---
slug: one-dimension-too-many
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: 108
confirmed: 85
wrong: 23
unverifiable: 0
observed: 0
stale: 0
---

This was the directed repair pass for one-dimension-too-many at a9812b7749: the fixer independently re-read the evidence for all 24 triage items, reproduced the browser sampler and numerical engines, fixed 15 WRONG and 8 MINOR items, and rejected X2 on the stated evidence. The record also carries the 84 CONFIRMED source checks from N2-codex.md, with their qualifications made explicit; those are the checkers' re-reads, not a claim that this fixer repeated every source fetch. MINOR fixes are included in wrong: 23 for counting and have no page correction lines. The 15 WRONG items share ten dated correction lines by underlying error. There are no STALE or listed items in this triage.

Repair review by Codex, 2026-10-14: the authored fixes and all ten correction lines remain supported. Of the 23 findings, 21 are fixed and two remain open: X20 and X24 still occur without correction annotations in the preserved There, Here corpus. Those copies are outside this worker's permitted layer. Both findings retain WRONG and use PENDING until the board issues note ids; the claim counts above are unchanged.

## Claims

- MINOR | X1: Pólya's Theorem: Random Walks Return in 2D, Not 3D | https://www.math.uchicago.edu/~lawler/srwbook.pdf; https://oeis.org/A086230/internal | Lawler and Limic, Theorem 4.1.1, distinguishes certain return from transience; OEIS A086230 gives a positive 3-D return probability. The title now explicitly qualifies certainty.
- CONFIRMED | X2: the wording is traditional; no primary source is on record | https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf | REJECTED: Durrett (2019), p. 289, attributes the joke to a UCLA colloquium but provides no dated transcript, recording, Kakutani-authored text, or explicit eyewitness account establishing the wording. This evidence does not establish a primary source. The unchanged phrase is recorded under the directed rejection, not as an exhaustive proof that no primary source exists. X4 separately corrects the broader denial of documented provenance.
- MINOR | X3: The drunk-man / drunk-bird line is universally attributed to Kakutani and the wording above is the standard one, but it carries no citation to a paper or dated talk | https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf; https://diglib.tugraz.at/download.php?id=5c80eb8c1d47f&location=browse | Durrett, p. 289, gives an undated UCLA attribution and uses will eventually find. The explanation now identifies the displayed wording as a traditional variant and cites Durrett; the displayed quotation is unchanged.
- WRONG | X4: Kakutani aphorism: traditional, provenance undocumented. | https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf | Durrett, p. 289, documents the UCLA attribution. The colophon now states that provenance and the limits of the passage, without inferring a talk date or original wording.
- MINOR | X5: Animated random walk. The path trails behind the walker; each time it returns to the starting point, that point flashes teal. | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | The 1-D branch calls draw1D and returns before flashAge is used. draw1D marks zero positions with teal dots. Accessible and surrounding descriptions now distinguish the display modes.
- WRONG | X6: In 3-D the walker usually crosses home a few times early, then wanders out of reach and the flashes stop. | https://oeis.org/A086230/internal; https://www.math.uchicago.edu/~lawler/srwbook.pdf | OEIS A086230 gives p=0.340537329550999..., so the fraction that never returns is approximately 0.659462670449001. Lawler and Limic, p. 75, gives finite visits almost surely in 3-D. The page now states both facts.
- WRONG | X7: so these numbers sit a little below the true probabilities. | research/polya-recurrence/assay-2026-10-14.mjs; research/polya-recurrence/verify.mjs; https://oeis.org/A086230/internal | Re-executing the actual page sampler with xorshift32 seed 3, 3000 trials, and horizon 4000 gives 0.35433333333333333, above p(3). Seeds 1 and 2 give 0.3396666666666667 and 0.3263333333333333. The page now separates finite-time probability from sampled fraction.
- WRONG | X8: while the 3-D bar stalls just under 0.34 no matter how long you wait. | https://oeis.org/A086230/internal | OEIS A086230 gives 0.340537329550999..., above 0.34 and one third. The horizon note now gives about 0.34054 and explains sampling variation.
- WRONG | X9: in 1939 G. N. Watson cracked the triple integral that u(3) hides inside, and found a closed form in Gamma functions: | https://www.ebi.ac.uk/europepmc/webservices/rest/search?query=EXT_ID:16592395&format=json&resultType=core; https://arxiv.org/pdf/1004.1435 | The Glasser and Zucker primary abstract, PMID 16592395, dates their Gamma-form publication to May 1977. Guttmann, p. 4, assigns Watson the 1939 elliptic-integral evaluation. Page, metadata, markdown, README, and both verifier copies now distinguish those attributions; the valid formula is unchanged.
- MINOR | X10: The d≥4 values have no simple closed form | https://arxiv.org/pdf/2311.11326; https://lib.ysu.am/open_books/418789_1.pdf | Gaunt, Nadarajah and Pogány, Theorem 2.1 and Remarks 2.2–2.5, give exact Lauricella expressions but report no further simplification for d≥4. The page and README now distinguish the numerical decimals from the general special-function expressions.
- MINOR | X11: The same recurrent-below-3, transient-at-3 threshold holds for Brownian motion (the continuous limit) | https://www.math.stonybrook.edu/~bishop/fractalbook.pdf | Bishop and Peres, p. 229, Propositions 7.8.2 and 7.8.4, establish planar point polarity and neighborhood recurrence. The page, markdown, and README now specify that meaning of Brownian recurrence.
- WRONG | X12: and for any walk with finite-variance, mean-zero steps | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Lawler and Limic, pp. 17, 26, 75, impose lattice and irreducibility hypotheses. Equal steps ±(1,0,0) remain recurrent despite ambient dimension 3. The page and README now restrict the lattice claim to simple symmetric walks on the full integer lattice, without making aperiodicity necessary for simple walk.
- WRONG | X13: with a finite horizon you never see those, so the measured fractions sit below the true 1, 1, and 0.3405. | research/polya-recurrence/assay-2026-10-14.mjs; research/polya-recurrence/verify.mjs; https://oeis.org/A086230/internal | The same actual-sampler reproduction as X7 exceeds p(3). The honest-edges paragraph and markdown now distinguish missed late returns from sampling variation.
- WRONG | X14: Extend the horizon and 1-D/2-D climb; 3-D does not, because its missing mass is genuinely at infinity, not merely late. | https://gdz.sub.uni-goettingen.de/iiif/image/gdz:PPN235181684_0084:00000153/full/1600,/0/default.jpg | Pólya, printed p. 150, states that finite-time hitting probability grows with the horizon. A first return at step 4002 is possible via 2001 +x steps then 2001 -x steps, with probability 6^(-4002). The page now allows later 3-D returns and reserves permanent missing mass for the infinite-time limit.
- WRONG | X15: all agree on 1.5163860592 | research/polya-recurrence/assay-2026-10-14.mjs; research/polya-recurrence/verify.mjs; https://oeis.org/A086230/internal | Executing the existing engines A–E gives series 1.5162351910436629, quadrature 1.5137785355070168, and Gamma 1.5163860591519769. The two comparison assertions allow absolute error 0.02. The HTML now gives separate approximations and the actual comparison tolerance.
- WRONG | X16: all agreeing on `1.5163860592`. | research/polya-recurrence/assay-2026-10-14.mjs; research/polya-recurrence/verify.mjs; https://oeis.org/A086230/internal | The same A–E execution as X15 contradicts ten-decimal agreement. Markdown and README now give the separate approximations and 0.02 tolerance; verifier comments and assertion labels state the actual comparison.
- MINOR | X17: All figures recomputed in-browser and offline by research/polya-recurrence/verify.mjs (29/29). | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html; git show f5e5f84f34:research/polya-recurrence/verify.mjs | The browser rows are literal constants; it computes sampled returns and bar widths. The verifier checks constants only through p(5). The colophon, metadata descriptions, and check-panel caption now describe selected coverage.
- MINOR | X18: Every number is recomputed in your browser and offline by `research/polya-recurrence/verify.mjs` | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html; git show f5e5f84f34:research/polya-recurrence/verify.mjs | The literal browser table and calls uD_integral(4,60) and uD_integral(5,34) do not recompute every displayed number. Markdown now separates browser simulations, sourced constants, and selected offline checks.
- MINOR | X19: every figure is checked offline (29/29). | git show f5e5f84f34:research/polya-recurrence/verify.mjs | No verifier calculation checks the displayed p(6), p(7), or p(8). The dek now promises selected calculations and retains the verified 29/29 count.
- WRONG | X20: And both end on the same self-referential note: | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html; git show f5e5f84f34:src/content/strata/dead-reckoning.md | The pinned and current layer ending compares dimensions. Dead Reckoning describes the language-model analogy. The relation now proposes the question as its own analogy instead of claiming both pages end with it.
- WRONG | X21: the same threshold holds for Brownian motion and any finite-variance walk, but not for heavy-tailed jumps or drift | https://www.math.uchicago.edu/~lawler/srwbook.pdf | The markdown omitted the lattice and nondegeneracy conditions and even the mean-zero condition. The axis-confined counterexample from X12 refutes the general statement. Markdown now restricts the lattice claim, qualifies Brownian recurrence, and says heavy tails or drift can change behavior.
- WRONG | X22: while the 3-D bar stalls just under a third; | https://oeis.org/A086230/internal | 0.340537329550999... exceeds 1/3. The markdown now describes finite-time probabilities approaching about 0.34054 with sampling variation.
- WRONG | X23: run a crowd to see the return fraction stall just under a third, | https://oeis.org/A086230/internal | 0.340537329550999... exceeds 1/3. The dek now describes sampling a finite-time probability approaching about 34.05%.
- WRONG | X24: and say plainly where it doesn't (the square-site threshold; the d≥4 walk constants), measuring it instead. | https://ir.cwi.nl/pub/1298/1298D.pdf; https://arxiv.org/pdf/2311.11326 | Gaunt, Nadarajah and Pogány (2024), Theorem 2.1 and Remark 2.2, give exact Lauricella expressions for u(d) and p(d), d≥3. The relation now separates the numerical square-site estimate from these exact walk expressions. This remains WRONG as directed, not MINOR or STALE.
- CONFIRMED | C001 (simple-walk-model): one step in a direction chosen by a fair coin, no memory of where you have been. | https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf | Checker re-read: Doyle and Snell, pp. 65–66, define an unbiased choice among 2d nearest neighbors. The browser uses four or six equiprobable choices; fair-coin language means unbiased randomness, not one flip per 3-D step.
- CONFIRMED | C002 (frontmatter-return): On a line or a flat grid you are certain to return to where you started; add a third dimension and that certainty breaks, and you come home only about 34% of the time. | https://www.math.uchicago.edu/~lawler/srwbook.pdf; https://oeis.org/A086230/internal | Checker re-read: Theorem 4.1.1 gives recurrence for d=1,2 and transience for d≥3. A086230 supplies the approximate 34% value.
- CONFIRMED | C003 (polya-theorem): George Pólya proved this in 1921: the simple symmetric random walk is recurrent in one and two dimensions and transient in three and above. | https://gdz.sub.uni-goettingen.de/iiif/image/gdz:PPN235181684_0084:00000153/full/1600,/0/default.jpg; https://api.crossref.org/works/10.1007/BF01458701 | Checker re-read: The checker visually read Pólya, printed p. 150; the dimension threshold is stated there. Crossref confirms the 1921 date. This fixer also visually re-read that page.
- CONFIRMED | C004 (certain-plane): return is certain, probability exactly one, though you may have to wait a very long time. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Lawler and Limic, p. 75, defines recurrence as probability-one return. This does not promise a short wait.
- CONFIRMED | C005 (return-34): The flying walker comes home only about 34% of the time. | https://oeis.org/A086230/internal | Checker re-read: A086230 gives p(3)=0.340537329550999..., approximately 34%.
- CONFIRMED | C006 (escape-two-thirds): Two times in three it drifts off and never returns. | https://oeis.org/A086230/internal | Checker re-read: The complement is approximately 0.659462670449001, roughly two thirds in this context, not exactly two thirds.
- CONFIRMED | C007 (frontmatter-date): That is Pólya's 1921 recurrence theorem | https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf | Checker re-read: Doyle and Snell, pp. 65–66, dates the result to Pólya in 1921.
- CONFIRMED | C009 (dimension-verdict): RETURN: d≤2 CERTAIN · d≥3 NOT | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: NOT qualifies CERTAIN in this compact recurrence/transience label.
- CONFIRMED | C010 (kakutani-quotation): A drunk man will find his way home, but a drunk bird may get lost forever. | https://diglib.tugraz.at/download.php?id=5c80eb8c1d47f&location=browse | Checker re-read: The TU Graz thesis reproduces this traditional wording. That establishes a variant in circulation, not Kakutani’s original utterance.
- CONFIRMED | C011 (kakutani-attribution): attributed to Shizuo Kakutani | https://diglib.tugraz.at/download.php?id=5c80eb8c1d47f&location=browse | Checker re-read: The thesis attributes the quotation to Kakutani; Durrett independently gives a UCLA colloquium attribution.
- CONFIRMED | C014 (aphorism-folklore): The aphorism is folklore. | https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf | Checker re-read: Durrett presents an informal joke. That supports the aphorism description, not an absence of documented provenance; see X4.
- CONFIRMED | C016 (walker-colour): A single random walk · returns to the origin marked in teal | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: stepWalk detects origin returns; both draw and draw1D use the teal color for return marks.
- CONFIRMED | C018 (forever-flashes): In 1-D and 2-D returns keep coming, forever | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The mathematical recurrence assertion is confirmed almost surely. The literal 1-D flash description is qualified under X5: 1-D uses dots, 2-D flashes.
- CONFIRMED | C019 (forever-count): leave it running and the return count never stops climbing. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Recurrence gives infinitely many returns almost surely. The code preserves the counter when trimming its stored path.
- CONFIRMED | C021 (one-walk-proof): One walk can't prove anything (a 3-D walk might get lucky) | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Recurrence is a property of a probability law. One finite trajectory cannot prove it, and 3-D has positive return probability.
- CONFIRMED | C022 (ensemble): Certainty is a statement about the whole ensemble | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Probability-one return is an ensemble statement.
- CONFIRMED | C023 (horizon-undercount): A finite horizon can only undercount | https://gdz.sub.uni-goettingen.de/iiif/image/gdz:PPN235181684_0084:00000153/full/1600,/0/default.jpg | Checker re-read: For the same paths, return by a finite horizon is contained in eventual return. This confirms the truncation statement, not an inequality between every sample and the population; see X7/X13.
- CONFIRMED | C025 (trials): run 3,000 walks each | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: The handler sets trials=3000 and calls the sampler for d=1,2,3.
- CONFIRMED | C026 (initial-horizon): horizon: 4,000 steps ▸ extend | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: The code starts at 4000 steps, doubles through 64000, then resets.
- CONFIRMED | C027 (gold-lines): Gold line = the true return probability. | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: The gold positions are 100%, 100%, and 34.05%, matching the exact low-dimensional values and the rounded 3-D value.
- CONFIRMED | C028 (horizon-low-d): the 1-D and 2-D bars climb toward 1 | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The limiting low-dimensional probabilities are 1. This confirms convergence of finite-time probabilities, not monotonicity of fresh samples.
- CONFIRMED | C030 (horizon-missing): That stall is transience: the missing two-thirds are gone for good. | https://oeis.org/A086230/internal | Checker re-read: Only the limiting missing probability, approximately two thirds, is permanent. Late finite-time returns are distinguished in X14.
- CONFIRMED | C031 (steps-even): only even times can return | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Simple walk is bipartite: coordinate-sum parity matches step-count parity. The source location is printed p. 11.
- CONFIRMED | C032 (recurrence-sum): The walk is recurrent exactly when the expected number of returns is infinite | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Theorem 4.1.1 makes escape probability the reciprocal of the total expected visits, with reciprocal zero for a divergent sum.
- CONFIRMED | C033 (sum-diverges): when Σ p₂ₙ diverges. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Odd-time return probabilities vanish, so the even-time sum gives the same recurrence criterion.
- CONFIRMED | C034 (lclt-formula): p₂ₙ ≈ 2 · ( d / (4πn) )d/2 | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The Gaussian local estimate with covariance I/d and the period-two factor gives 2(d/(4πn))^(d/2) at time 2n.
- CONFIRMED | C035 (decay): So the terms shrink like n−d/2. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The local central limit estimate gives the power n^(-d/2).
- CONFIRMED | C036 (decay-one): In 1-D that is n−1/2 and the sum diverges; | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: At d=1 the power is -1/2 and the corresponding series diverges.
- CONFIRMED | C037 (decay-two): in 2-D it is n−1 | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: At d=2 the power is -1.
- CONFIRMED | C038 (harmonic): the harmonic series, which still just barely diverges; | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The harmonic series diverges, consistent with Theorem 4.1.1.
- CONFIRMED | C039 (decay-three): in 3-D it is n−3/2, and that sum converges. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: At d=3 the power is -3/2 and the series converges.
- CONFIRMED | C040 (infinite-wait): Two dimensions is the knife-edge: the walker comes home with probability one, but the expected wait is infinite. | https://www.math.bgu.ac.il/~yadina/RWnotes.pdf | Checker re-read: Yadin, Example 5.1, p. 33, and the null-recurrence definition on p. 18 imply infinite mean return time for the recurrent infinite grid.
- CONFIRMED | C041 (infinite-wait-repeat): In 2-D the expected time to return is infinite (that harmonic-series knife-edge again). | https://www.math.bgu.ac.il/~yadina/RWnotes.pdf | Checker re-read: The same null-recurrence argument gives infinite expected return time despite almost-sure finite return.
- CONFIRMED | C042 (returns-finite): the total number of returns becomes finite. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Lawler and Limic, p. 75, gives a geometric visit count with finite mean in 3-D, hence finitely many visits almost surely.
- CONFIRMED | C043 (u-meaning): Its value u(3) = Σ p₂ₙ is the expected number of visits to the origin (counting the start) | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The visit count is summed from time zero and therefore includes the starting visit.
- CONFIRMED | C044 (p-from-u): the return probability is p = 1 − 1/u. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Escape probability is 1/u, so return probability is 1-1/u.
- CONFIRMED | C047 (gamma-formula): u(3) = (√6 / 32π³) · Γ(1/24) · Γ(5/24) · Γ(7/24) · Γ(11/24) | https://oeis.org/A086231/internal | Checker re-read: A086231, Finch p. 322, and Gaunt et al. equation (1.2) agree on this Gamma product.
- CONFIRMED | C048 (watson-integral): W = (1/π³) ∫∫∫[0,π]³ dx dy dz / (3 − cos x − cos y − cos z) | https://arxiv.org/pdf/1004.1435 | Checker re-read: The integral denominator is 1-(cos x+cos y+cos z)/3. Symmetry and normalization give the displayed W integral.
- CONFIRMED | C049 (u-three-w): = 3 · W, W = (1/π³) | https://arxiv.org/pdf/1004.1435 | Checker re-read: With the page’s denominator 3-cos x-cos y-cos z, the visit expectation is 3W.
- CONFIRMED | C050 (u-value): = 1.5163860592… | https://oeis.org/A086231/internal | Checker re-read: A086231 gives 1.516386059151978..., which rounds to 1.5163860592.
- CONFIRMED | C051 (p-value): p(3) = 1 − 1/u(3) = 0.3405373296… | https://oeis.org/A086230/internal | Checker re-read: A086230 rounds to 0.3405373296; 1-1/u agrees.
- CONFIRMED | C052 (escape-value): (escape: 0.6594626704…) | https://oeis.org/A086230/internal | Checker re-read: Subtracting p(3) from one gives approximately 0.659462670449001, rounded here to 0.6594626704.
- CONFIRMED | C053 (bird-percentage): So the drunk bird's odds of ever getting home are about 34.05% | https://oeis.org/A086230/internal | Checker re-read: 100p(3)=34.0537329550999..., rounded to 34.05%.
- CONFIRMED | C054 (probabilities-fall): The same machinery gives the odds in higher dimensions, and they keep falling: | https://arxiv.org/pdf/1004.1435; https://lib.ysu.am/open_books/418789_1.pdf | Checker re-read: The checker compared the displayed d=3 through d=8 values using OEIS and Finch Table 5.1; they decrease. This is the checked displayed range.
- CONFIRMED | C055 (table-3d): 0.3405373296 | https://oeis.org/A086230/internal | Checker re-read: A086230 rounds to the displayed ten-decimal value.
- CONFIRMED | C056 (table-4d): 0.1932016732 | https://oeis.org/A086232/internal | Checker re-read: A086232 gives 0.193201673224983..., rounding to the displayed value.
- CONFIRMED | C057 (table-5d): 0.1351786098 | https://oeis.org/A086233/internal | Checker re-read: A086233 gives 0.1351786098206552..., rounding to the displayed value.
- CONFIRMED | C058 (table-6d): 0.1047154956 | https://oeis.org/A086234/internal | Checker re-read: A086234 and Finch Table 5.1 support the displayed 0.1047154956.
- CONFIRMED | C059 (table-7d): 0.0858449341 | https://oeis.org/A086235/internal | Checker re-read: A086235 gives 0.085844934113379..., rounding to the displayed value.
- CONFIRMED | C060 (table-8d): 0.0729126499 | https://oeis.org/A086236/internal; https://lib.ysu.am/open_books/418789_1.pdf | Checker re-read: A086236 gives 0.072912649959384...; the page follows Finch’s ten-place truncation, not nearest rounding. No rounding rule is promised.
- CONFIRMED | C061 (table-low-d): 1-D and 2-D return with certainty (p = 1); | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The recurrence theorem establishes p=1 in d=1,2.
- CONFIRMED | C062 (table-transience): from 3-D on, the bird escapes. | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Escape here means eventual departure after finitely many returns, not zero probability of any return.
- CONFIRMED | C064 (higher-numerical): they are computed numerically | https://oeis.org/A086233/internal | Checker re-read: The OEIS evaluations and Finch Table 5.1 support the description of the displayed decimals as numerical values.
- CONFIRMED | C065 (lattice-model): Pólya's theorem is about steps on a perfect integer grid. | https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf | Checker re-read: Doyle and Snell define integer-coordinate vertices joined to nearest neighbors.
- CONFIRMED | C068 (dimension-not-grid): the dimension is what matters, not the grid | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Confirmed only within the relevant full-dimensional walk class and with the Brownian neighborhood qualification. X12/X21 remove the unrestricted ambient-dimension reading.
- CONFIRMED | C069 (heavy-tail-drift): but a walk with heavy-tailed jumps or a drift can behave differently. | https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf; https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Durrett, p. 296, gives a one-dimensional stable-law counterexample; a deterministic positive step also demonstrates that drift can change recurrence. The cautious can is supported.
- CONFIRMED | C070 (exact-scope): The claim is exactly: simple symmetric walk, finite variance, no drift. | https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf | Checker re-read: The actual equal-probability nearest-neighbor demonstration is symmetric, mean zero, and finite variance.
- CONFIRMED | C071 (later-returns): A walker that hasn't come home by step 4,000 might still return at step 40,000; | https://gdz.sub.uni-goettingen.de/iiif/image/gdz:PPN235181684_0084:00000153/full/1600,/0/default.jpg | Checker re-read: An allowed path with 20000 positive x-steps followed by 20000 negative x-steps first returns at step 40000 with positive probability.
- CONFIRMED | C072 (mc-bias): A finite horizon misses late returns. | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: Confirmed only as missed late returns and downward truncation bias. X7/X13 correct the stronger guarantee about every sampled fraction.
- CONFIRMED | C075 (recurrent-definition): "Recurrent" means return with probability one | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Lawler and Limic, p. 75, explicitly identifies recurrence with probability-one return at a positive time.
- CONFIRMED | C084 (node-no-deps): pure Node, no dependencies, run with node research/polya-recurrence/verify.mjs. | git show f5e5f84f34:research/polya-recurrence/verify.mjs | Checker re-read: The single verifier has no imports or dependencies and executes with Node.
- CONFIRMED | C088 (check-independent): This layer's check works the answer out again from scratch, independently of this page. | git show f5e5f84f34:research/polya-recurrence/verify.mjs | Checker re-read: The verifier calculates independently and does not read the page. This audit preserves that independence.
- CONFIRMED | C090 (check-limit): It can settle the finding; it cannot notice a figure that drifted on this page. | git show f5e5f84f34:research/polya-recurrence/verify.mjs | Checker re-read: Because the verifier has no page input, it cannot detect prose drift. Separate source checks are required.
- CONFIRMED | C091 (published-check): It is published, and it runs on its own: one file, and Node. | git show f5e5f84f34:public/checks/research/polya-recurrence/verify.mjs | Checker re-read: The checker fetched the published verifier and found it identical to the research copy; it ran as one file with Node. The two local copies remain identical after this fix.
- CONFIRMED | C093 (cite-polya): Pólya, Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz, Math. Ann. 84 (1921) 149–160. | https://api.crossref.org/works/10.1007/BF01458701 | Checker re-read: Crossref gives Pólya, the cited title, volume 84, pages 149–160, and 1921.
- CONFIRMED | C094 (cite-watson): Watson, Three triple integrals, Quart. J. Math. 10 (1939) 266–276. | https://api.crossref.org/works/10.1093/qmath/os-10.1.266 | Checker re-read: Crossref gives Watson, the cited title, volume 10, pages 266–276, and 1939.
- CONFIRMED | C095 (cite-oeis-p): Return constants: OEIS A086230 / A086231; | https://oeis.org/A086230/internal | Checker re-read: A086230 identifies the 3-D lattice return probability.
- CONFIRMED | C096 (cite-oeis-u): Return constants: OEIS A086230 / A086231; | https://oeis.org/A086231/internal | Checker re-read: A086231 supplies 1.516386059151978..., the page’s u=3W normalization.
- CONFIRMED | C097 (cite-finch): Finch, Mathematical Constants (CUP 2003) §5.9; | https://dl.icdst.org/pdfs/files/023b87febd629534a90c87b3c8c61b54.pdf; https://lib.ysu.am/open_books/418789_1.pdf | Checker re-read: The checker read Finch’s frontmatter, section 5.9, pp. 322–329, and Table 5.1, confirming title, date, section, and displayed constants.
- CONFIRMED | C098 (cite-lawler): Lawler & Limic, Random Walk: A Modern Introduction (CUP 2010). | https://assets.cambridge.org/97805215/19182/frontmatter/9780521519182_frontmatter.pdf | Checker re-read: The publisher’s frontmatter confirms the authors, title, and first publication in 2010.
- CONFIRMED | C099 (percolation-half): Percolation's threshold lives in the opening probability p (cross ½ and a giant cluster snaps into being) | https://arxiv.org/abs/math/0410359 | Checker re-read: The primary abstract states the square-bond threshold is 1/2 and percolation occurs above it. The relation explicitly scopes this model.
- CONFIRMED | C100 (percolation-exact): p_c = ½ on the square bond lattice | https://arxiv.org/abs/math/0410359 | Checker re-read: The primary paper confirms the exact square-bond threshold 1/2.
- CONFIRMED | C101 (percolation-page-values): Both pages give the exact value where one exists | git show f5e5f84f34:src/content/strata/percolation.md | Checker re-read: The linked page gives p_c=1/2; this page gives a Gamma expression for p(3). These are the highlighted exact cases.
- CONFIRMED | C102 (percolation-page-estimates): The percolation page gives a numerical square-site estimate and this page displays higher-dimensional walk values numerically. | git show f5e5f84f34:src/content/strata/percolation.md | Checker re-read: The checker confirmed only that the pages display numerical estimates in those cases. The broader denial of an exact walk expression is WRONG under X24, not confirmed.
- CONFIRMED | C103 (dead-reckoning-cone): There, a vessel that can't take a fix accumulates error and its cone of uncertainty widens without bound; | git show f5e5f84f34:src/content/strata/dead-reckoning.md | Checker re-read: The linked model describes a square-root-of-time uncertainty cone and a fix collapsing it. This is a description of that model.
- CONFIRMED | C104 (three-drift): here, a 3-D walker with no memory drifts out to infinity and never returns | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: The transient lattice walk eventually leaves every finite ball almost surely, though it may first return one or more times.
- CONFIRMED | C105 (two-engines-related): Both ask whether a wanderer comes home, and both answer with two engines that must agree | git show f5e5f84f34:src/content/strata/a-sextillion-ways-home.md | Checker re-read: The linked page compares a validated Monte Carlo estimate and an exact counting method. This does not claim its unfinished five-bell integer is known.
- CONFIRMED | C106 (bells-count): There the question is how many bell-ringings return to rounds (a hard integer, pinned where the machine can reach and estimated where it can't); | git show f5e5f84f34:src/content/strata/a-sextillion-ways-home.md | Checker re-read: The linked page validates the known four-bell count of 44 while leaving the five-bell exact count open and estimating it.
- CONFIRMED | C108 (related-ahead): Two truths about the same object | git show f5e5f84f34:src/content/strata/ahead-the-whole-game.md | Checker re-read: Ahead the Whole Game explicitly studies simple symmetric random walk.
- CONFIRMED | C109 (related-dimension): The two places where adding a dimension flips the behaviour of space. | git show f5e5f84f34:src/content/strata/all-edge-no-middle.md | Checker re-read: All Edge, No Middle discusses dimension-dependent ball-to-box ratios and concentration. The relation is thematic, not an identical-threshold claim.
- CONFIRMED | C114 (brownian-continuous-limit): Brownian motion (the continuous limit) | https://www.math.uchicago.edu/~lawler/srwbook.pdf | Checker re-read: Lawler and Limic, section 3.1, p. 63, supplies the functional central limit theorem connection, beyond the source record’s table-of-contents excerpt.
- CONFIRMED | C115 (dead-reckoning-world): a vessel that can't take a fix accumulates error and its cone of uncertainty widens without bound; | https://ntrs.nasa.gov/api/citations/19710016383/downloads/19710016383.pdf; git show f5e5f84f34:src/content/strata/dead-reckoning.md | Checker re-read: NASA appendix D supports accumulation of position error between fixes. The unbounded square-root-of-time cone is a claim about the linked model, not every navigation system.
- CONFIRMED | C117 (added-colophon-coverage): (added) Sources for every figure are in the colophon. | https://lib.ysu.am/open_books/418789_1.pdf; git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: The checker found the displayed constants in Finch pp. 322–323 and the named colophon sources. Source coverage does not imply all figures were recomputed.
- CONFIRMED | C118 (added-bar-recomputation): (added) recompute the bars from the values | git show f5e5f84f34:public/strata/one-dimension-too-many/index.html | Checker re-read: The browser computes bar widths from each quoted constant using r[1]*160; this limited calculation claim is correct.

## Findings

- [fixed] X1 (MINOR): Lawler and Limic, Theorem 4.1.1, distinguishes certain return from transience; OEIS A086230 gives a positive 3-D return probability. The title now explicitly qualifies certainty.
- [fixed] X3 (MINOR): Durrett, p. 289, gives an undated UCLA attribution and uses will eventually find. The explanation now identifies the displayed wording as a traditional variant and cites Durrett; the displayed quotation is unchanged.
- [fixed] X4 (WRONG): Durrett, p. 289, documents the UCLA attribution. The colophon now states that provenance and the limits of the passage, without inferring a talk date or original wording.
- [fixed] X5 (MINOR): The 1-D branch calls draw1D and returns before flashAge is used. draw1D marks zero positions with teal dots. Accessible and surrounding descriptions now distinguish the display modes.
- [fixed] X6 (WRONG): OEIS A086230 gives p=0.340537329550999..., so the fraction that never returns is approximately 0.659462670449001. Lawler and Limic, p. 75, gives finite visits almost surely in 3-D. The page now states both facts.
- [fixed] X7 (WRONG): Re-executing the actual page sampler with xorshift32 seed 3, 3000 trials, and horizon 4000 gives 0.35433333333333333, above p(3). Seeds 1 and 2 give 0.3396666666666667 and 0.3263333333333333. The page now separates finite-time probability from sampled fraction.
- [fixed] X8 (WRONG): OEIS A086230 gives 0.340537329550999..., above 0.34 and one third. The horizon note now gives about 0.34054 and explains sampling variation.
- [fixed] X9 (WRONG): The Glasser and Zucker primary abstract, PMID 16592395, dates their Gamma-form publication to May 1977. Guttmann, p. 4, assigns Watson the 1939 elliptic-integral evaluation. Page, metadata, markdown, README, and both verifier copies now distinguish those attributions; the valid formula is unchanged.
- [fixed] X10 (MINOR): Gaunt, Nadarajah and Pogány, Theorem 2.1 and Remarks 2.2–2.5, give exact Lauricella expressions but report no further simplification for d≥4. The page and README now distinguish the numerical decimals from the general special-function expressions.
- [fixed] X11 (MINOR): Bishop and Peres, p. 229, Propositions 7.8.2 and 7.8.4, establish planar point polarity and neighborhood recurrence. The page, markdown, and README now specify that meaning of Brownian recurrence.
- [fixed] X12 (WRONG): Lawler and Limic, pp. 17, 26, 75, impose lattice and irreducibility hypotheses. Equal steps ±(1,0,0) remain recurrent despite ambient dimension 3. The page and README now restrict the lattice claim to simple symmetric walks on the full integer lattice, without making aperiodicity necessary for simple walk.
- [fixed] X13 (WRONG): The same actual-sampler reproduction as X7 exceeds p(3). The honest-edges paragraph and markdown now distinguish missed late returns from sampling variation.
- [fixed] X14 (WRONG): Pólya, printed p. 150, states that finite-time hitting probability grows with the horizon. A first return at step 4002 is possible via 2001 +x steps then 2001 -x steps, with probability 6^(-4002). The page now allows later 3-D returns and reserves permanent missing mass for the infinite-time limit.
- [fixed] X15 (WRONG): Executing the existing engines A–E gives series 1.5162351910436629, quadrature 1.5137785355070168, and Gamma 1.5163860591519769. The two comparison assertions allow absolute error 0.02. The HTML now gives separate approximations and the actual comparison tolerance.
- [fixed] X16 (WRONG): The same A–E execution as X15 contradicts ten-decimal agreement. Markdown and README now give the separate approximations and 0.02 tolerance; verifier comments and assertion labels state the actual comparison.
- [fixed] X17 (MINOR): The browser rows are literal constants; it computes sampled returns and bar widths. The verifier checks constants only through p(5). The colophon, metadata descriptions, and check-panel caption now describe selected coverage.
- [fixed] X18 (MINOR): The literal browser table and calls uD_integral(4,60) and uD_integral(5,34) do not recompute every displayed number. Markdown now separates browser simulations, sourced constants, and selected offline checks.
- [fixed] X19 (MINOR): No verifier calculation checks the displayed p(6), p(7), or p(8). The dek now promises selected calculations and retains the verified 29/29 count.
- [open note:d08a91] X20 (WRONG): The pinned and current layer ending compares dimensions. Dead Reckoning describes the language-model analogy. The source relation now proposes the question as its own analogy, but public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still quote the false shared-ending claim without a correction annotation. Preserve the quotation and add an adjacent correction through research/lineage-edge-grammar/copy-corrections.mjs; those other-layer files are outside this worker's scope. Board note requested in the repair report.
- [fixed] X21 (WRONG): The markdown omitted the lattice and nondegeneracy conditions and even the mean-zero condition. The axis-confined counterexample from X12 refutes the general statement. Markdown now restricts the lattice claim, qualifies Brownian recurrence, and says heavy tails or drift can change behavior.
- [fixed] X22 (WRONG): 0.340537329550999... exceeds 1/3. The markdown now describes finite-time probabilities approaching about 0.34054 with sampling variation.
- [fixed] X23 (WRONG): 0.340537329550999... exceeds 1/3. The dek now describes sampling a finite-time probability approaching about 34.05%.
- [open note:199837] X24 (WRONG): Gaunt, Nadarajah and Pogány (2024), Theorem 2.1 and Remark 2.2, give exact Lauricella expressions for u(d) and p(d), d≥3. The source relation now distinguishes these expressions from the numerical square-site estimate, but public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still quote the false denial without a correction annotation. Preserve the quotation and add an adjacent correction through research/lineage-edge-grammar/copy-corrections.mjs; those other-layer files are outside this worker's scope. Board note requested in the repair report.

## Checks

The evidence reproduction is `node research/polya-recurrence/assay-2026-10-14.mjs`. The required verifier and claims gate results, exact correction lines, source links, and every before/after edit are recorded in `research/polya-recurrence/fix-report-2026-10-14.md`.

The subsequent repair review and check results are in `research/assay-line/queue-1014c/one-dimension-too-many/repair-report.md`, including the two board-note requests that supersede the fix report's closed status for X20 and X24.