---
slug: coupon-collector
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: 91
confirmed: 63
wrong: 28
unverifiable: 0
observed: 0
stale: 0
---

This pass implements the coupon-collector triage against c13e5951ed. All 28 triage items were re-read against primary sources or the authored code before editing. The 63 CONFIRMED claims carry forward the source rereads recorded by Codex in research/assay-line/queue-1014c/coupon-collector/N2-codex.md; this does not claim a fresh third reread of every one. The wrong total includes 16 WRONG and 12 MINOR lines, with eight of the latter confined to evidence-record repairs. The 28 triage items are fixed within this layer. Repair review leaves nine WRONG findings open for copies in other layers (X6, X9-X14 and X16-X17), with 19 findings fixed. The N1 source record remains a record of the pre-fix page, with its eight evidence gaps repaired.

## Claims
- MINOR | You stop the instant you hold a complete set. | https://www.randomservices.org/random/urn/Coupon.html | [X1] Definitions gives W_k=min{n:V_n=k}; k=m is the first complete collection. The page is true; its recorded excerpt omitted this definition.
- MINOR | And you need one of each; | https://www.randomservices.org/random/urn/Coupon.html | [X2] The same first-attainment definition at k=m requires all m distinct types. The source record needed that missing definition.
- MINOR | The clean answer rests on three assumptions, each named here. | https://www.randomservices.org/random/urn/Coupon.html | [X3] Definitions combines independent uniform sampling with replacement and first attainment of all m types. Those are the three page conditions; the record omitted the last condition.
- MINOR | The first few come easily; the album fills fast. Then it slows. | https://gwern.net/doc/statistics/order/1960-newman.pdf; https://www.randomservices.org/random/urn/Coupon.html | [X4] Decomposition as a Sum gives geometric p_i=(m-i+1)/m, so mean phase waits increase with i. The old hobby excerpt did not support the slowdown.
- MINOR | So the standard deviation grows like n·π/√6 ≈ 1.28 n | https://www.randomservices.org/random/special/ExtremeValue.html; https://www.randomservices.org/random/urn/Coupon.html | [X5] Moments gives Var(W_k)=sum (i-1)m/(m-i+1)^2. At k=m=n, reindexing gives Var(T)=n^2 sum_{j=1}^n 1/j^2-nH_n, hence SD(T)/n tends to pi/sqrt(6)=1.282549830161864. The old limiting-Gumbel variance excerpt alone was insufficient.
- WRONG | The distribution is sharp relative to its mean, but it has a long, heavy right tail: the runs that go on and on. | https://www.randomservices.org/random/special/ExtremeValue.html | [X6] Distribution Functions gives G(c)=exp(-exp(-c)); its survival function is asymptotic to exp(-c). Moments gives finite M(t)=Gamma(1-t) for positive t<1. Independently, the finite collector satisfies P(T>r)<=n(1-1/n)^r by the union bound. The technical heavy-tail classification is false.
- WRONG | rescale the collection time as (T − n ln n)/n and it converges, for every starting distribution of luck, onto the Gumbel distribution exp(−e^−c) (Erdős–Rényi, 1961). | https://static.renyi.hu/renyi_cikkek/1961_on_a_classical_problem_of_probability_theory.pdf | [X7] Visually re-read printed p. 215, assumptions and equation (2): independent uniform urn choices from an empty start; the universal quantifier is over real CDF arguments. A complete initial set has T=0 and standardized time -ln n, so arbitrary starting laws are not covered.
- WRONG | Real print runs deliberately make the last few rare; | https://paninigroup.com/en/int/ExtraStickers | [X8] Re-fetched the English manufacturer FAQ: Extra Stickers are unnecessary for completing the main collection and have no dedicated album space. Their rarity does not support a general claim about scarcity of required final stickers.
- WRONG | To collect the second half of the set | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | [X9] Visually re-read printed p. 216, section 4: C_j is the first time j distinct types have been collected and E[C_j]=m(H_m-H_(m-j)). Thus n(H_n-H_(n/2)) is the zero-to-n/2 expectation for even n, not the remaining-half expectation.
- WRONG | costs n·(Hn − Hn/2) draws on average, which tends to n·ln 2. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | [X10] The same p. 216 formula gives first-half cost n(H_n-H_(n/2)); dividing by n gives a limit of ln 2. The second half costs nH_(n/2). Independent exact sums at n=100 give 68.81721793101951 and 449.9205338329425 draws respectively.
- WRONG | collecting the second half of the set costs n·(H_n − H_{n/2}) → n·ln 2, the very sum that dooms the hundred prisoners. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | [X11] The dek repeats the first-half/second-half error. Flajolet-Gardy-Thimonier p. 216 defines C_j from an empty collection, so j=n/2 gives the first half.
- WRONG | For the collector it is the cost, scaled by n, of the SECOND half of the set: going from n/2 distinct coupons to all n takes n·(H_n − H_{n/2}) → n·ln 2 draws. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | [X12] The relation note explicitly reversed the state range. From p. 216, zero to n/2 costs n(H_n-H_(n/2)); n/2 to n costs nH_(n/2). Its concluding rare-last-coupons comparison was also wrong.
- WRONG | Collecting the *second half* of the set costs `n·(H_n − H_{n/2}) → n·ln 2` | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | [X13] The markdown bridge repeats the same false second-half label for the first-half formula on p. 216.
- WRONG | For the hundred prisoners it is the probability a doom-loop longer than fifty exists (so they survive with chance 1 − ln 2). | https://www.cl.cam.ac.uk/~gw104/Locker_Puzzle.pdf | [X14] Visually re-read printed p. 29, right column, and read p. 30: the pointer-following success probability for 100 is 1-sum_{k=51}^{100}1/k; 1-ln 2 is the large-even-population limit. Independent rational summation gives 0.3118278206898048, about 31.18%.
- MINOR | Both read γ off the gap H_n − ln n; | https://algo.inria.fr/flajolet/Publications/book.pdf; https://www.randomservices.org/random/special/ExtremeValue.html | [X15] Downloaded and visually re-read printed p. 724 (PDF page 740), equation (4): H_n=log n+gamma+1/(2n)-1/(12n^2)+O(n^-4). The old excerpt was a Gamma reflection proof on p. 745, not evidence for the true harmonic-gap claim.
- WRONG | a long heavy tail. | https://www.randomservices.org/random/special/ExtremeValue.html | [X16] The Gumbel CDF and finite positive exponential moments establish a light right tail, as in X6. The finite collector also has the exponential union bound; the relation note should say long right tail.
- WRONG | The spread is governed by n²·π²/6, so the standard deviation grows only like 1.28·n while the mean grows like n ln n: sharp relative to its size, but with a long heavy right tail. | https://www.randomservices.org/random/special/ExtremeValue.html | [X17] The dek repeats the false heavy-tail classification. The surrounding variance asymptotic and relative concentration are supported by the finite variance formula; only the tail classification needed correction.
- MINOR | Every number on this page is recomputed in your browser by the same routine the offline verifier runs. | public/strata/coupon-collector/index.html; research/coupon-collector/verify.mjs | [X18] Read the whole inline script and verifier: browser expT uses floating-point Hn, while the verifier expFormula uses BigInt fractions. The browser prints fixed 1.283 and 0.367 references and has no finite-variance calculation. The same-routine/every-number promise is false; triage assigns checking overclaims MINOR.
- MINOR | recompute every number in your browser. | public/strata/coupon-collector/index.html | [X19] The description and JSON-LD repeat every-number recomputation, while the browser uses fixed reference strings for sigma and the limiting median. Expected totals, phase costs and simulations are computed live.
- WRONG | Each draw lands on a random slot; a slot you already own just flashes and is wasted. | public/strata/coupon-collector/index.html | [X20] drawOne increments draws; new coupons get class slot fresh, but duplicates only set lastFresh=-1. CSS has no separate duplicate animation. A controlled 0,1,0,0 draw sequence confirmed that duplicate slot 0 stays slot got and the last draw changes no slot class.
- WRONG | The bar at the top counts draws; the slots count distinct coupons. | public/strata/coupon-collector/index.html | [X21] The authored DOM puts col-read after col-slots and controls; render writes textual draws and collected counts. CSS does not reorder it. There is no top draw-count bar.
- WRONG | The histogram is your simulated collection times, standardised to (T − n ln n)/n; the curve is the Gumbel limit exp(−e−c). | public/strata/coupon-collector/index.html; https://www.randomservices.org/random/special/ExtremeValue.html | [X22] The script normalizes the histogram as a density and plots Math.exp(-c-Math.exp(-c)). Random Services Distribution Functions distinguishes this density from the CDF exp(-exp(-c)). At c=1 they are 0.2546463800435825 and 0.6922006275553464.
- MINOR | Every phase cost, variance and limit computed live; | public/strata/coupon-collector/index.html; research/coupon-collector/verify.mjs | [X23] The browser computes expected totals, phase costs, simulation results and the density, but its standard-deviation coefficient is literal. Finite varFormula and varPhases are only in the separate verifier.
- MINOR | every figure checked offline in research/coupon-collector/verify.mjs (19/19). | research/coupon-collector/verify.mjs | [X24] The verifier reads no page file and checks selected one-copy expectations, variances, asymptotics and simulation quantities. It does not check every printed figure or the fixed-m multiple-copy asymptotic. The 19/19 result is retained; the half-collection error is corrected separately.
- WRONG | curve exp(−e−c) | public/strata/coupon-collector/index.html; https://www.randomservices.org/random/special/ExtremeValue.html | [X25] The generated readout repeats the CDF where the plotted density belongs. Re-read the pdf function and Random Services Distribution Functions, as in X22.
- WRONG | curve exp(−e−c) | public/strata/coupon-collector/index.html; https://www.randomservices.org/random/special/ExtremeValue.html | [X26] This duplicates X25 on the same generated readout. The plotted function is the Gumbel density exp(-c-exp(-c)), not the CDF exp(-exp(-c)).
- MINOR | standard deviation → n·π/√6 ≈ 1.283·n | https://www.randomservices.org/random/special/ExtremeValue.html; https://www.randomservices.org/random/urn/Coupon.html | [X27] The finite collection variance gives SD(T)/n tending to pi/sqrt(6)=1.282549830161864, so 1.283 is correctly rounded. The evidence needed the finite-variance calculation, not merely the limiting law variance.
- MINOR | (Euler–Mascheroni 0.577216) | https://algo.inria.fr/flajolet/Publications/book.pdf; https://www.randomservices.org/random/special/ExtremeValue.html | [X28] Random Services Moments gives Euler gamma approximately 0.5772156649, which rounds to 0.577216. Replaced the irrelevant Gamma reflection proof, leaving the correct displayed number alone.
- CONFIRMED | There are n distinct coupons. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [001 model-rules] Introduction: "a sequence of independent random variables" and "each uniformly distributed on \(D\)". The m types in the source are the n types on the page.
- CONFIRMED | Each draw gives you one of them, chosen uniformly at random, independent of everything before | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [002 model-uniform] Introduction: "a sequence of independent random variables" and "each uniformly distributed on \(D\)". The m types in the source are the n types on the page.
- CONFIRMED | Draws are independent, with replacement. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [004 model-replacement] Introduction: "a sequence of independent random variables" and "each uniformly distributed on \(D\)". The m types in the source are the n types on the page.
- CONFIRMED | every coupon equally likely. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [007 uniform-equally] Introduction: "a sequence of independent random variables" and "each uniformly distributed on \(D\)". The m types in the source are the n types on the page.
- CONFIRMED | This is not bad luck or a rigged print run. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [008 not-rigged] Read in the stated uniform model, not as an investigation of manufacturers. The source says each purchase is "equally likely to be any one of \(m\) types"; its increasing geometric waits explain the slowdown without unequal printing.
- CONFIRMED | Split the collection into phases: phase k begins the moment you hold k distinct coupons and ends when you find a new one. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [011 phase-definition] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | During phase k there are n−k coupons you still need, so each draw is new with probability (n−k)/n. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [012 phase-probability] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | so phase k costs n/(n−k) draws on average. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [013 phase-cost] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | The last single coupon costs, on average, a full n draws, because at the end only one in n draws helps you. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [014 last-cost] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | The first coupon costs exactly one draw | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [015 first-cost] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | The wait for a single success at probability p is, on average, 1/p | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [016 geometric-mean] Taking differences of the source formula "E(W_k) = sum(i=1..k) m/(m-i+1)" gives E(Z_i)=1/p_i. This is the positive-integer geometric waiting-time convention used by the page.
- CONFIRMED | Because the phases are independent, their variances add, and the total comes out to n²·(1 + ¼ + 1/9 + …) − n·Hn, whose leading term is n²·π²/6. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [017 variance-formula] Source: "var(W_k) = sum(i=1..k) (i-1)m/(m-i+1)^2". Setting k=m=n and changing index gives n^2 sum(j=1..n) 1/j^2 - n H_n. The finite upper limit is n; dividing by n^2 gives limit pi^2/6. Independent exact phase-variance check passes.
- CONFIRMED | the last ten coupons of a hundred-set cost more draws than the first ninety combined. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [018 last-ten-dek] The source mean sum gives last ten = 100 H_10 = 292.8968253968254 and first ninety = 100(H_100-H_10) = 225.84092636713663. Independently recomputed with exact rational sums; the stated inequality holds in expectation.
- CONFIRMED | A deck of 52 distinct cards: 236 draws. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [019 card-expectation] The source mean formula gives 52 H_52 = 235.97828543626736, which rounds to 236.
- CONFIRMED | A set of 100: 519. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [020 hundred-expectation] The source mean formula gives 100 H_100 = 518.737751763962, which rounds to 519.
- CONFIRMED | It is arithmetic, and the arithmetic is exact: it has a name, the coupon collector's problem, and an answer written in the harmonic numbers. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [021 name-answer] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | E[T] = n/n + n/(n−1) + n/(n−2) + … + n/1 | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [022 expectation-display] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | = n · ( 1/1 + 1/2 + … + 1/n ) = n · Hn | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [023 expectation-harmonic] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | That Hn = 1 + ½ + ⅓ + … + 1/n is the harmonic number | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [024 harmonic-definition] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | The collection time is not spread evenly; it is back-loaded, and brutally so. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [025 back-loaded] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | A standard die (n=6) takes 6·H6 = 14.7 rolls on average to show all six faces. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [026 die-expectation] The fair-die example explicitly gives "E(W) = 14.7". Exact independent sum is 147/10.
- CONFIRMED | like ln n + γ (γ ≈ 0.577, the Euler–Mascheroni constant). | https://algo.inria.fr/flajolet/Publications/book.pdf | Carried forward from the Codex N2 source reread: [027 harmonic-growth] Visually read printed p. 724, equation (4): "H_n = log n + gamma + 1/(2n) - 1/(12n^2) + eta_n", with eta_n=O(n^-4). Thus H_n grows like log n plus gamma, and n H_n has leading term n log n.
- CONFIRMED | only linearly, while the mean grows like n ln n. | https://algo.inria.fr/flajolet/Publications/book.pdf | Carried forward from the Codex N2 source reread: [029 mean-growth] Visually read printed p. 724, equation (4): "H_n = log n + gamma + 1/(2n) - 1/(12n^2) + eta_n", with eta_n=O(n^-4). Thus H_n grows like log n plus gamma, and n H_n has leading term n log n.
- CONFIRMED | Rescale the collection time by subtracting n ln n and dividing by n. | https://static.renyi.hu/renyi_cikkek/1961_on_a_classical_problem_of_probability_theory.pdf | Carried forward from the Codex N2 source reread: [031 gumbel-scaling] Visually read Erdos-Renyi p. 215, equation (2): the probability tends to "exp(-e^(-x)/(m-1)!)" after centering at log n+(m-1)loglog n. At m=1 this is exactly the stated uniform independent coupon-collector limit. The authoritative repository dates the paper to 1961.
- CONFIRMED | As n grows, that standardised quantity stops depending on n at all and converges to a single fixed shape | https://static.renyi.hu/renyi_cikkek/1961_on_a_classical_problem_of_probability_theory.pdf | Carried forward from the Codex N2 source reread: [032 gumbel-convergence] Visually read Erdos-Renyi p. 215, equation (2): the probability tends to "exp(-e^(-x)/(m-1)!)" after centering at log n+(m-1)loglog n. At m=1 this is exactly the stated uniform independent coupon-collector limit. The authoritative repository dates the paper to 1961.
- CONFIRMED | the Gumbel distribution, the law of extremes (Erdős & Rényi, 1961): | https://static.renyi.hu/renyi_cikkek/1961_on_a_classical_problem_of_probability_theory.pdf | Carried forward from the Codex N2 source reread: [033 gumbel-attribution] Visually read Erdos-Renyi p. 215, equation (2): the probability tends to "exp(-e^(-x)/(m-1)!)" after centering at log n+(m-1)loglog n. At m=1 this is exactly the stated uniform independent coupon-collector limit. The authoritative repository dates the paper to 1961.
- CONFIRMED | P( T ≤ n ln n + c·n ) → exp( −e−c ) | https://static.renyi.hu/renyi_cikkek/1961_on_a_classical_problem_of_probability_theory.pdf | Carried forward from the Codex N2 source reread: [034 gumbel-formula] Visually read Erdos-Renyi p. 215, equation (2): the probability tends to "exp(-e^(-x)/(m-1)!)" after centering at log n+(m-1)loglog n. At m=1 this is exactly the stated uniform independent coupon-collector limit. The authoritative repository dates the paper to 1961.
- CONFIRMED | The right tail is the one stubborn coupon. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [036 tail-stubborn] The source gives the last phase geometric success probability 1/n. Its long expected wait and exponential right tail support the informal description of the slow end, not a guarantee that every individual run is slow.
- CONFIRMED | with unequal probabilities the expected time only goes up, | https://files.ele-math.com/articles/jca-14-13.pdf | Carried forward from the Codex N2 source reread: [038 unequal-expectation] Nakata p. 178: "Thus it is harder to collect all kinds of coupons if there is some bias for the probability of appearance of coupons." Corollary 2 explicitly says the uniform expectation is no larger than E(X_p).
- CONFIRMED | and the formula becomes a sum over subsets. | https://files.ele-math.com/articles/jca-14-13.pdf | Carried forward from the Codex N2 source reread: [039 subset-formula] Nakata p. 177 equation (1) is the stated alternating sum over subsets J, with denominator 1-P_J. The record transcription agrees with the readable formula.
- CONFIRMED | the “double Dixie cup” problem (Newman & Shepp, 1960) | https://gwern.net/doc/statistics/order/1960-newman.pdf | Carried forward from the Codex N2 source reread: [040 dixie-name] Newman-Shepp pp. 58-59 title and definition concern collecting m sets; visually read Theorem 2: "E_m(n) = n[log n + (m-1) loglog n + C_m + o(1)] for m fixed". This matches the name and the fixed-m asymptotic quoted on the page.
- CONFIRMED | costs n(ln n + (m−1)·ln ln n + …), | https://gwern.net/doc/statistics/order/1960-newman.pdf | Carried forward from the Codex N2 source reread: [041 dixie-cost] Newman-Shepp pp. 58-59 title and definition concern collecting m sets; visually read Theorem 2: "E_m(n) = n[log n + (m-1) loglog n + C_m + o(1)] for m fixed". This matches the name and the fixed-m asymptotic quoted on the page.
- CONFIRMED | the cousin problem of collecting m copies of every coupon | https://gwern.net/doc/statistics/order/1960-newman.pdf | Carried forward from the Codex N2 source reread: [042 dixie-multiple-copies] Newman-Shepp pp. 58-59 title and definition concern collecting m sets; visually read Theorem 2: "E_m(n) = n[log n + (m-1) loglog n + C_m + o(1)] for m fixed". This matches the name and the fixed-m asymptotic quoted on the page.
- CONFIRMED | That very sum, Hn − Hn/2, is exactly the chance the hundred prisoners are doomed by an over-long loop. | https://www.cl.cam.ac.uk/~gw104/Locker_Puzzle.pdf | Carried forward from the Codex N2 source reread, with the author credit corrected during repair: [048 prisoners-exact] Curtin and Warshauer, p. 29, give success "1 - (1/51 + 1/52 + ... + 1/100) approximately .311828" for 100 players. Its complement is exactly H_100-H_50, the chance of a cycle longer than fifty under the pointer-following strategy.
- CONFIRMED | The same harmonic tail, H_n − H_{n/2} → ln 2, decides both. | https://www.cl.cam.ac.uk/~gw104/Locker_Puzzle.pdf; https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [050 shared-tail] The source proves "sum(k=1..n) 1/(n+k) -> ln 2" for prisoners. Flajolet p. 216 supplies the same sum for the collector FIRST half. The common harmonic sum exists; the erroneous identification with the second half is separately marked WRONG.
- CONFIRMED | There the expected count of records in a shuffled list is H_n | https://algo.inria.fr/flajolet/Publications/book.pdf | Carried forward from the Codex N2 source reread: [051 record-mean] Printed p. 141: "the expected number of records in a random permutation of size n equals Hn, the harmonic number." Independently H_1000=7.485470860550345, approximately 7.5. The comparison is between the two expected values.
- CONFIRMED | a thousand candidates → only ~7.5 records | https://algo.inria.fr/flajolet/Publications/book.pdf | Carried forward from the Codex N2 source reread: [052 record-thousand] Printed p. 141: "the expected number of records in a random permutation of size n equals Hn, the harmonic number." Independently H_1000=7.485470860550345, approximately 7.5. The comparison is between the two expected values.
- CONFIRMED | Two probability problems whose whole answer is a harmonic number. | https://algo.inria.fr/flajolet/Publications/book.pdf | Carried forward from the Codex N2 source reread: [053 related-harmonic] Printed p. 141: "the expected number of records in a random permutation of size n equals Hn, the harmonic number." Independently H_1000=7.485470860550345, approximately 7.5. The comparison is between the two expected values.
- CONFIRMED | here the expected time to collect n coupons is n·H_n and the same slow growth means a 100-set takes only ~519 draws, not thousands. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [054 related-hundred] The source mean formula gives 100 H_100 = 518.737751763962, which rounds to 519.
- CONFIRMED | Riffle shuffling needs about (3/2)·log₂ n shuffles to randomise a deck | https://www.stat.berkeley.edu/~aldous/157/Papers/bayer_diaconis.pdf | Carried forward from the Codex N2 source reread: [056 riffle-count] Visually read Bayer-Diaconis p. 294: "3/2 log_2 n + theta shuffles are necessary and sufficient to mix up n cards." Section 4, p. 309 says "Each deck size shows the cutoff phenomenon." The claim is about the Gilbert-Shannon-Reeds riffle model. Coupon-collector expectation grows like n log n.
- CONFIRMED | a sharp cutoff cliff; | https://www.stat.berkeley.edu/~aldous/157/Papers/bayer_diaconis.pdf | Carried forward from the Codex N2 source reread: [057 riffle-cutoff] Visually read Bayer-Diaconis p. 294: "3/2 log_2 n + theta shuffles are necessary and sufficient to mix up n cards." Section 4, p. 309 says "Each deck size shows the cutoff phenomenon." The claim is about the Gilbert-Shannon-Reeds riffle model. Coupon-collector expectation grows like n log n.
- CONFIRMED | Both ask how many random steps it takes to reach a target state on n items, and both answers hide in the logarithms. | https://www.stat.berkeley.edu/~aldous/157/Papers/bayer_diaconis.pdf | Carried forward from the Codex N2 source reread: [059 related-random-steps] Visually read Bayer-Diaconis p. 294: "3/2 log_2 n + theta shuffles are necessary and sufficient to mix up n cards." Section 4, p. 309 says "Each deck size shows the cutoff phenomenon." The claim is about the Gilbert-Shannon-Reeds riffle model. Coupon-collector expectation grows like n log n.
- CONFIRMED | Two faces of balls-into-bins: the birthday/collision question (how soon do two draws land in the same bin | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [060 birthday-bins] Flajolet p. 208 discusses repeated uniform birth dates and coupon types, and its abstract calls them "random allocation processes". These are the collision and occupancy interpretations of balls into bins.
- CONFIRMED | a hash's weakness | https://csrc.nist.gov/glossary/term/cryptographic_hash_function | Carried forward from the Codex N2 source reread: [061 hash-collision] NIST defines collision resistance: "It is computationally infeasible to find any two distinct inputs that map to the same output." The nearby-layer fragment identifies a collision as the event an attack seeks, not a claim that all cryptographic hashes are easily broken.
- CONFIRMED | while you hold k distinct coupons, each draw is new with probability (n−k)/n, so it costs n/(n−k) draws on average to find the next one, | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [062 dek-phase] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | derived by splitting the collection into independent geometric **phases** | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [063 body-independent] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | the total expected wait is the clean sum n/n + n/(n−1) + … + n/1 = n·H_n, the harmonic number times n. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [064 dek-total] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | The coupon collector's problem: to collect a full set of n coupons drawn at random, you need on average n·H_n draws, and almost all the agony is the end. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [065 meta-total] Visually read p. 216: "E{C_m} = m(1/m + 1/(m-1) + 1/(m-2) + ... + 1/1) = mH_m". Reversing the sum gives the page formula and increasing phase costs. The probabilistic claims concern expected waits.
- CONFIRMED | The answer is n times the harmonic number, so a die needs 14.7 rolls on average to show all six faces and a 52-card set takes 236 draws, with the single last coupon costing a full n draws by itself. | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [066 plain-expectation] Independent rational sums give "n=6: E[T]=14.70, n=52: E[T]=235.98, n=100: E[T]=518.74" in reader-check.log. These round to 14.7, 236, and 519. The geometric last-phase mean is n.
- CONFIRMED | A die needs 14.7 rolls to show all six faces; a 52-card set, 236 draws; a 100-set, 519. | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [067 dek-numbers] Independent rational sums give "n=6: E[T]=14.70, n=52: E[T]=235.98, n=100: E[T]=518.74" in reader-check.log. These round to 14.7, 236, and 519. The geometric last-phase mean is n.
- CONFIRMED | `Var[T] = n²·Σ1/k² − n·H_n → n²·π²/6` | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [069 body-variance] Source: "var(W_k) = sum(i=1..k) (i-1)m/(m-i+1)^2". Setting k=m=n and changing index gives n^2 sum(j=1..n) 1/j^2 - n H_n. The finite upper limit is n; dividing by n^2 gives limit pi^2/6. Independent exact phase-variance check passes.
- CONFIRMED | `P(T ≤ n ln n + c·n) → exp(−e^−c)` | https://static.renyi.hu/renyi_cikkek/1961_on_a_classical_problem_of_probability_theory.pdf | Carried forward from the Codex N2 source reread: [070 body-gumbel] Visually read Erdos-Renyi p. 215, equation (2): the probability tends to "exp(-e^(-x)/(m-1)!)" after centering at log n+(m-1)loglog n. At m=1 this is exactly the stated uniform independent coupon-collector limit. The authoritative repository dates the paper to 1961.
- CONFIRMED | The last few slots will resist for a long time | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [121 last-slots-wait] The source gives the last phase geometric success probability 1/n. Its long expected wait and exponential right tail support the informal description of the slow end, not a guarantee that every individual run is slow.
- CONFIRMED | the first coupon costs exactly one draw, the single last coupon costs a full n, | https://www.randomservices.org/random/urn/Coupon.html | Carried forward from the Codex N2 source reread: [122 first-last-dek] The source defines Z_i from i-1 to i distinct values, calls them "independent random variables", and gives "p_i = \frac{m - i + 1}{m}". With i=k+1, the mean is n/(n-k), first wait 1, final wait n.
- CONFIRMED | −ln(ln 2) = 0.367 | https://www.randomservices.org/random/special/ExtremeValue.html | Carried forward from the Codex N2 source reread: [123 median-prediction-math] The source quantile is "G^(-1)(p) = -ln[-ln(p)]". At p=1/2 this is -ln(ln 2)=0.36651292058166435, correctly rounded to 0.367. The browser prints this prediction as a constant.
- CONFIRMED | Erdős, P. & Rényi, A., “On a classical problem of probability theory,” Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961). | https://real.mtak.hu/201134/ | Carried forward from the Codex N2 source reread: [132 citation-erdos] The Academy repository names "Erdős, Pál and Rényi, Alfréd (1961)", the exact paper title, volume "6 (1-2)", and "pp. 215-220". The primary PDF title and authors agree.
- CONFIRMED | Newman, D. J. & Shepp, L., “The double Dixie cup problem,” Amer. Math. Monthly 67 (1960). | https://gwern.net/doc/statistics/order/1960-newman.pdf | Carried forward from the Codex N2 source reread: [133 citation-newman] The primary PDF title is "THE DOUBLE DIXIE CUP PROBLEM" and names "DONALD J. NEWMAN" and "LAWRENCE SHEPP"; the next page has the 1960 running date. Crossref confirms The American Mathematical Monthly 67(1), January 1960, DOI 10.2307/2308930. The source tool falsely reads the second initial J. as the first initial; D. J. is correct. Crossref also omits Shepp from its author array; the primary PDF supplies him.
- CONFIRMED | Feller, W., An Introduction to Probability Theory and Its Applications, Vol. I (the classic treatment). | https://uat.store.wiley.com/en-us/an-introduction-to-probability-theory-and-its-applications-volume-1-3rd-edition-p-9780471257080 | Carried forward from the Codex N2 source reread: [134 citation-feller] The publisher page says "An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Edition" and "William Feller". These match the page citation. "Classic" is not a separate checkable bibliographical datum.
- CONFIRMED | Flajolet, P., Gardy, D. & Thimonier, L., “Birthday paradox, coupon collectors, …,” Discrete Appl. Math. 39 (1992). | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [135 citation-flajolet] The PDF bibliographic line names Flajolet, Gardy and Thimonier, the title "Birthday paradox, coupon collectors, caching algorithms and self-organizing search", and Discrete Applied Mathematics 39 (1992), 207-229. The online DOI check matches 10.1016/0166-218X(92)90177-C.
- CONFIRMED | Erdős & Rényi (*Magyar Tud. Akad.*, 1961); | https://real.mtak.hu/201134/ | Carried forward from the Codex N2 source reread: [136 citation-erdos-short] The Academy repository names "Erdős, Pál and Rényi, Alfréd (1961)", the exact paper title, volume "6 (1-2)", and "pp. 215-220". The primary PDF title and authors agree.
- CONFIRMED | Newman & Shepp (the double Dixie cup problem, *Amer. Math. Monthly*, 1960); | https://gwern.net/doc/statistics/order/1960-newman.pdf | Carried forward from the Codex N2 source reread: [137 citation-newman-short] The primary PDF title is "THE DOUBLE DIXIE CUP PROBLEM" and names "DONALD J. NEWMAN" and "LAWRENCE SHEPP"; the next page has the 1960 running date. Crossref confirms The American Mathematical Monthly 67(1), January 1960, DOI 10.2307/2308930. The source tool falsely reads the second initial J. as the first initial; D. J. is correct. Crossref also omits Shepp from its author array; the primary PDF supplies him.
- CONFIRMED | Feller (*An Introduction to Probability Theory*, Vol. I); | https://uat.store.wiley.com/en-us/an-introduction-to-probability-theory-and-its-applications-volume-1-3rd-edition-p-9780471257080 | Carried forward from the Codex N2 source reread: [138 citation-feller-short] The publisher page says "An Introduction to Probability Theory and Its Applications, Volume 1, 3rd Edition" and "William Feller". These match the page citation. "Classic" is not a separate checkable bibliographical datum.
- CONFIRMED | Flajolet, Gardy & Thimonier (*Discrete Appl. Math.*, 1992). | https://algo.inria.fr/flajolet/Publications/FlGaTh92.pdf | Carried forward from the Codex N2 source reread: [139 citation-flajolet-short] The PDF bibliographic line names Flajolet, Gardy and Thimonier, the title "Birthday paradox, coupon collectors, caching algorithms and self-organizing search", and Discrete Applied Mathematics 39 (1992), 207-229. The online DOI check matches 10.1016/0166-218X(92)90177-C.

## Findings
- [fixed] X1 (MINOR): Extended N1 model-stop with the verbatim minimum definition and its locator; retained the true page sentence.
- [fixed] X2 (MINOR): Extended N1 model-one-each with the same definition and the k=m reasoning; retained the true page sentence.
- [fixed] X3 (MINOR): Extended N1 model-assumptions with the stopping-time definition and locator; retained the true page conditions.
- [fixed] X4 (MINOR): Replaced the insufficient N1 childhood-slowdown evidence with the relevant geometric-phase source sentence and locator.
- [fixed] X5 (MINOR): Replaced the insufficient N1 spread evidence with the finite-variance formula and added its reindexing and asymptotic derivation.
- [open note:62ac70] X6 (WRONG): Changed heavy right tail to long right tail in the HTML; shared the dated tail correction with X16 and X17. The Same Sum Three Times carries no heavy-tail copy (checked 2026-10-18). The only thing left is the preserved quotation in There, Here: the coupon-collector to seven-shuffles note (notes[228]) in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still calls the tail heavy and needs an adjacent correction that preserves the quotation.
- [fixed] X7 (WRONG): Replaced the arbitrary-start claim in the dek with an empty start and independent uniform sampling with replacement; added its dated correction.
- [fixed] X8 (WRONG): Removed the unsupported print-run generalization, kept the conditional unequal-probability mathematics, and added a correction limited to what the Panini FAQ establishes.
- [open note:62ac70] X9 (WRONG): Corrected the HTML bridge to the first half, zero to n/2, for even n. Corrected verifier Part 5 and its served copy, including direct checks of both state ranges; added the shared first-half correction. On 2026-10-18 the portal copies were corrected in The Same Sum Three Times (index.html bridge note and check line, markdown dek, both relation notes and body, and research/harmonic-numbers-portal/verify.mjs Part 4 with its served copy, now checked against the phase sum), with a dated Corrected 2026-10-18 line. The only thing left is the preserved quotation in There, Here: three stored relation notes (notes[227], [1935] and [1937]) in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still call it the second-half cost and need annotations that preserve the quotations.
- [open note:62ac70] X10 (WRONG): Kept the first-half harmonic formula and stated the scaled limit precisely: first-half cost divided by n tends to ln 2. Shares X9 correction; the portal copies were corrected on 2026-10-18, and the only thing left is the preserved quotation in There, Here (notes[227], [1935] and [1937] in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1).
- [open note:62ac70] X11 (WRONG): Corrected the dek to the first-half expected cost and its scaled limit. Shares X9 correction; the portal copies were corrected on 2026-10-18, and the only thing left is the preserved quotation in There, Here (notes[227], [1935] and [1937] in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1).
- [open note:62ac70] X12 (WRONG): Corrected the relation label, start/end states and concluding comparison to the first half. Shares X9 correction; the portal copies were corrected on 2026-10-18, and the only thing left is the preserved quotation in There, Here (notes[227], [1935] and [1937] in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1).
- [open note:62ac70] X13 (WRONG): Corrected the markdown body bridge to zero through n/2 for even n and the scaled limit. Shares X9 correction; the portal copies were corrected on 2026-10-18, and the only thing left is the preserved quotation in There, Here (notes[227], [1935] and [1937] in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1).
- [open note:62ac70] X14 (WRONG): Replaced the finite-100 survival value with 1-(H_100-H_50), about 31.18%, and explicitly reserved 1-ln 2 for the limit; added its dated correction. The Same Sum Three Times states 1-ln 2 only as the limit (checked 2026-10-18). The only thing left is the preserved quotation in There, Here: the coupon-collector to the-loop-that-saves-them note (notes[227]) in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 retains the false finite probability and needs an adjacent correction that preserves the quotation.
- [fixed] X15 (MINOR): Replaced N1 related-gamma with the verbatim extracted p. 724 equation, its correct locator and a visually checked mathematical reading.
- [open note:62ac70] X16 (WRONG): Changed the seven-shuffles relation phrase to long right tail. Shares X6 dated correction; The Same Sum Three Times holds no copy (checked 2026-10-18), and the only thing left is the preserved quotation in There, Here (notes[228] in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1).
- [open note:62ac70] X17 (WRONG): Changed the dek phrase to long right tail. Shares X6 dated correction; The Same Sum Three Times holds no copy (checked 2026-10-18), and the only thing left is the preserved quotation in There, Here (notes[228] in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1).
- [fixed] X18 (MINOR): Replaced the all-values/same-routine browser promise with the actual live quantities and separate verifier; also narrowed the markdown live-coverage introduction and check comment. No page correction line for MINOR.
- [fixed] X19 (MINOR): Narrowed both HTML description and JSON-LD description to expected totals, phase costs and simulated collections. No page correction line for MINOR.
- [fixed] X20 (WRONG): Removed the duplicate-flash assertion while retaining wasted draws, and added its dated correction.
- [fixed] X21 (WRONG): Changed top bar to the draw counter below the slots and added its dated correction.
- [fixed] X22 (WRONG): Labeled the histogram overlay as the Gumbel density with exp(-c-exp(-c)); added a dated correction shared with X25 and X26.
- [fixed] X23 (MINOR): Narrowed the footer to actual live quantities and the separate offline variance check. No page correction line for MINOR.
- [fixed] X24 (MINOR): Narrowed the footer and both verifier headers to independently checked one-copy calculations; retained 19/19. No additional page correction line for MINOR.
- [fixed] X25 (WRONG): Corrected the generated density-curve readout formula. Shares X22 dated correction.
- [fixed] X26 (WRONG): The same readout edit resolves this duplicate finding. Shares X22 dated correction; no duplicate text edit.
- [fixed] X27 (MINOR): Replaced N1 sigma-prediction-math evidence with the finite-variance formula and derivation; retained 1.283 on the page.
- [fixed] X28 (MINOR): Replaced N1 gamma-prediction-math evidence with the verbatim Euler-constant equation and the Moments locator; retained 0.577216 on the page.