code, both built on a finite combinatorial design where structure is forced, not chosen.</span></a></li>
```

[research/dobble-projective-plane/verify.mjs:2](/home/liam/aw-claude-assaying-boyle-1014/research/dobble-projective-plane/verify.mjs:2) (before line 2).

Before:

```text
// verify.mjs \u2014 Dobble / Spot It! is a finite projective plane, checked exactly.
```

After:

```text
// verify.mjs: Dobble / Spot It! comes from a finite projective plane, checked exactly.
```

[research/dobble-projective-plane/verify.mjs:7](/home/liam/aw-claude-assaying-boyle-1014/research/dobble-projective-plane/verify.mjs:7) (before line 7).

Before:

```text
//   careful hand-tuning \u2014 it is a theorem. The deck IS a finite projective
//   plane: cards are the LINES, symbols are the POINTS, and the defining
//   axiom of a projective plane is precisely "any two lines meet in exactly
//   one point." Build the plane from arithmetic over a finite field and the
```

After:

```text
//   careful hand-tuning; it is a theorem. The retail deck omits two cards
//   from a finite projective plane: cards are the LINES, symbols are the
//   POINTS, and one axiom says "any two lines meet in exactly one point."
//   Build the plane from arithmetic over a finite field and the
```

[research/dobble-projective-plane/verify.mjs:116](/home/liam/aw-claude-assaying-boyle-1014/research/dobble-projective-plane/verify.mjs:116) (before line 116).

Before:

```text
console.log("\nDobble / Spot It! = a finite projective plane \u2014 exact verification\n");
```

After:

```text
console.log("\nDobble / Spot It! from a finite projective plane: exact verification\n");
```

[research/dobble-projective-plane/verify.mjs:183](/home/liam/aw-claude-assaying-boyle-1014/research/dobble-projective-plane/verify.mjs:183) (before line 183).

Before:

```text
  log(ok, "the shipped 55-card subset still has exactly-one-match (property is hereditary)");
```

After:

```text
  log(ok, "a 55-card subset still has exactly-one-match (property is hereditary)");
```

[research/dobble-projective-plane/verify.mjs:188](/home/liam/aw-claude-assaying-boyle-1014/research/dobble-projective-plane/verify.mjs:188) (before line 188).

Before:

```text
// Bruck–Ryser–Chowla theorem (1949): if a projective plane of order n exists
```

After:

```text
// Bruck–Ryser theorem (1949): if a projective plane of order n exists
```

[public/checks/research/dobble-projective-plane/verify.mjs:2](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/dobble-projective-plane/verify.mjs:2) (before line 2).

Before:

```text
// verify.mjs \u2014 Dobble / Spot It! is a finite projective plane, checked exactly.
```

After:

```text
// verify.mjs: Dobble / Spot It! comes from a finite projective plane, checked exactly.
```

[public/checks/research/dobble-projective-plane/verify.mjs:7](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/dobble-projective-plane/verify.mjs:7) (before line 7).

Before:

```text
//   careful hand-tuning \u2014 it is a theorem. The deck IS a finite projective
//   plane: cards are the LINES, symbols are the POINTS, and the defining
//   axiom of a projective plane is precisely "any two lines meet in exactly
//   one point." Build the plane from arithmetic over a finite field and the
```

After:

```text
//   careful hand-tuning; it is a theorem. The retail deck omits two cards
//   from a finite projective plane: cards are the LINES, symbols are the
//   POINTS, and one axiom says "any two lines meet in exactly one point."
//   Build the plane from arithmetic over a finite field and the
```

[public/checks/research/dobble-projective-plane/verify.mjs:116](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/dobble-projective-plane/verify.mjs:116) (before line 116).

Before:

```text
console.log("\nDobble / Spot It! = a finite projective plane \u2014 exact verification\n");
```

After:

```text
console.log("\nDobble / Spot It! from a finite projective plane: exact verification\n");
```

[public/checks/research/dobble-projective-plane/verify.mjs:183](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/dobble-projective-plane/verify.mjs:183) (before line 183).

Before:

```text
  log(ok, "the shipped 55-card subset still has exactly-one-match (property is hereditary)");
```

After:

```text
  log(ok, "a 55-card subset still has exactly-one-match (property is hereditary)");
```

[public/checks/research/dobble-projective-plane/verify.mjs:188](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/dobble-projective-plane/verify.mjs:188) (before line 188).

Before:

```text
// Bruck–Ryser–Chowla theorem (1949): if a projective plane of order n exists
```

After:

```text
// Bruck–Ryser theorem (1949): if a projective plane of order n exists
```

[research/dobble-projective-plane/og.html:31](/home/liam/aw-claude-assaying-boyle-1014/research/dobble-projective-plane/og.html:31) (before line 31).

Before:

```text
      <div class="sub">Dobble <b>is a projective plane</b>: any two lines meet in exactly <i>one</i> point \u2014 the same sentence as one shared symbol.</div>
```

After:

```text
      <div class="sub">Dobble <b>comes from a projective plane</b>: any two lines meet in exactly <i>one</i> point, the same sentence as one shared symbol.</div>
```

[social/bluesky/queue/0015-every-two-cards.json:3](/home/liam/aw-claude-assaying-boyle-1014/social/bluesky/queue/0015-every-two-cards.json:3) (before line 3).

Before:

```text
  "text": "Why do any two Dobble cards share exactly one symbol? Not design \u2014 the deck IS a finite projective plane, where \"any two lines meet in one point\" is the same sentence as \"any two cards share one symbol.\" A full order-7 deck: 57 cards, 57 symbols. The box ships 55 (two dropped, a factory call).",
```

After:

```text
  "text": "Why do any two Dobble cards share exactly one symbol? The deck comes from a finite projective plane, where \"any two lines meet in one point\" is the same sentence as \"any two cards share one symbol.\" A full order-7 deck: 57 cards, 57 symbols. The box ships 55 (two dropped, a factory call).",
```

[social/bluesky/queue/0015-every-two-cards.json:7](/home/liam/aw-claude-assaying-boyle-1014/social/bluesky/queue/0015-every-two-cards.json:7) (before line 7).

Before:

```text
    "description": "Dobble is a finite projective plane. Build the deck live mod a prime; the one-shared-symbol rule is a theorem, not a filter. Verifier 51/51. Order 6 is impossible (Bruck-Ryser); order 12 nobody knows.",
```

After:

```text
    "description": "Dobble comes from a finite projective plane with two cards removed. Build the deck live mod a prime; the one-shared-symbol rule is a theorem, not a filter. Verifier 51/51. Order 6 is impossible (Bruck-Ryser); order 12 nobody knows.",
```

[assay/records/every-two-cards.md:1](/home/liam/aw-claude-assaying-boyle-1014/assay/records/every-two-cards.md:1). Before: absent. After: the requested dated claims record, 159 claim lines, 35 findings, and this complete fix report.