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"It Must Appear" Does Not Tell You When ── Checking R(5,5) by Exhaustion Would Take 10^271.8 Colourings, 191.8 Orders More Than There Are Atoms in the Universe ── Existence Can Be Proved; Size Is Not Given ── [Paper 302]

Yuuki Yamagishi
Aug 2026 · Zenodo (CERN European Organization for Nuclear Research)
Computability, Logic, AI Algorithms

Abstract

Ramsey’s theorem says that if it is large enough, order must appear. This paper asks how large “large enough” is──the answer is the theorem does not say. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): No new mathematical theorem and no new law is claimed──Ramsey’s theorem, the known values and ranges of Ramsey numbers, van der Waerden numbers, and Erdos’s probabilistic lower bound are all standard. We do not build combinatorics──all we use is one binomial coefficient and two logarithms. We do not prove Ramsey’s theorem──we merely quote it. We do not compute Ramsey numbers──no attempt is made to find R(5,5). We count only the effort it would take. We assert no values for the ranges──R(5,5) in [43,48] and R(6,6) in [102,160] are ranges known at one epoch and may be improved. The claim is that the range is not empty, not the endpoints. We do not say exhaustion is the only route──actual searches cut enormously by symmetry and pruning. 2^903 is a naive upper bound, not the necessary work. The point is the order of magnitude by which it remains out of reach. We do not use “Ackermann type” strictly──it indicates that the pre-Gowers bound was of a tower-of-growing-height kind. This paper asserts no classification of the bound, only its divergence from the true value. Relation to earlier papers: Paper 163 separated “a good code exists” from “here is a good code”──that is existence against construction; this is existence against quantity. It treats the case where one can construct yet cannot reach, so the cut differs. Paper 179 showed that “cannot be constructed” has distinct roots──this paper treats what can be constructed yet is out of reach. Paper 190 measured “rare” on a logarithmic scale──the 271.8 orders here are read logarithmically too. Paper 259 showed there are two roads to independence──this paper likewise asks what kinds of showing there are. What is added is computing the exhaustion for R(5,5) as 10^271.8 and measuring the gap to the atoms of the universe as 191.8 orders, lining up the jumps at R(3,3), R(4,4), R(5,5), computing that the probabilistic lower bound is one 24.94th at k=10, and placing the separator at existence against quantity. First, only two values are settled. R(3,3)=6 and R(4,4)=18; R(5,5) is pinned only to the range [43,48] (Section 2). Second, this is the core of the paper. Checking R(5,5)=43 by exhaustion needs 2^903=10^271.8 colourings, 191.8 orders more than the atoms in the universe (Section 3). Third, each step up jumps. R(3,3) is 32768 colourings, within reach by hand; R(4,4) is 1.14x10^46, which computers barely reached (Section 3). Fourth, the lower-bound proof does not give the value either. Erdos’s probabilistic method gives R(k,k)>2^k/2, which at k=10 is one 24.94th of the truth (Section 4). Fifth, the upper bound is further off still. W(3,3)=27, yet the bound from the classical proof was of Ackermann type (Section 5). Sixth, the separator is existence against quantity. A different cut from Paper 163’s existence against construction (Section 6). Ramsey’s theorem says that if it is large enough, order must appear. But it does not say how large. Only R(3,3)=6 and R(4,4)=18 are settled, and R(5,5) is pinned only to the range [43,48]──even settling R(4,4) took 65 years from the theorem. Counting the work of exhaustion shows why──R(5,5)=43 has 903 edges and 2^903=10^271.8 colourings, 191.8 orders more than the atoms in the observable universe. What is lacking is not the speed of computers but the quantity of matter. The lower bound is no better──the probabilistic method gives R(k,k)>2^k/2, one 24.94th of the truth at k=10, and the ratio widens with k. The upper bound is further off still──W(3,3) is truly 27, yet the classical bound could not be written down. A theorem’s upper bound measures the power of the proof’s tools, not the size of the object. One thing separates them──whether it exists, how large it is, and which one it is, are three different questions. The first may be “yes” while the other two stay open. The single word “proved” points at only one of the three. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- ラムゼーの定理は「十分大きければ必ず秩序が現れる」と言う。本稿が問うのは、「十分大きい」がどれだけかである──答は、定理は教えないである。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない──ラムゼーの定理、ラムゼー数の既知の値と範囲、ファン・デル・ヴェルデン数、エルデシュの確率論的下限は、いずれも標準的である。組合せ論を作らない──使うのは一つの二項係数と、二つの対数だけである。ラムゼーの定理を証明しない──引くだけである。ラムゼー数を計算しない──R(5,5) の値を求めようとはしていない。求めるのに要る手間だけを数える。範囲の値を主張しない──R(5,5) in [43,48]、R(6,6) in [102,160] はある時点で知られている範囲であり、改善されうる。本稿の主張は範囲が空でないことであって、端の値ではない。総当たりが唯一の道だと言わない──実際の探索は対称性と枝刈りで大幅に減らす。2^903 は素朴な上界であって、必要な計算量ではない。それでも到底届かない、という桁の話である。アッカーマン級という語を厳密に使わない──ガワーズ以前の上界が塔の高さが増える型だったことを指す。本稿は上界の形を主張せず、真の値との乖離だけを言う。既刊との関係:論文163 は「良い符号が在る」と「これが良い符号だ」を分けた──あちらは存在と構成、本稿は存在と定量である。構成できても大きさが分からない場合を扱うので、切り口が違う。論文179 は「構成できない」に別根があると示した──本稿は構成できるのに手が届かない場合である。論文190 は「稀」を対数の目盛りで測った──本稿の 271.8 桁も対数で読む。論文259 は独立性を示す道が二つあると示した──本稿も示し方の種類を問う。加えたのはR(5,5) の総当たりを 10^271.8 通りと計算し、宇宙の原子との差を 191.8 桁と測ったこと、R(3,3)・R(4,4)・R(5,5) で手間が跳ぶ段を並べたこと、確率論的下限が k=10 で 24.94 分の一だと計算したこと、分離子を「存在と定量」に置いたことである。 第一に、確定している値は二つだけである。 R(3,3)=6 と R(4,4)=18 で、R(5,5) は[43,48] の範囲にしか収まっていない(第2節)。 第二に、これが本稿の芯である。 R(5,5)=43 を総当たりで確かめるには 2^903=10^271.8 通りが要り、宇宙の原子より 191.8 桁多い(第3節)。 第三に、一段上がるごとに跳ぶ。 R(3,3) は 32768 通りで手が届き、R(4,4) は 1.14x10^46 通りで計算機がようやく届いた(第3節)。 第四に、下限の証明も値を教えない。エルデシュの確率論的方法は R(k,k)>2^k/2 を与えるが、k=10 で実際の 24.94 分の一である(第4節)。 第五に、上界はもっと外れる。 W(3,3)=27 なのに、古典的証明が与えた上界はアッカーマン級だった(第5節)。 第六に、分離子は「存在と定量」である。論文163 の「存在と構成」とは別の切り口である(第6節)。 ラムゼーの定理は「十分大きければ必ず秩序が現れる」と言う。だが「十分大きい」を教えない。確定している値は R(3,3)=6 と R(4,4)=18 の二つだけで、R(5,5) は[43,48] の範囲にしか収まっていない──R(4,4) の確定でさえ、定理から 65 年かかった。総当たりの手間を数えれば理由が分かる──R(5,5)=43 は辺が 903 本、塗り分けが 2^903=10^271.8 通りで、観測可能な宇宙の原子より 191.8 桁多い。足りないのは計算機の速さではなく、物質の量である。下限の側も同じである──確率論的方法は R(k,k)>2^k/2 を与えるが、k=10 で実際の 24.94 分の一であり、比は k とともに開いていく。上界はもっと外れる──W(3,3) の真の値は 27 なのに、古典的証明の上界は書き下せない大きさだった。定理の上界は、証明の道具の性能を測っているのであって、対象の大きさを測っていない。分けるものは一つ──存在するかと、いくつかと、どれかは、三つの別の問いである。一つ目が「はい」でも、残り二つは開いたままでありうる。「証明された」という一語が、三つのうち一つしか指していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。

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The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA William J. Dwyer, MD, MPH, FAAP — Department of Mathematics and Statistics, University of Massachusetts Lowell. ORCID 0009-0004-0855-7222. Concept DOI (always resolves to the latest version): 10.5281/zenodo.21908169. What this is The reproducibility deposit for the deflated-Welch statistic T_BB, a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA (the Behrens–Fisher problem for k ≥ 3 groups). Welch's test becomes liberal under skew and unstable variance weights at small samples; T_BB = Q(s²)·exp(−R) keeps the ordinary group means and buys a guaranteed level by deflating the Welch quadratic by a Berger–Boos scale-inflation radius R. Three operating points are provided: a fixedcalibrated radius (κ_s), a design-adaptive near-guarantee radius (closed-form polygamma Cornish–Fisher with a finite-nkurtosis guard), and a fully proved smallest-eigenvalue radius R_eig (Gaussian, extended under bounded kurtosis). What the deposit contains Manuscript (author + anonymized) and a derivations supplement (DA1–DA13) plus a long-form derivations companion, covering: why Welch fails under skew in closed form; the Berger–Boos deflation and its exact worst-case radius; the polygamma-cumulant Cornish–Fisher radius with saddlepoint-exact normal backbone; the excess-kurtosis tail term with its finite-n upper-confidence guard; the imbalance correction; the fully proved smallest-eigenvalue radius (with the k-group multiplicity fix, free-β optimization, and the proved-under-bounded-kurtosis widening); and the k-sample Behrens–Fisher null distribution. Interactive demonstrator rerun_cochran/honest_anova.html — computes raw-mean Welch, the fixed / adaptive / proved T_BB radii, the estimand-changing transform routes, and the full routing receipt in the browser, reproducing the deposited Python. Its engine is extracted as a standalone Node module (m01A_anova_engine.js) and checked cell-by-cell against Python across an 84-design taxonomy (verify_anova_engine_taxonomy.py/.js, max |Δp| = 0.00000). Reproducibility scripts (rerun_cochran/, rerun/) — every reported number traces to a named, deterministically-seeded script (size/power/surface, the calibration and information-limit decompositions, the proved-radius verification, the imbalance calibration, the skew-router branch, and the figures). Real-data evidence — anova_flip_scan.py scans 2,783 public one-way layouts (254 datasets): guaranteed T_BBwithholds ~41% of Welch-significant calls, concentrated where the weight-instability screen fires, and never manufactures significance (Table 7 / Figure 15). Figures and the deterministic deposit builder (fixed timestamps → stable md5). All evaluation is simulation-based; the one empirical component is the public-dataset scan, which uses only openly distributed data. Code is released under the MIT License; text and figures under CC BY 4.0. Version history (consolidated changelog) Published version DOIs are marked ✅; the concept DOI above always resolves to the latest. Staged versions were rolled into the next published one unless noted. v1.0.77 ✅ 10.5281/zenodo.22167690 (2026-08-30): CSDA guide-for-authors conformance — abstract trimmed to 247 words (from 284), keywords cut to 7 (from 11), the withholding highlight shortened to ≤85 characters, and the arXiv PDF/source regenerated. No change to methods, results, figures, or code. v1.0.76 ✅ 10.5281/zenodo.22167536 (2026-08-30) — AI-disclosure heading aligned to Elsevier. The manuscript's declaration heading is now "Declaration of generative AI and AI-assisted technologies in the manuscript preparation process" (was "Use of generative AI"); the disclosure body is unchanged. Prepared alongside an Elsevier-compliant cover-letter variant and an EM suggested-reviewer sheet (both kept outside the deposit). docx/pdf rebuilt; deterministic md5 refreshed. v1.0.75 ✅ 10.5281/zenodo.22167304 (2026-08-30) — Submission-sharpening pass. Graphical abstract + Elsevier Highlights; figures and tables renumbered into reading order with per-table Source clauses; the validity–power frontier (Figure 8) now carries the proved R_eig operating point (100% validity, size-adjusted power 0.613, merge_tbb_proved_frontier.py); new Section 7 "Recovering power by design" + Table 8 (rc_anova_power_by_design.py); and a live required-n calculator in honest_anova.html (per-group and total n for 80% power, "power now @ total n"), with a numeric-heading CSS fix and the engine re-verified against Python at 0.00000. v1.0.74 ✅ 10.5281/zenodo.22165892 (2026-08-29) — Proved-under-bounded-kurtosis radius (DA12.6). The proved non-normal widening now keys on excess kurtosis, √(1 + κ̂·(n−1)/(2n)), from the exact Var(s²/σ²) = 2/(n−1) + κ/n, so symmetric heavy tails (Student-t) are covered where the old skew form √(1 + 0.75·skew²) under-covered; tbbProvedswitched to the kurtosis form across the demonstrator, engine, and Python truth (re-verified JS-vs-Python at 0.00000); new rc_anova_kurtosis_proof.py + deep-dive. v1.0.73 ✅ 10.5281/zenodo.22165709 (2026-08-29) — Reconstructed & verified demonstrator engine (standalone Node module + taxonomy verifier, max |Δp| = 0.00000 across 84 designs; Yuen zero-variance fix; T_BB-routed presets both directions); series-impact deep-dive (the corrected R_eig k-group multiplicity gap also reaches m03 and m01t). v1.0.72 (2026-08-29) — Title set to "The deflated-Welch statistic…"; corrected + optimized proved radius R_eig (β/k multiplicity fix + β-optimization, DA12); real-data Welch-vs-T_BB flip scan (2,783 layouts; Table 7 / Figure 15) + demonstrator imbalance-factor fix; long-form derivations companion. v1.0.71 / v1.0.70 (2026-08-21) — Zhang normal-reference comparator benchmarked on the efficiency frontier (valid on only 24% of designs, in the calibrated-liberal cluster); k = 2 adaptive-radius case-study fold (design-scaling vs shape-keying distinction). v1.0.69 ✅ 10.5281/zenodo.22035826 (2026-08-20) — HTML R1/R2 presentation pass + Figure 9 adaptive per-cluster label merge. v1.0.68 ✅ 10.5281/zenodo.22033737 (2026-08-20) — Companion consolidation into a single six-column Table 6; Figures 11–14 harmonized into one story. v1.0.67 / v1.0.65 / v1.0.60 (2026-08-19/20) — Guarded-reference naming-collision fix; the 40,000-replication expanded-frontier pin (Table 3 + Figure 8) with the symmetric-heteroscedastic skew-router branch; the mean-preserving lightened-R_eig do-not-use fallback. v1.0.59 ✅ 10.5281/zenodo.21995320 (2026-08-18) — Reporting standard + honest_anova.html demonstrator re-aligned to the current T_BB methods paper. v1.0.57 ✅ 10.5281/zenodo.21986847 (2026-08-17) — Reviewer-comprehension pass (multi-paragraph abstract, contributions list, trimmed captions); proved radius R_eig added as a Table 3 scorecard row; corner tail-index correction (N−k)/2 (low-order moments exist in every deployed design). v1.0.56–v1.0.49 (2026-08-16) — The k-sample Behrens–Fisher corner-distribution program: two-moment scaled-χ² corner reference, derived corner cumulants, the secular-eigenvalue law + closed CGF + power-law tail, consolidated into derivations DA13 with a prior-art/novelty audit. v1.0.48 ✅ 10.5281/zenodo.21963458 (2026-08-16) — The unifying λ(z) correction (a smooth instability-keyed deflation strength). v1.0.45 ✅ 10.5281/zenodo.21962965 (2026-08-16) — Atomic sparsity index + bootstrap-t edge hardening + shape-aware pooled standardized-residual bootstrap (SA-PSRB); multivariate transfer to m03. v1.0.44–v1.0.41 (2026-08-16) — Shape-moment re-injection order (skew is the sweet spot), validated and hardened pooled standardized-residual bootstrap, atomic weight-noise probes. v1.0.40 ✅ 10.5281/zenodo.21961667 (2026-08-16) — Log-domain weight-stabilization probe (negative for stabilization; clarifies the size-adjusted oracle ceiling); includes the oracle-power gap decomposition (≈92% conservatism, ≈8% estimation). v1.0.37 ✅ 10.5281/zenodo.21961327 (2026-08-16) — Residual-bootstrap qualification of the shoot-out + the first proved Gaussian smallest-eigenvalue radius R_eig (DA12, the p = 1 specialization of the m03 theorem). v1.0.36 (2026-08-15) — Figure 11 T_BB-region colour fix (amber, matching the routing figures). v1.0.27 ✅ 10.5281/zenodo.21908170 — Earlier published baseline of the deposit. Provenance: every number traces to a named, deterministically-seeded script listed in the manuscript Declarations; the demonstrator engine reproduces the deposited Python to max |Δp| = 0.00000 across the taxonomy verification. License. Code and scripts in the deposit are released under the MIT License; text and figures under CC BY 4.0. Reuse is permitted with attribution to the author and citation of the concept DOI above. How to cite. Dwyer, W. J. The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA. Reproducibility deposit, Zenodo. https://doi.org/10.5281/zenodo.21908169

William Dwyer · 2 citations

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