--- slug: the-four-that-finally-moved passed: 2026-10-18 by: the assay line (research/assay-line/queue-1018/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex checked: 96 confirmed: 91 wrong: 4 unverifiable: 0 observed: 0 stale: 1 --- This pass decided all five triage items before editing the layer at cd6ebcd061, independently re-read their evidence, then applied four fixes and one dated update using the assigned assay date, 2026-10-18. The 91 CONFIRMED lines below retain the source re-reads and evidence from N2-codex.md, whose own header is dated 2026-10-19 and whose source record is N1-sources.json at a77c45dd68; they are attributed to that checker, not represented as 91 new source fetches by the fixer. The wrong count includes one WRONG and three MINOR lines. The two rounding findings share one error and one correction line. No finding was rejected or left open. ## Claims - MINOR | The exact lane rounds λk to an integer l and reports the actual ratio. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | X1 (finite-lower-clamp). Re-read public/strata/the-four-that-finally-moved/index.html:970 at cd6ebcd061: "var l = Math.max(2,Math.min(k,Math.round(k * lambda)));". Evaluating that expression gives l=2 at k=4, lambda=0.25, although lambda*k=1. The note at line 492 omits the clamp. Keep MINOR and cheaply disclose the range 2 through k; this is the same omitted clamp as X2, so both share one error label. The control note now discloses the range 2 through k. MINOR fixed without a separate correction line. - WRONG | For the finite lane, l is the nearest integer to λk. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | X2 (finite-lower-clamp). Re-read the authored page code at public/strata/the-four-that-finally-moved/index.html:970, unchanged from cd6ebcd061: "var l = Math.max(2,Math.min(k,Math.round(k * lambda)));". The prose at line 609 says "l is the nearest integer to λk". Executing the extracted expression gives l=2 when k=4 and lambda=0.25, while lambda*k=1 and its nearest integer is 1; k=4, lambda=0.5 gives 2 as a control. A sweep reproduces 18 affected settings among 8,892. Keep WRONG: disclose rounding followed by the range 2 through k in both prose copies and call the readout finite l. Add one Corrected line for this shared error, without changing the correct calculation. Both prose copies now disclose rounding followed by the range 2 through k, and the output label says finite l. One Corrected line records this error in the HTML and markdown. - STALE | arXiv lists only version 1 from 26 July 2024 | https://arxiv.org/abs/2407.19026; git show a77c45dd68:research/the-four-that-finally-moved/README.md | X3 (arxiv-version-history). Re-read https://arxiv.org/abs/2407.19026, Submission history: "[v2] Sat, 29 Aug 2026 18:58:26 UTC (78 KB)". The page at line 618 explicitly says "As checked on 29 July 2026", and research/the-four-that-finally-moved/README.md records that earlier only-v1 observation. Preserve that true dated history. Group all STALE items for this layer (X3 alone) into ONE Updated 2026-10-18 line, mirrored in the markdown body, recording the July observation and the later v2 date with a source link and assay-line credit. Do not claim any change to the theorem or its proof. The dated July observation remains. One Updated line in the HTML, mirrored in markdown, records v2 submitted on 29 August 2026. - MINOR | preprint, not peer-reviewed | https://www.math.mcgill.ca/snorin/papers.html | X4 (unsupported-review-status). Re-read https://www.math.mcgill.ca/snorin/papers.html, Extremal Combinatorics item 50: "Optimizing the CGMS upper bound on Ramsey numbers" links to arXiv, and the item ends after the coauthors with no journal venue. Other entries on that list include journals. This establishes the listed preprint status, not a history of no peer review. Keep MINOR and minimally replace "preprint, not peer-reviewed" with "preprint" in the page badge and layer README, with no dated correction line. The page badge and layer README now say preprint, with no unsupported review-history claim and no dated correction line. - MINOR | (added) research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs | https://artwaste.land/research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs; https://artwaste.land/checks/research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs | X5 (verifier-reader-access). Re-read the page at lines 629-630 and 1040 and the complete layer verifier. The page names the verifier but has no verifier href or run command. Fresh curl requests to both URLs in this item return HTTP 404 (4,404 bytes each). The verifier reads "public/strata/the-four-that-finally-moved/index.html" relative to its repository location and uses only Node built-ins. A small local fix can accept an explicit HTML input path, copy the same verifier into this layer's public directory, and add download links plus an exact Node command. Keep MINOR, fix locally, and record it without a dated correction line; do not edit the shared publisher or placard. The verifier accepts an explicit HTML path. A byte-identical public copy in this layer directory, page-input link, and curl/Node commands make the check available after landing; no shared publisher or placard was edited. - CONFIRMED | Operate the first improvement to the exponential base of the diagonal Ramsey upper bound since 1935. | https://annals.math.princeton.edu/2026/203-3/p04 | Entry 1, first-base-change. The Annals abstract says "the first exponential improvement" and "proved in 1935". - CONFIRMED | Diagonal Ramsey Upper Bounds From 4 to 3.7992 | https://arxiv.org/html/2407.19026v1 | Entry 2, title-numeric-endpoints. Theorem 1 and its diagonal consequence print "(3.7992\ldots)^{k+o(k)}"; 4 is the classical limiting base. - CONFIRMED | Sweep from off-diagonal to diagonal, watch the browser assemble the peer-reviewed 2023 constant and the stronger 2024 preprint curve, then test the finite excess-edge identity that powers the method. | https://arxiv.org/html/2407.19026v1; https://arxiv.org/pdf/2303.09521v1; https://annals.math.princeton.edu/2026/203-3/p04 | Entry 3, peer-reviewed-2023-and-2024. The supplied excerpt supports the 2024 curve. The composite claim is completed by CGMS Theorem 1.1, its epsilon paragraph, the Annals publication, and GNNW Lemma 2. In the original 2023 PDF, p. 2 already says "the second with ε = 2−7". - CONFIRMED | Sweep from off-diagonal to diagonal, watch the browser assemble the peer-reviewed 2023 constant and the stronger 2024 preprint curve, then test the finite excess-edge identity that powers the method. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 4, instrument-description. The quoted math block implements "binomialBigInt", "gnnwBase", "cgmsBase", and "excessIdentity". The surrounding control listeners execute them; reproduced with node reproduce.mjs. - CONFIRMED | This page shows how a famous upper limit in mathematics finally got smaller. | https://annals.math.princeton.edu/2026/203-3/p04 | Entry 5, plain-upper-limit. The primary abstract calls the result "the first exponential improvement". - CONFIRMED | You can move its controls, compare the old and new limits, and test a small piece of the newer method yourself. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 6, plain-controls. The supplied math block is supplemented by the actual control listeners at lines 1021-1028. The reproduced instrument has 16 working toggles: "ALL 16 TOGGLES PASS". - CONFIRMED | The full live instrument is at [The Four That Finally Moved](/strata/the-four-that-finally-moved/). | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 7, markdown-location. The source canonical href is "https://artwaste.land/strata/the-four-that-finally-moved/". Curl returned "HTTP 200" at that address. - CONFIRMED | It separates exact finite recurrence bounds from asymptotic base comparisons, then lets the reader test the later preprint's finite excess-edge identity on an editable colouring. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 8, markdown-functionality. The functions "binomialBigInt" and "excessIdentity" implement the finite calculations; "gnnwBase" supplies the separately labelled asymptotic comparison. The default outputs were reproduced. - CONFIRMED | For decades, better diagonal Ramsey bounds shaved factors from around an exponential base of 4. | https://arxiv.org/html/2303.09521v2 | Entry 9, decades-base-four. The manuscript describes a "polynomial factor", then a "super-polynomial factor", and gives Sah a correction of order (log k)^2 in the exponent. - CONFIRMED | In 2023, a theorem moved the base itself. | https://arxiv.org/abs/2303.09521; https://arxiv.org/pdf/2303.09521v1 | Entry 10, theorem-2023. The arXiv history says "[v1] Thu, 16 Mar 2023 17:38:08 UTC". The original v1 PDF contains Theorem 1.1 and the positive epsilon choices. - CONFIRMED | Put the ratio on the diagonal and watch the browser assemble that peer-reviewed landmark from its published epsilon, then the stronger value from a later preprint. | https://arxiv.org/html/2303.09521v2 | Entry 11, landmark-versus-later. The manuscript gives "the second with" epsilon 2^-7 and credits the later optimisation to Gupta, Ndiaye, Norin and Wei. Annals metadata supplies the journal status. - CONFIRMED | This page keeps finite bounds, asymptotic theorems, and scale illustrations in separate lanes. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 12, separate-lanes. The source computes "finite = binomialBigInt(k + l - 2,l - 1)" separately from the exponential bases and "pureBaseRatio(k)". - CONFIRMED | The exact finite lane uses integer l. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 13, finite-integer-l. The source uses "Math.max(2,Math.min(k,Math.round(k * lambda)))", which always yields an integer for the slider inputs. - CONFIRMED | The curve uses the continuous ratio from the asymptotic theorem. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 14, continuous-ratio. The plot samples "lambda = xMin + (xMax - xMin) * i / steps" and sends it directly to the asymptotic base functions, without finite-l rounding. - CONFIRMED | The three diagonal bases are drawn on a logarithmic vertical scale | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 15, plot-logarithmic. The source maps values through "yScale(Math.log(fn(lambda)))" and the landmark through "yScale(Math.log(cgmsBase()))". - CONFIRMED | live browser arithmetic | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 16, browser-arithmetic. "updateBaseLab" recomputes the displayed arithmetic and is attached to the input listeners. The full inline script ran successfully in the scratch DOM harness. - CONFIRMED | Used for the exact finite binomial and the scale-only ratio. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 17, k-usage. The code uses k in "binomialBigInt(k + l - 2,l - 1)" and "pureBaseRatio(k)". - CONFIRMED | 1935 entropy base | https://annals.math.princeton.edu/2026/203-3/p04; https://numdam.org/item/CM_1935__2__463_0.pdf | Entry 19, label-1935. The Annals abstract dates the classical bound to "1935". The original paper p. 466 supplies the binomial bound; its logarithmic asymptotic exponent gives the displayed base. - CONFIRMED | 2023 diagonal landmark | https://arxiv.org/abs/2303.09521 | Entry 20, label-2023. The first arXiv version is dated "16 Mar 2023" and already states the exponential improvement. - CONFIRMED | 2024 preprint curve | https://arxiv.org/abs/2407.19026 | Entry 21, label-2024. The history retains "[v1] Fri, 26 Jul 2024 18:08:35 UTC". The displayed curve explicitly uses that 2024 version; the later v2 does not invalidate this label. - CONFIRMED | published input: ε = 2^-7 | https://arxiv.org/html/2303.09521v2 | Entry 22, published-epsilon. CGMS gives the second explicit choice epsilon=2^-7. This is present in both the 2023 v1 PDF and the cited v2. - CONFIRMED | includes a cited o(k) term | https://arxiv.org/html/2407.19026v1 | Entry 23, o-k-label. Theorem 1 explicitly contains "G(\ell/k)k+o(k)". - CONFIRMED | Exact finite Erdős-Szekeres bound, indexing convention R(k,l) ≤ C(k+l-2,l-1) | https://numdam.org/item/CM_1935__2__463_0.pdf | Entry 24, finite-erdos-szekeres. Manually read rendered PDF p. 466, equation (3): "m_2(k+1,l+1)" equals binom(k+l,k). Relabel k and l by one to obtain C(k+l-2,l-1). The preceding recurrence and boundary values establish the Ramsey upper bound. - CONFIRMED | The 2023 and 2024 expressions are asymptotic. | https://arxiv.org/html/2303.09521v2 | Entry 25, asymptotic-2023. CGMS Theorem 1.1 says "for all sufficiently large" k. This is not a numerical certificate for the slider values. - CONFIRMED | The 2023 and 2024 expressions are asymptotic. | https://arxiv.org/html/2407.19026v1 | Entry 26, asymptotic-2024. GNNW Theorem 1 contains "o(k)" in the exponent. - CONFIRMED | This instrument does not turn either one into a certified upper bound at the finite k selected above. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 27, no-finite-certificate. The generated text says "scale-only ratio" and prints bases rather than a claimed finite Ramsey guarantee. The exact finite output is separately computed with BigInt. - CONFIRMED | The further result is structural: edit a finite colouring and test the exact excess-edge identity used by the later paper's candidate argument. | https://arxiv.org/html/2407.19026v1 | Entry 28, candidate-argument. GNNW Lemma 2 and its proof give the excess inequality and its square remainder. The final equality is the one implemented; the candidate is defined immediately above. - CONFIRMED | This identity is a local shadow of the method, not a new proof of its asymptotic theorem. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 29, identity-limited-demonstration. The extracted functions compute a finite identity only. Positive control: the block contains "excessIdentity" and the base formulas, not an implementation of the induction proving Theorem 1. - CONFIRMED | Each cell is one edge from X to Y. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 30, cells-as-edges. The render loop runs four rows and four columns and labels each button "Edge x" followed by its endpoints. - CONFIRMED | Click to change red to blue. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 31, toggle-colours. The handler is "matrix[rowIndex][columnIndex] = !matrix[rowIndex][columnIndex]". All sixteen click handlers changed the edge count in the expected direction. - CONFIRMED | The paper measures red edges above a chosen baseline density p. | https://arxiv.org/html/2407.19026v1 | Entry 32, paper-baseline. The paper defines the excess as the amount of red edges "when compared to density" p. - CONFIRMED | all 16 edges editable | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 33, sixteen-edges. The two render loops each use "< 4". The harness counted "EDITABLE EDGES 16" and all sixteen handlers worked. - CONFIRMED | p is an exact multiple of 1/20 in this instrument. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 34, p-grid. The slider has integer values 1 through 19; the code sets "var denominator = 20". This is the exact rational grid intended by the display; internal calculations use Number. - CONFIRMED | f_p(X,Y) = e_R - p|X||Y| | https://arxiv.org/html/2407.19026v1 | Entry 35, excess-formula. The definition in section 2 is "f_{p}(X,Y)=e_{R}(X,Y)-p|X||Y|". - CONFIRMED | Σ_v f_p(X,N_R(v)∩Y) | https://arxiv.org/html/2407.19026v1 | Entry 36, identity-left. Lemma 2 sums "f_{p}(X,N_{R}(v)\cap Y)" over v in X. - CONFIRMED | Σ_y (deg_R(y,X)-p|X|)^2 | https://arxiv.org/html/2407.19026v1 | Entry 37, identity-remainder. The final line of the Lemma 2 proof is the sum over y in Y of "(|N_R(y)∩X|-p|X|)^2" (linear transcription of the displayed formula). - CONFIRMED | left - base = remainder | https://arxiv.org/html/2407.19026v1 | Entry 38, identity-equality. The first and last lines of the Lemma 2 proof equate the excess difference to the square sum. Independently, sum_y d_y(d_y-p|X|)-p|X|(sum_y d_y-p|X||Y|)=sum_y(d_y-p|X|)^2. - CONFIRMED | Every green value below is recomputed by this page from the current controls. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 39, green-recomputed. The supplied excerpt contains the actual assignments to "check-cgms", "check-gnnw", "check-binomial", and "check-identity". The harness reproduced each output. - CONFIRMED | Published constants enter only where they are named as published inputs. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 40, published-inputs. The epsilon is labelled "Published input" in the generated check. The coefficients appear in the displayed G(1) formula and are attributed by "Source of G(λ)" in the source list. - CONFIRMED | this page writes R(k,l) for a red K_k or blue K_l, and uses the finite recurrence bound C(k+l-2,l-1). | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 41, indexing. The finite call is "binomialBigInt(k + l - 2,l - 1)". CGMS also defines R(k,l) using a red K_k or blue K_l, agreeing with the page convention. - CONFIRMED | The papers sometimes shift indices or abbreviate R(k,k) as R(k). | https://arxiv.org/html/2303.09521v2 | Entry 42, papers-indexing. CGMS explicitly writes "R(k,k)=R(k)". Sah uses "R(k+1,k+1)". - CONFIRMED | every displayed base is per k. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 43, bases-per-k. The code exponentiates H(lambda) and H(lambda)+G(lambda) as per-k bases. The exact binomial lane is kept separate. - CONFIRMED | The entropy exponent is (1+λ)log(1+λ)-λlog λ, with natural logarithms. | https://arxiv.org/html/2407.19026v1 | Entry 44, entropy-exponent. GNNW section 4 includes "(\lambda+1)\log(\lambda+1)-\lambda\log(\lambda)" in F(lambda); the exponentials are base e. This also follows from Stirling expansion of the binomial. - CONFIRMED | the exact BigInt lane is finite. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 45, finite-bigint. The binomial function starts with "var row = [1n]" and uses exact integer additions for finite n. - CONFIRMED | The base curves suppress subexponential terms. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 46, suppress-subexponential. "gnnwBase" computes only "Math.exp(entropyExponent(lambda) + gnnwCorrection(lambda))". The page separately warns about the omitted o(k) term. - CONFIRMED | The pure-base ratio is only a scale illustration. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 47, ratio-illustration. The generated output explicitly says "scale-only ratio" and the function returns "Math.pow(cgmsBase() / 4, k)". - CONFIRMED | the binomial is exact BigInt arithmetic. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 48, exact-binomial. The binomial loop uses "1n" and BigInt addition throughout; all reachable inputs matched the independently written multiplicative reference. - CONFIRMED | Exponentials, logarithms, and the plotted curve use browser double precision. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 49, double-precision. The source uses "Math.exp", "Math.log", and ordinary Number arithmetic for the bases and plot, alongside explicit BigInt literals for the binomial. - CONFIRMED | λ is a reader choice. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 50, lambda-choice. The source has a range input "id="lambda-control"" with values from 0.25 to 1 and an input listener. - CONFIRMED | The graph partition has four vertices in X and four in Y. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 52, partition-sizes. The source declares "Array.from({ length:4 }" with "new Array(4).fill(false)" for the two parts. - CONFIRMED | Its colouring and p are reader choices. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 53, colour-p-choices. The matrix click handler toggles each Boolean; "pControl.addEventListener" recomputes the identity for the selected density. - CONFIRMED | The peer-reviewed Campos, Griffiths, Morris and Sahasrabudhe theorem says sufficiently large k but supplies no usable numerical threshold here. | https://arxiv.org/html/2303.09521v2 | Entry 54, cgms-threshold. Theorem 1.1 says "for all sufficiently large" k, with no numerical threshold in that statement. The theorem phrase is a positive control for locating the relevant passage; no claim of an exhaustive absence from all mathematics is made. - CONFIRMED | Sah's peer-reviewed pre-2023 bound uses an absolute c greater than zero. | https://arxiv.org/html/2005.09251v1; https://api.crossref.org/works/10.1215%2F00127094-2022-0048 | Entry 55, sah-constant. Sah Theorem 1.1 says "There is an absolute constant" c>0. The manuscript header is "19 May 2020"; Crossref supplies its later 2023 Duke publication. - CONFIRMED | This page does not assign it a numerical value. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 56, no-sah-value. The page mentions "an absolute c greater than zero"; the shipped math contains only the named CGMS and GNNW functions. No numerical Sah curve is implemented or displayed. - CONFIRMED | The Gupta, Ndiaye, Norin and Wei formula contains o(k). | https://arxiv.org/html/2407.19026v1 | Entry 57, gnnw-error-term. The supplied Theorem 1 excerpt explicitly includes "o(k)". - CONFIRMED | Its size is not supplied by the asymptotic statement. | https://arxiv.org/html/2407.19026v1 | Entry 58, gnnw-error-size. Theorem 1 gives an asymptotic "o(k)" without a numeric error bound. The later v2 explains the uniform little-o meaning but retains no finite numerical value in the statement. - CONFIRMED | The 2024 optimisation remains a preprint. | https://www.math.mcgill.ca/snorin/papers.html | Entry 59, gnnw-preprint-status. Norin lists "Optimizing the CGMS upper bound on Ramsey numbers" and links to arXiv without a journal venue. His page is titled "Publications and Preprints". This supports treating the item as a preprint, not asserting a known peer-review history. - CONFIRMED | an author publication list gives no journal venue. | https://www.math.mcgill.ca/snorin/papers.html | Entry 62, author-venue-list. The complete list item ends with "(with P. Gupta, N. Ndiame and L. Wei)". Positive control from the same list: "Proper minor-closed families are small" includes "J. Comb. Theory B 96 (2006), 754-757." - CONFIRMED | The exponential growth of diagonal Ramsey numbers is not determined. | https://arxiv.org/html/2303.09521v2; https://arxiv.org/html/2407.19026v2 | Entry 63, growth-undetermined. CGMS reports "still a large gap"; its introduction compares a lower exponential base sqrt(2) with upper bases near 4. GNNW v2 retains the 3.7992 upper value and does not close this gap. - CONFIRMED | The cited lower and upper bounds leave an exponential gap. | https://arxiv.org/html/2303.09521v2 | Entry 64, exponential-gap. The introduction gives "R(k)" at least 2^(k/2), while the upper result has a fixed base greater than sqrt(2). The cited bounds thus leave an exponential gap. - CONFIRMED | Neither the Annals theorem nor the later preprint determines exact large Ramsey numbers. | https://arxiv.org/html/2303.09521v2; https://arxiv.org/html/2407.19026v1 | Entry 65, no-large-exact-values. CGMS Theorem 1.1 and GNNW Theorem 1 are inequalities, not exact values. Both apply asymptotically to unbounded k. - CONFIRMED | The dependency-free verifier extracts these shipped functions, sweeps every reachable k and l, every λ step, and every one of the 65,536 matrix colourings at every p step against separately written references. | git show a77c45dd68:research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs | Entry 66, verifier-sweeps. Read the complete verifier and ran the pinned copy: "PASS every reachable edge colouring and p step matches independent reference: 1,245,184 states" and "76 slider positions". It checks all k=4..120 and l=2..k, a superset of reachable rounded pairs. 65,536*19=1,245,184. - CONFIRMED | It also mutates the extracted ε calculation in memory and requires that the tests catch the fault. | git show a77c45dd68:research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs | Entry 67, verifier-mutation. The source replaces "return 1 / (1 << 7);" with "return 1 / 127;" and asserts the mutation is caught. The recreated run prints "PASS mutation test catches a perturbed shipped epsilon". - CONFIRMED | Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe, An exponential improvement for diagonal Ramsey , Annals of Mathematics 203(3), 869-932, May 2026. DOI 10.4007/annals.2026.203.3.4. | https://annals.math.princeton.edu/2026/203-3/p04; https://annals.math.princeton.edu/2026/203-3 | Entry 68, citation-cgms. The Annals article names all four authors and gives "Pages 869-932 from Volume 203 (2026), Issue 3" and the stated DOI. The separate issue page supplies May. - CONFIRMED | Annals of Mathematics 203(3), 869-932, May 2026. | https://annals.math.princeton.edu/2026/203-3 | Entry 69, annals-may. The issue page heading is "Volume 203, Issue 3, May 2026". - CONFIRMED | The journal records acceptance on 30 June 2025 and online publication on 1 December 2025. | https://annals.math.princeton.edu/2026/203-3/p04 | Entry 70, annals-milestones. The primary milestones are "Accepted: 30 June 2025" and "Published online: 1 December 2025". - CONFIRMED | The same result at arXiv:2303.09521v2 , first submitted 16 March 2023 and revised 4 August 2025. | https://arxiv.org/abs/2303.09521 | Entry 71, citation-cgms-manuscript. The arXiv history has "[v1] Thu, 16 Mar 2023" and "[v2] Mon, 4 Aug 2025". The manuscript and journal have the same title and theorem. - CONFIRMED | The manuscript supplies the explicit unoptimised choices 2^-10 and 2^-7. | https://arxiv.org/html/2303.09521v2 | Entry 72, unoptimised-epsilons. The manuscript explicitly gives epsilon=2^-10 for the first proof and epsilon=2^-7 for the second. The original v1 PDF corroborates both choices. - CONFIRMED | The second is used here. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 73, chosen-epsilon. The function "cgmsEpsilon" returns "1 / (1 << 7)". - CONFIRMED | Paul Erdős and George Szekeres, A combinatorial problem in geometry , Compositio Mathematica 2, 463-470, 1935. NUMDAM identifier CM_1935__2__463_0. | https://numdam.org/item/CM_1935__2__463_0/ | Entry 74, citation-erdos. NUMDAM gives "Erdös, P. ; Szekeres, G.", the exact title, "Tome 2 (1935), pp. 463-470", and identifier CM_1935__2__463_0. - CONFIRMED | Source of the classical finite recurrence bound. | https://numdam.org/item/CM_1935__2__463_0.pdf | Entry 75, classical-source. Manually read PDF p. 466, equations (1)-(3), including "By this recurrence-formula and the initial values". Equation (3) gives the binomial after shifting both indices by one. - CONFIRMED | Ashwin Sah, Diagonal Ramsey via effective quasirandomness , Duke Mathematical Journal 172(3), 545-567, 2023. DOI 10.1215/00127094-2022-0048. | https://api.crossref.org/works/10.1215%2F00127094-2022-0048 | Entry 76, citation-sah. Crossref returns "Ashwin" "Sah", the exact title, "Duke Mathematical Journal", volume 172, issue 3, pages "545-567", publication date [2023,2,15], and the stated DOI. - CONFIRMED | The theorem gives a subexponential improvement while retaining limiting base 4. | https://arxiv.org/html/2005.09251v1 | Entry 77, sah-subexponential. Sah Theorem 1.1 has exp(-c(log k)^2) times C(2k,k). Since (log k)^2/k tends to zero, its limiting exponential base remains 4. - CONFIRMED | Parth Gupta, Ndiamé Ndiaye, Sergey Norin, and Louis Wei, Optimizing the CGMS upper bound on Ramsey numbers , arXiv:2407.19026v1, 26 July 2024. | https://arxiv.org/abs/2407.19026 | Entry 78, citation-gnnw. The arXiv page names Gupta, Ndiaye, Norin and Wei and records "[v1] Fri, 26 Jul 2024". Its current default is v2; the citation explicitly names the still-accessible v1. - CONFIRMED | Source of G(λ), the 3.7992 diagonal value, and the excess identity used in layer two. | https://arxiv.org/html/2407.19026v1; https://arxiv.org/html/2407.19026v2 | Entry 79, gnnw-source-components. Read Theorem 1, its 3.7992 consequence, and Lemma 2 at the versioned v1 URL. V2 preserves these results while correcting the numerical justification of the earlier proof. - CONFIRMED | Lawrence C. Paulson, Formalising New Mathematics in Isabelle: Diagonal Ramsey , 16th International Conference on Interactive Theorem Proving, LIPIcs 352, 18:1-18:18, 2025. DOI 10.4230/LIPIcs.ITP.2025.18. | https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.18; https://api.datacite.org/dois/10.4230/LIPIcs.ITP.2025.18 | Entry 80, citation-paulson. DROPS "Cite As" matches the author, title, conference, volume and page range. Crossref HTTP 404 is a registry mismatch: DataCite returns the DOI, author, title, publicationYear 2025, and container "LIPIcs, Volume 352, ITP 2025". - CONFIRMED | This independently documents a formalisation of the 2023 result. | https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.18 | Entry 81, paulson-formalisation. The abstract says "This paper reports the formalisation" of a major new Ramsey result, citing "Marcelo Campos et al., 2023"; its title identifies Isabelle. - CONFIRMED | The graph lab verifies a published finite identity on the colouring you made. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 82, footer-identity. The source computes "difference:left - baseline" and independently computes the remainder. The pinned verifier checked all 1,245,184 reachable matrix/density states against a separate reference. - CONFIRMED | It does not reconstruct the book algorithm or certify the asymptotic bound by itself. | git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html | Entry 83, footer-scope. The supplied math block implements the finite identity and formula evaluation only. Positive control: it contains "excessIdentity"; the full GNNW induction is in the external paper, not these eight math functions. - CONFIRMED | This layer's check reads the bytes this page serves and asserts against them. | git show a77c45dd68:research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs | Entry 104, placard-reads. The verifier reads "public/strata/the-four-that-finally-moved/index.html" using "fs.readFileSync" and extracts the inline math. Curl confirms these math bytes are also live. - CONFIRMED | It is reading what you are reading. | git show a77c45dd68:research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs | Entry 106, placard-reading. The verifier extracts the math between "BEGIN SHIPPED MATH" and "END SHIPPED MATH" from the same HTML served live. It tests this code, not every sentence on the page. - CONFIRMED | That layer makes exact small Ramsey cases operable. This one begins where exact enumeration stops and keeps finite calculations separate from the asymptotic change in the exponential upper-bound base. | git show a77c45dd68:src/content/strata/complete-disorder-is-impossible.md | Entry 109, related-complete-disorder-is-impossible. The related layer says "all 32,768 two-colourings of six people, none escaping" and describes its interactive small Ramsey cases. This matches the relationship description. - CONFIRMED | The portal treats Ramsey numbers as a finite but formidable resource wall. This layer opens one precise movement in that wall and shows why an asymptotic base change matters even though exact large values remain unknown. | git show a77c45dd68:src/content/strata/the-limits-of-knowing.md | Entry 110, related-the-limits-of-knowing. The related layer explicitly says "resource, computability, proof" and describes Ramsey as bounded and limited by the search size. - CONFIRMED | Both layers turn entropy from a slogan into a computed exponent. Here the binomial entropy expression supplies the classical off-diagonal Ramsey base before the newer correction is applied. | git show a77c45dd68:src/content/strata/entropy-not-disorder.md | Entry 111, related-entropy-not-disorder. The related layer displays "S/k = log C(N,n)" and exact microstate counts. This page computes H(lambda), the logarithmic binomial exponent. The relationship is supported. - CONFIRMED | peer-reviewed journal article | https://annals.math.princeton.edu/2026/203-3/p04 | Entry 112, cgms-journal-status. The primary journal article has a recorded acceptance date and final Annals volume, issue, pages, and DOI. The excerpt is journal metadata, not an explicit description of the referee process. - CONFIRMED | peer-reviewed journal article | https://numdam.org/item/CM_1935__2__463_0/; https://www.cambridge.org/core/journals/compositio-mathematica | Entry 113, erdos-journal-status. NUMDAM identifies the research article in Compositio. The publisher describes its contributions as "carefully screened byexperts in the field" and links pre-1997 papers to NUMDAM. - CONFIRMED | peer-reviewed journal article | https://api.crossref.org/works/10.1215%2F00127094-2022-0048; https://www.dukeupress.edu/duke-mathematical-journal | Entry 114, sah-journal-status. Crossref identifies a "journal-article" in Duke Mathematical Journal. The publisher instructs authors to use its "manuscript peer-review system". - CONFIRMED | author manuscript | https://arxiv.org/abs/2303.09521 | Entry 115, manuscript-status. The arXiv entry has author names, a submission/revision history, and the same research title and result as the journal publication. - CONFIRMED | peer-reviewed proceedings paper | https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.18; https://www.dagstuhl.de/en/publishing/series/details/lipics | Entry 117, paulson-proceedings-status. DROPS identifies the conference paper. The LIPIcs publisher explicitly says "LIPIcs volumes are peer-reviewed". - CONFIRMED | "datePublished":"2026-07-29" | git show a77c45dd68:src/content/strata/the-four-that-finally-moved.md | Entry 118, date-published-metadata. The pinned frontmatter says "date: 2026-07-29". Git history also contains this layer on 29 July 2026, consistent with the declared publication date. - CONFIRMED | (added) Choose a red clique size k and a ratio λ = l/k. | https://arxiv.org/html/2407.19026v1; git show a77c45dd68:public/strata/the-four-that-finally-moved/index.html (line 475) | Added A1. GNNW Theorem 1 explicitly uses "G(\ell/k)"; the page distinguishes its continuous chosen ratio from the rounded finite pair. The exact ratio definition was omitted from the record. The claim search found the existing finite-bound entries as its positive control. ## Findings - [fixed] X1 (MINOR, finite-lower-clamp): the lambda control note now says the rounded l is limited to 2 through k. Recorded here without a separate MINOR correction line. - [fixed] X2 (WRONG, finite-lower-clamp): the controls explanation now states rounding followed by the range 2 through k, and the live label says finite l. The calculation is unchanged. The HTML Corrections note and markdown body carry one Corrected 2026-10-18 line naming the original claim, the actual rule, the k=4 and lambda=0.25 example, and the assay line. - [dated] X3 (STALE, arxiv-version-history): preserved the observation as checked on 29 July 2026 and grouped all STALE findings (X3 alone) into one Updated 2026-10-18 line, mirrored in markdown, linking the submission history for v2 on 29 August 2026 and naming the assay line. - [fixed] X4 (MINOR, unsupported-review-status): narrowed the source badge and README status from preprint, not peer-reviewed to preprint. The author list supplies no journal venue and cannot establish absence of peer review. No dated correction line. - [fixed] X5 (MINOR, verifier-reader-access): added an optional HTML-path argument to the source verifier, copied it verbatim into public/strata/the-four-that-finally-moved/, and added page/verifier downloads plus explicit curl and Node commands. The README documents the run command and how to refresh the copy. No dated correction line. ## Verification `node research/the-four-that-finally-moved/verify-the-four-that-finally-moved.mjs` exited 0 with all 14 assertions passing, including 76 lambda positions and 1,245,184 identity states. The public verifier and page were then downloaded from a temporary local static server into an initially empty directory; `node verify-the-four-that-finally-moved.mjs index.html` also exited 0 with all 14 assertions passing. The downloaded verifier is byte-identical to the source. This tests the proposed reader route locally, before landing. `node scripts/check-claims-pass.mjs` exited 1 solely for the other layer chaitin-omega, whose third finding had an invalid disposition line at the time of the run. That record was not edited. Calling the gate's exported `checkRecord` on this layer returned no errors: checked 96, confirmed 91, wrong 4, stale 1, open 0. The required triage validation printed `5 items 0 incomplete` before the first layer edit and again after the fixes. The shipped mathematical block and generated check placard match cd6ebcd061 byte for byte. The HTML and markdown contain identical correction/update text, one line of each kind, and the HTML notes precede the placard. The JSON-LD revision date is 2026-10-18. Scoped `git diff --check` passed, and the inline JavaScript parses.