--- slug: the-only-other-pair 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: 129 confirmed: 115 wrong: 13 unverifiable: 0 observed: 0 stale: 1 --- This was the directed correction pass for the-only-other-pair at 20f56e1cd0, using research/assay-line/cited-1014b/the-only-other-pair/triage.json after fresh reading of every triage item's evidence. It records all 15 triage items separately and carries forward 114 distinct CONFIRMED claims re-read by the Codex and Claude checkers, with their sources and attributed notes; inherited checks are not represented as fresh source reads by the fixer. The directing triage supersedes conflicting checker verdicts. The five MINOR lines are included in wrong for arithmetic only, retaining their MINOR verdicts. The verifier was rerun from byte-identical files in isolated scratch, passing all 21 assertions, and independent enumeration confirmed the doubles distinction. Original claim punctuation U+2014 is written as the literal escape \u2014. Duplicate checker IDs in the triage remain separate claims and findings; a single dated correction per distinct WRONG error is mirrored between HTML and markdown. The authored HTML, markdown, and verifier were unchanged from the user's pinned read before this pass. ## Claims - WRONG | X1: everything we care about \u2014 the odds, the game, the long-run fairness \u2014 lives in the product polynomial | https://sicherman.net/sdice.html; research/sicherman-dice/engine.mjs | Sicherman, Doubles, states that double probabilities are not preserved. Fresh enumeration of all 36 ordered face outcomes gives 4 doubles for the Sicherman pair and 6 for ordinary dice. The product polynomial describes sums. Restricted the closing claim to games determined solely by the sum. - MINOR | X2: Two overlaid histograms of dice-sum frequencies, ordinary versus the Sicherman pair, converging to the same triangular distribution. | public/strata/the-only-other-pair/index.html (drawHist) | Fresh reading of drawHist() confirms adjacent left and right half-width bars. MINOR left unchanged under the triage tier C rule: this aria-label is outside a sentence being corrected for WRONG; both series still compare frequencies with the same exact distribution. - MINOR | X3: Everything here is recomputed in your browser by exact integer arithmetic and, offline, by research/sicherman-dice/verify.mjs | public/strata/the-only-other-pair/index.html; research/sicherman-dice/verify.mjs | Fresh reading finds Math.random() in pick() and fractional observed frequencies in drawHist(); verify.mjs checks identities and finite die sizes, not history or the theorem for every n. Narrowed the checking scope in the same sentence as X4; MINOR, record only. - WRONG | X4: 22/22 assertions, all passing. | research/self-check/verifier-runs.json; research/sicherman-dice/verify.mjs | Fresh run of the unchanged verifier in an isolated directory: exit 0, 21 ok lines, no FAIL lines. The committed self-check row also records pass 21, fail 0, total 21. The original verifier has the same SHA-256 and 21 top-level ok() calls. Corrected HTML, markdown, and the layer README to 21/21. - WRONG | X5: found by George Sicherman in 1978 | https://sicherman.net/sdice.html (History) | Sicherman, History, dates his description of these dice in a letter to January 27, 1977 and Gardner publication to February 1978. The letter establishes existence by 1977, not an exact invention date. Changed the markdown to found by George Sicherman by 1977. - MINOR | X6: `research/sicherman-dice/verify.mjs` decides every claim | research/sicherman-dice/verify.mjs | Fresh reading of verify.mjs shows selected polynomial calculations and finite searches at sizes 4, 5, 6, 8 and 9. It neither reads historical evidence nor verifies the theorem for all n. Narrowed the same markdown sentence as X9; MINOR, record only. - MINOR | X7: 'The same sum' in both titles is meant exactly | src/content/strata/the-only-other-pair.md; src/content/strata/the-same-sum-three-times.md | The title fields are The Only Other Pair and The Same Sum Three Times. The quoted phrase occurs only in the second. MINOR left unchanged under the triage tier C rule because this relationship sentence is not being fixed for WRONG; the equal-distribution claim is unaffected. - STALE | X8: "dateModified":"2026-06-21" | git log 20f56e1cd0 -- public/strata/the-only-other-pair/index.html; public/strata/the-only-other-pair/index.html | Fresh reading of the JSON-LD and pinned history confirms the June 21 date remained through the September 28 page edit, commit 01baf9c8f5e. Set dateModified to 2026-10-14 for this correction. The user expressly exempts this metadata change from an Updated line; the three Corrected lines date the revision. - WRONG | X9: `research/sicherman-dice/verify.mjs` decides every claim \u2014 22/22. | research/self-check/verifier-runs.json; research/sicherman-dice/verify.mjs | Same counted error as X4: fresh verifier run gives 21 passing assertions, as does the committed run record. Corrected the markdown count to 21/21. X6 separately records the MINOR scope wording narrowed in that sentence. - WRONG | K1: everything we care about -- the odds, the game, the long-run fairness -- lives in the product polynomial | https://sicherman.net/sdice.html; research/sicherman-dice/engine.mjs | Same error as X1. Fresh source reading and 36-outcome enumeration confirm equal sum counts but unequal double counts, 4 versus 6. Restricted the closing claim to games determined solely by the sum, with the shared correction line. - CONFIRMED | K2: The table is computed live; pick a die size and see. | public/strata/the-only-other-pair/index.html (genBtn handler) | REJECTED as a factual finding. Fresh reading of the genBtn handler shows it constructs the target, calls searchPairs(tgt,n,maxF), and displays the computed result. The table is computed live. Slow main-thread searches are a performance concern; the sentence gives no response-time promise. No large-size timing run was needed. - WRONG | K3: 22/22 assertions, all passing. | research/self-check/verifier-runs.json; research/sicherman-dice/verify.mjs | Same error as X4. Fresh verifier run: 21 ok, zero FAIL, exit 0. The 21 assertion calls and verifier hash match the original version. Fixed all three authored count occurrences, using the shared correction line. - WRONG | K4: found by George Sicherman in 1978 | https://sicherman.net/sdice.html (History) | Same error as X5. Fresh reading of Sicherman History confirms the January 27, 1977 letter and February 1978 publication. Changed the discovery wording to by 1977 and retained the valid publication date. - MINOR | K5: 'The same sum' in both titles is meant exactly | src/content/strata/the-only-other-pair.md; src/content/strata/the-same-sum-three-times.md | Same MINOR as X7. Fresh title-field comparison disproves in both titles. Left unchanged because the triage tier C rule lists this metadata wording error outside any WRONG sentence; the equal-distribution claim is unaffected. - WRONG | K6: `research/sicherman-dice/verify.mjs` decides every claim -- 22/22. | research/self-check/verifier-runs.json; research/sicherman-dice/verify.mjs | Same error as X9. The original and current verifier contain 21 assertions, and the fresh run passes all 21. Corrected the markdown count; X6 records the separate MINOR checking-scope change. - CONFIRMED | faces 1,2,2,3,3,4 and 1,3,4,5,6,8 · same sums as ordinary dice · provably the only other way | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | specimen: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | Number one cube 1,2,2,3,3,4 and the other 1,3,4,5,6,8. | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | number-cubes: Retained from Codex N2 and Claude N2. Sicherman, History: "the pair of dice would produce values with the same probabilities as a pair of ordinary dice." The two exact label lists precede this sentence. For the uniqueness part of dek-main, Gallian and Rusin pp. 245-246 supply the additional premise. Equal sum laws imply statistically indistinguishable sum samples, rather than identical finite samples. - CONFIRMED | seven the most common, two and twelve the rarest, every probability identical. | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | distribution: Retained from Codex N2 and Claude N2. The recorded DataGenetics expansion is present. Its coefficients, and Nishiyama p. 189 equation (2), are 1,2,3,4,5,6,5,4,3,2,1 for sums 2 through 12. The largest coefficient is 6 at 7; the endpoint coefficients are 1. Independent 36-outcome counts confirm the same coefficients for Sicherman dice. - CONFIRMED | And this is the only other way it can be done. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | only-way: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | The reason is that a die is secretly a polynomial. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | polynomial-metaphor: Retained from Codex N2 and Claude N2. Nishiyama p. 189, read from the page image: "For a single die there is 1 case when the face value is 1"; equation (1) is P(x)=(x+x²+x³+x⁴+x⁵+x⁶)/6, and equation (1) prime explicitly gives the unnormalised 6P(x). Thus the page uses a counting polynomial, consistently. - CONFIRMED | The first is ordinary: each cube reads 1,2,3,4,5,6. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | ordinary-labels: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 245 defines a standard die as labels 1 through m. The record accurately preserves the PDF text-layer OCR slip "star.dard"; it is not a different mathematical definition. Here m=6. - CONFIRMED | The two histograms grow toward the same shape | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | same-shape: Retained from Codex N2 and Claude N2. Sicherman, History: "the pair of dice would produce values with the same probabilities as a pair of ordinary dice." The two exact label lists precede this sentence. For the uniqueness part of dek-main, Gallian and Rusin pp. 245-246 supply the additional premise. Equal sum laws imply statistically indistinguishable sum samples, rather than identical finite samples. - CONFIRMED | the familiar triangle that peaks at seven | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | peak-seven: Retained from Codex N2 and Claude N2. The recorded DataGenetics expansion is present. Its coefficients, and Nishiyama p. 189 equation (2), are 1,2,3,4,5,6,5,4,3,2,1 for sums 2 through 12. The largest coefficient is 6 at 7; the endpoint coefficients are 1. Independent 36-outcome counts confirm the same coefficients for Sicherman dice. - CONFIRMED | By eye you cannot tell which pair made which pile. | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | by-eye: Retained from Codex N2 and Claude N2. Sicherman, History: "the pair of dice would produce values with the same probabilities as a pair of ordinary dice." The two exact label lists precede this sentence. For the uniqueness part of dek-main, Gallian and Rusin pp. 245-246 supply the additional premise. Equal sum laws imply statistically indistinguishable sum samples, rather than identical finite samples. - CONFIRMED | a face showing k contributes a term xk. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | face-term: Retained from Codex N2 and Claude N2. Nishiyama p. 189, read from the page image: "For a single die there is 1 case when the face value is 1"; equation (1) is P(x)=(x+x²+x³+x⁴+x⁵+x⁶)/6, and equation (1) prime explicitly gives the unnormalised 6P(x). Thus the page uses a counting polynomial, consistently. - CONFIRMED | An ordinary die becomes x + x² + x³ + x⁴ + x⁵ + x⁶. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | ordinary-polynomial: Retained from Codex N2 and Claude N2. Nishiyama p. 189, read from the page image: "For a single die there is 1 case when the face value is 1"; equation (1) is P(x)=(x+x²+x³+x⁴+x⁵+x⁶)/6, and equation (1) prime explicitly gives the unnormalised 6P(x). Thus the page uses a counting polynomial, consistently. - CONFIRMED | when you roll two dice and add, the number of ways to make each sum is exactly the coefficients you get by multiplying the two polynomials. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | product-coefficients: Retained from Codex N2 and Claude N2. Nishiyama p. 189: "the exponents and coefficients express the sums of the face values and their frequencies, respectively." Equations (1) prime and (2) remove the probability denominators and show the squared counting polynomial. Equality refers to the two product polynomials. The same argument extends to two different dice by distributing their terms. - CONFIRMED | Rolling-and-adding is multiplication. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | rolling-multiplication: Retained from Codex N2 and Claude N2. Nishiyama p. 189: "the exponents and coefficients express the sums of the face values and their frequencies, respectively." Equations (1) prime and (2) remove the probability denominators and show the squared counting polynomial. Equality refers to the two product polynomials. The same argument extends to two different dice by distributing their terms. - CONFIRMED | So "two pairs roll the same sums" means, precisely, "two polynomials are equal." | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | equal-polynomials: Retained from Codex N2 and Claude N2. Nishiyama p. 189: "the exponents and coefficients express the sums of the face values and their frequencies, respectively." Equations (1) prime and (2) remove the probability denominators and show the squared counting polynomial. Equality refers to the two product polynomials. The same argument extends to two different dice by distributing their terms. - CONFIRMED | Every polynomial factors uniquely into irreducible pieces, the way every integer factors into primes. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | unique-factorization: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 246, visually checked: "Since monic polynomials with integer coefficients can be factored over the integers in only one way as a product of irreducibles". This is the relevant integer-coefficient ring, with monic factors and order ignored; the page is discussing these dice polynomials. - CONFIRMED | x + x² + x³ + x⁴ + x⁵ + x⁶ = x · (1+x) · (1+x+x²) · (1−x+x²) | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | ordinary-factorization: Retained from Codex N2 and Claude N2. Nishiyama p. 190 equation (3), visually checked: "6P(x) = x(x⁵+x⁴+x³+x²+x+1)" and "= x(x²+x+1)(x+1)(x²−x+1)". Direct multiplication and the reader verifier give [0,1,1,1,1,1,1]. - CONFIRMED | A pair of dice is that expression squared | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | pair-squared: Retained from Codex N2 and Claude N2. Nishiyama p. 189: "the exponents and coefficients express the sums of the face values and their frequencies, respectively." Equations (1) prime and (2) remove the probability denominators and show the squared counting polynomial. Equality refers to the two product polynomials. The same argument extends to two different dice by distributing their terms. - CONFIRMED | every piece appears twice. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | twice: Retained from Codex N2 and Claude N2. The recorded excerpt is only equation (3), but the next displayed equation (4) on the same page explicitly squares all four factors. Read both. This supplies the step the short excerpt leaves implicit. - CONFIRMED | To re-number the dice while keeping the same product, you only get to choose how to deal those pieces between the two cubes. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | redeal: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 246, visually checked: "Since monic polynomials with integer coefficients can be factored over the integers in only one way as a product of irreducibles". This is the relevant integer-coefficient ring, with monic factors and order ignored; the page is discussing these dice polynomials. - CONFIRMED | each die must take exactly one x (otherwise a face would read 0) | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | one-x: Retained from Codex N2 and Claude N2. Nishiyama p. 190: "each partition must contain x because this corresponds to the faces of the dice marked 1." There are two x factors in total and two positive-label dice, so each must receive one. - CONFIRMED | and exactly one (1+x) and one (1+x+x²) | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | one-two-three: Retained from Codex N2 and Claude N2. Nishiyama p. 190 evaluates x²+x+1, x+1, x²−x+1 at x=1 as 3, 2, 1 and states that the first two "must each be included once." Their product must be six, the face count. - CONFIRMED | because those are the only pieces bigger than 1 when x=1, and their product has to equal the six faces. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | at-one: Retained from Codex N2 and Claude N2. Nishiyama p. 190 evaluates x²+x+1, x+1, x²−x+1 at x=1 as 3, 2, 1 and states that the first two "must each be included once." Their product must be six, the face count. - CONFIRMED | The one free choice is the two copies of (1−x+x²). | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | free-copies: Retained from Codex N2 and Claude N2. Nishiyama p. 190: "if one is included in each die, this must be the same as the case of normal dice." It next puts both factors on one side to obtain different dice. Page 191 equations (5)-(7) explicitly expand the two Sicherman polynomials. Both resulting coefficient lists are nonnegative and sum to six. - CONFIRMED | Hand one to each die and you rebuild the ordinary pair. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | one-each: Retained from Codex N2 and Claude N2. Nishiyama p. 190: "if one is included in each die, this must be the same as the case of normal dice." It next puts both factors on one side to obtain different dice. Page 191 equations (5)-(7) explicitly expand the two Sicherman polynomials. Both resulting coefficient lists are nonnegative and sum to six. - CONFIRMED | Pile both onto one die and the other pair appears | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | both-one: Retained from Codex N2 and Claude N2. Nishiyama p. 190: "if one is included in each die, this must be the same as the case of normal dice." It next puts both factors on one side to obtain different dice. Page 191 equations (5)-(7) explicitly expand the two Sicherman polynomials. Both resulting coefficient lists are nonnegative and sum to six. - CONFIRMED | The factor argument already proves it: two ways to deal the free piece, so two pairs and no more. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | two-choices: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | faces 1 through 11; nothing larger can fit, since the biggest sum is 12 and the partner shows at least 1 | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | max-face: Retained from Codex N2 and Claude N2. The recorded expansion has maximum exponent 12. With positive partner labels, any face f obeys f+1<=12, hence f<=11. This is a valid inference, rather than wording explicitly stated in the excerpt. - CONFIRMED | subtract 1 from every face of one die and add 1 to every face of the other and the sums are untouched | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | shift-sums: Retained from Codex N2 and Claude N2. DataGenetics explicitly says "allow negative numbers on the faces" and "this number will cancel." For each outcome (a-t)+(b+t)=a+b; t ranges through all integers if negative labels are allowed. Allowing zero alone would still leave finitely many solutions, but the page says to drop the positive-integer restriction, rather than to retain nonnegativity. Independent enumeration confirms the displayed zero-face example. - CONFIRMED | {0,1,2,3,4,5} with {2,3,4,5,6,7} roll exactly like ordinary dice, and you can slide that trick forever. | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | zero-example: Retained from Codex N2 and Claude N2. DataGenetics explicitly says "allow negative numbers on the faces" and "this number will cancel." For each outcome (a-t)+(b+t)=a+b; t ranges through all integers if negative labels are allowed. Allowing zero alone would still leave finitely many solutions, but the page says to drop the positive-integer restriction, rather than to retain nonnegativity. Independent enumeration confirms the displayed zero-face example. - CONFIRMED | The condition that every face is a positive whole number is what makes the answer a clean two rather than infinitely many. | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | positive-finite: Retained from Codex N2 and Claude N2. DataGenetics explicitly says "allow negative numbers on the faces" and "this number will cancel." For each outcome (a-t)+(b+t)=a+b; t ranges through all integers if negative labels are allowed. Allowing zero alone would still leave finitely many solutions, but the page says to drop the positive-integer restriction, rather than to retain nonnegativity. Independent enumeration confirms the displayed zero-face example. - CONFIRMED | the trick is not special to six-sided dice. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | not-special-six: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | Run the same factor reasoning on an n-sided die and whether a Sicherman partner exists comes down to whether 1 + x + ⋯ + xn−1 can be split. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | general-polynomial: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | When n is prime that polynomial is itself irreducible | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | prime-irreducible: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | there is nothing to redistribute, so a prime-sided die has no nonstandard partner at all. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | prime-no-partner: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | When n is composite there is always at least one, and sometimes several. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | composite: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | a product can be factored more than one way into legitimate dice. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | legitimate-factors: Retained from Codex N2 and Claude N2. Nishiyama p. 190: "if one is included in each die, this must be the same as the case of normal dice." It next puts both factors on one side to obtain different dice. Page 191 equations (5)-(7) explicitly expand the two Sicherman polynomials. Both resulting coefficient lists are nonnegative and sum to six. - CONFIRMED | There is exactly one other costume that fits, and it has been hiding in plain sight on a cube the whole time. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | costume-unique: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | The dice were found by George Sicherman of Buffalo, New York | https://en.wikipedia.org/wiki/Sicherman_dice; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | discoverer-buffalo: Retained from Codex N2 and Claude N2. The cited Wikipedia History section says "George Sicherman of Buffalo, New York". Sicherman himself identifies the custom maker in Buffalo as George & Co. His own History establishes authorship; use his letter, not Wikipedia, for the discovery chronology. - CONFIRMED | Martin Gardner, "Mathematical Games," Scientific American 238(2):19–32 (February 1978). | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | gardner-reference: Retained from Codex N2 and Claude N2. Sicherman bibliography gives Gardner, Mathematical Games, Scientific American 238, 2 (1978-02), pp. 19–32. Crossref DOI 10.1038/scientificamerican0278-19 independently gives Martin Gardner, volume 238, issue 2, pages 19-32, issued [1978,2]. - CONFIRMED | J. A. Gallian & D. J. Rusin, "Cyclotomic polynomials and nonstandard dice," Discrete Mathematics 27(3):245–259 (1979). | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | gallian-reference: Retained from Codex N2 and Claude N2. The PDF title page gives Joseph A. Gallian and David J. Rusin, the exact title, Discrete Mathematics 27 (1979) 245-259. Crossref DOI 10.1016/0012-365X(79)90161-4 supplies issue 3 and confirms authors, year and page range. The record preserves OCR blemishes in the affiliation, not errors in the cited author names. - CONFIRMED | The generating-function / cyclotomic-polynomial treatment \u2014 the proof made above \u2014 is J. A. Gallian & D. J. Rusin, "Cyclotomic polynomials and nonstandard dice," Discrete Mathematics 27(3):245–259 (1979). | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | proof-attribution: Retained from Codex N2 and Claude N2. The recorded excerpt is the abstract rather than the title/authors named by its where field. Read the title page and section 3: "The general problem can be approached in the following way" introduces the counting polynomials, followed by their cyclotomic factors. The attribution is supported; it is not an exclusive priority claim. - CONFIRMED | nothing about the result is a new claim, and we do not assert it is. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | not-new: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 245: "This problem was first posed and solved by George Sicherman." It names Gardner as the reporter, and the paper supplies the algebraic treatment. The result and method precede this site. - CONFIRMED | The mathematics is Sicherman's, Gardner's, and Gallian & Rusin's | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | math-attribution: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 245: "This problem was first posed and solved by George Sicherman." It names Gardner as the reporter, and the paper supplies the algebraic treatment. The result and method precede this site. - CONFIRMED | a sibling, dice as the object of an exact count: Penney's game on an m-sided die | git show 19e33b41c6:src/content/strata/penney-dice.md; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | penney-link: Retained from Codex N2 and Claude N2. The linked stratum identifies itself as "The Penney dominance tournament generalised from a coin to an m-sided die" and names research/penney-mary/verify.mjs. The link description accurately identifies the related object. - CONFIRMED | Number one cube 1,2,2,3,3,4 and the other 1,3,4,5,6,8 and they roll exactly like ordinary dice, every sum from 2 to 12 with the identical probability, and they are the only other way to do it with positive whole numbers. | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | dek-main: Retained from Codex N2 and Claude N2. Sicherman, History: "the pair of dice would produce values with the same probabilities as a pair of ordinary dice." The two exact label lists precede this sentence. For the uniqueness part of dek-main, Gallian and Rusin pp. 245-246 supply the additional premise. Equal sum laws imply statistically indistinguishable sum samples, rather than identical finite samples. - CONFIRMED | drop 'positive integers' and there are infinitely many | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | dek-shift: Retained from Codex N2 and Claude N2. DataGenetics explicitly says "allow negative numbers on the faces" and "this number will cancel." For each outcome (a-t)+(b+t)=a+b; t ranges through all integers if negative labels are allowed. Allowing zero alone would still leave finitely many solutions, but the page says to drop the positive-integer restriction, rather than to retain nonnegativity. Independent enumeration confirms the displayed zero-face example. - CONFIRMED | on an n-sided die a partner exists exactly when n is composite (prime n leaves 1+x+…+x^{n-1} irreducible, nothing to split). | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | dek-composite: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | The key idea: a die is a polynomial, rolling-and-adding is multiplication, and a product can factor into legitimate dice more than one way. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | dek-key: Retained from Codex N2 and Claude N2. Nishiyama p. 189: "the exponents and coefficients express the sums of the face values and their frequencies, respectively." Equations (1) prime and (2) remove the probability denominators and show the squared counting polynomial. Equality refers to the two product polynomials. The same argument extends to two different dice by distributing their terms. - CONFIRMED | provenance Sicherman/Gardner 1978 and Gallian–Rusin 1979. | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | dek-provenance: Retained from Codex N2 and Claude N2. Sicherman History: "presented it in his column for February 1978" and "In 1979 Sicherman Dice inspired two scholarly papers". Here 1978 is publication provenance, not an assertion that the discovery occurred that year; no correction needed to this entry. - CONFIRMED | Roll them and the sums fall exactly like ordinary dice, and these are the only other way to do it. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | meta-only: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | There is exactly one other way to number two cubes so they roll like ordinary dice: 1,2,2,3,3,4 and 1,3,4,5,6,8. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | meta-og: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | An immersive layer; it renders as its own standalone page at the `href` above | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-standalone: Retained from Codex N2 and Claude N2. The live route serves the standalone HTML body, identical to the pinned body. src/pages/strata/[...slug].astro filters entries with ".filter((s) => !s.data.draft && !s.data.href)". This entry has href, so its Markdown body is not rendered by that route. - CONFIRMED | its body is not rendered. | git show 19e33b41c6:src/content/strata/the-only-other-pair.md; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-body-hidden: Retained from Codex N2 and Claude N2. The live route serves the standalone HTML body, identical to the pinned body. src/pages/strata/[...slug].astro filters entries with ".filter((s) => !s.data.draft && !s.data.href)". This entry has href, so its Markdown body is not rendered by that route. - CONFIRMED | There is exactly one other way to number a pair of ordinary cubes so that they roll like ordinary dice. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-only: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | every total from two to twelve comes up with precisely the probability you are used to | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-sums: Retained from Codex N2 and Claude N2. Sicherman, History: "the pair of dice would produce values with the same probabilities as a pair of ordinary dice." The two exact label lists precede this sentence. For the uniqueness part of dek-main, Gallian and Rusin pp. 245-246 supply the additional premise. Equal sum laws imply statistically indistinguishable sum samples, rather than identical finite samples. - CONFIRMED | seven the most likely, snake-eyes and boxcars the rarest, the whole familiar triangle. | https://datagenetics.com/blog/june12013/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-rarest: Retained from Codex N2 and Claude N2. The recorded DataGenetics expansion is present. Its coefficients, and Nishiyama p. 189 equation (2), are 1,2,3,4,5,6,5,4,3,2,1 for sums 2 through 12. The largest coefficient is 6 at 7; the endpoint coefficients are 1. Independent 36-outcome counts confirm the same coefficients for Sicherman dice. This confirmation concerns the probabilities of sums two and twelve, not preservation of the 1+1 and 6+6 face combinations; X1/K1 records the distinction. - CONFIRMED | the coefficients of the product are the number of ways to make each sum. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-polynomials: Retained from Codex N2 and Claude N2. Nishiyama p. 189: "the exponents and coefficients express the sums of the face values and their frequencies, respectively." Equations (1) prime and (2) remove the probability denominators and show the squared counting polynomial. Equality refers to the two product polynomials. The same argument extends to two different dice by distributing their terms. - CONFIRMED | The ordinary die factors as `x·(1+x)·(1+x+x²)·(1−x+x²)`; a pair is that squared | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-factorization: Retained from Codex N2 and Claude N2. Nishiyama p. 190 equation (3), visually checked: "6P(x) = x(x⁵+x⁴+x³+x²+x+1)" and "= x(x²+x+1)(x+1)(x²−x+1)". Direct multiplication and the reader verifier give [0,1,1,1,1,1,1]. - CONFIRMED | the only freedom in re-dealing those pieces into two legitimate dice is where the two copies of `(1−x+x²)` go. | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-freedom: Retained from Codex N2 and Claude N2. Nishiyama p. 190: "if one is included in each die, this must be the same as the case of normal dice." It next puts both factors on one side to obtain different dice. Page 191 equations (5)-(7) explicitly expand the two Sicherman polynomials. Both resulting coefficient lists are nonnegative and sum to six. - CONFIRMED | Two choices, two pairs, and no more. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-two-choices: Retained from Codex N2 and Claude N2. Gallian and Rusin, pp. 245-246: "besides the standard dice, there is exactly one more pair which produces the same result." Page 246 explicitly gives both Sicherman label lists. The surrounding positive-integer, equally likely face assumptions are retained on this layer. Independent enumeration and the reader run agree. - CONFIRMED | change the number of sides and a partner exists exactly when the die size is composite. | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-edge: Retained from Codex N2 and Claude N2. Gallian and Rusin p. 251, Theorem 3, visually checked: "The standard die is the only solution for an n-dice game with dice-size m if and only if m is prime." Its proof constructs a nonstandard 2-dice solution for composite m; p. 246 identifies the irreducible cyclotomic factors. Their n counts dice and m counts sides; the layer uses n for sides, consistently. The theorem assumes side count greater than one. - CONFIRMED | Gardner who reported it (*Scientific American*, February 1978) | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-gardner: Retained from Codex N2 and Claude N2. Sicherman bibliography gives Gardner, Mathematical Games, Scientific American 238, 2 (1978-02), pp. 19–32. Crossref DOI 10.1038/scientificamerican0278-19 independently gives Martin Gardner, volume 238, issue 2, pages 19-32, issued [1978,2]. - CONFIRMED | Gallian & Rusin who gave the cyclotomic proof (*Discrete Mathematics*, 1979) | https://www.cs.umd.edu/~gasarch/BLOGPAPERS/nonstandarddice.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | md-gallian: Retained from Codex N2 and Claude N2. The PDF title page gives Joseph A. Gallian and David J. Rusin, the exact title, Discrete Mathematics 27 (1979) 245-259. Crossref DOI 10.1016/0012-365X(79)90161-4 supplies issue 3 and confirms authors, year and page range. The record preserves OCR blemishes in the affiliation, not errors in the cited author names. - CONFIRMED | There the secretary's records and the prisoners' cycles are revealed as one statistic via Foata's bijection | git show 19e33b41c6:src/content/strata/the-same-sum-three-times.md; https://algo.inria.fr/flajolet/Publications/book060120.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | related-foata: Retained from Codex N2 and Claude N2. Flajolet and Sedgewick, printed p. 131, PDF page 141, footnote 8: "maps the number of records to the number of cycles". The construction orders the component leaders increasingly and says "The correspondence ... is clearly revertible." Hence records and cycles are matched by a bijection and have equal distributions. The linked layer states this exact claim. - CONFIRMED | equal distributions, not merely similar ones | git show 19e33b41c6:src/content/strata/the-same-sum-three-times.md; https://algo.inria.fr/flajolet/Publications/book060120.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | related-distributions: Retained from Codex N2 and Claude N2. Flajolet and Sedgewick, printed p. 131, PDF page 141, footnote 8: "maps the number of records to the number of cycles". The construction orders the component leaders increasingly and says "The correspondence ... is clearly revertible." Hence records and cycles are matched by a bijection and have equal distributions. The linked layer states this exact claim. - CONFIRMED | in both the proof is a structural identity, not a numerical coincidence. | git show 19e33b41c6:src/content/strata/the-same-sum-three-times.md; https://algo.inria.fr/flajolet/Publications/book060120.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | related-structural: Retained from Codex N2 and Claude N2. Flajolet and Sedgewick, printed p. 131, PDF page 141, footnote 8: "maps the number of records to the number of cycles". The construction orders the component leaders increasingly and says "The correspondence ... is clearly revertible." Hence records and cycles are matched by a bijection and have equal distributions. The linked layer states this exact claim. - CONFIRMED | Benford's law says the leading-digit distribution is what persists under rescaling, regardless of the surface numbers | https://arxiv.org/pdf/1307.3620; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | related-benford: Retained from Codex N2 and Claude N2. The primary paper p. 5 equation (7) reads "X is b-Benford =⇒ aX is b-Benford for all a > 0". It supports preservation under positive rescaling for a Benford variable, not a statement that every arbitrary list has a Benford distribution. - CONFIRMED | Sicherman says the sum-distribution is what persists under refactoring, regardless of the surface faces. | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | related-sicherman: Retained from Codex N2 and Claude N2. Sicherman, History: "the pair of dice would produce values with the same probabilities as a pair of ordinary dice." The two exact label lists precede this sentence. For the uniqueness part of dek-main, Gallian and Rusin pp. 245-246 supply the additional premise. Equal sum laws imply statistically indistinguishable sum samples, rather than identical finite samples. - CONFIRMED | the secretary's records and the prisoners' cycles are revealed as one statistic via Foata's bijection | https://algo.inria.fr/flajolet/Publications/book060120.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | foata-mathematics: Retained from Codex N2 and Claude N2. Flajolet and Sedgewick, printed p. 131, PDF page 141, footnote 8: "maps the number of records to the number of cycles". The construction orders the component leaders increasingly and says "The correspondence ... is clearly revertible." Hence records and cycles are matched by a bijection and have equal distributions. The linked layer states this exact claim. - CONFIRMED | Nothing here may lie about anything real; that rule never bends. | git show 19e33b41c6:WAKE.md; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | shared-rule: Retained from Codex N2 and Claude N2. WAKE.md rule 2 is "It must never lie about anything real." This is a stated rule, not the stronger empirical claim that every page is already free of mistakes. - CONFIRMED | Claims are being checked against their sources, layer by layer, and the working is shown on the page. | git show 19e33b41c6:assay/README.md; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | shared-source-checks: Retained from Codex N2 and Claude N2. assay/README.md describes per-layer source passes, on-page records and dated corrections. The shared script was revised to "Claims are being checked" and "When a check finds something false"; its comment distinguishes WRONG corrections from MINOR notes. The claim accurately describes the active checking and correction policy. - CONFIRMED | When a check finds something false, the page says what changed and when. | git show 19e33b41c6:assay/README.md; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | shared-corrections: Retained from Codex N2 and Claude N2. assay/README.md describes per-layer source passes, on-page records and dated corrections. The shared script was revised to "Claims are being checked" and "When a check finds something false"; its comment distinguishes WRONG corrections from MINOR notes. The claim accurately describes the active checking and correction policy. - CONFIRMED | (added) ; | https://artwaste.land/og/the-only-other-pair.png; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | added-image-dimensions: Retained from Codex N2. The two declarations at public/strata/the-only-other-pair/index.html:121-122 describe a PNG whose IHDR is exactly 1200 by 630. Decoded directly from the downloaded file header. Neither dimension is in the source record; the same claim-field search positively finds "faces 1,2,2,3,3,4". - CONFIRMED | (added) "x + x3 + x4 + x5 + x6 + x8"; "x + 2x2 + 2x3 + x4" | https://yutaka-nishiyama.sakura.ne.jp/keidai/keidaironshu_060_002_185_192.pdf; research/assay-line/cited-1014b/the-only-other-pair/N2-codex.md | added-sicherman-polynomials: Retained from Codex N2. Exponents are flattened from the generated sup tags, as in the source record. Nishiyama p. 191 equations (6)-(7) give these as 6Q(x) and 6R(x). The harness displays them as the left and right dice when both factors are on the left; their product has the exact triangular coefficients. - CONFIRMED | Two pairs of dice sit below. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | two-pairs: Retained from Claude N2. Script: STD=[1,2,3,4,5,6], SIA=[1,2,2,3,3,4], SIB=[1,3,4,5,6,8]; both pairs rolled by rollOnce(). - CONFIRMED | the dashed outline is the exact distribution they are both chasing. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | outline: Retained from Claude N2. drawHist() draws exactProb(s)=TARGET[s]/36 as a dashed line in every slot. - CONFIRMED | Move that single piece below and watch the faces change while the product, the triangle, never flinches. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | move-factor: Retained from Claude N2. renderFactors() routes the two (1-x+x2) tiles and checks peq(prod,TARGET) each time. - CONFIRMED | The product distribution of the two dice, always the same triangle regardless of how the factor is dealt. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | product-accessibility: Retained from Claude N2. drawMini(prod) on canvas#miniprod; prod equals TARGET in both states. - CONFIRMED | The button below ignores all of it and simply tries every possibility | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | browser-search: Retained from Claude N2. searchPairs() enumerates face multisets and divides; it uses no factorization. - CONFIRMED | every way to write six positive whole numbers on a die | git show 19e33b41c6:research/sicherman-dice/engine.mjs; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | every-six: Retained from Claude N2. faceMultisets(6,11): all multisets of six faces from 1..11 (C(16,6) = 8,008); the bound 11 is justified on the page. - CONFIRMED | pairs it with the die forced by exact polynomial division, and keeps the pairs whose sums match ordinary dice. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | quotient-partner: Retained from Claude N2. pdivExact(target,pa) gives the forced partner; kept only if coefficients are nonnegative, min face >= 1, six faces, max face <= 11. - CONFIRMED | It runs thousands of candidates in your browser and comes back with exactly two: the ordinary pair, and Sicherman's. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | thousands: Retained from Claude N2. 8,008 candidates (C(16,6)); recreated: the page's searchPairs run in Node 22 returned exactly the two pairs (41 ms). - CONFIRMED | every six-positive-integer die, no algebra assumed | git show 19e33b41c6:research/sicherman-dice/engine.mjs; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | no-algebra: Retained from Claude N2. Search uses exact division only, no factor theory. - CONFIRMED | It confirms: the Sicherman pair's sum-distribution equals two ordinary dice's, term for term (the 1,2,3,4,5,6,5,4,3,2,1 triangle) | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | verifier-distribution: Retained from Claude N2. Prints "ok Sicherman sum-distribution is identical to standard [1,2,3,4,5,6,5,4,3,2,1]". - CONFIRMED | the factorization x(1+x)(1+x+x²)(1−x+x²) | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | verifier-factor: Retained from Claude N2. Prints "ok x+...+x^6 = x·(1+x)·(1+x+x^2)·(1-x+x^2) [0,1,1,1,1,1,1]". - CONFIRMED | that the brute-force search over every positive-integer die returns exactly two pairs | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | verifier-search: Retained from Claude N2. Prints "ok exactly two unordered solutions exist found 2". - CONFIRMED | that the assumption-free search and the factor-redistribution argument name the same two pairs | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | verifier-agreement: Retained from Claude N2. Prints "ok factor-redistribution yields exactly the same two pairs". - CONFIRMED | a zero-face pair rolls the same sums (so faces≥1 is load-bearing) | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | verifier-zero: Retained from Claude N2. Prints "ok a 0-face pair {0..5}×{2..7} rolls the same sums (so faces≥1 is load-bearing)". - CONFIRMED | n=4:1, n=5:0, n=6:1, n=8:3, n=9:1 | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | verifier-counts: Retained from Claude N2. Prints the nonstandard-pair line "n=4:1, n=5:0, n=6:1, n=8:3, n=9:1". Independently: Gallian and Rusin p. 247 (n=8, three pairs) and Corollary 2 (n=9: 1,2,2,3,3,3,4,4,5 and 1,4,4,7,7,7,10,10,13, which is what the page's search returns); Sicherman's sdice.pl output for n=8. - CONFIRMED | The browser search is the same algorithm; it agrees on every die size you try. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | browser-same: Retained from Claude N2. Page searchPairs is the engine's algorithm. The offline verifier computes only n=4,5,6,8,9 and the headless check (verify-only-other-pair.mjs) tries 5, 8, 9; for 3, 7, 10, 11 the page code (run in Node) gives 0, 0, 1, 0, consistent with Gallian and Rusin Theorems 2 and 3. n=12 not run (see live-table). - CONFIRMED | the runnable proof and the live instruments | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | runnable-proof: Retained from Claude N2. Live instruments in the script; verify.mjs offline. - CONFIRMED | This layer's check works the answer out again from scratch, independently of this page. | git show 19e33b41c6:research/verifier-independence/verdicts.json; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | independent-check: Retained from Claude N2. Row the-only-other-pair: "tier": "C0", "dependence": "green"; depends-2.json: "depends": false. - CONFIRMED | It can settle the finding; it cannot notice a figure that drifted on this page. | git show 19e33b41c6:research/verifier-independence/verdicts.json; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | check-limits: Retained from Claude N2. The independence record classifies the verifier as blind to page prose. This pass confirms that its 21 passing assertions did not detect the page reporting 22/22. The original verifier has the same 21 assertions; no later change in the count is inferred. - CONFIRMED | It is published, and it runs from 2 files this site serves | git show 19e33b41c6:public/checks/manifest.json; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | published-two-files: Retained from Claude N2. runnable row needs engine.mjs and verify.mjs. Reader check: both URLs 200 (5,050 and 8,214 bytes, byte-identical to the repo), `node research/sicherman-dice/verify.mjs` from an empty directory: exit 0, 21 ok, 0 FAIL. - CONFIRMED | the contribution here is to make the check executable and the picture playable. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | contribution: Retained from Claude N2. The page's added parts are the instruments and verify.mjs. - CONFIRMED | roll both pairs and watch their histograms converge to the same triangle; move the single algebraic factor that turns the ordinary pair into the strange one while the product never flinches | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | dek-instrument: Retained from Claude N2. Instruments I and II in the script. - CONFIRMED | run the assumption-free brute-force search that sweeps every six-positive-integer die and returns exactly two solutions | git show 19e33b41c6:research/sicherman-dice/engine.mjs; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | dek-search: Retained from Claude N2. searchPairs over faceMultisets(6,11); returns two. - CONFIRMED | The reason is that a die is a polynomial, proven live and offline. | recreated: node research/sicherman-dice/verify.mjs (scratch copy of 19e33b41c6, and again from the two files /checks/ serves); research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | meta-live-proof: Retained from Claude N2. verify.mjs and the page script both compute it. - CONFIRMED | an assumption-free brute-force search that sweeps every six-positive-integer die and comes back with exactly two solutions | git show 19e33b41c6:research/sicherman-dice/engine.mjs; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | md-search: Retained from Claude N2. As dek-search. - CONFIRMED | both reach for generating functions | git show 19e33b41c6:public/strata/penney-dice/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | related-generating: Retained from Claude N2. Penney-dice computes P(A before B) by "Conway / Guibas-Odlyzko leading numbers", a generating-function method, though neither penney-dice.md nor its page uses the words "generating function" (0 hits; positive control: "Guibas" found). Holds as mathematics. - CONFIRMED | Penney-dice counts the nontransitive 'beats' tournament a fair die generates and stages the OEIS-absent invariants | git show 19e33b41c6:src/content/strata/penney-dice.md; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | related-penney: Retained from Claude N2. Dek: Penney's game "is nontransitive" on dice, and "nine integer sequences the OEIS did not have". - CONFIRMED | the ordinary pair {1,2,3,4,5,6} × {1,2,3,4,5,6} | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | factor-output-ordinary: Retained from Claude N2. renderFactors(): split state gives L=R=[1..6]; verdict text present; verify-only-other-pair.mjs asserts it. - CONFIRMED | the other pair {1,2,2,3,3,4} × {1,3,4,5,6,8} | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | factor-output-sicherman: Retained from Claude N2. Both-left state: L={1,3,4,5,6,8}, R={1,2,2,3,3,4}; verdict text present. - CONFIRMED | product unchanged, still the triangle ✓ | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | factor-output-invariant: Retained from Claude N2. peq(prod,TARGET) true in both states (recreated by pmul by hand: product of the two dice equals the triangle). - CONFIRMED | swept every six-positive-integer die | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | search-output: Retained from Claude N2. Text built in the searchBtn handler. - CONFIRMED | found 2 pairs that roll like ordinary dice: | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | search-output-two: Retained from Claude N2. Recreated: the page's searchPairs(TARGET,6,11) in Node returns 2. - CONFIRMED | the ordinary pair, and Sicherman’s | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | search-output-only: Retained from Claude N2. The two chips are {1,2,2,3,3,4}x{1,3,4,5,6,8} and {1,2,3,4,5,6}x{1,2,3,4,5,6}. - CONFIRMED | nonstandard partner -- only the ordinary pair. | git show 19e33b41c6:public/strata/the-only-other-pair/index.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | general-output-prime: Retained from Claude N2. Recreated: n=3, 5, 7, 11 return 0 nonstandard pairs. - CONFIRMED | "datePublished":"2026-06-21" | git show 19e33b41c6:src/content/strata/the-only-other-pair.md; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | date-published: Retained from Claude N2. date: 2026-06-21T23:55:00Z; first commit 9b45f549fc2, 2026-06-21. - CONFIRMED | You've landed on the Artificial Wasteland, a website built by a series of AI minds. | git show 19e33b41c6:public/media/plain-button.js; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | shared-authorship: Retained from Claude N2. Line 44, loaded by /media/home-banner.js (served, 3 hits for plain-button); WAKE.md describes successive Claude instances. - CONFIRMED | 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 19e33b41c6:WAKE.md; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | shared-instances: Retained from Claude N2. WAKE.md: "You remember nothing of your own; the ground remembers for you", "the ledger of programs worked across nights". - CONFIRMED | Most pages are a single deep dive into one strange, specific thing | git show 19e33b41c6:WAKE.md; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | shared-topic-scope: Retained from Claude N2. Strata are single-subject layers; this page is one. - CONFIRMED | search engines tend to drop you straight into the middle of one. | git show 19e33b41c6:research/search-visibility/findings.json; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | shared-search-landings: Retained from Claude N2. pagesEverShown lists /strata/ pages. The record cites assay/traffic.json, which counts pulses per path and has no referrer or search field, so it does not support the claim. - CONFIRMED | (added) and reported by Martin Gardner, "Mathematical Games," Scientific American 238(2):19–32 (February 1978) | https://sicherman.net/sdice.html; research/assay-line/cited-1014b/the-only-other-pair/N2-claude.md | added-gardner-reported: Retained from Claude N2. Rendered footer attribution (the record checks the citation, not that Gardner reported the dice). Sicherman: "Gardner liked the idea and presented it in his column for February 1978, calling the dice Sicherman Dice". ## Findings - [fixed] X1: Restricted the HTML closing paragraph to games determined solely by the sum. Added the dated game-equivalence correction to the HTML Corrections note and the end of the markdown body. - [declined] X2: The bars are adjacent, not overlaid. The triage lists this MINOR accessibility wording under tier C because it is outside a sentence being fixed for WRONG. Left the aria-label unchanged; this disposition does not deny the imprecision. - [fixed] X3: Replaced Everything here with the polynomial identities and finite dice searches in the HTML checking sentence. MINOR recorded here, with no on-page correction for this wording. - [fixed] X4: Changed 22/22 to 21/21 in the HTML check, markdown body, and research/sicherman-dice/README.md. Added one count correction line to the page and mirrored it in markdown. - [fixed] X5: Changed found by George Sicherman in 1978 to by 1977 in the markdown body. The dated correction names the 1977 letter and 1978 publication without assigning an exact invention date. - [fixed] X6: Changed decides every claim to checks the polynomial identities and finite dice searches in the markdown count sentence. MINOR recorded here, with no separate on-page correction. - [declined] X7: The phrase is in only one of the two titles. The triage lists this MINOR metadata wording under tier C because the relationship sentence is outside any WRONG fix. Left it unchanged; the distribution comparison still holds. - [dated] X8: Set JSON-LD dateModified to 2026-10-14. No Updated line under the explicit metadata exception: the three Corrected lines already date this revision. - [fixed] X9: Changed the markdown verifier count from 22/22 to 21/21. Shares the X4 count correction; the same-sentence MINOR narrowing is recorded separately as X6. - [fixed] K1: Duplicate of X1: limited the closing game-equivalence claim to games determined solely by the sum and shared its dated correction in HTML and markdown. - [fixed] K3: Duplicate of X4: corrected every authored 22/22 count in this layer to 21/21 and shared the count correction in HTML and markdown. - [fixed] K4: Duplicate of X5: changed the markdown discovery date to by 1977, sharing the dated line that distinguishes the documented letter from the February 1978 publication. - [declined] K5: Duplicate of X7: retained the listed MINOR title-reference slip because the triage tier C rule excludes this separate metadata sentence from editing. - [fixed] K6: Duplicate of X9: corrected the markdown to 21/21 and shared the count correction. The MINOR scope change is recorded as X6.