the assay office / record

The Pattern That Has Not Arrived

Written 2026-07-26. Claims re-read against their sources on 2026-10-01: 72 checked, 69 confirmed, 2 wrong, 0 unverifiable, 1 true when written and overtaken since (dated, not corrected). By the assay line (research/assay-line/tier-a-1001/), Codex fixer directed by claude-assaying-codex-0929; source record by Codex, checked by Codex.

This was the directed 2026-10-01 triage and repair pass for grundy-periodicity at f74f299d9d. All fourteen triage items were reread against their cited evidence. The record also carries forward all 58 CONFIRMED claims from research/assay-line/tier-a-1001/grundy-periodicity/N2-codex.md, with their evidence notes explicitly attributed to N2. The eleven source-excerpt findings are rejected as page errors and recorded CONFIRMED. One WRONG statement is corrected, one MINOR interaction defect is repaired and counted in wrong, and one STALE modification date is revised. This is a directed repair and consolidation, not a new blind audit of the layer.

Claims

• CONFIRMED Dawson's chess repeats forever after a finite certificate checks 89 equalities.
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X1 [lead-dawson]: REJECTED as a page error. The full Theorem 2.4 on p. 180 supplies the finite-octal window n0 <= n < 2*n0+p+k, with n0 and p positive, and extends its equality to every n >= n0. Here n0=52, p=34, k=3 gives [52,141), and the verifier rerun passes all 89 equalities. N1 stops before the theorem; that source-record defect does not make the page claim false.
• CONFIRMED Dawson's chess settles into a pattern that a finite Guy-Smith certificate, 89 equalities, proves repeats forever.
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X2 [plain-certificate]: REJECTED as a page error. Theorem 2.4 and the 89 passing equalities support the statement; see X1. The N1 excerpt stops before the theorem, a source-record limitation rather than false mathematics.
• CONFIRMED One finite window certifies Dawson's period forever
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X3 [social-dawson]: REJECTED as a page error. Theorem 2.4 and the 89 passing equalities support the social description; see X1. The N1 excerpt stops before the supporting theorem.
• CONFIRMED here unequal splitting produces a catalogued sequence whose eventual periodicity remains unknown.
  https://oeis.org/A002188/internal; https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : X4 [related-catalogued]: REJECTED as a page error. A002188 identifies the unequal-split Grundy sequence, and the cited Octal-Games table lists Grundy among unsettled games. N1 supplied only the unsuccessful-period-search excerpt, omitting the catalogue identification. The page claim is supported by the full sources.
• CONFIRMED There XOR is the complete winning strategy for heaps
  https://library.slmath.org/books/Book70/files/1006.pdf : X5 [related-nim-strategy]: REJECTED as a page error. Theorem 2.1 on pp. 177-178 identifies normal-play Nim losses by zero XOR and its proof constructs a move to zero XOR from a nonzero XOR. N1 stopped before that theorem. The related page explicitly uses normal play.
• WRONG Wythoff's losing positions sit on exact golden-ratio rays
  https://www.gabrielnivasch.org/fun/combinatorial-games/wythoff : X6 [related-wythoff]: FIXED within this layer. Nivasch gives (floor(phi*n), floor(phi^2*n)) and its reversal, for n >= 0. At n=1 the pair (1,2) has slope 2, not phi. The related note now says floor-function pairs generated by the golden ratio. The HTML and markdown carry the dated correction; copies in other layers are outside the authorized scope.
• CONFIRMED A nim-value is not a score. It is the one Nim heap equivalent to a game position.
  https://library.slmath.org/books/Book70/files/1006.pdf : X7 [nim-equivalence]: REJECTED as a page error. Theorem 1.10 on p. 176 gives equivalence to a Nim heap in the finite impartial normal-play setting of this page. The summary on p. 177 identifies the unique integer. N1 stopped at the theorem heading rather than including its statement.
• CONFIRMED Dawson's original pawn problem used the opposite last-move convention, so the two results must not be conflated.
  https://library.slmath.org/books/Book42/files/elkies.pdf : X8 [dawson-original]: REJECTED as a page error. Elkies, p. 65, footnote 5, states that Dawson proposed the last-player-loses convention. The page distinguishes that from its normal-play .137 analysis. N1 cuts off the footnote before the supporting statement.
• CONFIRMED For an octal game whose last nonzero digit is at position k, one finite equality window is enough:
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X9 [finite-window]: REJECTED as a page error. Theorem 2.4 supplies exactly this finite-octal window; see X1. The page applies positive n0=52 and p=34. N1 clipped the theorem from its excerpt.
• CONFIRMED if G(n+p)=G(n) for n0 <= n < 2*n0+p+k, then G(n+p)=G(n) for every n >= n0
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X10 [finite-window-formula]: REJECTED as a page error. Theorem 2.4 states this inequality and conclusion for finite octal games with positive n0 and p; those conditions hold for the displayed application. N1 clipped the theorem, not evidence against the formula.
• CONFIRMED Its 89 passing equalities do not make a persuasive picture; the theorem converts them into a proof for every later heap.
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X11 [dawson-equalities-proof]: REJECTED as a page error. Theorem 2.4 converts the 89 verified equalities to the entire tail; see X1. The clipped N1 excerpt is the limitation, not the proof claim.
• CONFIRMED The passing Dawson window certifies period 34 for every n from 52 onward.
  https://library.slmath.org/books/Book70/files/1006.pdf; research/grundy-periodicity/verify-grundy-periodicity.mjs : X12 [ledger-theorem]: REJECTED as a page error. Theorem 2.4, n0=52, p=34, k=3 and the 89 passing equalities certify the stated tail; see X1. N1 stopped before the supporting theorem.
• STALE "dateModified":"2026-07-26"
  git show 115036144b:public/strata/grundy-periodicity/index.html; git show 2e60bcb5dd:public/strata/grundy-periodicity/index.html : X13 [metadata-modification]: FIXED as a metadata revision, recorded STALE. Both the original page at 115036144b and the revised page at 2e60bcb5dd have dateModified 2026-07-26. Read-only history dates the latter revision 2026-10-01. This pass sets dateModified to 2026-10-01; the dated WRONG correction already records this revision, so no separate Updated line is added.
• MINOR 512/512 fail. Hover or focus a period to read its first unequal pair.
  public/strata/grundy-periodicity/index.html; git show f74f299d9d:public/strata/grundy-periodicity/index.html : X14 [live-scan]: FIXED as MINOR. Before the change, renderScan created plain spans with textContent and title but no tabindex or focus handler. Chromium confirms the original chips cannot receive focus. Every chip now has tabindex 0 and an aria-describedby reference; focus and mouse movement display its computed first unequal pair in a sticky live readout. All 512 mismatch pairs agree with a direct unequal-split recurrence. No page correction line is added for this MINOR repair.
• CONFIRMED Grundy's game, its unequal-splitting cousin, has been computed through at least 284,944,236,543 heaps without a known period.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [lead-frontier] The primary computation account says "finally reached G(284944236543) on 2012-11-25" and "But there shows none." The introduction specifies splitting a heap into two unequal heaps. The current octal table also lists Grundy among the unsettled games.
• CONFIRMED Grundy's game, where a move splits one heap into two unequal heaps, has been computed through at least 284,944,236,543 heaps with no period found.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [plain-frontier] The primary computation account says "finally reached G(284944236543) on 2012-11-25" and "But there shows none." The introduction specifies splitting a heap into two unequal heaps. The current octal table also lists Grundy among the unsettled games.
• CONFIRMED Grundy's unequal-split game remains open after at least 284 billion computed terms.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [social-grundy] The primary computation account says "finally reached G(284944236543) on 2012-11-25" and "But there shows none." The introduction specifies splitting a heap into two unequal heaps. The current octal table also lists Grundy among the unsettled games.
• CONFIRMED A live Guy-Smith certificate beside the open periodicity problem in Grundy's game.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html; git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [twitter-open] The pinned page computes its Guy-Smith certificate; the primary tables still list Grundy among unsettled games. The N1 excerpt alone covers only the unsuccessful search for a period.
• CONFIRMED The cited 2012 frontier is kept separate from the prefix reproduced in the browser.
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [markdown-frontier-separated] The cited-observations card explicitly says "are not reproduced by this browser run"; TARGET=1000003 bounds the local computation, while the historical frontier is printed separately.
• CONFIRMED Two pages compute Grundy sequences by mex and stop at an honest open edge.
  git show e172a4fb8f:src/content/strata/one-word-apart.md : N2 source check: [related-mex] The other pinned layer says "watch the browser recompute every value live by the mex rule" and distinguishes proved results from finite checks. This layer applies mex to both recurrences and expressly limits its Grundy conclusion.
• CONFIRMED Nim supplies the arithmetic under both pages.
  git show e172a4fb8f:src/content/strata/the-strategy-that-counts-in-binary.md : N2 source check: [related-nim-arithmetic] The linked pinned Nim layer identifies "the bitwise XOR of the heap sizes"; the audited page takes XOR of child nim-values before mex.
• CONFIRMED here each take-and-break option becomes the XOR of its child heaps before mex assigns the parent its nim-value.
  https://math.colgate.edu/~integers/dg1/dg1.pdf : N2 source check: [related-xor-mex] The primary paper defines mex as the least excluded nonnegative integer and states "G(G) = G(H)" followed by the XOR operator and "G(K)" for G=H+K. Its equations justify XOR of child values followed by mex for the parent.
• CONFIRMED Grundy's nim-values offer no proved eventual pattern despite a computation hundreds of billions of terms long.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [related-no-pattern] The primary computation account says "finally reached G(284944236543) on 2012-11-25" and "But there shows none." The introduction specifies splitting a heap into two unequal heaps. The current octal table also lists Grundy among the unsettled games.
• CONFIRMED These are the impartial and partizan sides of combinatorial game value.
  git show e172a4fb8f:src/content/strata/worth-half-a-move.md : N2 source check: [related-impartial-partizan] The linked pinned page describes "a partizan game, where each player owns a colour"; the audited games allow identical moves for either player.
• CONFIRMED Grundy values collapse an impartial position to a nimber joined by XOR
  https://math.colgate.edu/~integers/dg1/dg1.pdf : N2 source check: [related-nimber] The primary paper defines mex as the least excluded nonnegative integer and states "G(G) = G(H)" followed by the XOR operator and "G(K)" for G=H+K. Its equations justify XOR of child values followed by mex for the parent.
• CONFIRMED Hackenbush assigns signed dyadic numbers until a green edge makes number-valued play fail.
  https://arxiv.org/pdf/1202.5654 : N2 source check: [related-hackenbush] Read the complete paper, including its abstract about a star-based dyadic rational, p.2 identifying a green string with a Nim heap, and p.5: "a red-blue String is a number". The linked layer specifically uses finite blue-red stalks and one lone green edge. N1 also copied irrelevant p.4 text, but the relevant passages support the claim in that context.
• CONFIRMED Both pages ask a finite certificate to support a claim about an entire legal future.
  git show e172a4fb8f:src/content/strata/the-one-move-that-changed-the-game.md : N2 source check: [related-finite-futures] The other pinned layer explicitly compares a finite certificate with an entire legal future, then limits its successful trace comparison to three synchronized depths. This is a comparison of the questions asked, not a claim that both certify all future moves.
• CONFIRMED For any legal move, XOR the nim-values of the heaps left behind;
  https://math.colgate.edu/~integers/dg1/dg1.pdf : N2 source check: [nim-xor-options] The primary paper defines mex as the least excluded nonnegative integer and states "G(G) = G(H)" followed by the XOR operator and "G(K)" for G=H+K. Its equations justify XOR of child values followed by mex for the parent.
• CONFIRMED the current heap receives the mex, the smallest non-negative integer missing from those options.
  https://math.colgate.edu/~integers/dg1/dg1.pdf : N2 source check: [nim-mex] The primary paper defines mex as the least excluded nonnegative integer and states "G(G) = G(H)" followed by the XOR operator and "G(K)" for G=H+K. Its equations justify XOR of child values followed by mex for the parent.
• CONFIRMED Octal code .137 gives three removal rules. Remove one only when that empties the heap; remove two and leave zero or one heap; remove three and leave zero, one, or two heaps.
  https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [octal-rules] The source assigns bit 2^k to permission to leave k nonempty heaps. Thus 1 permits zero, 3 permits zero or one, and 7 permits zero, one or two; the digit position is the number removed.
• CONFIRMED This is Guy and Smith's normal-play analysis.
  https://library.slmath.org/books/Book70/files/1006.pdf : N2 source check: [guy-smith-normal] The quoted history explicitly attributes octal games and normal-play analysis to the 1956 Guy and Smith paper; the fetched Cambridge metadata confirms both authors and 1956.
• CONFIRMED The upper endpoint is 2*52+34+3=141, excluded, so the test contains exactly 89 equalities.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs; https://library.slmath.org/books/Book70/files/1006.pdf : N2 source check: [certificate-endpoint] Arithmetic: 2*52+34+3=141 and 141-52=89. The runtime certificate counter is 89. The half-open interval includes precisely n=52..140.
• CONFIRMED The cited computation reaches heap sizes above two billion while new record nim-values still appear.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [record-values-beyond-billion] The source reports "The last maximal value is G(4563802297)=291" and its record table includes still later maxima. 4,563,802,297 exceeds two billion.
• CONFIRMED A finite rule does not supply bounded state for free.
  https://arxiv.org/abs/1407.2823; https://arxiv.org/pdf/1407.2823 : N2 source check: [finite-rule-not-state] Fox's full paper, Theorem 6, constructs a ternary aperiodic nim sequence with an explicit rule for its subtraction set. Its set is infinite; this supports the broad distinction between a finitely describable rule and bounded state, not a counterexample with a finite subtraction set.
• CONFIRMED This is not a Guy-Smith certificate for Grundy's game, because unequal splitting cannot be expressed by standard octal notation.
  https://oeis.org/A002188/internal : N2 source check: [non-octal] A002188 explicitly says standard octal notation cannot capture the "unequal" splitting rule. Therefore the cited theorem for finite octal games does not certify this game merely from such a window.
• CONFIRMED A passing window would not prove its tail.
  https://oeis.org/A002188/internal : N2 source check: [passing-window-limit] A002188 explicitly says standard octal notation cannot capture the "unequal" splitting rule. Therefore the cited theorem for finite octal games does not certify this game merely from such a window.
• CONFIRMED A failed window does conclusively reject that candidate on the stated prefix.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [failed-window-rejection] A single recorded G(n+p) != G(n) falsifies equality on that stated window. Recreated first mismatches for every p=1..512; no conclusion about later starting points follows.
• CONFIRMED Flammenkamp's computational table reports octal .106 settling in 2002 with period 328,226,140,474 after preperiod 465,384,263,797.
  https://wwwhomes.uni-bielefeld.de/achim/octal.html; https://library.slmath.org/books/Book70/files/1006.pdf : N2 source check: [octal-106] Read the table headers and the .106 row: period 328226140474, preperiod 465384263797, solved 2002-05-21. The primary page explains its finite certification criterion; Siegel also lists .106 with this period.
• CONFIRMED Astronomical periods can be found and certified.
  https://wwwhomes.uni-bielefeld.de/achim/octal.html; https://library.slmath.org/books/Book70/files/1006.pdf : N2 source check: [astronomical-periods] Read the table headers and the .106 row: period 328226140474, preperiod 465384263797, solved 2002-05-21. The primary page explains its finite certification criterion; Siegel also lists .106 with this period.
• CONFIRMED Delete the lowest bit of a Grundy nim-value and count the one-bits that remain. Even parity marks a sparse value; odd parity marks a common value.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [sparse-definition] The source explicitly uses even parity of the binary representation after deleting the lowest digit for sparse values, with the complementary set common.
• CONFIRMED XOR preserves this parity relation, so the split is not decoration.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [xor-preserves-parity] The online NOT FOUND was an entity-decoding artifact. Curl plus HTML entity decoding finds the entire recorded excerpt (FOUND); the same locate function finds "Known Sparse SGV of Grundy's Game" as its positive control. The displayed parity equation and bitwise arithmetic give q(x XOR y)=q(x) XOR q(y).
• CONFIRMED It is an exact way to partition the mex search.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [partition-mex] Inspected the algorithm: the sparse-side pass finds every common option; mex is at most the first absent common value; the second pass resolves only absent sparse values below that ceiling. Discarded larger options cannot affect mex. Its first 4,097 results also match the independent direct recurrence.
• CONFIRMED G=10 = 1010₂
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [binary-ten] Direct arithmetic: 10=8+2 has binary digits 1010; deleting the last bit gives 101, which has two ones. The parity helper returns sparse.
• CONFIRMED Drop the low bit: 101 has two ones. Even parity.
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [binary-ten-parity] Direct arithmetic: 10=8+2 has binary digits 1010; deleting the last bit gives 101, which has two ones. The parity helper returns sparse.
• CONFIRMED Common value G=8 = 1000₂
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [binary-eight] Direct arithmetic: 8 has binary digits 1000; deleting the last bit gives 100, which has one one. The parity helper returns common.
• CONFIRMED Drop the low bit: 100 has one one. Odd parity.
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [binary-eight-parity] Direct arithmetic: 8 has binary digits 1000; deleting the last bit gives 100, which has one one. The parity helper returns common.
• CONFIRMED Flammenkamp's self-published computational record reports 1,287 sparse occurrences through the 2012 run, with the last known one at G(48,399,022)=259.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [sparse-record] The primary ledger prints "sparse=1287 last=48399022" and "G(48399022)=259". Its introduction and 2012 update support these as cited historical observations, not results recreated by this page.
• CONFIRMED if sparse occurrences stop, the infinite sequence must become periodic
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [sparse-stop-theorem] Flammenkamp states the claimed conditional implication. Its octal-games discussion explains the bounded-memory argument after the last rare value; for this unequal-split rule, once n exceeds twice the last sparse index, all sparse-side splits are unequal.
• CONFIRMED No source establishes whether it ends.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [sparse-end-unknown] Read as the status of the cited literature, this is supported: the primary record leaves the disappearance of sparse values conditional, and the current table explicitly includes Grundy among unsettled games. The source gives 1,287 observed sparse occurrences, not a proof that there are only 1,287.
• CONFIRMED Flammenkamp's 2012 endpoint, 1,287-record sparse ledger, last known sparse occurrence, and `.106` period data are not reproduced by this browser run.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [cited-not-reproduced] FLAMMENKAMP_RECORD holds the frontier and sparse ledger as literal historical constants; the computed Grundy horizon is 1,000,003. Positive control: the same source inspection finds the actual computeGrundy call and its computed assertions.
• CONFIRMED 284,944,236,543 ~= 2^38.1
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [frontier-power] Recalculation gives log2(284944236543)=38.0518886559..., rounding to 38.1. The pinned verifier prints "2^38.052 ~= 2^38.1".
• CONFIRMED last cited sparse: G(48,399,022)=259
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [last-sparse-output] The primary ledger prints "sparse=1287 last=48399022" and "G(48399022)=259". Its introduction and 2012 update support these as cited historical observations, not results recreated by this page.
• CONFIRMED No toy model is used.
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [no-toy-model] The complete .137 move rule and exact unequal-split mex are implemented. The finite horizon limits the extent of the computation, not the move rules.
• CONFIRMED The cited 2012 run reached at least its stated endpoint without a known period.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [ledger-observed] The primary computation account says "finally reached G(284944236543) on 2012-11-25" and "But there shows none." The introduction specifies splitting a heap into two unequal heaps. The current octal table also lists Grundy among the unsettled games.
• CONFIRMED Sparse XOR common is common; common XOR common and sparse XOR sparse are sparse.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [xor-classes] With q linear over XOR, the parity sums are 0+1=1, 1+1=0 and 0+0=0 modulo 2. These are exactly the three stated class identities.
• CONFIRMED Once the first missing common value is known, mex cannot exceed it.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [mex-ceiling] Once the first common value c is absent, the least excluded value cannot exceed c. Later searches need only settle smaller missing sparse values.
• CONFIRMED This partitions the complete move set without changing it.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [complete-partition] Inspected the algorithm: the sparse-side pass finds every common option; mex is at most the first absent common value; the second pass resolves only absent sparse values below that ceiling. Discarded larger options cannot affect mex. Its first 4,097 results also match the independent direct recurrence.
• CONFIRMED R. K. Guy and C. A. B. Smith, "The G-values of various games" , Proc. Cambridge Philos. Soc. 52 (1956), 514-526.
  https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0305004100031509 : N2 source check: [citation-guy-smith] Cambridge identifies Richard K. Guy and Cedric A. B. Smith, volume 52 issue 3, July 1956, pages 514-526, and the stated title. Crossref DOI check reports MATCH. The 2008 online-publication date is not the original article year.
• CONFIRMED OEIS A002187 and A002188
  https://oeis.org/A002187/internal : N2 source check: [citation-oeis-dawson] Opened both OEIS entries and read their titles and %S/%T/%U terms. Each corresponding 104-term fixture agrees exactly with the fetched entry; no table is used to generate either recurrence.
• CONFIRMED OEIS A002187 and A002188
  https://oeis.org/A002188/internal : N2 source check: [citation-oeis-grundy] Opened both OEIS entries and read their titles and %S/%T/%U terms. Each corresponding 104-term fixture agrees exactly with the fetched entry; no table is used to generate either recurrence.
• CONFIRMED Achim Flammenkamp, "Sprague-Grundy Values of Grundy's Game" and "Octal-Games" , University of Bielefeld. These are self-published computational records.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [citation-flammenkamp-grundy] Read both primary pages, their respective headings and Achim Flammenkamp bylines at the University of Bielefeld personal-site URLs. "Octal-Games" is the HTML title; the longer visible heading begins "Sprague-Grundy Values".
• CONFIRMED Achim Flammenkamp, "Sprague-Grundy Values of Grundy's Game" and "Octal-Games" , University of Bielefeld. These are self-published computational records.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [citation-flammenkamp-octal] Read both primary pages, their respective headings and Achim Flammenkamp bylines at the University of Bielefeld personal-site URLs. "Octal-Games" is the HTML title; the longer visible heading begins "Sprague-Grundy Values".
• CONFIRMED These are self-published computational records.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html; https://wwwhomes.uni-bielefeld.de/achim/octal.html : N2 source check: [self-published] Read both primary pages, their respective headings and Achim Flammenkamp bylines at the University of Bielefeld personal-site URLs. "Octal-Games" is the HTML title; the longer visible heading begins "Sprague-Grundy Values".
• CONFIRMED D. G. Horrocks and R. J. Nowakowski, "Regularity in the G-sequences of octal games with a pass" , Integers 3 (2003), G01, for the finite periodicity-test framing.
  https://eudml.org/doc/123843; https://math.colgate.edu/~integers/dg1/dg1.pdf : N2 source check: [citation-horrocks] EuDML identifies both authors, the title, Integers 3 (2003), Paper G01. The downloaded primary paper matches; its p.4 states the finite-octal periodicity criterion before the pass-game theorems.
• CONFIRMED for the finite periodicity-test framing.
  https://math.colgate.edu/~integers/dg1/dg1.pdf : N2 source check: [horrocks-framing] Read beyond N1's generic excerpt: p.4 says the underlying finite octal game is periodic if the period persists from e+1 through 2e+2p+k. With e=n0-1, these are the heap indices needed for the equality window on the audited page.
• CONFIRMED The dedicated verifier is research/grundy-periodicity/verify-grundy-periodicity.mjs.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [dedicated-verifier] The named pinned file exists and imports only node:fs, so it needs no external package installation. It recomputes both arrays, but also reads the local page HTML; the adjacent placard discloses that repository-file requirement.
• CONFIRMED It has no dependencies and recomputes both sequences from their rules.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [verifier-dependencies] The named pinned file exists and imports only node:fs, so it needs no external package installation. It recomputes both arrays, but also reads the local page HTML; the adjacent placard discloses that repository-file requirement.
• CONFIRMED This layer's check reads the bytes this page serves and asserts against them.
  git show e172a4fb8f:research/grundy-periodicity/verify-grundy-periodicity.mjs : N2 source check: [placard-bytes] The verifier calls readFileSync on the page and asserts the cited Flammenkamp constants. This statement does not establish full prose coverage: the required headline mutation survives, recorded separately as a HIGH finding.
• CONFIRMED "A bounded sequence from a finite rule must cycle."
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [hypothetical-dismissal-one] Confirmed the framing only: this is an unattributed hypothetical dismissal, immediately answered by the following paragraph. It is not asserted as true or attributed as a verbatim quotation from an external speaker.
• CONFIRMED "Either it looks periodic or you have not gone far enough."
  git show e172a4fb8f:public/strata/grundy-periodicity/index.html : N2 source check: [hypothetical-dismissal-two] Confirmed the framing only: this is an unattributed hypothetical dismissal, immediately answered by the following paragraph. It is not asserted as true or attributed as a verbatim quotation from an external speaker.
• CONFIRMED (added) Two impartial heap games hide very different kinds of order.
  https://www.gabrielnivasch.org/fun/combinatorial-games/wythoff; https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [added-impartial-games] (added) Coverage omission at public/strata/grundy-periodicity/index.html:475 and src/content/strata/grundy-periodicity.md:16. Wythoff's source explicitly calls it "an impartial game played with two piles of tokens"; Grundy's source gives the same unequal-split move to either player. The order comparison follows from the proved Wythoff floor formulas and the unsettled Grundy tail.
• CONFIRMED (added) It makes a run past 2.8*10^11 terms more than casual eyeballing.
  https://wwwhomes.uni-bielefeld.de/achim/grundy.html : N2 source check: [added-frontier-scale] (added) Coverage omission at public/strata/grundy-periodicity/index.html:348. Checkable numerical part: 2.8*10^11=280,000,000,000 < 284,944,236,543, the cited endpoint. The evaluative phrase is not a factual finding.

What was done

• [fixed] X6, WRONG: src/content/strata/grundy-periodicity.md now describes Wythoff's losing positions as floor-function pairs generated by the golden ratio. A Corrected 2026-10-01 line is added above the HTML check placard and mirrored at the end of the markdown body, naming the assay line and citing Nivasch. Other-layer copies are listed in the scope note below, not edited.
• [dated] X13, STALE: public/strata/grundy-periodicity/index.html dateModified changes from 2026-07-26 to 2026-10-01. The original value describes the original build; the page has since changed. Per the directed rule for JSON-LD revised alongside a WRONG correction, the Corrected line dates this revision and no separate Updated line is needed.
• [fixed] X14, MINOR: public/strata/grundy-periodicity/index.html now makes all 512 period chips focusable and exposes the selected mismatch on focus and hover in a sticky live readout. The mismatch data are unchanged. The MINOR repair is recorded here without a page correction line.