--- slug: khinchin 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 and a Claude agent; repair review by Codex, 2026-10-14 checked: 209 confirmed: 128 wrong: 81 unverifiable: 0 observed: 0 stale: 0 --- This was the directed Khinchin repair pass for queue-1014c, read at f5e5f84f34. The fixer re-read the evidence for all 82 triage items, corrected Khinchin's own text for 56 WRONG and 25 MINOR findings, and rejected X45 with evidence. The claims below preserve the pre-fix statements and each triage ID, followed by the checkers' remaining CONFIRMED readings with their original qualifications and provenance. Confirmations superseded by triage are omitted. MINOR findings count in wrong here and have no page correction line. Eighteen dated correction lines cover the distinct WRONG errors and repeated assertions within Khinchin; no STALE item was directed. The repair review confirmed three unannotated preserved relations in There, Here: their ten finding entries (eight WRONG and two MINOR, retaining triage levels) remain open with note:PENDING; the other 71 findings are fixed. This record does not resolve the checkers' separate untriaged concerns. ## Claims - WRONG | X1: "The exceptional set also contains everything you've ever heard of." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Wieting pp. 1 and 7 proves the explicit Z2 construction; Bailey p. 9 separates numerical evidence for pi from a limit proof. Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - MINOR | X2: "The sequence of integers it produces is full of structure for the constants we know \u2014 and full of randomness for the constants we don't." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; public/strata/khinchin/index.html | The theorem concerns measure, not familiarity; Z2 is deliberately structured and has the required mean. Replaced the known/unknown dichotomy with the particular patterns of phi, sqrt(2) and e. - WRONG | X3: "The numbers you know don't." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Wieting pp. 1 and 7 proves the explicit Z2 construction; Bailey p. 9 separates numerical evidence for pi from a limit proof. Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - WRONG | X4: "The page computes its CF exactly (rational arithmetic to as many terms as the input affords) and plots the running geometric mean of the coefficients. The dashed line is K0." | public/strata/khinchin/index.html; research/khinchin/verify-assay.mjs | Reproduced spurious terms for 22/7 and 355/113, and rounding of coefficient 9007199254740993. Exact Euclidean division gives [3;7], [3;7,16], and [0;9007199254740993]. Added direct BigInt rational CF extraction, retained exact integer coefficients through display and convergents, and tested the actual input handler. Zero denominators are rejected. - MINOR | X5: "The most common CF digit is 1 \u2014 over forty percent of all reals have it." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; research/khinchin/verify.mjs | Gauss density gives log2(4/3); uniform first-digit probability is the interval length 1/2. Specified limiting coefficient frequency in the page and Gauss measure in the verifier; the verifier also states the uniform probability 1/2. - WRONG | X6: "The convergence to K0 is real but slow (Khinchin–Lévy: the variance is infinite, so no √n rate)." | https://arxiv.org/pdf/1903.00255 | Equation (3) defines X_n=log(a_n); Lemma 3.2 gives finite moments, including the second. Removed the infinite-variance explanation and copied logarithmic-rate assertions without claiming a universal replacement rate. - MINOR | X7: "The famous numbers we have names for nearly all carry structure in their continued fractions, and structure breaks Khinchin's average in named ways. Three failure modes, three witnesses:" | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | The displayed witnesses are valid, but no population defines nearly all famous constants, and structured Z2 reaches K0. Named the three specific failures and distinguished the unresolved fourth card. - WRONG | X8: "The CF is the maximum-entropy answer to the question \"what can I say without saying anything\"; φ's answer is \"1\"." | https://www.cs.yale.edu/homes/yry/readings/general/shannon1948.pdf | Section 6 gives entropy zero for a distribution concentrated on one outcome. Phi has only digit 1. Replaced maximum entropy with zero entropy while preserving the quoted question and answer. - WRONG | X9: "Every quadratic irrational has an eventually periodic CF (Lagrange, 1770), so its geometric mean is always a specific algebraic number \u2014 never K0." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; research/khinchin/verify-assay.mjs | Bailey p. 9 gives an algebraic limiting mean and a conditional possible equality with K0; it does not prove universal inequality. The original sqrt(4) probe returned [2] and no fractional mean. Retained periodicity and algebraicity, removed universal exclusion in body and verifier, and made terminating square inputs take the rational verdict. Only the established sqrt(2), sqrt(3), sqrt(7) cases say they miss K0. - WRONG | X10: "Euler discovered the pattern in 1737: a3k = 2k, the others 1." | https://oeis.org/A003417/internal; https://scholarlycommons.pacific.edu/euler-works/71/ | OEIS uses a(0)=2 and a(3k-1)=2k; Euler Archive gives written 1737 and published 1744. Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - WRONG | X11: "a3k = 2k, the others 1." | https://oeis.org/A003417/internal; https://scholarlycommons.pacific.edu/euler-works/71/ | OEIS uses a(0)=2 and a(3k-1)=2k; Euler Archive gives written 1737 and published 1744. Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - WRONG | X12: "So every third term grows linearly, and the geometric mean of n terms grows like (n/3·e-1)1/3 \u2014 slowly, but unboundedly." | https://oeis.org/A003417/internal; research/khinchin/verify-assay.mjs | Blocks (1,2k,1) give product 2^m*m! at n=3m; Stirling gives (2n/(3e))^(1/3). Recomputed GM_300=4.2355519296. Restored the missing factor 2 and specified fractional coefficients; divergence is retained. - WRONG | X13: "We know the answer for almost every real; we don't know it for any particular real we have a name for." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | The real defined by Z2 is proved to have the required mean in section 5. Removed the blanket no-named-proof claim from body, summary, related notes and verifier; pi remains an explicitly open example. - WRONG | X14: "The orbit of x under iteration of T is a dynamical system whose n-th coordinate carries the n-th CF coefficient of x as ⌊1/Tn(x)⌋." | https://arxiv.org/pdf/1903.00255 | After equation (2), a_n(x)=a_1(G^(n-1)(x)); x=3/7 gives a1=2, while the printed exponent n gives 3. Changed the exponent to n-1. - MINOR | X15: "With respect to this measure, T is ergodic \u2014 a result Kuzmin (1928) and Lévy (1929) supplied independently." | https://arxiv.org/pdf/1903.00255; https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | These sources establish ergodicity separately from the Gauss-Kuzmin convergence history. The original attribution conflated the statements. Omitted the imprecise historical attribution, retaining ergodicity and the separately cited distribution history. - WRONG | X16: "It is a measure-theoretic recoding of \"doesn't lock onto any cycle.\"" | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | Page 2 defines ergodicity via invariant sets. Swapping two equally weighted points is ergodic and cyclic, a direct counterexample to the alleged equivalence. Explicitly said ergodicity does not mean the quoted absence of cycles, leaving the quotation intact. - WRONG | X17: "The set of measure 1 on which it works excludes every CF with structure \u2014 every rational, every quadratic irrational, e and its cousins, every Liouville number" | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Z2 has explicit structure and reaches K0. Periodic algebraic means alone do not prove inequality with K0. Replaced the universal structure criterion with rationals, phi, sqrt(2), and e and their separate arguments. - WRONG | X18: "because each of these violates Birkhoff's hypothesis by belonging to an explicit T-invariant set of measure zero." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | Pages 2-3 state transformation, measure and integrability hypotheses and an almost-everywhere conclusion. Null-set membership violates none of those hypotheses. Said a specified input requires its own argument; removed the false null-set explanation. - MINOR | X19: "What the page draws live: the sum on the right is run in your browser and watched converging to log K0; the left side is what Instrument I above measures for the number you give it." | public/strata/khinchin/index.html | setCfsumN exponentiates the log sum; Instrument I exponentiates the average of log coefficients. Explained that both widgets display exponentials of the formula sides. - MINOR | X20: "The convergence is logarithmic in N. A million terms gives ~five correct digits \u2014 slower than almost anything else in analysis." | research/khinchin/verify-assay.mjs; public/strata/khinchin/index.html | Independent log1p summation at N=1000000 gives 2.6853946022, error about 5.7399e-5; the log-sum tail is asymptotic to (log N+1)/(N log 2). Replaced the ambiguous logarithmic-rate and five-digit wording with the tail expression and measured error. - MINOR | X21: "Eight assertions that re-run in your browser as the page loads." | public/strata/khinchin/index.html | runLiveChecks contains nine calls; the browser reports 9/9. Changed Eight to Nine, retaining nine checks. - MINOR | X22: "They restate, in code that you can read, every load-bearing claim above." | public/strata/khinchin/index.html; research/khinchin/verify.mjs | The code covers selected examples and finite samples; 43 offline checks include four unconditional scope notes and do not prove every claim. Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - WRONG | X23: "Khinchin, A. Ya., Continued Fractions, §16–19 (Chicago, 1964; trans. Wirth/Eagle of the 3rd Russian edition, 1961)." | https://archive.org/download/khinchin-continued-fractions/Khinchin%20-%20Continued%20Fractions_djvu.txt | The contents end at section 16, Average values. The copyright page credits Scripta Technica translation and Herbert Eagle editing. Corrected the section and credits; retained Chicago 1964 and the 1961 third Russian edition. - WRONG | X24: "The original paper is \"Metrische Kettenbruchprobleme,\" Compositio Mathematica, 1, 1934, pp. 361–382." | https://www.numdam.org/item/CM_1935__1__361_0/ | NUMDAM dates volume 1 to 1935; the paper was received in March 1934. Changed the bibliography and verifier source citation to 1935 while retaining the theorem date 1934. - WRONG | X25: "Bailey, Borwein, Crandall computed 100 000 digits using accelerated series in 1997 (Math. Comp. 66, 219, 417–431)" | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.jonborwein.org/jmbpapers/ | Paper p. 2 reports 7350 decimal places. The authors' bibliography lists volume 66, issue 217, pp. 417-431; Crossref ends at 432. Changed 100000 to 7350 decimal places and issue 219 to 217. Kept the author-supported 417-431 pagination. - WRONG | X26: "Lévy, P., \"Sur le développement en fraction continue d'un nombre choisi au hasard,\" Compositio Math. 3 (1929), 286–303." | https://www.numdam.org/item/CM_1936__3__286_0/ | NUMDAM assigns the cited title, volume 3 and pp. 286-303 to 1936. Corrected the citation year and preserved the quoted title. - WRONG | X27: "Pattern a3k=2k." | https://oeis.org/A003417/internal; https://scholarlycommons.pacific.edu/euler-works/71/ | OEIS uses a(0)=2 and a(3k-1)=2k; Euler Archive gives written 1737 and published 1744. Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - MINOR | X28: "Lagrange, J.-L., \"Additions au mémoire sur la résolution des équations numériques,\" 1770, §V \u2014 the classical biconditional." | https://fr.wikisource.org/wiki/M%C3%A9moires_extraits_des_recueils_de_l%E2%80%99Acad%C3%A9mie_royale_de_Berlin/Additions_au_M%C3%A9moire_sur_la_r%C3%A9solution_des_%C3%A9quations_num%C3%A9riques | Section II, Remarque II starts at no. 32 and concerns quadratic roots; periodicity occurs in nos. 32-36, with the converse preceding it. Replaced section V with that locator and retained the quoted title. - MINOR | X29: "Re-runs every numeric claim on the page from first principles, in BigInt where possible." | public/strata/khinchin/index.html; research/khinchin/verify.mjs | The code covers selected examples and finite samples; 43 offline checks include four unconditional scope notes and do not prove every claim. Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - WRONG | X30: "It does not show the convergence for any named constant \u2014 none is known to converge to K0." | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | The real defined by Z2 is proved to have the required mean in section 5. Removed the blanket no-named-proof claim from body, summary, related notes and verifier; pi remains an explicitly open example. - WRONG | X31: "It runs the closed form for K0 live; it computes the CF of any real you give it as far as the precision lets it; it counts the empirical Gauss–Kuzmin frequencies on a stream of random reals; it shows the four exceptional families failing in the three ways they fail;" | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Wieting pp. 1 and 7 proves the explicit Z2 construction; Bailey p. 9 separates numerical evidence for pi from a limit proof. Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - MINOR | X32: "Histogram of first 10 CF digits across sampled reals" | public/strata/khinchin/index.html | renderMC sets bars=14 and bins values 1 through 14; mcAdd pools up to 15 fractional positions. Labelled the visible coefficient-value bins 1 to 14. - MINOR | X33: "most common first digit" | public/strata/khinchin/index.html | mcAdd pools coefficients across fractional positions, not only a1. Changed the modal label and subscript to pooled coefficients and a_i; also corrected the browser verifier's matching label. - MINOR | X34: "mean GM, across samples" | public/strata/khinchin/index.html | Only finite per-sample means strictly between 1 and 100 enter sumGMcompat and finiteGMs. Disclosed the 1=1, other positive-index terms 1; retained the supported 1737 date. - WRONG | K8: [e-asymptotic-gm] So every third term grows linearly, and the geometric mean of n terms grows like (n/3·e-1)1/3 -- slowly, but unboundedly. | https://oeis.org/A003417/internal; research/khinchin/verify-assay.mjs | Blocks (1,2k,1) give product 2^m*m! at n=3m; Stirling gives (2n/(3e))^(1/3). Recomputed GM_300=4.2355519296. Restored the missing factor 2 and specified fractional coefficients; divergence is retained. - WRONG | K9: [gauss-map-index] The orbit of x under iteration of T is a dynamical system whose n-th coordinate carries the n-th CF coefficient of x as ⌊1/Tn(x)⌋. | https://arxiv.org/pdf/1903.00255 | After equation (2), a_n(x)=a_1(G^(n-1)(x)); x=3/7 gives a1=2, while the printed exponent n gives 3. Changed the exponent to n-1. - MINOR | K10: [ergodicity-kuzmin-levy-attribution] With respect to this measure, T is ergodic -- a result Kuzmin (1928) and Lévy (1929) supplied independently. | https://arxiv.org/pdf/1903.00255; https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | These sources establish ergodicity separately from the Gauss-Kuzmin convergence history. The original attribution conflated the statements. Omitted the imprecise historical attribution, retaining ergodicity and the separately cited distribution history. - WRONG | K11: [structured-exclusions] The set of measure 1 on which it works excludes every CF with structure -- every rational, every quadratic irrational, e and its cousins, every Liouville number | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Z2 has explicit structure and reaches K0. Periodic algebraic means alone do not prove inequality with K0. Replaced the universal structure criterion with rationals, phi, sqrt(2), and e and their separate arguments. - WRONG | K12: [invariant-zero-set-violates-hypothesis] because each of these violates Birkhoff's hypothesis by belonging to an explicit T-invariant set of measure zero. | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf | Pages 2-3 state transformation, measure and integrability hypotheses and an almost-everywhere conclusion. Null-set membership violates none of those hypotheses. Said a specified input requires its own argument; removed the false null-set explanation. - MINOR | K13: [eight-live-assertions] Eight assertions that re-run in your browser as the page loads. | public/strata/khinchin/index.html | runLiveChecks contains nine calls; the browser reports 9/9. Changed Eight to Nine, retaining nine checks. - MINOR | K14: [checks-cover-every-claim] They restate, in code that you can read, every load-bearing claim above. | public/strata/khinchin/index.html; research/khinchin/verify.mjs | The code covers selected examples and finite samples; 43 offline checks include four unconditional scope notes and do not prove every claim. Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - WRONG | K15: [book-citation] Khinchin, A. Ya., Continued Fractions, §16–19 (Chicago, 1964; trans. Wirth/Eagle of the 3rd Russian edition, 1961). | https://archive.org/download/khinchin-continued-fractions/Khinchin%20-%20Continued%20Fractions_djvu.txt | The contents end at section 16, Average values. The copyright page credits Scripta Technica translation and Herbert Eagle editing. Corrected the section and credits; retained Chicago 1964 and the 1961 third Russian edition. - WRONG | K16: [original-paper-citation] The original paper is "Metrische Kettenbruchprobleme," Compositio Mathematica, 1, 1934, pp. 361–382. | https://www.numdam.org/item/CM_1935__1__361_0/ | NUMDAM dates volume 1 to 1935; the paper was received in March 1934. Changed the bibliography and verifier source citation to 1935 while retaining the theorem date 1934. - WRONG | K17: [high-precision-citation] Bailey, Borwein, Crandall computed 100 000 digits using accelerated series in 1997 (Math. Comp. 66, 219, 417–431) | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.jonborwein.org/jmbpapers/ | Paper p. 2 reports 7350 decimal places. The authors' bibliography lists volume 66, issue 217, pp. 417-431; Crossref ends at 432. Changed 100000 to 7350 decimal places and issue 219 to 217. Kept the author-supported 417-431 pagination. - WRONG | K18: [levy-citation] Lévy, P., "Sur le développement en fraction continue d'un nombre choisi au hasard," Compositio Math. 3 (1929), 286–303. | https://www.numdam.org/item/CM_1936__3__286_0/ | NUMDAM assigns the cited title, volume 3 and pp. 286-303 to 1936. Corrected the citation year and preserved the quoted title. - WRONG | K19: [euler-index-apparatus] Pattern a3k=2k. | https://oeis.org/A003417/internal; https://scholarlycommons.pacific.edu/euler-works/71/ | OEIS uses a(0)=2 and a(3k-1)=2k; Euler Archive gives written 1737 and published 1744. Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - MINOR | K20: [offline-all-numeric-claims] Re-runs every numeric claim on the page from first principles, in BigInt where possible. | public/strata/khinchin/index.html; research/khinchin/verify.mjs | The code covers selected examples and finite samples; 43 offline checks include four unconditional scope notes and do not prove every claim. Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - WRONG | K21: [functions-summary] It runs the closed form for K0 live; it computes the CF of any real you give it as far as the precision lets it; it counts the empirical Gauss–Kuzmin frequencies on a stream of random reals; it shows the four exceptional families failing in the three ways they fail; | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Wieting pp. 1 and 7 proves the explicit Z2 construction; Bailey p. 9 separates numerical evidence for pi from a limit proof. Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - MINOR | K22: [histogram-ten-digits] Histogram of first 10 CF digits across sampled reals | public/strata/khinchin/index.html | renderMC sets bars=14 and bins values 1 through 14; mcAdd pools up to 15 fractional positions. Labelled the visible coefficient-value bins 1 to 14. - WRONG | K23: [dek-4] The exceptional set (the reals that miss) has measure zero, and contains every named constant we have. | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Wieting pp. 1 and 7 proves the explicit Z2 construction; Bailey p. 9 separates numerical evidence for pi from a limit proof. Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - WRONG | K24: [dek-5] The page is the live witness: type any number, watch its CF and its running average emerge to ~50 exact terms; sample random reals and watch the Gauss–Kuzmin distribution snap into shape; meet the four exceptional families, φ, √n, e, and the unproved-yet (π, ln 2, γ), in three named failure modes. | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | Wieting pp. 1 and 7 proves the explicit Z2 construction; Bailey p. 9 separates numerical evidence for pi from a limit proof. Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - WRONG | K25: [relation-1-1] There: √2 is irrational because the Euclidean subtraction never halts. | git show 37137afa5c:src/content/strata/incommensurable.md | Section II assumes a lowest-terms fraction and derives both numerator and denominator even. Changed only Khinchin's relation to describe the parity proof. - MINOR | K26: [relation-1-2] Here: √2's CF [1; 2, 2, 2, …] is periodic, so its geometric mean is exactly 2 -- failing Khinchin in the same direction, by the same Lagrange theorem (CF eventually periodic ⟺ quadratic irrational, 1770) that lets √2's irrationality be proved at all. | git show 37137afa5c:src/content/strata/incommensurable.md; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | The parity proof does not require Lagrange; the periodic-CF classification is valid independently. Removed the necessity implication while retaining sqrt(2)'s mean and Lagrange's classification. - WRONG | K27: [relation-3-0] Both turn on a load-bearing ergodic-theoretic equals-sign with empirical evidence and no proof for the numbers we name. | git show 37137afa5c:src/content/strata/harmonics-of-the-primes.md | The linked layer describes the Riemann-von Mangoldt explicit prime/zero formula, not this ergodic average. Rewrote Khinchin's relation around exact identities and open questions, and regenerated its nearby teaser from that source. - WRONG | K28: [relation-4-1] There: a function whose values are known for the first five inputs and uncomputable thereafter. | git show 37137afa5c:src/content/strata/the-longest-finite-race.md | The linked page separates the unknown sixth value from a function with no uniform computing algorithm; a fixed finite integer is a computable constant. Separated the unresolved sixth value and the lack of a uniform algorithm in Khinchin's relation. - WRONG | K29: [preset-decimal-2] 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642742746239069428753868416326497742651603378279158633244744671066077395307552524833043770497627 | https://oeis.org/A001113/b001113.txt; research/khinchin/assay-digits.json; research/khinchin/verify-assay.mjs | The cached OEIS digit file and independent rational bounds on the factorial series agree. The old preset first differs at decimal 105. Replaced all 191 preset decimal places and the copied 110-place E_DEC verifier input with same-length correct prefixes. - WRONG | K30: [preset-decimal-6] 2.64575131106459059050161575363926042571025918308245018036833445920106882323028362776039288647232246527959675937795396625495329008701036768272275411653527953884937709825472298886770174815605457681 | https://oeis.org/A010465/b010465.txt; research/khinchin/assay-digits.json; research/khinchin/verify-assay.mjs | The cached OEIS file and exact integer square root of 7*10^420 agree. The old preset first differs at decimal 93. Replaced all 194 supplied decimal places, preserving the exact periodic-CF routine. - WRONG | K31: [preset-decimal-9] 2.685452001065306445309714835481795693820382293994629230503714 | https://oeis.org/A002210/b002210.txt; research/khinchin/assay-digits.json; research/khinchin/verify-assay.mjs | The OEIS b-file shows the old strings first differ at decimal 49. Full string comparisons are now used. Replaced the page's 60-place constant and preset plus the verifier's 63-place constant. Refreshed the published verifier verbatim. - MINOR | K32: (added) Page source comment above the presets: "Sources: OEIS A000796 (pi), A001113 (e), A001622 (phi), A002193 (sqrt 2), A002194 (sqrt 3), A002392 (ln 2), A001620 (Euler-Mascheroni gamma)." | https://oeis.org/A002162/internal; public/strata/khinchin/index.html | A002162 contains the ln(2) decimal expansion; A002392 is ln(10). Corrected the source-comment identifier while leaving the valid ln(2) digits unchanged. - WRONG | K33: (added) The published verifier prints "~41% of all reals have a_1 = 1 (the most common starting CF digit)" (research/khinchin/verify.mjs line 306, served at /checks/research/khinchin/verify.mjs). | https://www.reed.edu/math-stats/wieting/essays/AKhinchinSequence.pdf; research/khinchin/verify.mjs | Gauss density gives log2(4/3); uniform first-digit probability is the interval length 1/2. Specified limiting coefficient frequency in the page and Gauss measure in the verifier; the verifier also states the uniform probability 1/2. - CONFIRMED | "Pick a real number. Strip off the integer part. Take the reciprocal of what's left, and strip the integer of that. You'll get a long sequence of positive integers \u2014 the continued fraction." | https://archive.org/download/khinchin-continued-fractions/Khinchin%20-%20Continued%20Fractions_djvu.txt | [001 cf-algorithm] The supplied Theorem 14 excerpt distinguishes finite rational and infinite irrational expansions. Its exact words include "finite" and "infinite". The floor algorithm selects the canonical rational expansion; the final-1 alternative does not invalidate that qualified uniqueness. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "For almost every real you could have picked, the geometric mean of those integers approaches the same constant. It does not depend on the number you chose." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [002 almost-every-real] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Khinchin proved this in 1934." | https://www.numdam.org/item/CM_1935__1__361_0.pdf | [003 theorem-date] The record excerpt gives the 1935 publication year, which alone does not settle when the theorem was proved. The primary paper, p. 382, says "(Eingegangen den 29. März 1934.)". Retain the theorem date 1934. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The exceptional set \u2014 the reals for which it fails \u2014 has measure zero." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [004 exception-measure] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The continued fraction of a real number is a kind of true name: the unique expansion you reach by repeated floor and reciprocal, exact for rationals (it terminates) and infinite for irrationals." | https://archive.org/download/khinchin-continued-fractions/Khinchin%20-%20Continued%20Fractions_djvu.txt | [006 cf-unique-rational-irrational] The supplied Theorem 14 excerpt distinguishes finite rational and infinite irrational expansions. Its exact words include "finite" and "infinite". The floor algorithm selects the canonical rational expansion; the final-1 alternative does not invalidate that qualified uniqueness. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Almost every real hides a number, 2.685452…, in its continued fraction." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [008 almost-every-hidden-average] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "π \u2014 empirically near K0, never proven." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; git show 37137afa5c:public/strata/khinchin/index.html | [011 pi-note] The supplied Bailey discussion distinguishes numerical agreement from proof. For pi it reports 17,001,303 coefficients and GM 2.686393, not an infinite-limit theorem. The browser likewise labels these finite prefixes "consistent with K₀ (no proof)". The record supports this qualitative statement, not the separately claimed billion-term attributions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "For sqrt(N), cbrt(N), p/q, append directly (e.g. 22/7, sqrt(7))." | git show 37137afa5c:public/strata/khinchin/index.html | [013 input-syntax] parseInput has branches for p/q, sqrt(N), and cbrt(N). Recreated "sqrt(7)" and "cbrt(2)" successfully; p/q is accepted but numerically broken as entry 10 records. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The tail of CF terms shown in gold are the unusually large ones (ai > 100)." | git show 37137afa5c:public/strata/khinchin/index.html | [014 large-terms-gold] The rendering branch tests "a > 100" and assigns the gold class. Pi supplies the positive control 292. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Almost every real is structureless." | https://arxiv.org/pdf/1903.00255 | [015 almost-every-structureless] Read as the informal typicality claim explained immediately by the Gauss-Kuzmin distribution, not as a definition of algorithmic randomness. The supplied theorem states its frequency law for almost every real; the stronger all-structured exclusion is separately wrong at entry 64. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "What the CF expansion of a random real looks like, on average, is a specific distribution discovered by Gauss in a letter to Laplace (1812) and proven by Kuzmin a century later:" | https://www.ihes.fr/~rzhang/files/teaching/useminar_s26_papers/useminar_s26_lee.pdf; https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1928.6/ICM1928.6.ocr.pdf | [016 gauss-kuzmin-history] The supplied Lee history identifies the letter of "30 January 1812". Kuzmin p. 83 independently starts from Gauss's letter to Laplace and gives its limiting distribution; the article ends with the convergence result on p. 89. Kuzmin was presented at Bologna in 1928; the proceedings were printed in 1932. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "the probability that the k-th CF coefficient equals n approaches a fixed limit, independent of k for large enough k:" | https://arxiv.org/pdf/1903.00255 | [017 gauss-kuzmin-limit] The supplied source gives the limiting Gauss measure. Integrating its density over (1/(n+1),1/n] gives log2((n+1)^2/(n(n+2))). This is positive and decreases for n>=1. This verifies the limit, rather than an exact law for the first digit under uniform sampling. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "P(ai = n) → log2((n+1)2 / n(n+2)) \u2014 the Gauss–Kuzmin distribution, valid almost everywhere." | https://arxiv.org/pdf/1903.00255 | [018 gauss-kuzmin-formula] The supplied source gives the limiting Gauss measure. Integrating its density over (1/(n+1),1/n] gives log2((n+1)^2/(n(n+2))). This is positive and decreases for n>=1. This verifies the limit, rather than an exact law for the first digit under uniform sampling. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The next most common is 2, then 3, with the tail falling off but never reaching zero." | https://arxiv.org/pdf/1903.00255 | [020 digit-probability-tail] The supplied source gives the limiting Gauss measure. Integrating its density over (1/(n+1),1/n] gives log2((n+1)^2/(n(n+2))). This is positive and decreases for n>=1. This verifies the limit, rather than an exact law for the first digit under uniform sampling. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Below: draw a sample of random reals and watch their distribution snap into shape." | git show 37137afa5c:public/strata/khinchin/index.html | [021 sample-distribution] Inspected mcAdd: "const cf = cfDoubleLossy(x, 16)" and the loop starts at i=1, giving up to 15 fractional coefficients in double precision. Recreated the histogram in the local browser. Sampling noise is expected; the bars and theoretical line use the stated coefficient frequencies. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Each bar: empirical fraction of CF coefficients equal to n, across all sampled reals (15 CF terms each, double-precision). Dashed gold: theoretical Gauss-Kuzmin probability." | git show 37137afa5c:public/strata/khinchin/index.html | [022 histogram-measure] Inspected mcAdd: "const cf = cfDoubleLossy(x, 16)" and the loop starts at i=1, giving up to 15 fractional coefficients in double precision. Recreated the histogram in the local browser. Sampling noise is expected; the bars and theoretical line use the stated coefficient frequencies. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "At small n, the mean GM is biased upward \u2014 finitely many rare large ai dominate the product." | https://arxiv.org/pdf/1903.00255 | [023 finite-gm-bias] The supplied moment bound is not by itself a bias proof. Independent check: under stationary Gauss measure, E[(1/n)sum log(ai)]=log(K0); Jensen gives E[exp((1/n)sum log(ai))]>=K0. Heavy positive tails explain the finite-prefix upward bias. This does not imply the false variance statement in entry 24. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "[1; 1, 1, 1, 1, 1, 1, 1, 1, 1, …]" | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [026 phi-cf] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The golden ratio φ = (1+√5) / 2" | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [027 phi-formula] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "All coefficients are 1, so the geometric mean is exactly 1 at every step." | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [028 phi-gm-exact] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The golden ratio is the most irrational number \u2014 Hurwitz (1891) proved its rational approximations are the worst possible \u2014 and it gets there by carrying no information after the first digit." | https://link.springer.com/article/10.1007/BF01206656 | [029 phi-hurwitz-worst] Hurwitz's 1891 result has sharp constant 1/sqrt(5); phi and its continued-fraction equivalents attain the extremal approximation behavior. The all-1 tail also gives GM=1. "Worst" is read in this standard asymptotic Hurwitz sense, not as uniqueness among all reals. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "[1; 2, 2, 2, 2, 2, 2, 2, 2, 2, …]" | https://oeis.org/A040000/internal; git show 37137afa5c:public/strata/khinchin/index.html | [031 sqrt2-cf] A040000 gives "1, 2, 2, 2" followed by 2s. Omitting the integer coefficient gives GM=2 at every finite length. Recreated the pinned browser output. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Every quadratic irrational has an eventually periodic CF (Lagrange, 1770)" | https://fr.wikisource.org/wiki/M%C3%A9moires_extraits_des_recueils_de_l%E2%80%99Acad%C3%A9mie_royale_de_Berlin/Additions_au_M%C3%A9moire_sur_la_r%C3%A9solution_des_%C3%A9quations_num%C3%A9riques | [033 lagrange-1770] The original 1770 Additions, section II, Remarque II, proves periodicity for quadratic irrational roots; no. 34 says "sera toujours nécessairement périodique". The converse is derived immediately before that remark. The record's Rutgers excerpt gives the same historical attribution. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "√2's coefficients past the first are all 2;" | https://oeis.org/A040000/internal; git show 37137afa5c:public/strata/khinchin/index.html | [034 sqrt2-digits] A040000 gives "1, 2, 2, 2" followed by 2s. Omitting the integer coefficient gives GM=2 at every finite length. Recreated the pinned browser output. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "√3's are 1, 2, 1, 2;" | https://oeis.org/A040001/internal | [035 sqrt3-digits] A040001 gives sqrt(3)=[1;1,2,1,2,...]. The periodic product is 2, so the limiting mean is sqrt(2). Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "√7's are 1, 1, 1, 4." | https://oeis.org/A010121/internal; git show 37137afa5c:public/strata/khinchin/index.html | [036 sqrt7-digits] A010121 gives sqrt(7)=[2;1,1,1,4,1,1,1,4,...]. The period product is 4 and its fourth root is sqrt(2), finite. Independently recreated the exact quadratic-surd algorithm. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "They are all on a measure-zero set the theorem warns about." | https://oeis.org/A040000; https://oeis.org/A040001; https://oeis.org/A010121 | [037 quadratics-measure-zero] For the immediately listed sqrt(2), sqrt(3), and sqrt(7), the means are 2, sqrt(2), sqrt(2), all below K0. This finite set is null. This does not validate the earlier universal assertion about every quadratic irrational. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …]" | https://oeis.org/A003417/internal | [038 e-cf] A003417 gives "2, 1, 2, 1, 1, 4, 1, 1, 6". The repeated fractional block is (1,2k,1); for n=3m the product is 2^m m!, whose (3m)-th root tends to infinity. This confirms divergence, not the page's incorrect index or asymptotic constant. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, …]" | https://oeis.org/A001203/internal; git show 37137afa5c:public/strata/khinchin/index.html | [043 pi-cf] A001203 starts "3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14". This agrees with both the visible prefix and the recreated page engine. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "consistent with K0, never proven" | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; git show 37137afa5c:public/strata/khinchin/index.html | [044 pi-consistent-unproved] The supplied Bailey discussion distinguishes numerical agreement from proof. For pi it reports 17,001,303 coefficients and GM 2.686393, not an infinite-limit theorem. The browser likewise labels these finite prefixes "consistent with K₀ (no proof)". The record supports this qualitative statement, not the separately claimed billion-term attributions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "π's CF looks structureless." | https://oeis.org/A001203/internal | [045 pi-structureless] The source gives an irregular observed prefix and an unresolved infinite-limit question. "Looks" is explicitly an empirical characterization, not a proof of randomness or normality. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "No proof. Not for π, not for eπ, not for ln 2, not for the Euler-Mascheroni γ, not for any of the famous irrational constants we don't know to be structured." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.ihes.fr/~rzhang/files/teaching/useminar_s26_papers/useminar_s26_lee.pdf | [047 pi-other-constants-no-proof] The supplied Bailey excerpt specifically qualifies its open examples as numbers "cast in terms of fundamental constants". The current Lee treatment also identifies the familiar-constant gap. This supports the restricted familiar unstructured cases, unlike the page's unrestricted claim that no named example exists. Gamma's irrationality is itself unproved, as the page separately states. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The full theorem is one of ergodic theory's load-bearing applications." | https://arxiv.org/pdf/1903.00255 | [049 ergodic-application] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The argument has three steps; the page can fairly draw the first two." | git show 37137afa5c:public/strata/khinchin/index.html | [050 proof-three-steps] The pinned page labels exactly "1. The Gauss map.", "2. The invariant measure.", and "3. Birkhoff's theorem." The numerical widgets implement the two sides of the average formula after exponentiation. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Let T(x) = {1/x} \u2014 the fractional part of 1/x \u2014 defined on (0,1)." | https://arxiv.org/pdf/1903.00255 | [051 gauss-map-definition] The source defines the Gauss shift on irrational x in (0,1) and gives G([a1,a2,...])=[a2,a3,...]. The formula {1/x} is correct; rational orbits terminate at zero and must stop there. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "T is the operation that, applied to a real, strips off its next CF coefficient." | https://arxiv.org/pdf/1903.00255 | [052 gauss-map-shift] The source defines the Gauss shift on irrational x in (0,1) and gives G([a1,a2,...])=[a2,a3,...]. The formula {1/x} is correct; rational orbits terminate at zero and must stop there. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Gauss (in a 1812 letter to Laplace) wrote down a probability measure that T preserves: dμ = (1/log 2)·dx/(1+x)." | https://arxiv.org/pdf/1903.00255; https://www.ihes.fr/~rzhang/files/teaching/useminar_s26_papers/useminar_s26_lee.pdf | [054 gauss-invariant-1812] The invariant density is 1/((1+x)log(2)); its integral over (0,1) is 1 and it decreases with x. Lee cites the 1812 letter; Kuzmin p. 83 reproduces its logarithmic limiting CDF. Entry 54's long excerpt is real, but its locator "p. 12, reference 9" does not accurately describe the many pages quoted. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "a probability measure that T preserves: dμ = (1/log 2)·dx/(1+x)." | https://arxiv.org/pdf/1903.00255; https://www.ihes.fr/~rzhang/files/teaching/useminar_s26_papers/useminar_s26_lee.pdf | [055 gauss-invariant-measure] The invariant density is 1/((1+x)log(2)); its integral over (0,1) is 1 and it decreases with x. Lee cites the 1812 letter; Kuzmin p. 83 reproduces its logarithmic limiting CDF. Entry 54's long excerpt is real, but its locator "p. 12, reference 9" does not accurately describe the many pages quoted. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Roughly, more weight near 0, where 1/x carries violently across many integers." | https://arxiv.org/pdf/1903.00255; https://www.ihes.fr/~rzhang/files/teaching/useminar_s26_papers/useminar_s26_lee.pdf | [056 gauss-weight-near-zero] The invariant density is 1/((1+x)log(2)); its integral over (0,1) is 1 and it decreases with x. Lee cites the 1812 letter; Kuzmin p. 83 reproduces its logarithmic limiting CDF. Entry 54's long excerpt is real, but its locator "p. 12, reference 9" does not accurately describe the many pages quoted. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Ergodic means: the only sets T sends to themselves are the trivial ones (measure 0 or 1)." | https://bibliotekanauki.pl/articles/1388218.pdf | [058 ergodic-definition] Ryll-Nardzewski defines indecomposability by invariant sets having measure 0 or 1. The precise condition is T^(-1)E=E modulo null sets, which is the standard meaning of invariant in this explanation. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "For an ergodic dynamical system, time averages equal space averages on a set of full measure." | https://arxiv.org/pdf/1903.00255 | [060 birkhoff-averages] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Apply it to the function f(x) = log⌊1/x⌋:" | https://arxiv.org/pdf/1903.00255 | [061 birkhoff-log-function] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "limn→∞ (1/n) Σ log ai(x) = ∫(0,1) log⌊1/x⌋ dμ = Σk≥1 log k · log2((k+1)2/(k(k+2))) = log K0." | https://arxiv.org/pdf/1903.00255 | [062 birkhoff-integral-sum] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Exponentiating gives Khinchin's theorem." | https://arxiv.org/pdf/1903.00255 | [063 exponentiation] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "They are the same number \u2014 provided you give Khinchin's theorem a real on which it holds." | https://arxiv.org/pdf/1903.00255 | [067 two-limits-equal] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Continued fractions, the Gauss map, and Khinchin's constant are all standard material." | https://archive.org/download/khinchin-continued-fractions/Khinchin%20-%20Continued%20Fractions_djvu.txt | [071 standard-material] Khinchin's monograph, the primary papers, and the current continued-fraction literature treat these standard definitions and theorems. The supplied source supports this limited descriptive claim. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "OEIS A002210." | https://oeis.org/A002210/internal | [075 oeis-k-citation] A002210 is titled "Decimal expansion of Khinchin's constant." Its digits agree with the specimen and ordinary double-precision constant, though the longer preset string has a separate error. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Kuzmin, R. O., \"Sur un problème de Gauss,\" Atti del Congresso Internazionale dei Matematici, Bologna 1928, vol. VI, pp. 83–89;" | https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1928.6/ICM1928.6.ocr.pdf; /home/liam/.assay-line/queue-1014c/khinchin/N1/evidence/kuzmin.txt | [077 kuzmin-citation] Read the primary article text pp. 83-89 in the supplied cached source and independently opened the ICM volume. The cached OCR has "SUR UN P R O B L E M E DE GAUSS", R. Kuzmin, and the stated page span. The front matter says "BOLOGNA 3-10 SETTEMBRE 1928" and "TOMO VI"; imprint 1932 is the publication year, while 1928 correctly dates the congress. The large PDF fetch was stopped and its partial bulk file deleted. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Euler, L., \"De fractionibus continuis dissertatio,\" Comm. Acad. Sci. Petrop. 9 (1737), 98–137." | https://scholarlycommons.pacific.edu/euler-works/71/; https://scholarlycommons.pacific.edu/euler-commentarii/policies.html | [079 euler-citation] The Euler Archive confirms this title, volume 9, and pp. 98-137; it lists written 1737 and publication 1744. Its journal catalogue explicitly says "Volume 9: 1737, published 1744." The parenthesized nominal volume year is therefore valid. A full bibliography can additionally clarify publication in 1744. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "OEIS A003417." | https://oeis.org/A003417/internal | [081 e-oeis-citation] A003417 gives "2, 1, 2, 1, 1, 4, 1, 1, 6". The repeated fractional block is (1,2k,1); for n=3m the product is 2^m m!, whose (3m)-th root tends to infinity. This confirms divergence, not the page's incorrect index or asymptotic constant. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "OEIS A001203;" | https://oeis.org/A001203/internal; git show 37137afa5c:public/strata/khinchin/index.html | [082 pi-oeis-citation] A001203 starts "3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14". This agrees with both the visible prefix and the recreated page engine. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "convergents A002485/A002486." | https://oeis.org/A002485/internal; https://oeis.org/A002486/internal | [083 pi-convergents-citation] After the two conventional seed terms, A002485 gives numerator prefix "3,22,333,355,103993" and A002486 denominator prefix "1,7,106,113,33102". Pairwise these are the listed convergents; the two OEIS links have the right roles. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "OEIS A001203; convergents A002485/A002486." | https://oeis.org/A002485/internal; https://oeis.org/A002486/internal | [084 pi-denominator-citation] After the two conventional seed terms, A002485 gives numerator prefix "3,22,333,355,103993" and A002486 denominator prefix "1,7,106,113,33102". Pairwise these are the listed convergents; the two OEIS links have the right roles. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Hurwitz, A., \"Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche,\" Math. Ann. 39 (1891), 279–284." | https://link.springer.com/article/10.1007/BF01206656 | [086 hurwitz-citation] Opened the Springer article metadata directly: title, A. Hurwitz, Mathematische Annalen 39, pages 279-284, 1891, DOI 10.1007/BF01206656 all match. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The constant 1/√5 is sharp." | https://link.springer.com/article/10.1007/BF01206656 | [087 hurwitz-sharp-constant] Hurwitz's 1891 result has sharp constant 1/sqrt(5); phi and its continued-fraction equivalents attain the extremal approximation behavior. The all-1 tail also gives GM=1. "Worst" is read in this standard asymptotic Hurwitz sense, not as uniqueness among all reals. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "It does not prove Khinchin's theorem; the proof needs ergodicity, which is real measure theory." | git show 37137afa5c:research/khinchin/verify.mjs; https://bibliotekanauki.pl/articles/1388218.pdf | [090 does-not-prove-theorem] The code performs finite arithmetic and sample checks; it does not implement a proof of the almost-everywhere theorem. The displayed ergodic proof requires integrability, measure preservation, and ergodicity as in the primary source. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Sister of The Most Irrational Number (φ's CF as the slowest converger), Incommensurable (√2's irrationality reproved as a sonnet), and The Common Measure (the same Euclidean-subtraction engine, walked end to end on the spine)." | git show 37137afa5c:src/content/strata/most-irrational.md; git show 37137afa5c:src/content/strata/incommensurable.md; git show 37137afa5c:src/content/strata/the-common-measure.md | [096 sister-layers] Read all three linked layer records at the pin: most-irrational treats phi and Hurwitz; incommensurable presents a sonnet and parity proof; the-common-measure describes the shared continued-fraction/Euclidean engine. Titles and this summary match. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "This layer's check opens this page in a real browser and drives it." | git show 37137afa5c:research/khinchin/verify-page.mjs | [097 browser-check-drives-page] The pinned verify-page.mjs launches Chromium, navigates to the page, clicks presets, enters "0.5", and checks DOM values at 390px and 1100px. Recreated locally with only dependency, static-root, executable, and port adaptations; 42 assertions pass. The local browser was not pointed at the live site. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "It is reading what you are reading." | git show 37137afa5c:research/khinchin/verify-page.mjs | [099 checker-reads-same-page] The pinned verify-page.mjs launches Chromium, navigates to the page, clicks presets, enters "0.5", and checks DOM values at 390px and 1100px. Recreated locally with only dependency, static-root, executable, and port adaptations; 42 assertions pass. The local browser was not pointed at the live site. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Running geometric mean of CF coefficients" | git show 37137afa5c:public/strata/khinchin/index.html | [103 gm-plot-label] renderGMPlot skips a0, accumulates "Math.log(coeffs[i])", and draws "Math.exp(sumLog / i)". Its label matches the quantity drawn. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "theoretical (log2 4/3)" | git show 37137afa5c:public/strata/khinchin/index.html; https://arxiv.org/pdf/1903.00255 | [106 theoretical-probability] Independent arithmetic gives log2(4/3)=0.41503749927884376, which rounds to the displayed 0.41504. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "a1 (first CF digit)" | https://oeis.org/A001203/internal; git show 37137afa5c:public/strata/khinchin/index.html | [107 initial-a1] A001203 begins 3,7; skipping integer coefficient a0 gives a1=7. Recreated the pi default at both browser widths. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "max ai so far" | https://oeis.org/A001203/internal; git show 37137afa5c:public/strata/khinchin/index.html | [108 max-coefficient] The recreated initial 60-entry pi prefix has largest coefficient 292; its maximum readout is 292. The code takes the maximum after a0. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "For almost every real, you get the same constant: K_0 = 2.6854520010..." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [109 meta-description-4] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Khinchin's theorem (1934), made playable." | https://www.numdam.org/item/CM_1935__1__361_0.pdf | [110 meta-description-5] The record excerpt gives the 1935 publication year, which alone does not settle when the theorem was proved. The primary paper, p. 382, says "(Eingegangen den 29. März 1934.)". Retain the theorem date 1934. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Type any number; watch its mean converge, or not." | git show 37137afa5c:public/strata/khinchin/index.html | [111 meta-description-6] The page has an editable number input and dynamically recomputes finite-prefix geometric means. The named controls visibly produce the displayed differing behaviors. The general numerical exactness promise is assessed separately at entry 10. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "A constant, 2.6854520010..., that almost every real number hides in its continued fraction." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [114 meta-og-description-1] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Khinchin's 1934 theorem, played: type a number, watch its mean converge, and watch the famous ones miss in three different ways." | https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A000012; https://oeis.org/A040000; https://oeis.org/A003417 | [115 meta-og-description-2] The theorem date is supported by the 1934 receipt; the displayed named failure examples phi, sqrt(2), e have means 1,2,infinity. This metadata sentence is read in that explicit example context, not as an assertion that pi fails. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Khinchin (1934): the geometric mean of the continued-fraction coefficients of almost every real is the same constant, 2.685452..." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [116 meta-twitter-description-1] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Type a number; watch it work, or fail in a named way." | git show 37137afa5c:public/strata/khinchin/index.html | [117 meta-twitter-description-2] The page has an editable number input and dynamically recomputes finite-prefix geometric means. The named controls visibly produce the displayed differing behaviors. The general numerical exactness promise is assessed separately at entry 10. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Khinchin's constant: take almost any real number, write it as a continued fraction, and the geometric mean of the coefficients approaches the same value, K0 = 2.6854520010, whichever number you picked." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [118 meta-aw-plain-1] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Type a number to watch its continued fraction and running average emerge, and meet the known exceptions, the golden ratio, square roots and e, then pi and ln 2, where nobody has proved it either way." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; git show 37137afa5c:public/strata/khinchin/index.html | [119 meta-aw-plain-2] The described examples have directly computed means: phi=1, the listed square roots finite periodic means, and e divergent. The text distinguishes pi and ln2 as unresolved; the corresponding two presets and labels are present. The larger blanket quadratic claim is assessed at entry 32. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Take the geometric mean of the coefficients." | https://arxiv.org/pdf/1903.00255; git show 37137afa5c:public/strata/khinchin/index.html | [120 dek-2] In context these are the positive fractional coefficients a1 onward. Both the theorem and the implementation omit a0. This matters for x in (0,1), whose integer coefficient is zero. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "For almost every real you could have picked, the answer approaches the same constant: K₀ = 2.6854520010… (Khinchin, 1934)." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://oeis.org/A002210 | [121 dek-3] The supplied introduction states "for almost all positive irrationals". Its displayed product is the geometric-mean limit; the complement is null. A002210 and independent arithmetic confirm 2.6854520010. For the 1934 attribution, the primary paper ends "(Eingegangen den 29. März 1934.)". Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Birkhoff's ergodic theorem on the Gauss map, made visible." | https://arxiv.org/pdf/1903.00255 | [124 dek-6] Kamieński p. 2 identifies Khinchin's theorem as a consequence of the Birkhoff pointwise ergodic theorem. With f=log(a1), integrability holds, and the source has an(ω)=a1(G^(n-1)(ω)). Partitioning (0,1) into coefficient intervals gives the displayed series for log K0; exponentiation gives the geometric mean. This confirms the mathematical equalities, not the surrounding universal exclusions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Two pieces on the same engine." | git show 37137afa5c:src/content/strata/the-common-measure.md; git show 37137afa5c:src/content/strata/khinchin.md | [125 relation-0-0] Read the related layer at the pin. The common engine is Euclidean subtraction/reciprocal continued fractions; the-common-measure uses sqrt(2), phi, and the fifth via its pitch ratio in octaves. The Khinchin page applies that shift to a time average. No additional theorem is asserted by the sibling/sequel labels. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "There: φ's CF is all 1s \u2014 the slowest possible rational approximation, the most irrational number." | https://link.springer.com/article/10.1007/BF01206656 | [126 relation-0-1] Hurwitz's 1891 result has sharp constant 1/sqrt(5); phi and its continued-fraction equivalents attain the extremal approximation behavior. The all-1 tail also gives GM=1. "Worst" is read in this standard asymptotic Hurwitz sense, not as uniqueness among all reals. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Here: that same all-1s CF makes φ miss Khinchin's average from below (GM = 1, K₀ = 2.685)." | https://oeis.org/A000012; https://oeis.org/A002210 | [127 relation-0-2] All fractional coefficients of phi are 1, so GM=1<2.6854520010. The rounded K0=2.685 in this related-note comparison is correct. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The reason φ is hardest to approximate is exactly why it fails the universal average \u2014 both follow from the same line of its continued fraction." | https://link.springer.com/article/10.1007/BF01206656 | [128 relation-0-3] Hurwitz's 1891 result has sharp constant 1/sqrt(5); phi and its continued-fraction equivalents attain the extremal approximation behavior. The all-1 tail also gives GM=1. "Worst" is read in this standard asymptotic Hurwitz sense, not as uniqueness among all reals. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The sequel to √2 in the language of CFs." | git show 37137afa5c:src/content/strata/the-common-measure.md; git show 37137afa5c:src/content/strata/khinchin.md | [129 relation-1-0] Read the related layer at the pin. The common engine is Euclidean subtraction/reciprocal continued fractions; the-common-measure uses sqrt(2), phi, and the fifth via its pitch ratio in octaves. The Khinchin page applies that shift to a time average. No additional theorem is asserted by the sibling/sequel labels. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The portal's spine \u2014 anthyphairesis on √2, on the Pythagorean fifth, on φ \u2014 is the same Gauss map T(x) = {1/x} this stratum sits on top of." | git show 37137afa5c:src/content/strata/the-common-measure.md; git show 37137afa5c:src/content/strata/khinchin.md | [132 relation-2-0] Read the related layer at the pin. The common engine is Euclidean subtraction/reciprocal continued fractions; the-common-measure uses sqrt(2), phi, and the fifth via its pitch ratio in octaves. The Khinchin page applies that shift to a time average. No additional theorem is asserted by the sibling/sequel labels. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The portal walks the engine three ways to show irrationality; this stratum points the same engine at Birkhoff and asks what almost every orbit averages to." | git show 37137afa5c:src/content/strata/the-common-measure.md; git show 37137afa5c:src/content/strata/khinchin.md | [133 relation-2-1] Read the related layer at the pin. The common engine is Euclidean subtraction/reciprocal continued fractions; the-common-measure uses sqrt(2), phi, and the fifth via its pitch ratio in octaves. The Khinchin page applies that shift to a time average. No additional theorem is asserted by the sibling/sequel labels. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "There: an exact dual between primes and zeros, with the Riemann Hypothesis controlling its tightness." | git show 37137afa5c:src/content/strata/harmonics-of-the-primes.md | [135 relation-3-1] The linked page describes the explicit prime/zero formula and the Riemann Hypothesis's role in error estimates. This source-record comparison accurately reflects that mathematical subject. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Two places where math holds the door open for almost everyone except the people we've heard of." | git show 37137afa5c:src/content/strata/khinchin.md | [137 relation-3-3] These are relationship descriptions rather than separate quantified mathematical results. The referenced pages do concern distinct open problems. The concrete assertions beside these descriptions are checked individually at entries 134-140; no stylistic finding is raised. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Sibling open problems in the Number/Pattern register." | git show 37137afa5c:src/content/strata/khinchin.md | [138 relation-4-0] These are relationship descriptions rather than separate quantified mathematical results. The referenced pages do concern distinct open problems. The concrete assertions beside these descriptions are checked individually at entries 134-140; no stylistic finding is raised. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "Different shapes of 'we know what happens, just not where.'" | git show 37137afa5c:src/content/strata/khinchin.md | [141 relation-4-3] These are relationship descriptions rather than separate quantified mathematical results. The referenced pages do concern distinct open problems. The concrete assertions beside these descriptions are checked individually at entries 134-140; no stylistic finding is raised. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "The piece is built so a stranger can\ntype any decimal and watch their number either join the universal average or\nmiss it in one of three named ways \u2014 and so a reader who knows the engine\ncan see Birkhoff's ergodic theorem instantiated against a CF whose first\ntwenty terms are recomputed in BigInt rational arithmetic in their browser." | git show 37137afa5c:public/strata/khinchin/index.html; https://oeis.org/A001203 | [143 md-three-failures] The private metadata body describes finite-prefix interaction, which the recreated page supplies. Its decimal engine uses BigInt Euclidean divisions and displays at most 60 terms; pi's first twenty agree with A001203. This is a numerical illustration of the theorem, not an infinite-limit proof. Exactness for all inputs is separately wrong at entry 10. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "π \u2014 empirically near K₀, never proven." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; git show 37137afa5c:public/strata/khinchin/index.html | [145 preset-note-1] The supplied Bailey discussion distinguishes numerical agreement from proof. For pi it reports 17,001,303 coefficients and GM 2.686393, not an infinite-limit theorem. The browser likewise labels these finite prefixes "consistent with K₀ (no proof)". The record supports this qualitative statement, not the separately claimed billion-term attributions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "e \u2014 Euler's CF pattern (1, 2k, 1) makes its GM diverge." | https://oeis.org/A003417/internal | [146 preset-note-2] A003417 gives "2, 1, 2, 1, 1, 4, 1, 1, 6". The repeated fractional block is (1,2k,1); for n=3m the product is 2^m m!, whose (3m)-th root tends to infinity. This confirms divergence, not the page's incorrect index or asymptotic constant. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "φ \u2014 all CF digits are 1; GM = 1 exactly." | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [147 preset-note-3] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "√2 \u2014 CF [1; 2,2,2,…]; GM = 2." | https://oeis.org/A040000/internal; git show 37137afa5c:public/strata/khinchin/index.html | [148 preset-note-4] A040000 gives "1, 2, 2, 2" followed by 2s. Omitting the integer coefficient gives GM=2 at every finite length. Recreated the pinned browser output. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "√3 \u2014 CF [1; 1,2,1,2,…]; GM → √2." | https://oeis.org/A040001/internal | [149 preset-note-5] A040001 gives sqrt(3)=[1;1,2,1,2,...]. The periodic product is 2, so the limiting mean is sqrt(2). Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "√7 \u2014 CF [2; 1,1,1,4,1,1,1,4,…]; periodic, GM finite." | https://oeis.org/A010121/internal; git show 37137afa5c:public/strata/khinchin/index.html | [150 preset-note-6] A010121 gives sqrt(7)=[2;1,1,1,4,1,1,1,4,...]. The period product is 4 and its fourth root is sqrt(2), finite. Independently recreated the exact quadratic-surd algorithm. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "ln 2 \u2014 CF looks structureless; convergence to K₀ unproven." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf; git show 37137afa5c:public/strata/khinchin/index.html | [151 preset-note-7] The supplied Bailey discussion distinguishes numerical agreement from proof. For pi it reports 17,001,303 coefficients and GM 2.686393, not an infinite-limit theorem. The browser likewise labels these finite prefixes "consistent with K₀ (no proof)". The record supports this qualitative statement, not the separately claimed billion-term attributions. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "γ \u2014 Euler-Mascheroni, irrationality itself unknown." | https://arxiv.org/pdf/1303.1856 | [152 preset-note-8] Lagarias's survey states "Conjecture 1.0.1. Euler’s constant is irrational." and calls it a long-standing open problem. The page and the source record correctly keep this open question separate from the continued-fraction limit. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "K₀ \u2014 empirically its own CF's GM is consistent with K₀; same gap." | https://www.davidhbailey.com/dhbpapers/khinchine.pdf | [153 preset-note-9] Bailey reports the first 7182 elements from its 7350-digit K0 calculation and a geometric mean compatible numerically with K0, without proving the infinite limit. This supports the empirical note; it does not validate the page's erroneous K0 decimal preset. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "GM = 1 \u2014 misses K₀" | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [154 verdict-1] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "GM diverges \u2014 Euler pattern" | https://oeis.org/A003417/internal | [156 verdict-3] A003417 gives "2, 1, 2, 1, 1, 4, 1, 1, 6". The repeated fractional block is (1,2k,1); for n=3m the product is 2^m m!, whose (3m)-th root tends to infinity. This confirms divergence, not the page's incorrect index or asymptotic constant. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "rational \u2014 CF terminates" | https://archive.org/download/khinchin-continued-fractions/Khinchin%20-%20Continued%20Fractions_djvu.txt | [157 verdict-4] The canonical continued fraction of a rational terminates, as the supplied Theorem 14 states. This is a true mathematical label; the parser's incorrect expansion of rationals is separately recorded at entry 10. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "too few CF terms" | git show 37137afa5c:public/strata/khinchin/index.html | [158 verdict-5] The verdict branch uses "cf.length < 8" to show this finite-prefix advisory. It is not presented as a statistical confidence bound. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "random real \u2014 should be Khinchin (measure 1)." | git show 37137afa5c:public/strata/khinchin/index.html; https://arxiv.org/pdf/1903.00255 | [159 random-should-be-khinchin] The random control generates 200 decimal digits and shows a finite prefix. Interpreted as a simulation of a uniform real, the measure-one statement is the theorem, not a proof about this finite rational encoding. No claim that a computer literally samples the continuum is inferred. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "consistent with K₀ (no proof)" | git show 37137afa5c:public/strata/khinchin/index.html | [160 compatible-verdict] The code explicitly retains "no proof" or "at small n" in these finite-prefix compatibility labels. A finite CF prefix alone does not decide the infinite limit; the label makes no stronger inference. The implementation's threshold is a display heuristic, not a proved statistical confidence interval. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "K₀ = 2.6854520010653062 matches OEIS A002210" | https://oeis.org/A002210/internal; git show 37137afa5c:public/strata/khinchin/index.html | [162 live-check-1] The displayed decimal 2.6854520010653062 is the JavaScript double approximation used by the page. It agrees with A002210 to ordinary double precision; the separate long preset error starts much later. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "φ: CF = [1, 1, 1, 1, 1, 1, …], all 1s" | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [163 live-check-2] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "GM(φ, 20 terms) = 1 exactly" | https://oeis.org/A000012/internal; git show 37137afa5c:public/strata/khinchin/index.html | [164 live-check-3] A000012 gives the all-1 continued fraction. Solving x=1+1/x gives the positive root (1+sqrt(5))/2. Every product of the fractional coefficients is 1, hence every geometric mean is 1. Recreated the page's phi outputs. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "GM(√2, 20 terms) = 2 exactly" | https://oeis.org/A040000/internal; git show 37137afa5c:public/strata/khinchin/index.html | [165 live-check-4] A040000 gives "1, 2, 2, 2" followed by 2s. Omitting the integer coefficient gives GM=2 at every finite length. Recreated the pinned browser output. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "GM(e, 21 a_i) = 1.891 (growing \u2014 diverges in the limit)" | https://oeis.org/A003417/internal; git show 37137afa5c:public/strata/khinchin/index.html | [166 live-check-5] For 21 fractional terms of e there are seven factors 2k, so GM=(2^7*7!)^(1/21), which rounds to 1.891. The growth conclusion follows from the Euler pattern. Recreated the displayed check. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "π CF prefix = [3; 7, 15, 1, 292] matches OEIS A001203" | https://oeis.org/A001203/internal; git show 37137afa5c:public/strata/khinchin/index.html | [167 live-check-6] A001203 starts "3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14". This agrees with both the visible prefix and the recreated page engine. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "sum to N=100 000: log K = 0.98767 → target 0.98785" | git show 37137afa5c:public/strata/khinchin/index.html; https://www.davidhbailey.com/dhbpapers/khinchine.pdf | [169 live-check-8] Independent stable summation confirms log(partial product at 100000)=0.98767 when rounded to five decimals and log(K0)=0.98785. The live check prints exactly those rounded values. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "π convergent #4 = 355/113 (Zu Chongzhi, 5th c.)" | https://oeis.org/A002485/internal; https://oeis.org/A002486/internal; https://mathworld.wolfram.com/PiContinuedFraction.html | [170 live-check-9] The numerator/denominator sequences yield the fourth convergent 355/113. MathWorld's historical description credits Zu Chongzhi, whose fifth-century lifetime matches the label. The page engine reproduces the fraction. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "π convergent #4 = 355/113 (Zu Chongzhi, 5th c.)" | https://oeis.org/A002485/internal; https://oeis.org/A002486/internal; https://mathworld.wolfram.com/PiContinuedFraction.html | [175 zu-chongzhi-fifth-century] The numerator/denominator sequences yield the fourth convergent 355/113. MathWorld's historical description credits Zu Chongzhi, whose fifth-century lifetime matches the label. The page engine reproduces the fraction. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "K0 = 2.6854520010 6530644530 9714835481 …" | https://oeis.org/A002210/internal | [176 specimen-constant-digits] A002210 begins 2.685452001065306445309714835481; removing the page's digit-group spaces gives the same prefix. This specimen is correct even though the separate K0 preset is not. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "For almost every real number x, the geometric mean of the coefficients of its continued fraction expansion converges to the same constant: Khinchin's constant K_0 = 2.6854520010... A live, playable account of the 1934 theorem \u2014 type any number, watch its mean converge to K_0, and see why φ, √2, e fail in three different ways. Offline proof in research/khinchin/verify.mjs (43 checks PASS)." | https://www.numdam.org/item/CM_1935__1__361_0.pdf; https://www.davidhbailey.com/dhbpapers/khinchine.pdf; git show 37137afa5c:research/khinchin/verify.mjs | [178 structured-description] The metadata combines the correct almost-everywhere theorem, correct displayed constant, supported 1934 theorem date, directly verified three examples, and the reproduced 43-check result. In context "Offline proof" refers to numeric checking; the body explicitly disclaims a proof of the theorem. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196" | https://oeis.org/A000796/internal | [181 preset-decimal-1] The record excerpt supplies the correct initial prefix. Independently recomputed the entire displayed string at 230 decimal digits with mpmath 1.3.0 and compared every supplied digit: exact prefix agreement. Evidence: evidence/digits.json, including the full independently computed comparison string. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "1.61803398874989484820458683436563811772030917980576286213544862270526046281890244970720720418939113748475408807538689175212663386222353693179318006076672635443338908659593958290563832266131992829" | https://oeis.org/A001622/internal | [183 preset-decimal-3] The record excerpt supplies the correct initial prefix. Independently recomputed the entire displayed string at 230 decimal digits with mpmath 1.3.0 and compared every supplied digit: exact prefix agreement. Evidence: evidence/digits.json, including the full independently computed comparison string. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206" | https://oeis.org/A002193/internal | [184 preset-decimal-4] The record excerpt supplies the correct initial prefix. Independently recomputed the entire displayed string at 230 decimal digits with mpmath 1.3.0 and compared every supplied digit: exact prefix agreement. Evidence: evidence/digits.json, including the full independently computed comparison string. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "1.73205080756887729352744634150587236694280525381038062805580697945193301690880003708114618675724857567562614141540670302996994509499895247881165551209437364852809323190230558206797482010108467492" | https://oeis.org/A002194/internal | [185 preset-decimal-5] The record excerpt supplies the correct initial prefix. Independently recomputed the entire displayed string at 230 decimal digits with mpmath 1.3.0 and compared every supplied digit: exact prefix agreement. Evidence: evidence/digits.json, including the full independently computed comparison string. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "0.69314718055994530941723212145817656807550013436025525412068000949339362196969471560586332699641868754200148102057068573368552023575813055703267075163507596193072757082837143519030703862389167347" | https://oeis.org/A002162/internal | [187 preset-decimal-7] The record excerpt supplies the correct initial prefix. Independently recomputed the entire displayed string at 230 decimal digits with mpmath 1.3.0 and compared every supplied digit: exact prefix agreement. Evidence: evidence/digits.json, including the full independently computed comparison string. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "0.57721566490153286060651209008240243104215933593992359880576723488486772677766467093694706329174674951463144724980708248096050401448654283622417399764492353625350033374293733773767394279259525824" | https://oeis.org/A001620/internal | [188 preset-decimal-8] The record excerpt supplies the correct initial prefix. Independently recomputed the entire displayed string at 230 decimal digits with mpmath 1.3.0 and compared every supplied digit: exact prefix agreement. Evidence: evidence/digits.json, including the full independently computed comparison string. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | "current sample ${gm.toFixed(3)} \u2014 still compatible at small n" | git show 37137afa5c:public/strata/khinchin/index.html | [190 fallback-compatibility-verdict] The code explicitly retains "no proof" or "at small n" in these finite-prefix compatibility labels. A finite CF prefix alone does not decide the infinite limit; the label makes no stronger inference. The implementation's threshold is a display heuristic, not a proved statistical confidence interval. Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-codex.md. - CONFIRMED | [verifier-published-standalone] verify.mjs is published, and it runs on its own: one file, and Node. | recreated: curl https://artwaste.land/checks/research/khinchin/verify.mjs into an empty dir; node verify.mjs | 200, 25,278 bytes, byte-identical to the repository file; runs alone under Node, exit 0 Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. - CONFIRMED | [md-standalone] An immersive layer; it renders as its own standalone page at the `href` above (source: `public/strata/khinchin/index.html`). This file exists so the layer appears on the ground with the others | git show 37137afa5c:public/strata/khinchin/index.html | standalone page exists at that path Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. - CONFIRMED | [live-check-0] ALL PASS -- 9/9 live checks | public/strata/khinchin/index.html runLiveChecks (recreated: python reimplementation of the nine check() conditions) | all nine hold, so 9/9 Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. - CONFIRMED | [card-result-0] GM = 1 exactly | git show 37137afa5c:public/strata/khinchin/index.html | GM = 1 exactly Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. - CONFIRMED | [card-result-1] GM = 2.0000 → 2 misses K0 below | https://oeis.org/A040000 (recreated) | GM 2.0000 Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. - CONFIRMED | [card-result-2] GM = 2.592 → ∞ diverges (Euler's pattern) | https://oeis.org/A003417 (recreated: python, 60 terms [2;1,2,1,...,40,1,1]) | GM 2.592 Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. - CONFIRMED | [card-result-3] GM ≈ 2.751, K₀ = 2.685 consistent with K0, never proven | https://oeis.org/A001203 (recreated: python, 60 CF terms of the pi preset) | GM 2.751 Confirmation carried from research/assay-line/queue-1014c/khinchin/N2-claude.md. ## Findings - [fixed] X1 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] X2 (MINOR): Replaced the known/unknown dichotomy with the particular patterns of phi, sqrt(2) and e. - [fixed] X3 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] X4 (WRONG): Added direct BigInt rational CF extraction, retained exact integer coefficients through display and convergents, and tested the actual input handler. Zero denominators are rejected. - [fixed] X5 (MINOR): Specified limiting coefficient frequency in the page and Gauss measure in the verifier; the verifier also states the uniform probability 1/2. - [fixed] X6 (WRONG): Removed the infinite-variance explanation and copied logarithmic-rate assertions without claiming a universal replacement rate. - [fixed] X7 (MINOR): Named the three specific failures and distinguished the unresolved fourth card. - [fixed] X8 (WRONG): Replaced maximum entropy with zero entropy while preserving the quoted question and answer. - [fixed] X9 (WRONG): Retained periodicity and algebraicity, removed universal exclusion in body and verifier, and made terminating square inputs take the rational verdict. Only the established sqrt(2), sqrt(3), sqrt(7) cases say they miss K0. - [fixed] X10 (WRONG): Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - [fixed] X11 (WRONG): Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - [fixed] X12 (WRONG): Restored the missing factor 2 and specified fractional coefficients; divergence is retained. - [fixed] X13 (WRONG): Removed the blanket no-named-proof claim from body, summary, related notes and verifier; pi remains an explicitly open example. - [fixed] X14 (WRONG): Changed the exponent to n-1. - [fixed] X15 (MINOR): Omitted the imprecise historical attribution, retaining ergodicity and the separately cited distribution history. - [fixed] X16 (WRONG): Explicitly said ergodicity does not mean the quoted absence of cycles, leaving the quotation intact. - [fixed] X17 (WRONG): Replaced the universal structure criterion with rationals, phi, sqrt(2), and e and their separate arguments. - [fixed] X18 (WRONG): Said a specified input requires its own argument; removed the false null-set explanation. - [fixed] X19 (MINOR): Explained that both widgets display exponentials of the formula sides. - [fixed] X20 (MINOR): Replaced the ambiguous logarithmic-rate and five-digit wording with the tail expression and measured error. - [fixed] X21 (MINOR): Changed Eight to Nine, retaining nine checks. - [fixed] X22 (MINOR): Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - [fixed] X23 (WRONG): Corrected the section and credits; retained Chicago 1964 and the 1961 third Russian edition. - [fixed] X24 (WRONG): Changed the bibliography and verifier source citation to 1935 while retaining the theorem date 1934. - [fixed] X25 (WRONG): Changed 100000 to 7350 decimal places and issue 219 to 217. Kept the author-supported 417-431 pagination. - [fixed] X26 (WRONG): Corrected the citation year and preserved the quoted title. - [fixed] X27 (WRONG): Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - [fixed] X28 (MINOR): Replaced section V with that locator and retained the quoted title. - [fixed] X29 (MINOR): Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - [fixed] X30 (WRONG): Removed the blanket no-named-proof claim from body, summary, related notes and verifier; pi remains an explicitly open example. - [fixed] X31 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] X32 (MINOR): Labelled the visible coefficient-value bins 1 to 14. - [fixed] X33 (MINOR): Changed the modal label and subscript to pooled coefficients and a_i; also corrected the browser verifier's matching label. - [fixed] X34 (MINOR): Disclosed the 1 incommensurable quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still attributes Euclidean subtraction to that proof; it needs an annotation in the other layer, preserving the quotation. - [open note:94a266] X39 (MINOR): Removed the necessity implication from Khinchin while retaining sqrt(2)'s mean and Lagrange's classification. The preserved khinchin -> incommensurable quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still implies Lagrange is necessary for the irrationality proof; it needs an annotation in the other layer, preserving the quotation. - [open note:81436a] X40 (WRONG): Rewrote Khinchin's relation around exact identities and open questions, and regenerated its nearby teaser. The preserved khinchin -> harmonics-of-the-primes quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still calls both identities ergodic and denies proofs for named reals; it needs an annotation in the other layer, preserving the quotation. - [open note:81436a] X41 (WRONG): Removed the blanket no-named-proof claim from Khinchin's body, summary, related notes and verifier; pi remains an explicitly open example. The preserved khinchin -> harmonics-of-the-primes quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still denies a proof for any named real, despite Wieting's Z2 construction; it needs an annotation in the other layer, preserving the quotation. - [open note:9109c3] X42 (WRONG): Separated the unresolved sixth value and the lack of a uniform algorithm in Khinchin's relation. The preserved khinchin -> the-longest-finite-race quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still calls every later value uncomputable; it needs an annotation in the other layer, preserving the quotation. - [open note:9109c3] X43 (WRONG): Removed the blanket no-named-proof claim from Khinchin's body, summary, related notes and verifier; pi remains an explicitly open example. The preserved khinchin -> the-longest-finite-race quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still denies a proof for every named real, despite Wieting's Z2 construction; it needs an annotation in the other layer, preserving the quotation. - [fixed] X44 (WRONG): Retained periodicity and algebraicity, removed universal exclusion in body and verifier, and made terminating square inputs take the rational verdict. Only the established sqrt(2), sqrt(3), sqrt(7) cases say they miss K0. - [fixed] X46 (WRONG): Replaced all 191 preset decimal places and the copied 110-place E_DEC verifier input with same-length correct prefixes. - [fixed] X47 (WRONG): Replaced all 194 supplied decimal places, preserving the exact periodic-CF routine. - [fixed] X48 (WRONG): Replaced the page's 60-place constant and preset plus the verifier's 63-place constant. Refreshed the published verifier verbatim. - [fixed] X49 (MINOR): Relabelled distance as signed difference without changing the calculation. - [fixed] K1 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] K2 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] K3 (MINOR): Specified up to 60 coefficients and precision limits; removed the guarantee from comments and labelled the heuristic prefix count. - [fixed] K4 (WRONG): Removed the infinite-variance explanation and copied logarithmic-rate assertions without claiming a universal replacement rate. - [fixed] K5 (MINOR): Named the three specific failures and distinguished the unresolved fourth card. - [fixed] K6 (WRONG): Retained periodicity and algebraicity, removed universal exclusion in body and verifier, and made terminating square inputs take the rational verdict. Only the established sqrt(2), sqrt(3), sqrt(7) cases say they miss K0. - [fixed] K7 (WRONG): Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - [fixed] K8 (WRONG): Restored the missing factor 2 and specified fractional coefficients; divergence is retained. - [fixed] K9 (WRONG): Changed the exponent to n-1. - [fixed] K10 (MINOR): Omitted the imprecise historical attribution, retaining ergodicity and the separately cited distribution history. - [fixed] K11 (WRONG): Replaced the universal structure criterion with rationals, phi, sqrt(2), and e and their separate arguments. - [fixed] K12 (WRONG): Said a specified input requires its own argument; removed the false null-set explanation. - [fixed] K13 (MINOR): Changed Eight to Nine, retaining nine checks. - [fixed] K14 (MINOR): Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - [fixed] K15 (WRONG): Corrected the section and credits; retained Chicago 1964 and the 1961 third Russian edition. - [fixed] K16 (WRONG): Changed the bibliography and verifier source citation to 1935 while retaining the theorem date 1934. - [fixed] K17 (WRONG): Changed 100000 to 7350 decimal places and issue 219 to 217. Kept the author-supported 417-431 pagination. - [fixed] K18 (WRONG): Corrected the citation year and preserved the quoted title. - [fixed] K19 (WRONG): Fixed the body and apparatus to a0=2, a_(3k-1)=2k for k>=1, other positive-index terms 1; retained the supported 1737 date. - [fixed] K20 (MINOR): Narrowed page-authored live and offline coverage copy and verifier comments. The shared check placard is untouched. - [fixed] K21 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] K22 (MINOR): Labelled the visible coefficient-value bins 1 to 14. - [fixed] K23 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [fixed] K24 (WRONG): Restricted the body, dek, descriptions and summary to proved failures and unresolved presets; explicit constructions are acknowledged. - [open note:94a266] K25 (WRONG): Changed Khinchin's relation to describe the parity proof. The preserved khinchin -> incommensurable quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still attributes Euclidean subtraction to that proof; it needs an annotation in the other layer, preserving the quotation. - [open note:94a266] K26 (MINOR): Removed the necessity implication from Khinchin while retaining sqrt(2)'s mean and Lagrange's classification. The preserved khinchin -> incommensurable quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still implies Lagrange is necessary for the irrationality proof; it needs an annotation in the other layer, preserving the quotation. - [open note:81436a] K27 (WRONG): Rewrote Khinchin's relation around exact identities and open questions, and regenerated its nearby teaser. The preserved khinchin -> harmonics-of-the-primes quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still calls both identities ergodic and denies proofs for named reals; it needs an annotation in the other layer, preserving the quotation. - [open note:9109c3] K28 (WRONG): Separated the unresolved sixth value and the lack of a uniform algorithm in Khinchin's relation. The preserved khinchin -> the-longest-finite-race quotation in public/strata/there-here/index.html:250 and research/lineage-edge-grammar/page.json:1 still calls every later value uncomputable; it needs an annotation in the other layer, preserving the quotation. - [fixed] K29 (WRONG): Replaced all 191 preset decimal places and the copied 110-place E_DEC verifier input with same-length correct prefixes. - [fixed] K30 (WRONG): Replaced all 194 supplied decimal places, preserving the exact periodic-CF routine. - [fixed] K31 (WRONG): Replaced the page's 60-place constant and preset plus the verifier's 63-place constant. Refreshed the published verifier verbatim. - [fixed] K32 (MINOR): Corrected the source-comment identifier while leaving the valid ln(2) digits unchanged. - [fixed] K33 (WRONG): Specified limiting coefficient frequency in the page and Gauss measure in the verifier; the verifier also states the uniform probability 1/2.