---
slug: seven-shuffles
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
checked: 92
confirmed: 59
wrong: 33
unverifiable: 0
observed: 0
stale: 0
---

This was the directed repair of the 32 triage items for seven-shuffles at f5e5f84f34, using the Codex source record and checker reread in research/assay-line/queue-1014c/seven-shuffles/. I re-read the relevant paper pages and authored code before editing, reproduced the numerical counterexamples with exact integer ratios, fixed 24 WRONG and seven MINOR items, and rejected X30 with evidence. The 58 CONFIRMED claims from the checker are retained below with their qualifications and source-record provenance; wording clarified by a related fix is identified. MINOR counts in wrong. Fourteen dated correction lines cover the distinct WRONG errors and are mirrored in the markdown body; MINOR fixes have no page correction lines. This record covers the directed work list and retained confirmations, not the checker's separate OBSERVED or UNVERIFIABLE entries.

Repair review on 2026-10-14 added two WRONG source copies of X5: R1 in the layer's share-image template is fixed; R2 in the shared SEO title table remains open for the directing instance. The record now has 59 CONFIRMED, 26 WRONG and seven MINOR claims, with 32 fixed findings and one pending board note. The 14 dated corrections and their markdown mirrors were checked against the evidence and retained.

## Claims

- WRONG | X1: "In the early 1950s Gilbert and Shannon, and independently Reeds" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Diaconis survey, section 2A, printed pp. 2 and 5: the Gilbert-Shannon work is reported by Gilbert (1955); Reeds is independently credited in 1981.
- MINOR | X2: "some orders are far more probable than others" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf ; research/seven-shuffles/verify-assay.mjs | Exact integer-ratio evaluation gives a relative maximum-minus-minimum probability spread of 2.0920591206461233e-27 after 100 riffles. Nonuniformity survives, but a large difference need not. Removed far only.
- MINOR | X3: "across all possible orders, the largest gap in probability" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Levin-Peres-Wilmer p. 47 maximizes over events, hence sets of orders. The existing half-sum computation is correct; clarified only the prose.
- MINOR | X4: "drawn as the bracketed segments." | git show f5e5f84f34:public/strata/seven-shuffles/index.html | The inline risingSegments function assigns a sequence ID to every card; brackets restart when adjacent IDs differ. For (3,1,4,2), two rising sequences produce four fragments. The caption now names contiguous fragments.
- WRONG | X5: "indistinguishable from random." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Levin-Peres-Wilmer pp. 113-114 gives TV about 0.3341 after seven. Independent exact evaluation gives 0.334060999468. The page now identifies its chosen half-distance crossing and remaining bias.
- WRONG | X6: "The standard convention [...] once the distance has dropped below one-half" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Levin-Peres-Wilmer pp. 54-55 uses one-quarter as its default, while explaining that the choice is arbitrary. The page now owns its one-half choice.
- WRONG | X7: "it is the same fact as the mixing time, seen from the other side." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The inline likelihood loop estimates one of counts 1 through 11. This is a different task from the event-probability comparison that defines total variation on textbook p. 47.
- WRONG | X8: "Detection fails at precisely the point the deck is mixed" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf ; research/seven-shuffles/verify-assay.mjs | At seven, P(r<=25)=0.649993904972; under uniform it is 0.315932905503. Detectable information remains after the chosen threshold is crossed.
- WRONG | X9: "While the deck is under-shuffled the guess lands on the nose" | git show f5e5f84f34:public/strata/seven-shuffles/index.html ; research/seven-shuffles/verify-assay.mjs | Exact likelihood comparison gives MLE five at r=19, which occurs after six with probability 0.028742507752. Overall six-riffle count accuracy is 0.376164180682. The guarantee is false.
- WRONG | X10: "A deck the best detector can't read is, by definition, random." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The instrument compares eleven finite shuffle-count hypotheses and has no uniform hypothesis. Its failed count estimate cannot certify uniformity.
- WRONG | X11: "dealers' shoes and shuffling machines do considerably more than seven." | https://chance.dartmouth.edu/course/topics/winning_number.html ; https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf ; https://www.stat.berkeley.edu/~aldous/157/Papers/shuffling_machines.pdf | Kolata, 1990-01-09, How Casinos Do It reports four to seven in Las Vegas. Diaconis survey p. 6 describes an unrepeated shelf-shuffler operation. Diaconis-Fulman-Holmes section 2.1 p. 2 distinguishes pressure shufflers with perhaps seven to ten passes from shelf methods. These do not establish a universal count or 2026 practice.
- WRONG | X12: "including the published Aldous–Diaconis table" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey p. 3, Theorem 1 explicitly credits Bayer-Diaconis for the exact table; Aldous-Diaconis 1986 p. 344 instead gives upper bounds for counts 10 through 14. Both scanned pages were read visually and by OCR.
- WRONG | X13: "The cutoff phenomenon and the canonical distance table:" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The 1986 p. 344 table is an upper-bound table, unlike the exact table in survey Theorem 1. The bibliography now retains 1986 for earlier cutoff work and credits the exact table to 1992.
- MINOR | X14: "Everything numeric on this page [...] in exact BigInt arithmetic," | git show f5e5f84f34:public/strata/seven-shuffles/index.html | BigInt computes the rational TV sum. The inline Math.log and Math.log2 calls, plotting conversion and rounded decimals use floating point; prose figures are static. Narrowed the checking claim.
- MINOR | X15: "that the GSR probabilities sum to exactly 1 for every k" | git show f5e5f84f34:research/seven-shuffles/verify.mjs | research/seven-shuffles/verify.mjs tests normalization only at n=52, k=1 through 12. The general identity remains true; the description now states the finite tested range.
- WRONG | X16: "trail the dovetail shuffle to its lair," | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf | The Bayer-Diaconis preprint title on PDF p. 3 begins Trailing. Restored its intact words in the dek, with surrounding syntax adjusted. Checker level MINOR; triage level WRONG.
- WRONG | X17: "trails the dovetail shuffle to its lair" | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf | The same title page begins Trailing, not trails. Restored the intact title in the markdown body and verifier detail. Checker level MINOR; triage level WRONG.
- WRONG | X18: "canonical Aldous–Diaconis table" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The markdown dek repeated the exact-table attribution contradicted by survey p. 3 and the distinct upper-bound table in Aldous-Diaconis p. 344. Corrected to Bayer-Diaconis 1992.
- WRONG | X19: "goes blind at exactly the point the deck is mixed, because those are the same statement." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The metadata repeated the equivalence between count-estimation failure and mixing. The instrument only estimates one of eleven counts; textbook pp. 47 and 114 define and quantify the separate distributional question.
- WRONG | X20: "its maximal increasing runs" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf ; research/seven-shuffles/verify-assay.mjs | Textbook p. 107 defines rising sequences using consecutive values in relative order. In (3,1,4,2), rising sequences number two while ordinary increasing runs number three. Checker level MINOR; triage level WRONG.
- WRONG | X21: "canonical Aldous–Diaconis table" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The markdown body repeated the exact-table attribution. Survey p. 3 credits Bayer-Diaconis; the 1986 table gives upper bounds. Corrected the body copy.
- WRONG | X22: "a permutation's maximal increasing runs" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf ; research/seven-shuffles/verify-assay.mjs | The related-layer note repeated the different-statistic equivalence. The counterexample (3,1,4,2) separates the counts even though their distributions over uniform permutations share Eulerian weights. Checker level MINOR; triage level WRONG.
- MINOR | X23: "the riffle shuffle's distance-to-random is pinned at 1" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf ; research/seven-shuffles/verify-assay.mjs | Survey p. 3 describes proximity to the maximum. Exact TV(4)=0.999999533431, rounded to 1.000. In the qualified finite-deck analogy, changed pinned at 1 to stays near 1.
- WRONG | X24: "the order is essentially gone" | git show f5e5f84f34:public/strata/seven-shuffles/index.html ; research/seven-shuffles/verify-assay.mjs | P(r>=14 after four)=0.723952349521 while TV(4)=0.999999533431. The count-based branch could announce lost order for such decks; it now avoids a uniformity certificate.
- WRONG | X25: "blind (deck is mixed)" | git show f5e5f84f34:public/strata/seven-shuffles/index.html ; research/seven-shuffles/verify-assay.mjs | At six, r=20 occurs with probability 0.068510891092 and MLE is five. TV(6)=0.613549596566, above the page threshold. The actual reveal code was exercised with this possible outcome and now reports a missed estimate.
- WRONG | X26: "the detector can't tell seven shuffles from a thousand" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf ; research/seven-shuffles/verify-assay.mjs | The exact law gives P(r<=25)=0.649993904972 at seven and 0.315932905503 at a thousand, to twelve decimals. The statistic distinguishes the distributions; the implemented candidates also stop at eleven.
- WRONG | X27: "under-shuffled: the fingerprint still pins the count" | git show f5e5f84f34:public/strata/seven-shuffles/index.html ; research/seven-shuffles/verify-assay.mjs | At r=19 the inline likelihood estimates five, yet that count has positive probability after six. The readout now calls its result an estimate. Checker level MINOR; triage level WRONG.
- MINOR | X28: "that the GSR probabilities sum to exactly 1 for every k" | git show f5e5f84f34:research/seven-shuffles/verify.mjs | The markdown verifier description repeated every k, while the loop checks n=52, k=1 through 12. Changed only the claimed scope.
- WRONG | X29: "close enough that no betting strategy has a meaningful edge." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Levin-Peres-Wilmer p. 114 gives a playable solitaire example with about 0.801 win probability after seven GSR riffles versus exactly 0.5 for a uniform deck. This refutes the blanket no-meaningful-edge claim.
- CONFIRMED | X30: "the rising-sequence count has saturated near its random value (~26)" | https://www.scirp.org/journal/paperinformation?paperid=92119 ; research/seven-shuffles/verify-assay.mjs | REJECTED: Silverman section 2.5 discusses about thirteen shuffles for the asymptotic result; Figure 7 gives limiting mean 26.5. Exact means are 24.710367638668 at seven and 25.603445877611 at eight. The qualified near ~26 phrase is defensible and remains; separate claims of blindness or uniformity are corrected.
- WRONG | X31: "Press a shuffle count and read the distance." | git show f5e5f84f34:public/strata/seven-shuffles/index.html | Instrument 2 has the i2-n input and i2-recompute button handlers; shuffle counts are canvas labels without click handlers. The instruction now points to the deck-size slider, which was exercised.
- WRONG | X32: "both are about the geometry of all 52!" | git show f5e5f84f34:src/content/strata/a-sextillion-ways-home.md | The linked layer plain description and body at f5e5f84f34 concern five-bell Hamiltonian tours through 5!=120 orders. This layer has 52 cards and 52! orders. Corrected the source relation and regenerated its served teaser.

- CONFIRMED | C001: "A brand-new deck is in perfect order." | https://ambitiouswithcards.com/new-deck-order/ | N1/N2 retain a direct card-handling account of standard USPCC new-deck order; this confirms the ordered starting premise, not one universal manufacturer layout.
- CONFIRMED | C002: "After one shuffle it is obviously still ordered. After two, still." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorems 1 and 2: the rounded first two distances are 1.000 and the support has at most two and four rising sequences.
- CONFIRMED | C003: "Almost nothing, for several shuffles [...] and then, all at once, random. The edge of that cliff is at seven." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The checkers confirmed seven under the page's chosen TV-below-one-half convention. The current opening states that convention and the remaining distance explicitly.
- CONFIRMED | C004: "The randomization does not creep in. It arrives suddenly." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 1 supports the qualitative cutoff description, with a soft transition for a finite deck.
- CONFIRMED | C005: "When you riffle a deck [...] the casino calls it a dovetail [...] you cut it roughly in half and let the two halves fall back together in a rough interleave." | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf | The primary paper names the riffle/dovetail method and its two-packet interleaving procedure.
- CONFIRMED | C007: "That single rule reproduces the statistics of real human riffles remarkably well" | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The primary paper and survey support empirical fit with shuffler dependence; textbook p. 114 contrasts Reeds's results and Diaconis's finer interleaving. The page retains its model qualification.
- CONFIRMED | C009: "A GSR riffle is what's called a 2-shuffle (the deck is cut into 2 packets); do it again and the composition is exactly a 4-shuffle; k riffles in a row is precisely one 2k-shuffle." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey pp. 3-4 gives the composition of an a-shuffle and b-shuffle as an ab-shuffle.
- CONFIRMED | C010: "There are 52! possible orderings [...] a number near 8×1067" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey p. 5 gives the order of magnitude; integer multiplication confirms 52! has 68 digits and begins 8065817517.
- CONFIRMED | C011: "more than the estimated atoms in our galaxy." | https://www2.imm.dtu.dk/courses/02222/Spring_2010/W8L2/security_02222.pdf | N1/N2 identify the 10^67 estimate on DTU slide 21. Retained as an order-of-magnitude comparison from that lecture, not a precision astronomical census.
- CONFIRMED | C012: "A perfectly shuffled deck is one where all 52! orders are equally likely." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey p. 2 defines uniform probability as 1/n! for every order.
- CONFIRMED | C015: "It runs from 1 (a dead giveaway [...] you could win a bet about the order) down to 0 (truly indistinguishable from random)." | https://www.stat.berkeley.edu/~aldous/Papers/shuffling.pdf | The definition bounds TV between zero and one. These are abstract endpoints; finite full-support uniform reference does not make the deck's distance exactly one.
- CONFIRMED | C017: "Pick up the shuffled deck and find where card 1 sits, then card 2, then 3 [...] every time the next card is somewhere after the last, you're still in the same rising run; when it jumps backward, a new run begins." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Textbook p. 107 defines consecutive values in relative order; the inverse-position counter matches.
- CONFIRMED | C018: "A fresh deck is a single rising sequence." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The definition gives one rising sequence for the identity order.
- CONFIRMED | C019: "one riffle can at most double the number of rising sequences." | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | Cutting can split each original consecutive-value sequence into at most two pieces; interleaving preserves each packet's order. Survey Theorem 2 also gives the corresponding support bound from an ordered start.
- CONFIRMED | C020: "So after k shuffles there are at most 2k of them [...] one shuffle, at most 2; seven shuffles, at most 128." | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | Starting at one rising sequence and repeating the bound gives at most 2^k; the physical bound of 52 is tighter than 128 at seven but both upper bounds are true.
- CONFIRMED | C021: The fingerprint helps estimate your shuffles. | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | Retained only as a statistical fingerprint for estimating a count. X9/X27 remove any guaranteed-count reading; the current page explicitly says helps estimate.
- CONFIRMED | C022: "Bayer and Diaconis titled their paper, perfectly, "Trailing the dovetail shuffle to its lair."" | https://api.crossref.org/works/10.1214/aoap/1177005705 | The preprint title page and N1/N2 Crossref metadata agree on the intact title.
- CONFIRMED | C024: "Watch the count: it can at most double per shuffle, and it heads for about half the deck (~26) once the deck is mixed, at which point the fingerprint can no longer tell you much." | https://www.scirp.org/journal/paperinformation?paperid=92119 | Silverman section 2.5 and Figure 7 give limiting mean 26.5; exact means at seven and eight support approximate nearness.
- CONFIRMED | C026: As shuffling continues, the rising-sequence distribution approaches that of a uniform deck. | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Confirmed as convergence of the count distribution toward its uniform-deck distribution. The edited paragraph now states this limit without certifying an individual deck.
- CONFIRMED | C027: "Bayer and Diaconis (1992) proved that after a 2k-shuffle, the probability of ending in a particular order depends on nothing about that order except its number of rising sequences r:" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 2 explicitly attributes the order probability law to Bayer-Diaconis.
- CONFIRMED | C028: "P(order) = C(2k + 52 − r, 52) / 252k" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Substitute n=52 and a=2^k into survey Theorem 2; the HTML exponents implement that expression.
- CONFIRMED | C029: "Every order with the same number of rising sequences is exactly equally likely; orders with more rising sequences are rarer." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The binomial term depends only on r and decreases on its positive support; beyond the support the probabilities are all zero, so globally the relation is non-increasing.
- CONFIRMED | C030: "Count how many orders have each value of r [...] those counts are the famous Eulerian numbers [...] and you can sum up the total-variation distance to the perfect deck exactly." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Textbook section 8.3 relates rising sequences to descents of the inverse; survey p. 5 weights the TV sum by Eulerian counts.
- CONFIRMED | C031: "No simulation, no approximation: a finite sum of whole-number ratios." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The finite TV sum is a rational number. BigInt keeps it exact until conversion for display and plotting.
- CONFIRMED | C032: "52! has 68 digits; 252·10 has 157" | git show f5e5f84f34:research/seven-shuffles/verify.mjs | Independent integer multiplication gives 68 digits for 52! and 157 digits for 2^520.
- CONFIRMED | C035: "For 52 cards the curve is the signature of a cutoff: it clings to 1 [...] totally unshuffled, in effect [...] through four riffles, then falls off a cliff, crossing the half-way line at the seventh." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Exact TV(4)=0.999999533431 and TV(7)=0.334060999468 support proximity to one and the first half crossing.
- CONFIRMED | C036: "a bigger deck needs more shuffles, but always with the same abrupt drop." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 1 gives a limiting profile in k=(3/2)log2(n)+c, supporting the changing deck-size scale.
- CONFIRMED | C037: "The center of the cliff sits near (3/2)·log₂n ≈ 8.5 for 52 cards; the deck is already half-mixed a little before that, at seven." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The survey's general-n scale evaluates to 8.550659577212 for n=52; the finite-deck half crossing is seven.
- CONFIRMED | C038: "This shape has a name: the cutoff phenomenon. The distance does not decay smoothly; it stays near its maximum, then collapses through a narrow window." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 1 and its remarks support a narrow cutoff window, not a literal discontinuity at finite n.
- CONFIRMED | C040: "Aldous and Diaconis found this same abrupt-transition behaviour across a whole family of shuffles and walks in the 1980s; the riffle is its most famous instance." | https://www.stat.berkeley.edu/~aldous/Papers/shuffling.pdf | N1/N2 re-read the 1986 paper pp. 334-335 on shuffles and random walks with a general cutoff phenomenon. This is distinct from the later exact table.
- CONFIRMED | C041: "For the first four riffles you are, to a very good approximation, not shuffling at all. The work happens in a window two shuffles wide." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The checkers read this as a statement about TV: near one through four, then 0.923733 at five to 0.334061 at seven. It does not assert unchanged physical order.
- CONFIRMED | C043: "The distance is still essentially 1 through four shuffles; it is 0.924 at five, 0.614 at six, and at seven it finally falls under a half, to 0.334." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 1 and the independent integer-ratio evaluation agree at five, six and seven.
- CONFIRMED | C044: "Each further shuffle then roughly halves it [...] 0.167, 0.085, 0.043 [...] but the qualitative job is done at seven." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The later rounded distances 0.167, 0.085 and 0.043 roughly halve; the page now keeps residual bias explicit instead of implying an exact stopping of information.
- CONFIRMED | C045: "That is the number Diaconis gave, and it is why a 1990 New York Times story ran under the headline that seven is the magic number for a deck of cards." | https://chance.dartmouth.edu/course/topics/winning_number.html | The Dartmouth archive dates the report to 1990-01-09 and gives the seven-is-winning-number headline.
- CONFIRMED | C046: "If a shuffled deck still carries the mark of how many times it was riffled, then in principle you can read that mark [...] guess the number of shuffles from the order alone." | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | The early rising-sequence structure supports inference about the count. A statistical guess is supported, without guaranteed recovery.
- CONFIRMED | C054: "Other shuffles are not so kind: the overhand shuffle [...] sliding small packets off the top [...] is dramatically worse, needing thousands of repetitions to mix a deck, because each pass moves so little." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey p. 5 describes the overhand model on a scale exceeding 2500 at n=52, supporting thousands under that model.
- CONFIRMED | C056: "for the riffle, the order of a 52-card deck survives almost intact for four shuffles, and is gone by seven [...] and the going is a cliff, not a slope." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The checkers confirmed the finite-deck crossing under the stated half convention. The edited copy spells out the near-one distance through four and nonzero residual at seven.
- CONFIRMED | C057: "every factual claim is checked, and every model is named a model." | git show f5e5f84f34:research/seven-shuffles/verify.mjs | N1/N2 read WAKE.md as stating the site's checking rule. This is a rule, not a certificate that every historical sentence is true.
- CONFIRMED | C058: "The shuffle here is the Gilbert–Shannon–Reeds idealization of a riffle [...] a model, chosen because it matches measured human shuffles well (Bayer–Diaconis 1992) and because it makes the problem exactly solvable; it is not a claim that every hand riffles identically." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | The primary law is exactly solvable; empirical fit varies by person as textbook p. 114 describes. The page explicitly calls GSR an idealization.
- CONFIRMED | C060: ""Seven" is the first shuffle whose distance falls below one-half; the asymptotic cutoff center is (3/2)·log₂52 ≈ 8.55." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 1 and exact arithmetic give the first half crossing at seven and the asymptotic scale about 8.55.
- CONFIRMED | C061: "Bayer, D. & Diaconis, P. (1992). "Trailing the dovetail shuffle to its lair." Annals of Applied Probability 2(2):294–313. doi:10.1214/aoap/1177005705." | https://api.crossref.org/works/10.1214/aoap/1177005705 | N1/N2 Crossref metadata confirm the 1992 article, journal, volume, issue and pages; the title was also read on the preprint.
- CONFIRMED | C062: "Aldous, D. & Diaconis, P. (1986). "Shuffling cards and stopping times." American Mathematical Monthly 93(5):333–348. doi:10.1080/00029890.1986.11971821." | https://api.crossref.org/works/10.1080/00029890.1986.11971821 | N1/N2 Crossref metadata confirm the 1986 article, journal, volume, issue and pages; its upper-bound table was separately read.
- CONFIRMED | C064: "Gilbert, E. (1955), Bell Labs technical memo; Reeds, J. (1981), unpublished [...] as described and formalized in Bayer & Diaconis (1992)." | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | N1/N2 bibliography entries and survey pp. 2 and 5 give Gilbert 1955 and Reeds 1981.
- CONFIRMED | C065: "Diaconis, P. (2003). "Mathematical developments from the analysis of riffle shuffling," in Groups, Combinatorics & Geometry, 73–97;" | https://api.crossref.org/works/10.1142/9789812564481_0005 | N1/N2 Crossref metadata give the March 2003 chapter and pp. 73-97; the downloaded Stanford text is its 2002 report version.
- CONFIRMED | C066: "Kolata, G. (1990). "In Shuffling Cards, 7 Is Winning Number," The New York Times, Jan 9, 1990." | https://chance.dartmouth.edu/course/topics/winning_number.html | The Dartmouth archive confirms the report's title, author and 1990-01-09 date.
- CONFIRMED | C067: "Graham, Knuth & Patashnik, Concrete Mathematics (2nd ed., 1994), §6.2." | https://www-cs-faculty.stanford.edu/~knuth/gkp.html | N1/N2 retain Knuth's author-page bibliographic record and the section 6.2 reading of the book; these support the second edition and 1994 attribution.
- CONFIRMED | C068: "Eulerian numbers (the count of orders by rising sequences) and Worpitzky's identity (which makes the probabilities sum to one)" | https://dokumen.pub/concrete-mathematics-a-foundation-for-computer-science-2nd-edition-2nbsped-0201558025-9780201558029.html | N1/N2 re-read section 6.2 equation 6.37. Worpitzky's identity and Eulerian symmetry yield the general normalization; the verifier tests a finite set of cases.
- CONFIRMED | C078: ""Random" means total-variation distance below one-half [...] a convention, not a law of nature; finer thresholds need more shuffles." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Retained as this page's declared half threshold. Textbook section 4.5 allows tolerance choices and defaults to one-quarter; smaller tolerances need at least as many shuffles.
- CONFIRMED | C089: "Bayer and Diaconis proved in 1992 that the answer is seven, and that it arrives as a cliff: almost nothing happens for four shuffles, then the order collapses inside a two-shuffle window." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The seven-shuffle statement is supported under the chosen half threshold. The metadata now states that threshold and the nonzero residual explicitly.
- CONFIRMED | C095: "GSR matches measured human shuffles well, and it has the magical property that *shuffles multiply* [...] k riffles in a row is exactly one 2ᵏ-shuffle." | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf | Survey pp. 3-4 gives multiplicative composition; textbook p. 114 supplies the empirical-fit qualification.
- CONFIRMED | C099: Both layers study permutations: 52 cards here, tours through the orders of five bells there. | git show f5e5f84f34:src/content/strata/a-sextillion-ways-home.md | The linked layer has five bells. Retained only the common permutation setting, with X32 explicitly separating its 120 order vertices and tours from this layer's 52! card orders.
- CONFIRMED | C101: "The cutoff phenomenon is a phase transition in time." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | The survey's general-n cutoff profile supports the limiting phase-transition analogy.
- CONFIRMED | C103: "Both are sudden transitions that only sharpen as the system grows [...] a finite deck has a soft cutoff, an infinite family of decks a perfectly sharp one, exactly as a finite lattice rounds percolation's threshold that an infinite lattice makes a knife-edge." | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | N1/N2 confirmed the qualitative finite/infinite analogy: the fixed-width shuffle window shrinks relative to log n, while a finite percolation system lacks a literal infinite cluster.
- CONFIRMED | C104: "Both ask how many random steps it takes to reach a target state on n items, and both answers hide in the logarithms." | git show f5e5f84f34:src/content/strata/coupon-collector.md | The pinned coupon-collector dek gives n ln n growth; this layer's asymptotic scale involves log2(n). The comparison concerns a logarithmic component, not identical rates.
- CONFIRMED | C106: Few rising sequences; the fingerprint is informative. | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | Retained only the early count's informative structure, not guaranteed recovery. The current readout explicitly calls the fingerprint informative.
- CONFIRMED | C107: "structure fading; the fingerprint is still informative" | https://stacks.stanford.edu/file/druid:pb950nq8223/pb950nq8223_EFS-NSF-329.pdf ; research/assay-line/queue-1014c/seven-shuffles/N1-sources.json | For r=5 through 13 the likelihood favors early counts. The statement concerns information, not exact count recovery.
- CONFIRMED | C116: "Nothing here may lie about anything real; that rule never bends." | git show f5e5f84f34:WAKE.md | N1/N2 quote the rule in pinned WAKE.md. Confirmed as the site's rule, not proof of universal compliance.
- CONFIRMED | C122: "that a 2k-shuffle gives an order of r rising sequences probability C(2k+n−r, n)/2kn, that riffles compose multiplicatively, the total-variation cutoff, and the "seven shuffles" conclusion for n=52:" | https://stacks.stanford.edu/file/druid:nj257wm2658/nj257wm2658_2002-16.pdf | Survey Theorem 2 supplies the probability law and composition; Theorem 1 supplies the finite distance table and limiting profile.

- WRONG | R1: "seven riffles, and not before." in research/seven-shuffles/shoot-og.mjs | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf ; research/seven-shuffles/verify-assay.mjs | The template repeated X5's unqualified randomization claim. Textbook pp. 113-114 and exact arithmetic give TV(7)=0.334060999468. Replaced it with the chosen one-half threshold and remaining distance. The current PNG and thumbnail do not display this sentence; no share image was rendered.
- WRONG | R2: "How Many Riffle Shuffles to Randomize a Deck? Seven" in scripts/seo-titles.tsv | https://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf ; scripts/seo-apply-titles.mjs | The shared title table repeats X5's unqualified answer, and the apply script overwrites the standalone HTML title from that entry. Seven leaves TV about 0.334. The required title is "How Many Riffle Shuffles? The Threshold at Seven"; the shared-file edit is outside this worker's authorized scope.

## Findings

- [fixed] X1: Separated the 1955 report of the Gilbert-Shannon work from Reeds's independent 1981 work in the page; dated correction mirrored in markdown.
- [fixed] X2: Removed far from the finite-riffle nonuniformity sentence. MINOR, record only.
- [fixed] X3: Specified sets of possible orders in the TV definition. MINOR, record only.
- [fixed] X4: Named the drawn brackets contiguous fragments of rising sequences in captions, markdown and code comments. MINOR, record only.
- [fixed] X5: Replaced finite-deck indistinguishability with the chosen half-distance crossing and residual TV in prose and metadata; correction shared with X26.
- [fixed] X6: Made one-half explicitly this page's choice in prose, chart and readout, and identified the textbook default; dated correction mirrored.
- [fixed] X7: Separated count estimation from testing uniformity in Instrument 3 prose; correction shared with X8, X10 and X19.
- [fixed] X8: Removed the asserted exact point of detection failure; the finite distance remains stated; shared dated correction mirrored.
- [fixed] X9: Removed the exact-recovery promise from the caption and duplicate fingerprint claims; a possible six-riffle outcome witnesses failure; correction shared with X27.
- [fixed] X10: Removed the claim that this instrument defines or certifies randomness; shared dated correction mirrored.
- [fixed] X11: Replaced the universal casino count with dated reporting from 1990 and the survey's shelf method, with source links; dated correction mirrored.
- [fixed] X12: Corrected the table attribution in prose, JSON-LD and verifier source, then regenerated its downloadable copy; correction shared with X13, X18 and X21.
- [fixed] X13: Retained Aldous-Diaconis for earlier cutoff and upper-bound work and assigned the exact table to Bayer-Diaconis; shared dated correction mirrored.
- [fixed] X14: Narrowed exactness to rational TV computation in the page, dek and verifier comment, noting rounded display and floating-point logarithms. MINOR, record only.
- [fixed] X15: Stated n=52, k=1 through 12 in the page check description and verifier headings; regenerated the public verifier. MINOR, record only.
- [fixed] X16: Restored the intact Trailing title in the dek; dated correction shared with X17. Preserved triage WRONG and checker MINOR.
- [fixed] X17: Restored the intact title in the markdown body and verifier detail; shared dated correction mirrored. Preserved triage WRONG and checker MINOR.
- [fixed] X18: Corrected the dek's exact-table credit to Bayer-Diaconis 1992; shared dated correction mirrored.
- [fixed] X19: Corrected the detector/mixing equivalence in the dek and markdown body as well as HTML comments; shared dated correction mirrored.
- [fixed] X20: Defined rising sequences by consecutive values in relative order in the markdown and page caption; correction shared with X22. Preserved triage WRONG and checker MINOR.
- [fixed] X21: Corrected the markdown body's exact-table credit to Bayer-Diaconis 1992; shared dated correction mirrored.
- [fixed] X22: Corrected the related-layer definition and distinguished the statistics despite their shared Eulerian weights; shared dated correction mirrored. Preserved triage WRONG and checker MINOR.
- [fixed] X23: Changed pinned at 1 to stays near 1 in the percolation relation. MINOR, record only.
- [fixed] X24: Replaced the r>=14 order-loss certificate with a count description and limit on inference; dated correction mirrored and the actual readout exercised.
- [fixed] X25: Removed the r>=20 mixing classification and made the reveal report a correct or missed estimate; dated correction mirrored and the six-riffle witness exercised.
- [fixed] X26: Removed seven-versus-thousand blindness from the caption and readout; stated the finite candidate range and residual distance; shared dated correction mirrored.
- [fixed] X27: Removed the readout's guaranteed count recovery; shared dated correction mirrored and the r=19 witness exercised. Preserved triage WRONG and checker MINOR.
- [fixed] X28: Stated n=52, k=1 through 12 in the markdown check description. MINOR, record only.
- [fixed] X29: Replaced the no-edge guarantee with residual bias and the sourced solitaire example; dated correction mirrored.
- [fixed] X31: Replaced the nonexistent shuffle-count action with the existing deck-size slider; dated correction mirrored and both controls exercised.
- [fixed] X32: Separated the 52-card orders from tours through 120 five-bell orders in the markdown source and regenerated the nearby teaser; dated correction mirrored.
- [fixed] R1: Corrected research/seven-shuffles/shoot-og.mjs to say seven first crosses this page's one-half TV threshold, with about 0.334 remaining. Edited the template only; no image generator was run.
- [fixed] R2: On 2026-10-18 scripts/seo-titles.tsv:81 was changed from "How Many Riffle Shuffles to Randomize a Deck? Seven" to "How Many Riffle Shuffles? The Threshold at Seven", matching the corrected page title, so scripts/seo-apply-titles.mjs can no longer restore the unqualified answer; the copies pass found no other copy in the seven-shuffles layer; board note a1f1d5 is closed.