---
slug: look-then-leap
passed: 2026-10-14
by: the assay line (research/assay-line/cited-1014b/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex
checked: 92
confirmed: 64
wrong: 28
unverifiable: 0
observed: 0
stale: 0
---

This was the directed correction pass for look-then-leap at f5e5f84f34, using the cited-1014b source record and Codex checker rereads. The fixer rechecked the evidence before changing the 28 triaged items, retained all 64 checker-confirmed claims below, and checked the numerical counterexamples independently. The 18 WRONG and 10 MINOR lines are counted together in wrong; each has its own finding, while repeated WRONG errors share 12 dated correction lines. MINOR repairs have no page correction. Changes stay within this layer; its generated plain metadata, Nearby notes and downloadable verifier are rebuilt by research/look-then-leap/sync-layer.mjs. The Codex repair review on 2026-10-14 confirmed all 12 correction lines and reopened X18-X25 for remaining copies outside this layer: 20 findings are fixed and 8 await board note IDs. PENDING is a placeholder, not an issued note ID; see research/assay-line/cited-1014b/look-then-leap/repair-report.md.

## Claims

- MINOR | [X1 win-formula] "P(win" / "cutoff r) = (r−1)/n · ( 1/(r−1) + 1/r + … + 1/(n−1) )" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf; git show f5e5f84f34:public/strata/look-then-leap/index.html | Ferguson p. 283 separates r=1; the page already computes 1/n there.
- WRONG | [X2 factor-meaning] "The factor (r−1)/n is the chance the true best falls after your reconnaissance phase" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf; https://link.springer.com/article/10.1007/s10878-016-0050-8 | Ferguson Eq. (2.1) and the verifier use (1/n)((r-1)/(j-1)); (r-1)/n is the rejected-prefix probability.
- WRONG | [X3 sum-meaning] "The sum 1/(r−1) + … + 1/(n−1) is, position by position, the chance that the best of everything-so-far sat safely inside the reconnaissance and so didn't trigger an early, wrong leap." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf; https://link.springer.com/article/10.1007/s10878-016-0050-8 | The conditional prefix-maximum probability is (r-1)/(j-1). The unscaled harmonic tail can exceed one.
- WRONG | [X4 stopping-rule] "keep looking exactly while that tail sum is still above 1, and leap once it drops to 1 or below." | git show f5e5f84f34:research/look-then-leap/verify.mjs; https://link.springer.com/article/10.1007/s10878-016-0050-8 | Independent enumeration at n=3 gives r=2 a win rate of 1/2, versus 1/3 at r=3. The criterion starts at r.
- WRONG | [X5 markdown-stopping-rule] "the optimum is the least `r` with the tail sum `1/(r−1)+…+1/(n−1) ≤ 1`." | git show f5e5f84f34:research/look-then-leap/verify.mjs; https://link.springer.com/article/10.1007/s10878-016-0050-8 | The markdown repeats X4. The correct stopping tail starts at r; the probability formula still starts at r-1.
- MINOR | [X6 thirty-seven-plain] "The optimal rule is to reject the first 37 percent no matter how good, then take the next record-breaker, which wins about 37 percent of the time (1/e)." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson p. 283 gives a large-n approximation, not an exact fractional finite cutoff.
- MINOR | [X7 adversary] "An adversary who orders the candidates can defeat any strategy." | https://proceedings.neurips.cc/paper_files/paper/2020/file/5a378f8490c8d6af8647a753812f6e31-Paper.pdf | The NeurIPS paper p. 6 describes uniform random selection with probability 1/n per element; this survives every fixed order.
- MINOR | [X8 full-information] "you can win about 58% of the time (Gilbert & Mosteller, 1966), far better than 1/e." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson pp. 283 and 286 specify independent observations from a given distribution and a limiting success rate about 0.58016.
- MINOR | [X9 robbins-bounds] "its limiting value is only known to lie between about 1.908 and 2.329." | https://arxiv.org/abs/1606.03348; https://arxiv.org/pdf/1606.03348 | Meier and Sögner p. 1 report numerical upper value 2.32614 and lower bound 1.908; the original interval was loose but valid.
- MINOR | [X10 robbins-lower] "its limiting value is only known to lie between about 1.908 and 2.329." | https://arxiv.org/abs/1606.03348; https://arxiv.org/pdf/1606.03348 | Same sentence as X9. The lower bound is retained; the upper value and numerical qualification now match the cited paper.
- WRONG | [X11 lindley-first-print] "the first solution in print." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson p. 283 records a March 1960 solution outline before Lindley 1961 and qualifies journal priority.
- WRONG | [X12 bruss-special-case] "the odds theorem, of which the 1/e rule is the special case; and the bounds on Robbins' problem." | https://www.researchgate.net/publication/38351353_Sum_the_odds_to_one_and_stop; https://arxiv.org/pdf/1606.03348 | Bruss 2000 section 3 treats the secretary problem; section 5 bounds success probability. Meier and Sögner p. 1 report Robbins bounds.
- MINOR | [X13 browser-routine] "Every number on this page is recomputed in your browser by the same routine the offline verifier runs." | git show f5e5f84f34:public/strata/look-then-leap/index.html; git show f5e5f84f34:research/look-then-leap/verify.mjs | The HTML and standalone verifier implement the classical formulas separately; neither computes the cited variant constants.
- WRONG | [X14 cutoff-marks] "The tall line is your cutoff; the dotted line is n/e." | git show f5e5f84f34:public/strata/look-then-leap/index.html | The axis uses cutoff r, but the original guide and label used x(n/e+1). The executed drawing code now places both at x(n/e).
- WRONG | [X15 horizontal] "The dashed horizontal is the win probability 1/e ≈ 0.368. The optimum sits on it." | git show f5e5f84f34:public/strata/look-then-leap/index.html; https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Independent exact-fraction calculation gives P*(50)=0.37427501364792015, above 1/e=0.36787944117144233.
- WRONG | [X16 dek-peak] "the peak of the win curve sitting exactly on the 1/e line." | git show f5e5f84f34:public/strata/look-then-leap/index.html; https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | The dek repeats X15, and the markdown instrument description repeated it too. Both now describe convergence.
- MINOR | [X17 footer-computation] "Every record, curve and limit computed live; every figure checked offline in research/look-then-leap/verify.mjs (24/24)." | git show f5e5f84f34:research/look-then-leap/verify.mjs; git show f5e5f84f34:public/strata/look-then-leap/index.html | The verifier checks classical records and best choice, not the cited 3.8695, full-information success constant or Robbins bounds.
- WRONG | [X18 relation-prisoners] "There, a hundred prisoners survive 31% of the time because the killer-loop exists with chance H₁₀₀ − H₅₀ → 1 − ln 2" | https://mathworld.wolfram.com/100PrisonersProblem.html | MathWorld equation alt text gives survival 1-H100+H50; independent summation gives failure 0.6881721793101953.
- WRONG | [X19 relation-prisoners-limit] "H₁₀₀ − H₅₀ → 1 − ln 2" | https://mathworld.wolfram.com/100PrisonersProblem.html | The same source assigns 1-ln(2) to limiting survival. H(2m)-H(m) tends to ln(2).
- WRONG | [X20 relation-shuffle] "Here: shuffle almost any list and the best strategy wins a fixed 1/e, while the record positions obey 1/k regardless of what the numbers were." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | The exact optimum is 1/2 at n=3 and about 0.374275 at n=50; 1/e is the large-n limit.
- WRONG | [X21 relation-ergodic] "in both the hinge is the slow logarithmic growth of a sum (the harmonic series here, the Gauss map's ergodic average there)." | https://arxiv.org/pdf/1402.0208 | Cellarosi et al. p. 2, Eqs. (1.2)-(1.3), normalize the logarithmic sum by n; its average converges, so the sum grows linearly.
- MINOR | [X22 relation-friend] "The friendship paradox: pick a random person, then a random friend of theirs, and the friend is reliably more popular" | https://arxiv.org/pdf/2012.03991 | Cantwell et al. pp. 1-2, Eq. (3), give a nonnegative mean degree difference after isolated nodes are excluded; equality can occur.
- WRONG | [X23 relation-degree] "sampling proportional to degree, not uniformly." | https://arxiv.org/pdf/2012.03991 | Cantwell et al. Eq. (6) gives reciprocal-neighbor-degree weights. On path4 these yield (1/8,3/8,3/8,1/8), not degree weights (1/6,1/3,1/3,1/6).
- WRONG | [X24 relation-reweighted] "The secretary problem: pick a random candidate and you win 1/n of the time, but condition on 'first record after the 37% mark' and you win 1/e" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Independent enumeration at n=3 with one rejection gives policy success 1/2; a uniform candidate conditioned to be the first later record succeeds with probability 3/4.
- MINOR | [X25 nearby-coupon] "Two probability problems whose whole answer is a harmonic number." | git show f5e5f84f34:src/content/strata/coupon-collector.md | The pinned coupon source gives E[T]=n H_n. Both problems involve harmonic sums; their full answers need not equal a harmonic number.
- WRONG | [X26 nearby-mastermind] "Both problems separate the strategy that minimizes a uniform average from the one that protects a hard limit." | git show f5e5f84f34:src/content/strata/mastermind-five-guesses.md | The pinned Mastermind source distinguishes a five-row worst-case guarantee from mean performance; this layer compares best-choice probability and expected rank.
- WRONG | [X27 live-6-1-cut-read] "your cutoff: reject first 18 → win 37.43% · optimum: reject first 18 (38.0% of n) → win 37.43% · n/e = 18.39, 1/e = 36.79%" | git show f5e5f84f34:public/strata/look-then-leap/index.html | The page has optimum r=19 at n=50, so it rejects 18 and must display 36.0%. The actual page-code check now confirms this.
- WRONG | [X28 added-skip-fraction-output] (added) "n=100: skip-fraction=0.3800, P=0.3710" | git show f5e5f84f34:research/look-then-leap/verify.mjs | The verifier computes r/n=38/100 but called it a skipped fraction. Relabelled cutoff-fraction, keeping the valid convergence check and probability.
- CONFIRMED | [model] "The candidates arrive one by one. You can compare whoever is in front of you to everyone you've already seen" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: the stated model.
- CONFIRMED | [irreversible] "but the moment you reject someone they're gone, and the moment you accept someone you stop. No going back." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: the stated model.
- CONFIRMED | [single-best] "You want the very best of all n; second-best is a loss like any other." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: the stated model.
- CONFIRMED | [all-n-success] "close to 37% of the time, no matter how large n is" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson Eq. (2.3): large-n limits.
- CONFIRMED | [names] "the *secretary problem* (also the marriage problem, the sultan's-dowry problem, the game of googol)" | https://pmc.ncbi.nlm.nih.gov/articles/PMC12500721/ | The primary article introduction lists all three alternate names; the retrieved search excerpt agrees with N1.
- CONFIRMED | [records-definition] "Statisticians call these records, or left-to-right maxima." | https://faculty.wharton.upenn.edu/wp-content/uploads/2016/10/BBB.pdf | The Wharton paper p. 174 defines a record maximum in a left-to-right scan.
- CONFIRMED | [first-record] "The first item is always a record. After that they get rarer." | https://www.numdam.org/article/ASCFM_1962__8_2_7_0.pdf | Rényi pp. 7-8, Lemma 1; exact permutation enumeration agrees.
- CONFIRMED | [record-law] "In a uniformly random order, the probability that the item in position k is a record is exactly 1/k" | https://www.numdam.org/article/ASCFM_1962__8_2_7_0.pdf | Rényi pp. 7-8, Lemma 1; exact permutation enumeration agrees.
- CONFIRMED | [record-distribution] "Not the distribution the numbers came from, not their spread, not n." | https://www.numdam.org/article/ASCFM_1962__8_2_7_0.pdf | Rényi pp. 7-8, Lemma 1; exact permutation enumeration agrees.
- CONFIRMED | [symmetry-proof] "among the first k items, each is equally likely to be the largest, so the last of them is largest with probability 1/k." | https://faculty.wharton.upenn.edu/wp-content/uploads/2016/10/BBB.pdf | Wharton p. 174 gives the uniform-position symmetry argument.
- CONFIRMED | [expected-records] "the expected number of records in a list of n is Hn = 1 + ½ + ⅓ + … + 1/n" | https://faculty.wharton.upenn.edu/wp-content/uploads/2016/10/BBB.pdf | Wharton p. 174 gives the record/cycle bijection and linearity of expectation.
- CONFIRMED | [harmonic-asymptotic] "which grows like ln n + γ (γ ≈ 0.5772, the Euler–Mascheroni constant)" | https://dlmf.nist.gov/5.11; https://dlmf.nist.gov/5.4; https://dlmf.nist.gov/5.2 | DLMF 5.4.14 combined with 5.11.2 gives the asymptotic; the finite harmonic-sum check agrees.
- CONFIRMED | [thousand-records] "a list of a thousand items has, on average, only about 7.5 records, not hundreds." | https://dlmf.nist.gov/5.4 | Independent fraction summation gives H1000=7.485470860550345.
- CONFIRMED | [mutual-independence] "the record-or-not coin flips at the different positions are mutually independent." | https://www.numdam.org/article/ASCFM_1962__8_2_7_0.pdf | Rényi pp. 7-8, Lemma 1; exact permutation enumeration agrees.
- CONFIRMED | [independence-example] "Whether position 3 broke the record tells you nothing about whether position 9 will." | https://www.numdam.org/article/ASCFM_1962__8_2_7_0.pdf | Rényi pp. 7-8, Lemma 1; exact permutation enumeration agrees.
- CONFIRMED | [only-record] "accepting a non-record means knowingly taking someone you've already seen beaten, which can never be the overall best." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | An earlier strictly better candidate rules out global maximum.
- CONFIRMED | [early-late] "Early records are common but probably not the global best; late records are rare but, if they come, are likely the real thing." | https://faculty.wharton.upenn.edu/wp-content/uploads/2016/10/BBB.pdf; https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Uniform ordering gives conditional success k/n at a record at position k.
- CONFIRMED | [best-strategy] "The best possible strategy has a shape so plain it sounds like folk advice: look, then leap." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: threshold policy.
- CONFIRMED | [skip-rule] "Reject the first r − 1 candidates no matter how dazzling" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: threshold policy.
- CONFIRMED | [leap-rule] "Remember the best among them. Then leap at the very first later candidate who beats that benchmark." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: threshold policy.
- CONFIRMED | [no-later-record] "(If none does, you're stuck with the last" / "a loss.)" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf; https://link.springer.com/article/10.1007/s10878-016-0050-8 | Springer Proposition 1 specifies the last-candidate fallback; absent later records, the global best was rejected.
- CONFIRMED | [tail-log] "For large n the tail sum 1/(r−1) + … + 1/(n−1) is close to ln(n/r), so the rule becomes ln(n/r) = 1" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson Eq. (2.3): large-n limits.
- CONFIRMED | [cutoff-limit] "the optimal cutoff settles on n/e ≈ 0.368 n" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson Eq. (2.3): large-n limits.
- CONFIRMED | [win-limit] "Substitute that cutoff back and the win probability collapses to 1/e as well." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson Eq. (2.3): large-n limits.
- CONFIRMED | [twice] "So the same constant governs how long to look (a fraction 1/e of the line) and how often you win (a probability 1/e)." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson Eq. (2.3): large-n limits.
- CONFIRMED | [blind-guess] "wins about 1/n of the time (for 25 applicants, 4%)." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Uniform positions give 1/n; 1/25=0.04.
- CONFIRMED | [hundred-advantage] "With a hundred candidates the look-then-leap rule beats pure chance by a factor of 37, and the advantage only widens with n." | https://link.springer.com/article/10.1007/s10878-016-0050-8; git show f5e5f84f34:research/look-then-leap/verify.mjs | Independent fraction calculation gives 100 P*(100)=37.1042778712643; the asymptotic advantage grows with n.
- CONFIRMED | [four-assumptions] "This clean answer is paid for by four assumptions" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2 agrees; four groups are highlighted, with other model conditions elsewhere.
- CONFIRMED | [known-n] "You know n. The count of candidates is fixed and known in advance." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: the stated model.
- CONFIRMED | [unknown-horizon] "(If candidates arrive over an unknown horizon, the problem changes and so does the rule.)" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson p. 285 discusses random-horizon variants.
- CONFIRMED | [random-order] "Random order. The arrival order is a uniformly random shuffle" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: the stated model.
- CONFIRMED | [no-recall] "No recall. A rejected candidate is gone for good. Allow recalls and you can do better." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2 prohibits recall; free unrestricted recall permits choosing the final maximum.
- CONFIRMED | [all-or-nothing] "All-or-nothing. Only the single best counts; second-best scores zero." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §2: the stated model.
- CONFIRMED | [expected-rank] "the optimal rule is different and the achievable limit is a different constant, about 3.8695 (the expected rank you can guarantee as n→∞; Chow, Moriguti, Robbins & Samuels, 1964)." | https://link.springer.com/article/10.1007/BF02759948 | The 1964 publisher abstract gives the expected-rank limit about 3.8695; this is not a per-run guarantee.
- CONFIRMED | [robbins-definition] "The closely related question of minimizing expected rank with full information, called Robbins' problem" | https://arxiv.org/abs/2608.27419 | The August 2026 paper defines minimizing expected rank with observed values.
- CONFIRMED | [robbins-unsolved] "Robbins' problem, is still unsolved" | https://arxiv.org/html/2608.27419v1 | The August 2026 paper explicitly leaves optimal strategy and limiting value open.
- CONFIRMED | [citation-1] "Gardner, M. (1960). Mathematical Games. Scientific American 202(2) & (3)" | https://www.scientificamerican.com/article/mathematical-games-1960-02/; https://www.scientificamerican.com/article/mathematical-games-1960-03/; https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Both publisher pages identify Gardner, the title, year, volume 202 and issues 2 and 3.
- CONFIRMED | [citation-2] "Lindley, D. V. (1961). Dynamic programming and decision theory. Applied Statistics 10:39–51" | https://api.crossref.org/works/10.2307/2985407; https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson bibliography matches the stated reference.
- CONFIRMED | [citation-3] "Dynkin, E. B. (1963). The optimum choice of the instant for stopping a Markov process. Soviet Math. Dokl. 4" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson bibliography matches the stated reference.
- CONFIRMED | [citation-4] "Chow, Y. S., Moriguti, S., Robbins, H. & Samuels, S. M. (1964). Optimal selection based on relative rank. Israel J. Math. 2:81–90" | https://link.springer.com/article/10.1007/BF02759948 | The publisher lists the authors, title, year, volume 2 and pages 81-90.
- CONFIRMED | [citation-5] "Gilbert, J. P. & Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Stat. Assoc. 61:35–73" | https://api.crossref.org/works/10.1080/01621459.1966.10502008 | Crossref metadata matches the authors, title, year, volume 61 and pages 35-73.
- CONFIRMED | [citation-6] "Rényi, A. (1962). Théorie des éléments saillants d'une suite d'observations." | https://www.numdam.org/item/ASCFM_1962__8_2_7_0/ | The NUMDAM PDF identifies Rényi, the stated title, 1962, and pages 7-13.
- CONFIRMED | [citation-7] "Ferguson, T. S. (1989). Who solved the secretary problem? Statistical Science 4(3):282–296" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | The printed journal heading includes the discussion in its 282-296 range.
- CONFIRMED | [citation-8] "Bruss, F. T. (2000). Sum the odds to one and stop. Annals of Probability 28:1384–1391" | https://api.crossref.org/works/10.1214/aop/1019160340 | Crossref metadata and the author-posted paper match the citation.
- CONFIRMED | [gardner-popularized] "which popularised the problem." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §3 and the publisher pages document the 1960 columns.
- CONFIRMED | [dynkin-framing] "the optimal-stopping framing." | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf | Ferguson §3 describes the Markov-stopping formulation.
- CONFIRMED | [renyi-records-attribution] "the theory of records; P(record at k) = 1/k and their independence." | https://www.numdam.org/article/ASCFM_1962__8_2_7_0.pdf | Rényi pp. 7-8, Lemma 1; exact permutation enumeration agrees.
- CONFIRMED | [record-average] "The running count of records, averaged, climbs as the harmonic number Hn = 1 + ½ + ⅓ + … + 1/n" | git show f5e5f84f34:public/strata/look-then-leap/index.html; https://faculty.wharton.upenn.edu/wp-content/uploads/2016/10/BBB.pdf | Wharton p. 174 gives H_n; the browser averages counts over trials. Convergence need not be monotone.
- CONFIRMED | [cutoff-merge] "As n grows they merge:" | https://changyaochen.github.io/assets/pdfs/who_solved_secretary_problem.pdf; git show f5e5f84f34:public/strata/look-then-leap/index.html | Ferguson Eq. (2.3): large-n limits.
- CONFIRMED | [nothing-invented] "Nothing here is invented; the four assumptions and the open edges are named in §V." | git show f5e5f84f34:public/strata/look-then-leap/index.html | The named topics and sources exist and §V lists the assumptions. This does not certify every sentence.
- CONFIRMED | [md-small-optima] "exact small-`n` optima (n=3→1/2, n=4→11/24, n=7→ 29/70)" | git show f5e5f84f34:research/look-then-leap/verify.mjs; recreated: python3 arithmetic.py | Exact-fraction argmax gives 1/2, 11/24 and 29/70 with skips 1, 1 and 2.
- CONFIRMED | [relation-prisoners-blind] "Both take a problem the naive arithmetic calls hopeless ((½)¹⁰⁰; 1/n)" | https://mathworld.wolfram.com/100PrisonersProblem.html | MathWorld gives 2^(-100) for independent choices; uniform candidate selection gives 1/n.
- CONFIRMED | [relation-khinchin] "Khinchin: take almost any real, and the geometric mean of its continued-fraction terms tends to one fixed K₀." | https://arxiv.org/abs/1402.0208 | Cellarosi et al. abstract and Eq. (1.3) state almost-everywhere geometric-mean convergence.
- CONFIRMED | [nearby-harmonic] "Two impossible-looking choosing games that a single strategy turns winnable, and both answers are written in harmonic numbers." | git show f5e5f84f34:src/content/strata/the-loop-that-saves-them.md; https://link.springer.com/article/10.1007/s10878-016-0050-8 | The secretary and prisoner formulas both use harmonic sums, without equating their answers to a single H_n.
- CONFIRMED | [live-7-2-check] "E[records] = Hn; H1000 = 7.485 ≈ ln 1000 + γ = 7.485" | https://dlmf.nist.gov/5.4; git show f5e5f84f34:public/strata/look-then-leap/index.html | Independent summation gives 7.485470861; ln(1000)+gamma gives 7.484970944. Both round to 7.485.
- CONFIRMED | [live-7-3-check] "exact optima: n=3 → skip 1, P=1/2=0.5000; n=4 → skip 1, P=11/24=0.4583; n=7 → skip 2, P=0.4143" | git show f5e5f84f34:research/look-then-leap/verify.mjs; git show f5e5f84f34:public/strata/look-then-leap/index.html | Exact-fraction argmax gives 1/2, 11/24 and 29/70 with skips 1, 1 and 2.
- CONFIRMED | [live-7-4-check] "cutoff fraction → 1/e: at n=100 it's 0.380, at n=10⁵ it's 0.3679 (1/e=0.3679)" | git show f5e5f84f34:public/strata/look-then-leap/index.html; git show f5e5f84f34:research/look-then-leap/verify.mjs | The label is cutoff fraction: r=38 gives 0.380 at n=100; the live page preserves this correct distinction.
- CONFIRMED | [live-7-5-check] "win probability → 1/e: at n=100 P*=0.3710, at n=10⁵ P*=0.3679" | git show f5e5f84f34:public/strata/look-then-leap/index.html; git show f5e5f84f34:research/look-then-leap/verify.mjs | Independent fraction calculation gives P*(100)=0.371042778712643; the live panel and verifier agree.
- CONFIRMED | [live-7-6-check] "look-then-leap at n=100 beats blind guessing (1/100) by 37×" | git show f5e5f84f34:public/strata/look-then-leap/index.html; git show f5e5f84f34:research/look-then-leap/verify.mjs | 100 times the independently calculated optimum rounds to 37.
- CONFIRMED | [euler-gamma] "γ ≈ 0.5772, the Euler–Mascheroni constant" | https://dlmf.nist.gov/5.2 | DLMF defines Euler’s constant; the harmonic-sum computation confirms the 0.5772 rounding.
- CONFIRMED | [shared-rule] "Nothing here may lie about anything real; that rule never bends." | git show f5e5f84f34:WAKE.md | The pinned policy is explicit; this line reports the policy, not universal compliance.
- CONFIRMED | [shared-corrections] "When a check finds something false, the page says what changed and when." | git show f5e5f84f34:assay/README.md | The assay instructions require dated corrections for WRONG and distinguish record-only MINOR repairs.
- CONFIRMED | [added-harmonic-output] (added) "H_1=1/1, H_2=3/2, H_3=11/6, H_4=25/12, H_7=363/140" | git show f5e5f84f34:research/look-then-leap/verify.mjs | Independent fraction sums give exactly the listed harmonic numbers.
- CONFIRMED | [added-optima-output] (added) "n=5: skip 2, P=13/30≈0.4333; n=6: skip 2, P=77/180≈0.4278" | git show f5e5f84f34:research/look-then-leap/verify.mjs | Independent fraction argmax gives skips 2 and 2, with probabilities 13/30 and 77/180.

## Findings

- [fixed] X1 (MINOR): Added r≥2 and the r=1 probability 1/n to the displayed explanation; mirrored the special case in markdown. MINOR, record only.
- [fixed] X2 (WRONG): Replaced the reversed prefix-factor explanation with per-position joint probabilities; included in the shared X2/X3 correction.
- [fixed] X3 (WRONG): Explained the conditional factor (r-1)/(j-1) and the sum of products; included in the shared X2/X3 correction.
- [fixed] X4 (WRONG): Changed the HTML stopping criterion to the tail starting at r and specified taking the first record at or after r; shared X4/X5 correction.
- [fixed] X5 (WRONG): Changed the markdown stopping criterion to the tail starting at r; shared X4/X5 correction.
- [fixed] X6 (MINOR): Qualified the plain summary as approximately optimal for large n and regenerated its HTML metadata copy. MINOR, record only.
- [fixed] X7 (MINOR): Narrowed the adversarial-order statement to loss of the classical constant guarantee and retained the 1/n randomized fallback. MINOR, record only.
- [fixed] X8 (MINOR): Specified independent draws from a known continuous distribution and a large-n limit in HTML and markdown. MINOR, record only.
- [fixed] X9 (MINOR): Used the published numerical value about 2.32614, retained about 1.908, and cited Meier and Sögner. MINOR, record only.
- [fixed] X10 (MINOR): The X9 repair also resolves this duplicate sentence finding; no change to the valid lower bound. MINOR, record only.
- [fixed] X11 (WRONG): Changed Lindley to an early journal solution and added the correction identifying the earlier March 1960 outline.
- [fixed] X12 (WRONG): Removed Robbins bounds from the Bruss 2000 annotation, added the appropriate source, and added a dated attribution correction.
- [fixed] X13 (MINOR): Scoped the introductory and metadata promises to classical record and best-choice figures and to shared formulas. Narrowed the verifier header too. MINOR, record only.
- [fixed] X14 (WRONG): Moved the dotted guide and its n/e label from x(ne+1) to x(ne); added a dated correction.
- [fixed] X15 (WRONG): Changed the panel explanation to convergence toward 1/e; shared X15/X16/X20 correction.
- [fixed] X16 (WRONG): Changed the dek and markdown instrument description to convergence; shared X15/X16/X20 correction.
- [fixed] X17 (MINOR): Scoped the footer to classical record and best-choice figures, preserving 24/24. MINOR, record only.
- [open note:867d97] X18 (WRONG): Fixed locally: kept roughly 31% survival, gave the finite failure probability and its ln(2) limit in the relation; shared X18/X19 correction. The unannotated killer-loop quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[726]); preserve its words and add the failure/survival correction outside this worker's scope.
- [open note:867d97] X19 (WRONG): Fixed locally: replaced the failure/survival limit confusion with H(2m)-H(m) tending to ln(2); shared X18/X19 correction. The same unannotated quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[726]); it needs the X18 annotation.
- [open note:ac7941] X20 (WRONG): Fixed locally: changed the relation from a fixed success probability to convergence to 1/e for growing uniformly shuffled distinct lists; shared X15/X16/X20 correction. The unannotated fixed-1/e quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[727]); preserve its words and annotate finite probability versus limit.
- [open note:ac7941] X21 (WRONG): Fixed locally: rewrote the relation around the convergent average of logarithms and added a correction distinguishing it from harmonic growth. The unannotated shared-logarithmic-growth quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[727]); preserve its words and annotate the convergent average and linearly growing sum.
- [open note:62c825] X22 (MINOR): Fixed locally: qualified the friendship claim as a non-strict average among people with friends. The unannotated reliably-more-popular quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[728]); preserve its words and clarify the average scope. MINOR, record only; X23 covers the separate distribution error.
- [open note:62c825] X23 (WRONG): Fixed locally: kept uniform-person then uniform-friend sampling and described reciprocal-neighbor-degree weights; added the exact distribution in its correction. The unannotated degree-proportional quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[728]); preserve its words and annotate the actual sampling distribution.
- [open note:00bd27] X24 (WRONG): Fixed locally: replaced conditioning a uniform draw with applying the stopping policy and its asymptotic unconditional success; added a dated correction with the n=3 counterexample. The unannotated conditioning quotation remains in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 (notes[728]); preserve its words and annotate the different experiments.
- [open note:56594d] X25 (MINOR): Fixed locally: authored a coupon relation note about harmonic sums and regenerated the Nearby copy. On 2026-10-18 the authored sentence in src/content/strata/coupon-collector.md:11 and its Nearby copy at public/strata/coupon-collector/index.html:471 were changed to "Two probability problems whose answers involve harmonic sums." The only thing left is the preserved quotation in There, Here (public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1, notes[226]), which needs a clarifying annotation that leaves its words unchanged. MINOR, record only.
- [fixed] X26 (WRONG): Authored the correct objective-change comparison locally and regenerated the Nearby copy; added a dated correction.
- [fixed] X27 (WRONG): Changed the displayed rejected percentage to 100*(r-1)/n; shared X27/X28 correction.
- [fixed] X28 (WRONG): Renamed the verifier output cutoff-fraction and regenerated its downloadable copy; shared X27/X28 correction.