---
slug: bertrand-paradox
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: 93
confirmed: 86
wrong: 6
unverifiable: 0
observed: 0
stale: 1
---

This was a directed repair pass over the seven items in research/assay-line/queue-1014c/bertrand-paradox/triage.json, based on the layer read at f5e5f84f34. The fixer independently re-read each triage item's evidence, reproduced the drawing fault and checked the repaired coordinates. The 86 CONFIRMED claims and their source notes are carried from the checker's source reread in research/assay-line/queue-1014c/bertrand-paradox/N2-codex.md; those notes describe that checker's work at the pinned revision. The wrong count includes four MINOR items and two WRONG items. One WRONG remains open under the directing triage's shared placard note. X4's authored source is repaired, but its generated public verifier mirror still awaits the shared publisher, which cannot run within this task's write scope. Passed dates completion of this directed pass, including those recorded limits.

## Claims

- MINOR | `and the rim, with more room, gets the rest (1/4).` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [X1; rim-area-rest] Re-read Bertrand, chapter I, section 5, printed p. 5. The midpoint must lie inside radius R/2; the inner area ratio is (R/2)^2/R^2 = 1/4 and the remaining annulus has 3/4. The original paragraph already said "a quarter" but put its numerical label after "the rest". Moved the label next to the inner share.
- MINOR | `The translation-, rotation- and scale-invariant measure on lines makes the perpendicular distance uniform` | https://bayes.wustl.edu/etj/articles/well.pdf | [X2; line-measure] Re-read Jaynes sections 1 and 3-5, including equation (12), f(r,theta)=1/(2*pi*R*r). The Euclidean line measure d(rho)d(theta) is invariant under translations and rotations; dilation by s multiplies that measure by s. After conditioning on crossing the circle, d/R is uniform on [0,1] at every radius. The revised sentence distinguishes the measure from the normalized conditional law.
- WRONG | `Lines that miss the circle are dropped;` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [X3; discard-misses] Executed the actual sampler and drawing functions with controlled inputs. The original accepted distance-0.99 line was drawn at distance 1.1699999999999995 from the displayed circle. The sampler includes SCX and SCY in its endpoints; drawing added them again. Using the canvas origin once draws the witness at distance 0.99. All 84 tested accepted chords have both displayed endpoint radii equal to 1 within 4.45e-16; 24 misses are rejected. Sampling and counting are unchanged.
- MINOR | `The verifier shows the physical straw-drop converging to 1/2 as the throwing window grows, and unchanged when the circle is shifted or rescaled.` | git show f5e5f84f34:research/bertrand-paradox/verify.mjs | [X4; verifier-invariance] Re-read the complete verifier. R is fixed at 1; the sampled point adds cx and cy, and distance subtracts them again. Windows 4, 40 and 400 and a joint shift of circle and window are tested. Narrowed the page, markdown dek/body, verifier comments and test label to those experiments; retained the original calculations and 11 checks. MINOR under the directing triage own-checking-overclaim rule.
- MINOR | `the page demonstrates the invariance numerically rather than re-deriving his proof.` | git show f5e5f84f34:research/bertrand-paradox/verify.mjs | [X5; numerical-not-proof] The live sampler also uses one unit radius and a throwing window centred on the circle. Re-read it and the complete verifier: neither tests displacement of the circle within one fixed line distribution. The footer now names the simulated straw-drop, growing windows and joint shift. This is the same coverage overstatement as X4; Jaynes supplies the theoretical invariance result.
- WRONG | `Running it needs the repository around it.` | git show f5e5f84f34:research/bertrand-paradox/verify.mjs; https://artwaste.land/checks/research/bertrand-paradox/verify.mjs | [X6; repository-required] Re-read the complete self-contained verifier and the shared generator. The verifier has no imports, file reads, network calls or repository-path access. N2 also recorded a standalone download run with 11/11 passes. scripts/check-placard.mjs generates the false repository requirement for its read-only class. Retained WRONG and left the generated placard unchanged under the triage shared-class deferral to board note 81816d.
- STALE | `"dateModified":"2026-06-26"` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html; git log f5e5f84f34 -- public/strata/bertrand-paradox/index.html; git show 1227d0ac000:public/strata/bertrand-paradox/index.html | [X7; modification-date] Re-read pinned history: 1227d0ac000 changed the page on 2026-09-04 and 2e60bcb5dd1 changed it on 2026-10-01 after its original 2026-06-26 publication. Set dateModified to 2026-10-14 for this repair. The directing instructions exempt this metadata update from an Updated line because the WRONG correction already dates the revision.
- CONFIRMED | `The honest answer is 1/3, or 1/2, or 1/4` | https://bayes.wustl.edu/etj/articles/well.pdf | [three-answers] Jaynes, section 1: "These assignments lead to the results pA = 1=2, pB = 1=3, and pC = 1=4, respectively." His A is this page's radius method, and his B its endpoint method. Fractions are damaged in PDF text extraction; their mathematical values agree.
- CONFIRMED | `which one you get is decided entirely by what you meant by at random.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [random-definition] Bertrand, chapter I, section 5: "Aucune des trois n’est fausse, aucune n’est exacte, la question est mal posée." The preceding three constructions explain the dependence on the probability assignment.
- CONFIRMED | `Inside every circle sits one largest equilateral triangle, and its side has a fixed length:` | https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [largest-triangle] Grinstead and Snell, printed p. 49, visually inspected: "√3 is the length of the side of an inscribed equilateral triangle." Maximal side is fixed, while orientation can vary. The record's excerpt drops the radical.
- CONFIRMED | `for a circle of radius 1 it is exactly √3 ≈ 1.732.` | https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [triangle-unit-side] The visual p. 49 says "√3 is the length of the side of an inscribed equilateral triangle." Direct calculation gives 1.7320508075688772. The record's extracted "3" is missing its radical.
- CONFIRMED | `a chord is longer than the triangle's side exactly when its midpoint lies closer to the centre than half the radius.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [midpoint-threshold] Bertrand p. 5: "à une distance du centre plus petite que la moitié du rayon". Independently, 2√(R²-d²)>R√3 is equivalent to d<R/2.
- CONFIRMED | `Three reasonable people answer that three different ways, and each is internally flawless.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [three-valid-methods] Bertrand p. 5: "Aucune des trois n’est fausse". Each stated probability follows from its own uniform sampling rule; they are not competing answers to a fully specified experiment.
- CONFIRMED | `Bertrand's paradox (Joseph Bertrand, 1889)` | https://bayes.wustl.edu/etj/articles/well.pdf; https://www.e-rara.ch/zut/content/titleinfo/21172634 | [bertrand-name-date] Jaynes section 1: "Bertrand's problem (Bertrand, 1889)". The e-rara catalogue identifies "Bertrand, Joseph Louis François" and publication in 1889.
- CONFIRMED | `the phrase “at random” is not yet a question.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [undefined-random] Bertrand p. 5: "la question est mal posée." The page's explanation correctly concerns the unspecified sampling law.
- CONFIRMED | `a chord is “long” iff its midpoint lands in the inner half-radius disk (drawn faintly)` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634; git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [all-methods-threshold] Bertrand p. 5 supplies the half-radius criterion; pinned HTML line 373 draws "ctx.arc(cx, cy, R / 2, 0, TAU)". Both mathematical and drawing clauses were checked.
- CONFIRMED | `the three ways of randomising scatter the midpoints with completely different densities.` | https://bayes.wustl.edu/etj/articles/well.pdf | [different-densities] Jaynes section 1 distinguishes uniform distance, endpoint angle, and midpoint area. Independently their unit-radius distance densities are 1, 2/(π√(1-d²)), and 2d, respectively.
- CONFIRMED | `What settles each answer is how the midpoint's distance from the centre is distributed.` | https://bayes.wustl.edu/etj/articles/well.pdf | [radial-distance-determines] Jaynes section 2: "the distribution of chord lengths depends only on the radial distribution". The chord-length formula depends only on midpoint distance.
- CONFIRMED | `With random endpoints that distance leans toward the rim, and only a third of chords reach inside the inner disk (1/3).` | https://bayes.wustl.edu/etj/articles/well.pdf; https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [endpoints-rim-third] Jaynes section 1 gives the endpoint result 1/3. Grinstead and Snell p. 50 gives "2π/3 < α < 4π/3"; the distance density 2/(π√(1-d²)) increases toward the rim.
- CONFIRMED | `With a random radius the distance is spread evenly from centre to rim` | https://bayes.wustl.edu/etj/articles/well.pdf | [radius-even-distance] Jaynes section 1: "the linear distance between centers of chord and circle" is assigned uniform probability density. This is precisely method B on the page.
- CONFIRMED | `so exactly half fall inside (1/2), and by area that packs the dots toward the middle.` | https://bayes.wustl.edu/etj/articles/well.pdf | [radius-half-central-density] Jaynes equation 12 gives f(r,θ)=1/(2πRr), followed by "this has a singularity at the center". Half the uniform distance interval is favorable, although areal density increases inward.
- CONFIRMED | `Uniform-over-the-disk gives flat areal density, so the inner disk claims only its fair area share` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [disk-flat-density] Bertrand p. 5: "à l’intérieur d’un cercle quatre fois plus petit en surface." Uniform area assigns the inner disk its area share.
- CONFIRMED | `Long iff the midpoint sits in the inner half-radius disk, whose area is (1/2)² = 1/4 of the whole.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [disk-quarter-area] Bertrand p. 5 gives the fourfold area ratio. Independently π(R/2)²/(πR²)=1/4.
- CONFIRMED | `It was about which distribution the word “random” had silently named.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [distribution-not-chords] Bertrand p. 5: "la question est mal posée." Read with the three immediately preceding probability assignments, this supports the explanation.
- CONFIRMED | `Bertrand left it there.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [bertrand-left-it] The quoted conclusion is immediately followed by section 6, a different random-plane example. Jaynes PDF p. 2 also describes Bertrand's position as the problem having "no definite solution because it is ill posed"; this paraphrase is supported by the French original.
- CONFIRMED | `In 1973 E. T. Jaynes asked a sharper question: the problem never said how big the circle is or where it sits, so a well-posed answer ought not to depend on those either` | https://bayes.wustl.edu/etj/articles/well.pdf; https://api.crossref.org/works/10.1007/BF00709116 | [jaynes-question] Jaynes section 1: "Neither Bertrand's original statement nor our restatement in terms of straws specified the exact size of the circle, or its exact location." Publication year 1973 is independently confirmed by Crossref. PDF extraction loses some ligatures.
- CONFIRMED | `it should survive moving and rescaling the circle.` | https://bayes.wustl.edu/etj/articles/well.pdf | [moving-scaling] Jaynes section 1 says the solution "must be unchanged by a small change in the size or position of the circle." This is a conditional invariance requirement.
- CONFIRMED | `Only one of the three distributions has that invariance, and it is the 1/2 one.` | https://bayes.wustl.edu/etj/articles/well.pdf | [only-invariant-half] Jaynes sections 3-5 eliminate endpoint sampling under scale changes and uniform midpoint area under translations. Equation 12 is the uniform-distance law, with probability 1/2.
- CONFIRMED | `You can see it physically: drop long straight straws across the plane and keep the ones that happen to cross the circle.` | https://bayes.wustl.edu/etj/articles/well.pdf | [physical-straws] Jaynes formulates "A long straw is tossed at random onto a circle" conditional on intersection. Section 6 qualifies the physical prediction by a throwing uncertainty region large relative to the circle. The page's idealized-process and finite-window qualifications matter.
- CONFIRMED | `Their chords land on 1/2` | https://bayes.wustl.edu/etj/articles/well.pdf | [physical-half] Jaynes equation 13, integrated above reduced length √3/2, gives 1/2. Section 6 limits exact physical correspondence to the limiting experiment. The record's necessary-condition excerpt alone is insufficient; the full section supplies the qualification used here.
- CONFIRMED | `and they keep landing there no matter how you slide or shrink the circle underneath them.` | https://bayes.wustl.edu/etj/articles/well.pdf | [physical-any-circle] Jaynes sections 4-6 treat the same rain of straws and displaced or smaller circles. Confirmed for the ideal invariant process, with the large-window qualification. The record's excerpt alone does not imply this for arbitrary physical throwing distributions.
- CONFIRMED | `if you demand an answer blind to the circle's size and place, one exists and it is 1/2.` | https://bayes.wustl.edu/etj/articles/well.pdf | [conditional-unique] Jaynes section 5: "these requirements are compatible; and so the problem has one unique solution." Equation 12 supplies 1/2 under the stated requirements.
- CONFIRMED | `and the other answers are exactly as valid.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [other-answers-valid] Bertrand: "Aucune des trois n’est fausse". The other assignments remain valid for their specified mechanisms; the page explicitly makes that distinction.
- CONFIRMED | `a probability question isn't finished until the mechanism of the randomness is named.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [mechanism-needed] Bertrand's three different sampling arguments end "la question est mal posée." The page correctly requires a specified probability assignment or mechanism.
- CONFIRMED | `Inscribed equilateral triangle side = R√3;` | https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf; recreated: unit-circle geometry scaled by R | [triangle-general-side] Grinstead and Snell p. 49 visually gives "√3 is the length of the side of an inscribed equilateral triangle" for the unit circle. Scaling lengths by R gives R√3. The record dropped the radical from its copied excerpt.
- CONFIRMED | `a chord whose midpoint is distance d from the centre has length 2√(R²−d²), which exceeds R√3 iff d < R/2.` | https://bayes.wustl.edu/etj/articles/well.pdf | [chord-length-formula] Jaynes section 5 gives L=2(R²-r²)^(1/2). Squaring the nonnegative quantities independently gives L>R√3 exactly when r<R/2.
- CONFIRMED | `A) random endpoints → 1/3.` | https://bayes.wustl.edu/etj/articles/well.pdf | [a-probability] Jaynes section 1 gives endpoint-angle sampling 1/3; Grinstead and Snell p. 50 independently computes the favorable arc fraction 1/3.
- CONFIRMED | `Fix one endpoint; the chord is long iff the other lands on the opposite third of the rim.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634; https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [a-opposite-third] Grinstead and Snell p. 50: "2π/3 < α < 4π/3". Its width is 2π/3, the opposite third of the full rim. This also agrees with Bertrand's middle one of three equal angles.
- CONFIRMED | `B) random radius → 1/2.` | https://bayes.wustl.edu/etj/articles/well.pdf | [b-probability] Jaynes section 1 assigns 1/2 to uniform linear distance. His letter A denotes the page's method B, so there is no contradiction in the differing labels.
- CONFIRMED | `Midpoint at uniform distance along a random radius is long iff d < R/2` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [b-threshold] Bertrand p. 5: "dans la moitié la plus voisine du centre." The half-radius distance threshold and uniform distance give the stated result.
- CONFIRMED | `C) midpoint uniform in the disk → 1/4.` | https://bayes.wustl.edu/etj/articles/well.pdf | [c-probability] Jaynes section 1 gives pC=1/4; Bertrand p. 5 and direct disk-area division independently agree.
- CONFIRMED | `D) thrown straw → 1/2, and invariant.` | https://bayes.wustl.edu/etj/articles/well.pdf | [d-probability] Jaynes equation 12 and sections 4-6 support 1/2 for the ideal invariant line process. The finite-window code is an approximation, as disclosed in the page's check discussion.
- CONFIRMED | `The three exact probabilities (1/3, 1/2, 1/4) are theorems of plane geometry` | https://bayes.wustl.edu/etj/articles/well.pdf | [probabilities-theorems] The three exact answers follow from plane geometry combined with the three uniform sampling rules stated immediately above. Jaynes section 1 gives the same results. The claim is read in that explicit context, not as probabilities determined by geometry alone.
- CONFIRMED | `The claim that 1/2 is the unique translation-and-scale-invariant answer is Jaynes' result (1973);` | https://bayes.wustl.edu/etj/articles/well.pdf; https://api.crossref.org/works/10.1007/BF00709116 | [jaynes-result] Jaynes sections 4-5 derive the uniform-distance answer from the invariance requirement; Crossref dates the article to 1973. The page accurately attributes the conditional result.
- CONFIRMED | `not a fact forced by Bertrand's words;` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [choice-not-forced] Bertrand p. 5: "la question est mal posée." Jaynes explicitly treats his interpretation as another useful viewpoint; the page does not attribute his extra requirement to Bertrand.
- CONFIRMED | `Jaynes reported a physical broom-straw experiment agreeing with 1/2;` | https://bayes.wustl.edu/etj/articles/well.pdf | [reported-broom] Jaynes section 6 reports "tossing broom straws from a standing position onto a 5-in.-diameter circle" and "128 successful tosses confirmed Eq. (13)". The page accurately says Jaynes reported it, rather than claiming a new physical replication.
- CONFIRMED | `J. Bertrand, Calcul des probabilités (1889), the original statement.` | https://www.e-rara.ch/zut/content/titleinfo/21172634 | [bertrand-reference] The directly fetched e-rara catalogue says "Bertrand, Joseph Louis François: Calcul des probabilités. Paris : Gauthier-Villars, 1889." Chapter I, section 5 contains the chord problem.
- CONFIRMED | `E. T. Jaynes, “The Well-Posed Problem,” Foundations of Physics 3 (1973) 477–492, the invariance resolution.` | https://api.crossref.org/works/10.1007/BF00709116 | [jaynes-reference] Crossref returns "page":"477-492", "title":["The well-posed problem"], "volume":"3", author "E. T." "Jaynes", and publication [[1973,12]]. DOI MATCH is correct. The hosted retypeset PDF's header says 477-493, but the publisher-deposited bibliographic record supports the page's 477-492.
- CONFIRMED | `The inscribed-triangle threshold and the three densities are standard;` | https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [standard-threshold-densities] Grinstead and Snell's Example 2.6, pp. 47-50, develops all three coordinate choices and computes the half-radius threshold and three probabilities. The record's short excerpt alone does not show the full calculation; the surrounding pages do.
- CONFIRMED | `all three methods run live and pull apart in front of you;` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [live-three] Pinned HTML defines three distinct METHODS and updates "var p = cell.hits / cell.n;". The local Node VM execution of the actual page script initializes all three at 60 samples and successfully exercises their controls. No live browser was used.
- CONFIRMED | `a midpoint-cloud toggle shows why they differ` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [midpoint-toggle] The actual source binds "setMode('midpoints')" to the midpoint button. The local VM test verifies its pressed state and drawing branch.
- CONFIRMED | `inner half-radius disk (drawn faintly)` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [inner-disk-drawn] Pinned HTML line 373: "ctx.arc(cx, cy, R / 2, 0, TAU)" with a low-opacity fill. The local canvas-call capture confirms the half-radius disk.
- CONFIRMED | `crossing chords are coloured long / short.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [color-chords] Pinned source uses "var col = s.long ? '222,174,69' : '79,124,196';" in both panels. The length categories and colors are consistent. The separate spatial-registration fault is recorded under discard-misses, not treated as a color error.
- CONFIRMED | `The circle is shown off-centre on purpose` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [off-centre] Pinned source explicitly sets "var SCX = 0.32, SCY = -0.18;" and draws the circle with that offset. This claim about the circle's position is correct even though the lines receive an extra offset.
- CONFIRMED | `the fraction doesn't care.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [offcentre-fraction] The fraction uses "var signed = (px - SCX) * nx + (py - SCY) * ny;" after the window has been recentered on SCX, SCY. The offsets cancel, so the numerical fraction is independent of that drawing-centre choice. This is not a test of moving only the circle through one fixed rain of straws.
- CONFIRMED | `identical to method B.` | git show f5e5f84f34:research/bertrand-paradox/verify.mjs; https://bayes.wustl.edu/etj/articles/well.pdf | [same-as-b] The ideal line law in Jaynes equation 12 makes distance uniform, matching "const d = R * rnd();" in method B. The page immediately distinguishes finite-window straw approximations.
- CONFIRMED | `(A small finite-window bias is named there, not hidden.)` | git show f5e5f84f34:research/bertrand-paradox/verify.mjs | [finite-window-bias] The verifier explicitly says "a finite window biases the perpendicular-distance law slightly" and tests windows 4, 40, and 400. The source record accurately points to this qualification.
- CONFIRMED | `run it yourself; no network, no dependencies.` | git show f5e5f84f34:research/bertrand-paradox/verify.mjs | [no-network-dependencies] The full verifier has no imports, file reads, fetches, or dependencies. The single downloaded .mjs ran successfully in an otherwise empty directory. Positive control for code inspection: it contains the RNG, Math calls, and all eleven checks.
- CONFIRMED | `The live estimates above are the same simulation, drawn in your browser.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [same-live-simulation] Both page and verifier implement the same A/B/C probability laws and the same point-plus-angle straw sampler. They use different RNGs and window sizes, so "same simulation" is correctly read as the same sampling model, not identical samples.
- CONFIRMED | `Calling 1/2 “the” answer is a choice` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [choice-disclosed] The page explicitly says "This does not crown 1/2 as cosmically correct" and makes the invariance demand conditional. The qualification is present in the pinned and live text.
- CONFIRMED | `This layer's check works the answer out again from scratch, independently of this page.` | git show f5e5f84f34:research/verifier-independence/grading.json; git show f5e5f84f34:research/bertrand-paradox/verify.mjs | [check-independent] The full verifier reads no page file. The committed grading entry states "opened nothing that /strata/bertrand-paradox/ ships". Its answer calculation is independent of the page text.
- CONFIRMED | `It can settle the finding; it cannot notice a figure that drifted on this page.` | git show f5e5f84f34:research/verifier-independence/grading.json; git show f5e5f84f34:research/verifier-independence/screen-0.json | [settle-not-drift] Source inspection and the recorded corruption experiment agree: the verifier does not inspect displayed figures. This is an accurate limitation, not an undisclosed self-reading verifier failure.
- CONFIRMED | `It is published to read.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html; https://artwaste.land/checks/research/bertrand-paradox/verify.mjs | [published-check] The exact linked download returns HTTP 200 and 6,751 bytes. Its SHA-256 matches the pinned research verifier. The reader can read it and can also run it.
- CONFIRMED | `Two independent points on the rim, joined.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [method-a-label] Actual page source draws two separate uniform angles: "var a = Math.random() * TAU, b = Math.random() * TAU;" and joins their unit-circle points.
- CONFIRMED | `A random spoke; a uniform point on it is the midpoint.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [method-b-label] Actual source uses "var phi = Math.random() * TAU, d = Math.random();" and fromMidpoint(d cos phi,d sin phi), giving uniform angle and distance.
- CONFIRMED | `A point spread evenly over the whole disk is the midpoint.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [method-c-label] Actual source uses "var r = Math.sqrt(Math.random()), th = Math.random() * TAU;". The square root gives uniform disk area, not uniform radius.
- CONFIRMED | `0 chords each` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [counter-label] The static placeholder is zero and Reset sets "0 chords each". Initial script execution seeds 60 chords, or 400 with reduced motion, and updates the label accordingly. It is not a persistent claim that the running counter remains zero.
- CONFIRMED | `thrown straws → 1/2 (invariant)` | https://bayes.wustl.edu/etj/articles/well.pdf | [straw-target] Jaynes's invariant ideal process has probability 1/2. The caption is a theoretical target; the source names the finite-window approximation. The separate drawing fault does not change this theoretical value.
- CONFIRMED | `Pick a chord of a circle 'at random' and ask if it beats the inscribed triangle's side: the answer is 1/3, 1/2, or 1/4 depending on how you randomise.` | https://bayes.wustl.edu/etj/articles/well.pdf | [metadata-three-answers] Jaynes section 1 gives the same three probabilities and mechanisms. This agrees with the body and the independently recomputed geometry.
- CONFIRMED | `a thrown-straw construction singles out 1/2 as the only scale- and translation-invariant answer.` | https://bayes.wustl.edu/etj/articles/well.pdf | [metadata-only-invariant] Jaynes sections 4-5 identify the unique law under the stated invariance requirement, giving 1/2. The body explains that the requirement is a choice.
- CONFIRMED | `The honest answer is 1/3, or 1/2, or 1/4, and which one you get is fixed entirely by what 'at random' was supposed to mean.` | https://bayes.wustl.edu/etj/articles/well.pdf | [dek-three-answers] Jaynes section 1 and Bertrand p. 5 support the stated sampling dependence. All three headline fractions agree with the live page and verifier.
- CONFIRMED | `random endpoints pile midpoints at the rim → 1/3` | https://bayes.wustl.edu/etj/articles/well.pdf | [dek-endpoints-density] The endpoint midpoint-distance density is 2/(π√(1-d²)), with increasing weight toward the rim and probability 1/3 below d=1/2. The body carefully discusses distance; areal density also increases near the centre.
- CONFIRMED | `a random radius smears them along spokes → 1/2` | https://bayes.wustl.edu/etj/articles/well.pdf | [dek-radius-density] Jaynes's uniform linear-distance assignment gives 1/2. The actual page samples a random angle and a uniform distance, exactly as described.
- CONFIRMED | `uniform-over-the-disk gives the inner quarter its fair area → 1/4` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [dek-disk-density] Bertrand p. 5: "à l’intérieur d’un cercle quatre fois plus petit en surface." The inner half-radius disk has one quarter of total area.
- CONFIRMED | `a thrown-straw construction singles out 1/2 as the one answer invariant under sliding or rescaling the circle` | https://bayes.wustl.edu/etj/articles/well.pdf | [dek-straw-invariance] Jaynes equation 12 gives uniform distance, and sections 4-5 show the required consistency across translated and scaled circles. This confirms the conditional theoretical claim, independently of the deficient numerical invariance test.
- CONFIRMED | `Buffon's needle turns the throw into an estimate of π;` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634; https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [buffon-pi] Bertrand pp. 55-56 derives the needle-intersection probability; its OCR mangles the formula. Independently, Grinstead and Snell p. 45 prints P(E)=2/π for equal needle length and line spacing, and "compute π = 2/P(E)". Thus throws estimate π.
- CONFIRMED | `only the one a thrown line realises (perpendicular distance uniform) survives sliding and rescaling the circle.` | https://bayes.wustl.edu/etj/articles/well.pdf | [thrown-line-only] Jaynes sections 4-5 single out the uniform perpendicular-distance law under both requirements. Here "survives" refers to the same family of lines tested on different circles, not generating new chords separately inside each circle.
- CONFIRMED | `Buffon shows geometric probability can pin a constant exactly;` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634; https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [buffon-constant] The exact needle-intersection law involves π, as verified in Bertrand pp. 55-56 and the readable Grinstead-Snell equation P(E)=2/π. Actual finite tosses estimate rather than determine π exactly; the claim concerns the exact law.
- CONFIRMED | `Bertrand shows it can fail to have an answer at all until you say how you threw.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [throw-mechanism] Bertrand's three constructions end "la question est mal posée." Specifying the sampling experiment resolves that ambiguity.
- CONFIRMED | `The hundred prisoners look doomed at (1/2)^100 until you see the shuffle as cycles;` | https://aofa.cs.princeton.edu/online/slides/AA07-Perms.pdf | [prisoners-comparison] Princeton slides 28-29 say "the probability of success is on the order of 2−100" for the naive strategy and "Each prisoner "follows the cycle"" for the successful alternative. Independent guessing gives (1/2)^100.
- CONFIRMED | `the random chord looks like it should have one probability until you see that 'random' silently named a distribution over midpoints.` | https://bayes.wustl.edu/etj/articles/well.pdf | [random-names-midpoints] Jaynes section 1 explicitly distinguishes distance, angle, and area assignments; section 2 parameterizes the chord by its midpoint. This supports the page's explanation.
- CONFIRMED | `watch the empirical rate settle onto the exact number the page computed in front of you.` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html; git show f5e5f84f34:src/content/strata/the-loop-that-saves-them.md | [both-live-convergence] The Bertrand page computes "var p = cell.hits / cell.n;". The linked prisoners layer's pinned dek also explicitly describes a live convergence to the harmonic-number answer. This checks both layers intended by "Each" in the full relation text, not only the one cited in the record.
- CONFIRMED | `The Bias in the Sample shows a number bend under a selection rule;` | git show f5e5f84f34:src/content/strata/the-bias-in-the-sample.md | [bias-layer-claim] The pinned related layer describes "selecting a sample can forge an association between traits that are independent in the population" and gives its admission and threshold models. That is the selection-rule phenomenon claimed here.
- CONFIRMED | `Bertrand shows three honest sampling rules give three honest probabilities for one geometric question.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634 | [three-honest-rules] Bertrand's three cases and "Aucune des trois n’est fausse" support the statement when each rule is specified. Their probability spaces differ.
- CONFIRMED | `the moment you fix the mechanism of the randomness, the paradox dissolves into bookkeeping.` | https://www.e-rara.ch/zut/download/ftpack/plain/21172634; https://www.stat.yale.edu/~jtc5/251/readings/ProbabilityBook_GrinsteadSnell.pdf | [fix-mechanism] Bertrand's original comparison supplies three well-defined constructions but an ill-posed undifferentiated question. Grinstead and Snell pp. 47-50 explicitly calculates the probabilities after choosing coordinates.
- CONFIRMED | `<meta property="og:image" content="https://artwaste.land/og/bertrand-paradox.png" /> <meta property="og:image:width" content="1200" /> <meta property="og:image:height" content="630" />` | git show f5e5f84f34:public/og/bertrand-paradox.png | [social-image-dimensions] Directly decoded the pinned PNG IHDR: width 1200, height 630. The named artifact exists and the metadata dimensions are correct.
- CONFIRMED | `Nothing here may lie about anything real; that rule never bends.` | git show f5e5f84f34:WAKE.md | [site-no-lies-rule] WAKE.md explicitly prescribes "It must never lie about anything real." The sentence reports the rule, not a claim that this audit must find no mistakes.
- CONFIRMED | `The two classics of geometric probability` | https://bayes.wustl.edu/etj/articles/well.pdf | [two-classics] Jaynes calls Bertrand's problem "famous" and discusses the "similar Buffon needle problem"; Grinstead and Snell treats both in its continuous-probability chapter. The phrase describes these two canonical examples, not an assertion that no other classics exist.
- CONFIRMED | ``An immersive layer; it renders as its own standalone page at the `href` above`` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [immersive-page] The pinned standalone document begins "<!doctype html>", includes its own inline simulator, and the given live href returns HTTP 200 with matching reader text.
- CONFIRMED | ``the source is `public/strata/bertrand-paradox/index.html``` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [md-source-location] The specified pinned path contains the full standalone page and simulator; live text agrees apart from injected analytics scripts.
- CONFIRMED | `its body is not rendered.` | git show f5e5f84f34:src/pages/strata/[...slug].astro | [md-not-rendered] The pinned Astro route explicitly filters with ".filter((s) => !s.data.draft && !s.data.href)". The layer has href, so its Markdown body is not rendered through that route.
- CONFIRMED | `▶ Run all three +1 chord +1,000 Reset` | git show f5e5f84f34:public/strata/bertrand-paradox/index.html | [simulation-controls] The local execution of the actual page script verifies Run, +1, +1,000, midpoint-toggle, and Reset actions. See the six passing control checks in page-check.json.
- CONFIRMED | (added) `Use the menu or the search box up top to find your way around.` | git show f5e5f84f34:public/media/home-banner.js; git show f5e5f84f34:public/media/plain-button.js | [site-navigation] The source record omits this functional instruction from the same site-help paragraph it otherwise records. The actual banner creates an "aw-menu-btn" and a search form with action "/ask/", method "get", and input name "q". Live copies of both scripts return HTTP 200 and are byte-identical to the pinned copies. Positive control: the same coverage search finds the recorded sentence "Nothing here may lie about anything real; that rule never bends."

## Findings

- [fixed] X1 (MINOR): moved the 1/4 label from the outer remainder to the already stated inner quarter in public/strata/bertrand-paradox/index.html. No page correction line, per the MINOR rule.
- [fixed] X2 (MINOR): distinguished the translation- and rotation-invariant line measure from the scale-invariant conditional law of d/R in the page's check explanation. The 1/2 result is unchanged. No page correction line.
- [fixed] X3 (WRONG): changed the two straw endpoint drawing calls to use the canvas centre because the sampled coordinates already contain the circle offset. Added a Corrected 2026-10-14 line to the page's new Corrections note above the shared placard and mirrored it at the end of the markdown body. The line describes the false caption in the old display and the repaired endpoint positions, and credits the assay line, 2026-10-14.
- [fixed] X4 (MINOR), authored source: narrowed the numerical coverage description in the page and markdown dek/body, and the verifier's comments and shifted-centre test label, to growing windows and a joint shift at fixed radius. No page correction line. NOT FIXED in the generated copy public/checks/research/bertrand-paradox/verify.mjs: scripts/publish-checks.mjs is its writer and has no per-layer option; running it also rewrites shared public/checks/manifest.json, public/checks/index.html and other layers' mirrors. The task prohibits those writes and direct output edits, so mirror regeneration remains outstanding.
- [fixed] X5 (MINOR): replaced the footer's numerical-invariance claim with the actual straw-drop simulation and window experiments. This is the duplicate of X4 identified by triage. No page correction line.
- [open note:81816d] X6 (WRONG): the repository requirement remains false. The directing triage assigns this to the shared placard reach class; its author is scripts/check-placard.mjs, which this task prohibits editing. The named board note exists and holds that class. No local placard edit or correction line.
- [dated] X7 (STALE): changed JSON-LD dateModified from 2026-06-26 to 2026-10-14. The original date was valid when written. The task's explicit metadata exception requires no Updated line when the Corrected line already dates the repair.

Validation: node research/bertrand-paradox/verify.mjs exited 0 with 11/11 checks passing. The actual canvas-call regression in research/assay-line/queue-1014c/bertrand-paradox/check-straw.mjs passed 84 crossing chords and 24 rejected misses at two canvas sizes; it also reproduced the original fault before the repair. The production checkRecord function accepted this record with no errors and one open note. node scripts/check-claims-pass.mjs exited 1 solely for 31 other layers' records at the time of this run; bertrand-paradox was absent from its failures. See research/assay-line/queue-1014c/bertrand-paradox/fix-report.md for the file-level changes, exact correction text and command results.