---
slug: allowed-and-impossible
passed: 2026-10-14
by: the assay line (research/assay-line/queue-1014c/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex; repair review by Codex, 2026-10-14
checked: 69
confirmed: 56
wrong: 13
unverifiable: 0
observed: 0
stale: 0
minor: 6
---

This was the directed correction pass on the 14 triage items for allowed-and-impossible at f5e5f84f34. The fixer independently re-read the evidence for each item, fixed seven WRONG and six MINOR claims in the layer-owned text, and confirmed the rejection of X12. The 55 additional CONFIRMED claims and their evidence notes below are retained from the source checkers' N2-codex.md in research/assay-line/queue-1014c/allowed-and-impossible/; descriptions of their pinned runs and source visits remain their recorded observations, not new runs claimed by this fixer. Together with X12 they give 56 CONFIRMED claims. The gate's wrong count includes the six MINOR lines. Four dated corrections cover the repeated WRONG claims, with no dedicated MINOR correction lines. The fixer's edits, commands and validation are recorded in research/assay-line/queue-1014c/allowed-and-impossible/fix-report.md. Repair review confirmed the shared TV narration residual and unannotated quotations in There, Here. X8, X9 and X13 remain open for those copies, as detailed in research/assay-line/queue-1014c/allowed-and-impossible/repair-report.md.

## Claims

- MINOR | “Two Latin squares are orthogonal exactly when the first can be cut into six disjoint transversals” | https://webspace.maths.qmul.ac.uk/l.h.soicher/designtheory.org/library/encyc/latinsq/m/ | [X1; orthogonal-transversals] Re-read Cameron, Transversals: partition into n transversals characterizes existence of some orthogonal mate, not orthogonality of a specified pair. Corrected the order-6 statement.
- MINOR | “Gaston Tarry proved this by hand in 1900, sorting the order-6 squares into cases over many pages” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | [X2; tarry-citation] Re-read Lam, Thiel and Swiercz, pp. 1117 and 1122, reference 18: the result is dated 1900; the short account is pp. 122-123 (1900), the detailed account pp. 170-203 (1901). Kept 1900 for the announcement and added 1901 for the detailed account.
- WRONG | “leaves only the Hamming family and two Golay codes” | https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf | [X3; classification] Visually re-read Hill, printed p. 102, Theorem 9.5 and its preceding nonlinear-code examples. It classifies parameters of nontrivial perfect codes over prime-power alphabets. Added those qualifiers in the HTML.
- WRONG | “passes the deeper Lloyd test” | https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf; https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf | [X4; lloyd-test] Re-read Elkies, p. 1, and Hill, printed p. 103. The Lloyd polynomial is 2(x² - 91x + 2048). Independently recomputed 91² - 4*2048 = 89, a nonsquare, so no root is an integer. Changed passing to failing Lloyd.
- MINOR | “searching all 9408 order-6 Latin squares” | https://users.cecs.anu.edu.au/~bdm/data/latin.html | [X5; dek-unreduced] Re-read McKay, Reduced Latin squares: order 6 has 9408 reduced representatives. The page already explains reduction. Added reduced in the meta description, JSON-LD and search/status labels.
- MINOR | “Euler's 36 officers pass every count and still cannot be arranged.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [X6; dek-every-count] Re-read the authored elementary capacities paragraph and the verifier census, plus Lam p. 1117. Six symbols fit six rows and columns; these capacities do not decide orthogonality. Restricted the repeated officers wording to those counts.
- MINOR | “Order 5 works; order 6 is uniquely alone.” | https://users.cecs.anu.edu.au/~bdm/data/latin.html | [X7; dek-unique-six] Re-read McKay: order 2 is also excluded; the page already gives the exceptions 2 and 6. The checkers also re-read Bose, Shrikhande and Parker, Theorem 10, which covers every order above 6. Qualified the summary as the only impossible order above 2.
- WRONG | “and even clears Lloyd's deeper test” | https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf; https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf | [X8; frontmatter-lloyd] Re-read Elkies p. 1 and the perfect-codes relation. The same Lloyd polynomial has no integer roots. Corrected this frontmatter copy together with X4.
- WRONG | “yet van Lint–Tietäväinen leave only the Hamming family and two Golay codes” | https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf | [X9; frontmatter-classification] Visually re-read Hill, printed p. 102, Theorem 9.5: nontrivial codes, prime-power alphabets, and parameters are essential qualifiers. Corrected the relation together with X3.
- MINOR | “Every count recomputed live and offline (research/allowed-and-impossible/verify.mjs, 33/33)” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [X10; layer-check-count] Read the inline JavaScript and the offline verifier. The page runs Latin-square censuses for 4, 5 and 6, the order-5 construction, sphere arithmetic and the plane gate. The n=1..11 Langford loop is offline only. Distinguished these scopes without changing the 33-check count.
- WRONG | “Here are objects that pass every count you can throw at them and still cannot be built.” | https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf | [X11; all-counts] Visually re-read Hill, printed p. 104, Theorem 9.7. Double counting forces 3 times the cardinality of a set to equal 88. Since 88/3 is not an integer, the phantom fails a counting condition. Restricted the opening and verifier comment to the conditions actually passed.
- CONFIRMED | “It can rule a thing out for free; it can never, by itself, rule it in.” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html:242; https://dialectrix.com/langford.html | [X12; count-never-sufficient] REJECTED with evidence: re-read the surrounding authored passage. The sentence already says by itself, and the preceding Langford paragraph credits a separate Davies construction with proving sufficiency. In context the claim concerns a condition established only as necessary; no further qualification is needed. This is the triage reason, confirmed on re-reading.
- WRONG | “the [90,78,5] perfect-code phantom whose spheres tile the space exactly yet which cannot be built” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs; https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf | [X13; phantom-spheres-dek] Re-read Elkies p. 1 and verifier Part 3: the equation equates total sphere volumes, while the source rules out the proposed code and hence the tiling. Corrected the dek, Markdown body, deep-scales relation and related heading.
- WRONG | (added) “the whole space, tiled, no slack.” | https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf; git show f5e5f84f34:public/strata/allowed-and-impossible/index.html:595 | [X14; runtime-actual-tiling] Read the actual inline output and verifier assertion label. They described a tiling after computing only volume. Elkies p. 1 rules out the code. Changed both output labels to equal total sphere volume, with no tiling established.

- CONFIRMED | “In 1782 Leonhard Euler posed a parade-ground puzzle.” | https://scholarlycommons.pacific.edu/euler-works/530/ | [euler-date] The archive explicitly labels 1782 as the publication date. This supports the published posing; it does not establish that Euler first considered the problem in that year.
- CONFIRMED | “six ranks drawn from each of six regiments” [...] “and seat them in a 6×6 square so that every row and every column shows each rank once and each regiment once. No two officers in any line share a rank; none share a regiment.” | https://paperzz.com/doc/9425828/investigations-on-a-new-type-of-magic-square | [officers-problem] The supplied excerpt is present in section 1 of the Euler translation and states both the row and column conditions. curl received a challenge; web retrieval exposed the actual translation.
- CONFIRMED | “Euler tried, and failed, and conjectured it was impossible.” | https://paperzz.com/doc/9425828/investigations-on-a-new-type-of-magic-square | [euler-tried] Section 1 contains the supplied excerpt, including "we cannot give a rigorous proof". Section 144 describes his unsuccessful examination of many squares.
- CONFIRMED | “but for the wrong reason, and the proof would wait more than a century.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X0000986X; https://paperzz.com/doc/9425828/investigations-on-a-new-type-of-magic-square; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | [euler-wrong-reason] The supplied theorem alone does not establish the history. Euler section 144 extrapolates to 10, 14 and the other oddly even orders; the 1960 theorem disproves that extrapolation. Lam dates Tarry to 1900, 118 years after publication. The phrase about his reason is a supported inference.
- CONFIRMED | “What you cannot do is make all thirty-six officers distinct at the same time.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [officers-impossible] Lam p. 1117 states the nonexistence of an orthogonal pair at order 6. The pinned census independently returns zero mates among 9408 reduced squares. Repetition follows while both component squares remain Latin.
- CONFIRMED | “Two of them always collide: somewhere, the same rank from the same regiment appears twice, and fixing it breaks something else.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [collision] Lam p. 1117 states the nonexistence of an orthogonal pair at order 6. The pinned census independently returns zero mates among 9408 reduced squares. Repetition follows while both component squares remain Latin.
- CONFIRMED | “a transversal picks one cell per row, all in different columns, all different symbols” | https://webspace.maths.qmul.ac.uk/l.h.soicher/designtheory.org/library/encyc/latinsq/m/ | [transversal-definition] The definition in the supplied excerpt is present: one cell in each row, column and symbol class.
- CONFIRMED | “these are the cells the partner square would mark with a single symbol” | https://webspace.maths.qmul.ac.uk/l.h.soicher/designtheory.org/library/encyc/latinsq/m/ | [transversal-partner] The supplied excerpt is present. Labeling each part of a transversal partition with a different symbol constructs the partner.
- CONFIRMED | “And "can be cut into transversals" is unchanged when you permute rows, permute columns, or rename symbols” [...] “the very moves that reduce a Latin square to standard form (first row and first column in order). So checking every reduced square checks every square there is.” | https://users.cecs.anu.edu.au/~bdm/data/latin.html; https://webspace.maths.qmul.ac.uk/l.h.soicher/designtheory.org/library/encyc/latinsq/m/ | [reduction-invariance] The quoted sorting statement is present. The additional invariance follows directly: bijections of rows, columns and symbols preserve distinctness and disjointness. The source excerpt alone only states reduction.
- CONFIRMED | “There are exactly 9408 reduced Latin squares of order 6.” | https://users.cecs.anu.edu.au/~bdm/data/latin.html | [reduced-count] The entry "order 6 (9408)" occurs under "Reduced Latin squares". The pinned generator independently yields 9408.
- CONFIRMED | “Not one of the 9408 has an orthogonal partner, so no 6×6 Latin square does, so the thirty-six officers cannot be seated.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [zero-partners] Lam p. 1117 states the nonexistence of an orthogonal pair at order 6. The pinned census independently returns zero mates among 9408 reduced squares. Repetition follows while both component squares remain Latin.
- CONFIRMED | “Build the two squares from arithmetic: regiment (i+j) mod 5, rank (i+2j) mod 5. Because the map (i,j) ↦ (i+j, i+2j) is invertible mod 5, every one of the 25 rank-and-regiment pairs is different. All twenty-five officers stand:” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [order5-construction] The determinant is 1 modulo 5. Inverting gives j = rank - regiment and i = 2 regiment - rank. Direct enumeration gives two Latin squares and 25 distinct pairs.
- CONFIRMED | “In 1959–60, Bose, Shrikhande and Parker showed the officers stand at every order except 2 and 6” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X0000986X | [bsp-citation] Theorem 10 says "any order v > 6". The paper was received in August 1959 and published in 1960; its references identify the 1959 precursor papers. Small orders are separately checked by the census. The DOI warning about the abbreviated year and title is not a bibliographic error.
- CONFIRMED | “demolishing Euler's guess that the whole family ≡ 2 (mod 4) was impossible.” | https://scholarlycommons.pacific.edu/euler-works/530/; https://paperzz.com/doc/9425828/investigations-on-a-new-type-of-magic-square; https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X0000986X | [euler-conjecture] The archive excerpt is present and Euler section 144 extends the conjecture to all oddly even orders. The original 1960 paper supplies the counter-result.
- CONFIRMED | “An error-correcting code surrounds each message with a sphere of near-misses; the code is perfect when those spheres tile the whole space of strings with no gap and no overlap” [...] “which forces an exact identity, M · (volume of one sphere) = 2ⁿ” | https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf | [perfect-definition] For the binary space used here, disjoint correction balls covering every vector give M times the ball volume equal to 2^n. Harvard expresses exact coverage by the convolution identity on p. 1.
- CONFIRMED | “1 + 90 + C(90,2) = 4096 = 2¹²” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [phantom-arithmetic] Recomputed: 90 times 89 divided by 2 is 4005; adding 90 and 1 gives 4096, which equals 2^12. The assertion checks these numbers.
- CONFIRMED | “2⁷⁸ · 2¹² = 2⁹⁰” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [phantom-product] The BigInt assertion computes both sides exactly. Independently, 78 + 12 = 90. This is an equality of volumes, not a construction of disjoint balls.
- CONFIRMED | “The classification of perfect codes (van Lint 1971; Tietäväinen 1973)” | https://research.tue.nl/en/publications/nonexistence-theorems-for-perfect-error-correcting-codes; https://epubs.siam.org/doi/10.1137/0124010 | [classification-citation] The supplied van Lint title and 1971 date match Eindhoven. That excerpt does not establish the second date: the SIAM publisher identifies Aimo Tietäväinen and January 1973, pp. 88-96.
- CONFIRMED | “the perfect-code classification (van Lint, Tietäväinen)” | https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf | [tietavainen-citation] Hill Theorem 9.5 has the quoted attribution. Its accompanying paragraph dates completion to Tietäväinen in 1973 following van Lint contributions.
- CONFIRMED | “and provably cannot be built.” | https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf | [phantom-nonexistence] The Harvard argument on p. 1 derives the impossibility from the absence of integral Lloyd roots. Its concluding sentence matches the supplied excerpt.
- CONFIRMED | “A finite projective plane of order n is a perfectly balanced incidence of points and lines.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | [plane-incidence] Lam p. 1117 gives n²+n+1 points and lines, n+1 incidences at each point and line, and unique intersections and joins. The short excerpt represents a complete definition there.
- CONFIRMED | “The Bruck–Ryser–Chowla theorem supplies a counting gate: for n ≡ 1 or 2 (mod 4), a plane can exist only if n is a sum of two squares.” | https://www.erdosproblems.com/723 | [brc-gate] The Erdos problem page states the congruence condition and sum-of-two-squares requirement, explicitly naming 6 and 14 as excluded orders. Squares at most 6 are 0, 1 and 4; none sum to 6.
- CONFIRMED | “Order 6 is stopped at the gate: 6 is not a sum of two squares, so no plane of order 6 (consistent with the officers).” | https://www.erdosproblems.com/723 | [plane6] The Erdos problem page states the congruence condition and sum-of-two-squares requirement, explicitly naming 6 and 14 as excluded orders. Squares at most 6 are 0, 1 and 4; none sum to 6.
- CONFIRMED | “10 = 1² + 3²” [...] “and the gate is satisfied.” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [plane10-arithmetic] The assertion returns [1,3], and direct arithmetic gives 1+9=10. Also 10 is 2 modulo 4, so the stated gate applies.
- CONFIRMED | “and the gate is satisfied.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | [plane10-permitted] Lam p. 1117 explicitly says this example shows the Bruck-Ryser necessary condition is not sufficient. The arithmetic 10=1²+3² passes that condition.
- CONFIRMED | “Yet in 1989 Lam, Thiel and Swiercz ended a search of thousands of computer-hours with the verdict that no projective plane of order 10 exists.” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | [plane10-computer-hours] The 1989 paper by the named authors reports nonexistence. Its p. 1119 gives an estimated 2000 CRAY-1A hours and, separately, 800 VAX-equivalent CPU days, so "thousands" is supported.
- CONFIRMED | “the non-existence of the order-10 plane (Lam, Thiel & Swiercz, 1989)” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534 | [plane10-citation] The byline, volume XLI, year 1989 and DOI all match. Crossref flags low overlap with the exact title; the page makes an attribution, not a purported verbatim title.
- CONFIRMED | “nobody knows whether its plane exists” | https://www.erdosproblems.com/723 | [plane12-open] The source currently states the order-12 existence question is open. Retrieved 2026-10-14; this supports the date-sensitive claim.
- CONFIRMED | “Order 12 is past the gate too” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [plane12-gate] 12 is 0 modulo 4, so the stated test gives no obstruction. The inline branch reports this as silence, not as a construction or an existence proof.
- CONFIRMED | “lay 1,1,2,2,…,n,n so the two k's have k numbers between them” | https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-langfords-problem-ii/C8B96417FD2094A51224F3A5F36C3102 | [langford-definition] The Davies publisher extract states the separation condition and displays 41312432, which has the specified intervening counts.
- CONFIRMED | “exists exactly when the triangular number Tn = n(n+1)/2 is even, i.e. n ≡ 0 or 3 (mod 4).” | https://dialectrix.com/langford.html | [langford-iff] The selected excerpt states only necessity. The same source later supplies Davies construction for every valid n. Together they support iff. Direct residue arithmetic equates even n(n+1)/2 with n congruent to 0 or 3 modulo 4.
- CONFIRMED | “The parity is forced (necessary); a construction by Roy Davies (1959) covers every case it allows (sufficient).” | https://dialectrix.com/langford.html; https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-langfords-problem-ii/C8B96417FD2094A51224F3A5F36C3102 | [davies-construction] The construction section explicitly covers any valid n. The publisher dates Davies paper to December 1959. This supplies sufficiency, unlike the necessity-only excerpt used for the preceding entry.
- CONFIRMED | “Langford sufficiency (Davies, 1959)” | https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/on-langfords-problem-ii/C8B96417FD2094A51224F3A5F36C3102 | [davies-citation] Publisher byline: Roy O. Davies; issue date: December 1959; pages 253-255. Crossref has the same name. Its tool warning mistakenly treats O., the middle initial, as the first initial.
- CONFIRMED | “length n = 90, correct up to 2 errors” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [generated-phantom-length] The code sets n=90 and uses the radius-2 volume 1+n+C(n,2). This is the proposed code parameter set, not a claim that the code was constructed.
- CONFIRMED | “Bruck–Ryser–Chowla is silent here (it speaks only for n ≡ 1 or 2 mod 4).” | https://www.erdosproblems.com/723; git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [brc-silent-runtime] The runtime branch agrees with the stated Bruck-Ryser condition: only residues 1 and 2 require a sum of two squares. Order 12 executes the silence branch; 6 and 10 execute the obstruction and permission branches.
- CONFIRMED | “the count gives no obstruction either way.” | https://www.erdosproblems.com/723; git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [brc-no-obstruction] The runtime branch agrees with the stated Bruck-Ryser condition: only residues 1 and 2 require a sum of two squares. Order 12 executes the silence branch; 6 and 10 execute the obstruction and permission branches.
- CONFIRMED | “(mod 4): BRC requires n = a² + b².” | https://www.erdosproblems.com/723; git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [brc-formula-runtime] The runtime branch agrees with the stated Bruck-Ryser condition: only residues 1 and 2 require a sum of two squares. Order 12 executes the silence branch; 6 and 10 execute the obstruction and permission branches.
- CONFIRMED | “Everything quantitative here is recomputed from scratch and kept apart from what is cited.” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [all-quantities] Read in its check-section context, the arithmetic, census and order-5 construction are recomputed, with the external nonexistence/classification results expressly labeled as citations. The separate claim that every count runs live is too broad.
- CONFIRMED | “Cited, not re-derived: the perfect-code classification (van Lint, Tietäväinen); the non-existence of the order-10 plane (Lam, Thiel & Swiercz, 1989); the Bose–Shrikhande–Parker resolution of Euler's conjecture; Langford sufficiency (Davies, 1959)” [...] “each far beyond a from-scratch check.” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [cited-not-derived] The verifier separately labels the global classification and order-10 result as cited; its local search covers only small orders and Langford n through 11. The four named theorems are not proved by the 33 checks.
- CONFIRMED | “It can settle the finding; it cannot notice a figure that drifted on this page.” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [check-limit] The verifier contains no page read. It can establish its computed result but has no mechanism to compare a sentence on the page. This limitation is explicitly disclosed.
- CONFIRMED | “Both verifiers brute-force the small cases and match the parity predicate” | git show f5e5f84f34:research/allowed-and-impossible/verify.mjs; git show f5e5f84f34:research/langford/census.mjs; git show f5e5f84f34:verify-langford-pairings.mjs | [both-verifiers] The supplied source is a browser smoke test, not the claimed arithmetic census. The correct second source is research/langford/census.mjs: its n<=11 enumeration matches the parity predicate. All numeric checks pass; the full supplementary run has two separate Lean-toolchain failures, not failures of this finite comparison.
- CONFIRMED | “The same lesson in coding theory, and the page links straight to it.” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [related-code-page] The pinned page links to /strata/perfect-codes/ at lines 218 and 273. The linked page discusses the same sphere-volume condition and its insufficiency.
- CONFIRMED | “The Wasteland walks this one in full at A Message That Heals Itself.” | git show f5e5f84f34:public/strata/perfect-codes/index.html | [linked-phantom] The linked page has a dedicated phantom section with the volume equation, the Lloyd polynomial and the nonexistence conclusion. It was corrected on 2026-09-28; the stale copied errors here were false when originally written, not STALE observations.
- CONFIRMED | “find a quantity the allowed moves cannot change, and an impossibility falls out (curvature, Euler's odd-degree parity, the fifteen-puzzle bit, knot tricolourings)” | git show f5e5f84f34:public/strata/find-what-doesnt-change/index.html | [invariant-sibling] The linked page has sections on sphere curvature, odd-degree parity, the fifteen-puzzle invariant, and a 3-versus-9 knot-colouring count preserved by Reidemeister moves. This verifies the summary of that page.
- CONFIRMED | “A counting condition is always necessary for a combinatorial object to exist” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html:200 | [universal-count] The page defines a counting condition as something "a valid object is forced to satisfy". With that explicit definition it is necessary by definition; an unrestricted arbitrary counting predicate would not be.
- CONFIRMED | “Six ranks fit in six columns; six regiments fit in six rows; the books balance perfectly.” | git show f5e5f84f34:public/strata/allowed-and-impossible/index.html | [six-by-six-counts] 6 times 6 equals 36. Each component can independently have one of each symbol per row and column; these elementary cardinalities impose no contradiction. They do not settle simultaneous orthogonality.
- CONFIRMED | “An immersive layer; it renders as its own standalone full-screen page at the” | git show f5e5f84f34:src/content/strata/allowed-and-impossible.md | [markdown-standalone] The href points to the public standalone HTML, which curl fetched as HTTP 200. The Astro route excludes entries with an href.
- CONFIRMED | “its body is not rendered.” | git show f5e5f84f34:src/pages/strata/[...slug].astro | [markdown-not-rendered] The route filter is "!s.data.draft && !s.data.href". This entry has an href, so its Markdown body is excluded from that route.
- CONFIRMED | “This file exists so the layer appears on the ground with the others” | git show f5e5f84f34:src/pages/index.astro | [markdown-ground] The ground loads the strata collection, filters draft entries, and builds its nodes from their href, title and dek. This non-draft metadata file enters that collection.
- CONFIRMED | “Both pages keep scrupulously apart what the computation proves from what a theorem certifies.” | git show f5e5f84f34:public/strata/perfect-codes/index.html | [proof-separation] Both check sections identify local computations and cited global results separately. The linked perfect-code page specifically labels completeness as cited, not re-proved. False assertions about the theorem here are separately recorded.
- CONFIRMED | “and yet no such code exists.” | https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf; https://sites.math.rutgers.edu/~zeilberg/akherim/HillCoding/PartI.pdf; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [runtime-no-code] The cited verifier only checks volume and cites nonexistence. The independent Harvard proof and Hill Theorem 9.7 establish that no such binary code exists.
- CONFIRMED | “…and yet there is none (Lam, Thiel & Swiercz, 1989).” | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00002534; git show f5e5f84f34:research/allowed-and-impossible/verify.mjs | [runtime-order10-citation] The verifier comment is an attribution, not a proof. Opened the actual Lam, Thiel and Swiercz 1989 paper, whose introduction states the nonexistence conclusion.
- CONFIRMED | “Nothing here may lie about anything real; that rule never bends.” | git show f5e5f84f34:WAKE.md | [shared-truth-rule] WAKE rule 2 reads "It must never lie about anything real." The page states a rule, not a claim that every existing sentence has already been audited.
- CONFIRMED | (added) “A perfect code's correction-spheres must tile the message space exactly” [...] “a necessary counting identity, M·Vol = qⁿ.” | https://people.math.harvard.edu/~elkies/M256.13/lloyd.pdf; git show f5e5f84f34:src/content/strata/allowed-and-impossible.md:10 | [general-alphabet-volume] The source formulates perfect coverage over F^n; for an alphabet of size q the space has q^n vectors. The source record only recorded the binary 2^n formulation, not this general one.
- CONFIRMED | (added) “Two pages about computational non-existence and how carefully it has to be stated.” | git show f5e5f84f34:src/content/strata/boolean-pythagorean.md; git show f5e5f84f34:public/strata/allowed-and-impossible/index.html:304 | [nearby-boolean-proof] The linked page metadata describes a certificate of nonexistence at 7825 and contrasts that proof trace with this layer exhaustive census. This verifies the description of the link; it is not a separate audit of that layer.

## Findings

- [fixed] X1: MINOR: public/strata/allowed-and-impossible/index.html now states that an order-6 Latin square has an orthogonal mate exactly when it partitions into six disjoint transversals. Record only; no dedicated page correction line.
- [fixed] X2: MINOR: clarified the 1900 announcement and 1901 detailed account in the HTML and Markdown dek. The added year comes from Lam et al., reference 18. Record only; no dedicated page correction line.
- [fixed] X3: WRONG: corrected the HTML classification to parameters of nontrivial perfect codes over prime-power alphabets. A Corrected 2026-10-14 line covers X3 and X9, with the same text at the Markdown body end.
- [fixed] X4: WRONG: the HTML now says the phantom fails Lloyd and links to the proof. A Corrected 2026-10-14 line covers X4 and X8 and is mirrored in Markdown.
- [fixed] X5: MINOR: added reduced in the meta/JSON-LD descriptions, census button, progress/status text and census summary. Record only; no dedicated page correction line.
- [fixed] X6: MINOR: replaced unrestricted officers counts with elementary row and column counts in the og description, JSON-LD, Markdown dek and body. The WRONG universal claim X11 has its own correction; no MINOR-only correction was added.
- [fixed] X7: MINOR: the Markdown dek now says order 6 is the only impossible order above 2. Record only; no dedicated page correction line.
- [open note:9fe7a9] X8: WRONG: the perfect-codes relation now says the phantom fails Lloyd, covered by the same dated correction as X4. The preserved quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still claims it clears Lloyd's test without an adjacent correction. Elkies p. 1 proves failure. That other layer needs a correction annotation; its quotation and shared correction pipeline are outside this repair's edit scope.
- [open note:9fe7a9] X9: WRONG: the perfect-codes relation now states the parameter classification with the nontrivial and prime-power qualifiers, covered by the same dated correction as X3. The preserved quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still classifies the codes themselves without an adjacent correction. Hill, Theorem 9.5, classifies parameters. That other layer needs a correction annotation; its quotation and shared correction pipeline are outside this repair's edit scope.
- [fixed] X10: MINOR: the Markdown dek distinguishes the live computations from the offline Langford check through n=11. Record only; no dedicated page correction line.
- [fixed] X11: WRONG: restricted the HTML opening to particular necessary counting tests and the verifier comment to balanced sphere volumes. A dated correction cites Hill's counting contradiction, mirrored in Markdown. The published verifier copy matches the source.
- [open note:9fe7a9] X13: WRONG: replaced the asserted phantom tiling with the sphere-volume identity in the Markdown dek/body and deep-scales relation, and changed the HTML heading. A dated correction covers X13 and X14, mirrored in Markdown. The false tiling claim survives in shared public/tv/lib/narration/batch-12.js:234 and the unannotated deep-scales quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1. Elkies p. 1 rules out the code. These copies are outside this repair's edit scope; the narration needs corrected wording and the preserved quotation needs a correction annotation.
- [fixed] X14: WRONG: the inline calculator and both verifier copies now describe equal total sphere volume, not an established tiling. Covered by the same dated correction as X13. The shared TV narration residual is tracked under X13.