# Verification report — OWR-13677-003 (Durante, Fernández-Sánchez, Trutschnig: are purely singular copulas of second category in (C_d, D_1)?)

Verification date: 2026-10-14.

**Verdict.** The note gives a **complete answer** to the question as posed in the source; every statement it makes
is proved. The answer itself follows in a few lines from known results, and the note claims no priority for it.
- **Settled.** For every d >= 2 the purely singular d-copulas form a dense G_delta set in (C_d, D_1), where D_1 is
  the Markov-kernel metric of Trutschnig (2011) and Fernández-Sánchez and Trutschnig (2015); hence they are
  co-meager and of second category (Theorem 1.1). The answer is yes; the question asked only for the second
  category.
- **Not new: Theorem 1.1 in every dimension.** Its ingredients are known: the purely singular copulas form a G_delta
  set already for the uniform metric (standard; for d = 2, Theorem 4.1 of Durante, Fernández-Sánchez, Trutschnig,
  "A typical copula is singular", JMAA 2015, gives a dense G_delta subset); (C_d, D_1) is complete; and the check-min
  approximations B_N(A) are purely singular and converge to A in D_1. For d = 2 this density is classical (Brown 1966
  Theorem 2; Kim 1968 Theorem 2.2 with Lemma 2.2, whose approximants with translations are the B_N, N = 2^n;
  Iwanik 1980 Corollary 1; Kasper, Fuchs and Trutschnig, Bernoulli 2021, Theorem 3.2 and its proof), and the 2015
  paper itself states it before its Theorem 4.1 and uses it, citing Mikusiński and Taylor (Metrika 72 (2010),
  Theorem 6), in the proof of its Theorem 3.2; yet its Section 5 lists the question as open. For d >= 3 the density
  may be in Mikusiński and Taylor (2010), a paper on check-min, checkerboard, Bernstein and shuffle-of-Min
  approximations of n-copulas in arbitrary dimension that could not be read; their ∂-convergence of d-copulas is
  D_1-convergence for every conditioning coordinate (Remark 3.8 of the note).
- **New, to our knowledge.** Theorem 1.2: the copulas whose conditional distributions are singular and atomless for
  almost every value of the conditioning coordinate form a dense G_delta set in (C_d, D_1); Corollary 1.3: the same
  with full support added, so completely dependent copulas and copulas with a Lebesgue-null support are meager in
  D_1. Copulas with singular atomless conditional distributions were constructed by Trutschnig and
  Fernández-Sánchez (JMAA 2014); their typicality was not found in print. Smaller additions: the explicit bound
  D_1(A, B_N(A)) <= ||H_A - P_N H_A||_1 + (d - 1)/N and the identity ||P_N H_A - H_{B_N}||_1 = 1/(3N) for d = 2.
- **Statement in the corpus.** In version 1.7.0 (commit 372682f) the UnsolvedMath record restates the question with
  D_1 defined as the L^1 distance of the copula functions; in version 1.6.0 its statement was the source's sentence.
  The L^1 distance is not the source's metric; under it the question reduces to the known result for the uniform
  metric (Proposition 5.1).

The note is unrefereed.

## Statement checked
- **Primary source.** F. Durante (joint work with J. Fernández-Sánchez and W. Trutschnig), "Singular copulas",
  Oberwolfach Report 20/2015 ("Copulae: On the Crossroads of Mathematics and Economics", 12–18 April 2015),
  Oberwolfach Rep. 12 (2015), no. 2, pp. 1124–1127, doi:10.4171/OWR/2015/20 (issued 2016).
  - Read in the publisher's PDF (EMS Press, 54 pages, 579,050 bytes, sha256 `7abfc7ae…cecb65`), downloaded
    anonymously on 2026-10-14 (twice, identical files).
  - p. 1124: copulas on I^d with uniform univariate marginals; Lebesgue decomposition μ_C = μ_C^ac + μ_C^s.
  - p. 1125: the Markov kernel K_C(x, ·) of the last d − 1 coordinates given the first; ∂_1 C(x, y) = K_C(x, [0, y])
    for almost every x; for d = 2 the decomposition K = K^a + K^s + K^d (singular part without atoms, discrete part).
  - p. 1126: "the D1 metric in [15]" ([15] = Trutschnig, JMAA 384 (2011) 690–705) and "the M-convergence in [11]"
    ([11] = Mikusiński, Taylor, Metrika 72 (2010) 385–414); Theorem 2 (complete dependence copulas are closed in
    (C_2, D_1)); Theorem 3, proved in [4] = "A typical copula is singular": absolutely continuous copulas are of
    first category in (C_d, d_∞) and (C_d, D_1); "(purely) singular" copulas are co-meager in (C_d, d_∞); Baire
    notions defined. The question: it remains open whether the class of purely singular copulas is of second
    category in (C_d, D_1).
  - The source does not write D_1 for d >= 3; the multivariate version is Fernández-Sánchez and Trutschnig,
    J. Theoret. Probab. 28 (2015) 1311–1336, restated (with conditioning on the last coordinate) in Mroz, Fuchs,
    Trutschnig, Electron. J. Stat. 15 (2021) 1951–1992, p. 4.
- **Closest prior source.** Durante, Fernández-Sánchez, Trutschnig, "A typical copula is singular", JMAA 430 (2015)
  517–527, read in full in the publisher's version (saved by the author). Bivariate only (extension to d >= 3 "with
  simple and obvious modifications"). Section 5 states that it remains open whether C_sing is of second category in
  (C, D_1). It proves Corollary 3.1 (absolutely continuous copulas first category in D_1), Theorem 3.2 (checkerboards
  first category; its proof takes check-min copulas D_1-close to Π from Mikusiński–Taylor 2010, Theorem 6),
  Theorem 4.1 and Corollary 4.1 (C_sing co-meager in d_∞; before Theorem 4.1 it states the D_1-density of rational
  checkerboards and that "the same holds" for the rational check-min copulas), Theorem 4.2 and Remark 4.2 (shuffles
  of Min). Not stale: it contains no statement on C_sing in (C, D_1) other than the open question.
- **Corpus record.** ulamai/UnsolvedMath, OWR-13677-003 (upstream status open). Version 1.6.0 (commit c6e7f41):
  statement = the source's sentence. Version 1.7.0 (commit 372682f, the current one on 2026-10-14): restated
  ("clean") statement with D_1(C_1, C_2) = ∫|C_1(u) − C_2(u)| du and "nonmeager", status "corrected_verified"; this
  definition is not the source's.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Source: D_1 = Markov-kernel metric, purely singular = μ^ac = 0, every d >= 2 | yes: dense G_delta, co-meager, second category | Theorem 1.1 |
| D_1 with conditioning on another coordinate (as in Mroz–Fuchs–Trutschnig), sums or maxima of these, D_p, D_∞; Mikusiński–Taylor's ∂-convergence | yes | Remarks 3.7, 3.8 |
| (R1) D_1 = L^1 distance of the copula functions (corpus restatement, version 1.7.0) | equivalent to d_∞: d_∞^{d+1} <= 2(4d)^d L; the question is the known d_∞ result | Proposition 5.1, Remark 3.4 |
| (R2) "singular" = has a singular component (loose phrase on p. 1124) | trivial: complement of the absolutely continuous copulas, co-meager by the source's Theorem 3(i) | Section 5 |
| (R3) "purely singular" = support of Lebesgue measure zero | no: these copulas are meager (in D_1 and in d_∞); contradicts the source's Theorem 3(ii), so not the source's reading | Corollary 1.3, Section 5 |
| (R4) kernel level: K^a = K^d = 0 for almost every x (the "singular" copulas of Dietrich–Trutschnig) | yes in D_1; meager in (C_2, d_∞), so not the source's reading of Theorem 3(ii) | Theorem 1.2, Section 5 |

## Results in the paper
- **Lemma 3.1.** d_∞(A, B)^d <= 2(4(d − 1))^{d−1} D_1(A, B); D_1-topology finer than d_∞.
- **Lemma 3.2.** S_d = ∩_k G_k, with G_k = {A : μ_A(V) > 1 − 1/k for a finite union V of rational boxes with
  λ^d(V) < 1/k} open already for d_∞.
- **Lemma 3.3.** The diagonal checkerboard (check-min approximation) B_N(A) is a purely singular copula, and
  D_1(A, B_N(A)) <= ||H_A − P_N H_A||_1 + (d − 1)/N → 0.
- **Theorem 1.1** by the Baire category theorem; completeness of (C_d, D_1) proved in Appendix A.
- **Remark 3.4.** The same lemmas re-prove the d_∞ result for every d >= 2.
- **Remark 3.5.** For d = 2: ||P_N H_A − H_{B_N}||_1 = 1/(3N) for every copula A, hence
  |D_1(A, B_N(A)) − 1/(3N)| <= ||H_A − P_N H_A||_1; D_1(Π, B_N(Π)) = 1/(3N); B_N(Π) is the uniform mixture of the N
  circular shifts.
- **Remark 3.6.** The case d = 2 in the literature: strong operator convergence implies D_1-convergence; Brown,
  Kim, Iwanik; Kim's approximants with translations are the B_N; the proof of Kasper–Fuchs–Trutschnig, Theorem 3.2,
  applies to the B_{2^n}; the G_delta set of Theorem 4.1 of the closest source is D_1-dense (rational cell masses
  via Birkhoff's theorem).
- **Remark 3.7.** Other conditioning coordinates, sums and maxima, D_p, D_∞ (including completeness of D_∞).
- **Remark 3.8.** Mikusiński–Taylor's ∂-convergence of d-copulas is D_1-convergence for every conditioning
  coordinate; what open sources say about the check-min results of Mikusiński–Taylor (2010).
- **Lemmas 4.1–4.5, Theorem 1.2, Corollary 1.3.** Lebesgue decomposition commutes with disintegration;
  D_1-convergence gives a.e. weak convergence of kernels along a subsequence (with the upper-face condition for
  d >= 3); the largest atom is measurable and upper semicontinuous; Ψ(A) = ∫ a(K_A(x, ·)) dx is upper semicontinuous,
  so the atomless-kernel copulas form a G_delta; checkerboards with rotated Cantor noise give density; full support is
  a dense G_delta.
- **Proposition 5.1.** L(C_1, C_2) <= d_∞(C_1, C_2) and d_∞^{d+1} <= 2(4d)^d L.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of the note** (`check_note.py`, standard library, exact rationals, d = 2, 2 s; TOTAL failures: 0):
  D_1(Π, B_N) = 1/(3N) for N <= 12; for 58 random rational checkerboards (2 <= M <= 8, 1 <= N <= 10, also N not
  dividing M and N > M) the margins of B_N, the identity ||P_N H_A − H_{B_N}||_1 = 1/(3N) and the two-sided bound
  of Remark 3.5 (largest ratio of D_1(A, B_N) to the bound of Lemma 3.3(c): 0.547); Lemma 3.1 on 90 pairs (largest
  d_∞²/D_1 = 0.5); the kernel of B_N at 125 rational points; B_N(Π) = mixture of the N shifts for N <= 8; two
  deliberately wrong inputs rejected.
- **Programs of the first version** (`original/`): exact D_1(Π, B_N) = 1/(3N), N <= 12; midpoint-rule estimates
  for FGM(1/2), (M+Π)/2, Clayton(2) (d = 2, N <= 64) and FGM_3(1/2), (M_3+Π_3)/2, Π_3 (d = 3, N <= 32): all 39 lines
  below the bound; N D_1(A, B_N) about 0.15–0.32. The d = 3 values are midpoint estimates (256³ grid), 0.3–1 % too
  high (Π_3, N = 1: 0.31765 against 19/60 = 0.31667).
- **Verification run A** (`verification_run_A/`): exact d = 2 checks (Π for N <= 12; 36 random checkerboards, cell
  term always exactly 1/(3N); Lemma 3.1 on 195 pairs, largest d_∞²/D_1 = 0.5); numerics for completely dependent
  copulas (D_1(W, B_N) = 1/(2N)), for d = 3 (D_1(Π_3, M_3) = 19/60) and six random 3-dimensional checkerboards; the
  Cantor-noise approximants of Lemma 4.5 for d = 2 (N D_1(Π, B'_N) = 0.1047).
- **Verification run B** (`verification_run_B/`): an exact engine for D_1 in d = 2 validated on known values; the
  bound of Lemma 3.3(c) exactly on 84 cases (largest ratio 0.573), the per-x estimate at 360 points, deliberately
  wrong inputs; a floating-point cross-check of the engine (agreement 1.5e-5); high-accuracy values for FGM(1/2),
  Clayton(2) (d = 2) and Π_3, FGM_3(1/2), (M_3+Π_3)/2 (d = 3); Kim's approximants equal B_N exactly on 18 cases
  (N = 2, 4, 8).
- **Second round** (`independent_run_2/`): an exact program for d = 2 with a new method (refinement to a common grid,
  closed-form cell integrals): D_1(Π, B_N) = 1/(3N) for N <= 16; the identity of Remark 3.5 and both bounds on 64
  random rational checkerboards with all divisibility patterns of M and N (largest ratio to the bound of Lemma
  3.3(c): 0.637); B_N(M) = M, D_1(W, B_N(W)) = 1/(2N), D_1((M+Π)/2, B_N) = 1/(6N); Lemma 3.1 on 70 pairs; the
  estimate of Remark 3.6(d) on 50 pairs; wrong inputs detected. A floating-point program for d = 3: exact uniform
  margins of B_N, the cell estimate at every node and Lemma 3.3(c) on 12 random three-dimensional checkerboards
  (N ||P_N H_A − H_{B_N}||_1 between 0.305 and 0.327, against the bound 2), D_1(Π_3, M_3) = 0.31663 against
  19/60, Lemma 3.1 for d = 3 (32 pairs) and Proposition 5.1 for d = 2, 3 (58 pairs). TOTAL failures: 0 in both.
- **Re-runs.** On 2026-10-14 all programs of the first package with a recorded output were run again from a copy of
  the package (`reproducibility/RERUN_LOG.txt`) and, by the second round, once more from an extracted copy of the
  archive (`reproducibility/independent_run_2/rerun_first_package.txt`); in both re-runs every output was
  reproduced, up to lines that record running times.

## Independent verification runs
The results were first obtained in a written version with proofs and the two programs of `original/`. Two rounds
of independent verification, all AI-assisted, followed (2026-10-14); each run re-derived the proofs and wrote its
own programs. In the first round, run A concentrated on the statement and the proofs, run B on independent
computation, the literature and the bibliography; both examined the first written version. In the second round a
further run examined the text of the note.

| Item | Run A | Run B | Second round (text of the note) |
|---|---|---|---|
| Statement, conventions, corpus restatement against the source | CONFIRMED (wording fixes F1–F3) | CONFIRMED | CONFIRMED; the restatement is in version 1.7.0, not 1.6.0 |
| Lemma 3.1, Lemma 3.2, Lemma 3.3, Theorem 1.1, Appendix A | CONFIRMED (line by line, boundary cases included) | CONFIRMED (line by line) | CONFIRMED (line by line) |
| Remarks on other coordinates, D_p, D_∞ | CONFIRMED (wording fix F4) | CONFIRMED | CONFIRMED |
| Theorem 1.2 with Lemmas 4.1–4.5, Corollary 1.3 | CONFIRMED | CONFIRMED (a further reader advised) | CONFIRMED, Corollary 1.3 included |
| Identity ||P_N H_A − H_{B_N}||_1 = 1/(3N) for d = 2 | found and proved | consistent (exact cases) | re-derived; exact on 64 new cases |
| Cited results | — | Kim, Brown, Iwanik, KFT 2021, DT 2025 read | all compared with the texts read (Brown, Kim, Iwanik, KFT 2021, DT 2025, MFT 2021, FST 2026, MT 2009, the closest source) |
| Computations | re-run, identical outputs | re-run, identical outputs; d = 3 midpoint values 0.3–1 % high | own programs (d = 2 exact, d = 3 numeric); whole package re-run from the archive, outputs reproduced |
| Literature, novelty | closest source unread then (F7); FSS 2016 overlooked (F5) | d = 2 density classical; novelty low; EJOR 2015 and Metrika 2010 unread | Theorem 1.1 follows from known results in every dimension (MT 2010 probably contains the density for d >= 3); new: Theorem 1.2, Corollary 1.3 |
| Overall | complete answer | complete answer; credit for d = 2 required | complete answer; no priority for Theorem 1.1 |

No run found a wrong lemma or theorem, or a gap.

**Corrections required by runs A and B**, all applied in the note:
1. (A, F1) The source does not write D_1 for d >= 3; it cites the multivariate paper only for the kernel
   representation (Section 1.1, remark (i)).
2. (A, F2) Completeness is attributed to Trutschnig (2011) for d = 2 and to Mroz–Fuchs–Trutschnig (2021) for all d,
   with a self-contained proof (Section 2, Appendix A).
3. (A, F3) The readings (R2) and (R3) are added, and the sentence on "singular" versus "purely singular" is
   qualified (Section 1.1, Section 5).
4. (A, F4) "The construction A ↦ B_N commutes with permutations of the coordinates" (Remark 3.7(a)); "B_N(Π) is not
   completely dependent" is stated for N >= 2 (Remark 3.5).
5. (A, F5, F6) The Baire paper on exchangeable copulas (FSS 284 (2016)) is listed among the unread sources; the
   note states precisely what was read.
6. (A, F7; B) The closest source JMAA 430 (2015) 517–527 was read in full (the author saved it); it does not contain
   the D_1 statement. EJOR 247 (2015) and Metrika 72 (2010) were not available.
7. (A, F8) The note states that the corpus restatement mis-defines D_1 and that under it the question reduces to
   the known d_∞ result (abstract, Section 1.1, Proposition 5.1).
8. (B) Credit for d = 2 (Brown, Kim, Iwanik, Kasper–Fuchs–Trutschnig) and bibliographic data completed.
9. (B) Constant range "about 0.15–0.32"; the d = 3 values are labelled as midpoint estimates.

**When the note was written**, the closest source was read in full; Remark 3.6(d) (the G_delta set of its Theorem
4.1 is D_1-dense), the argument of Remark 3.6(b), Corollary 1.3 and the program `check_note.py` were added.

**Corrections required by the second round**, all applied in this version:
1. The statement of priority: Theorem 1.1 is presented, in every dimension, as following in a few lines from known
   results, with no priority claimed; Mikusiński–Taylor (2010) is named as unread in the abstract; the remaining
   new content is stated exactly (abstract, Section 1.3, Scope and priority).
2. The note says plainly that the closest source lists the question as open although the case d = 2 follows from
   its Theorem 4.1 and the D_1-density it states and uses (Section 1.3, Remark 3.6(d)).
3. Remark 3.8 relates Mikusiński–Taylor's ∂-convergence to D_1 and reports what open sources say about the
   check-min results of Mikusiński–Taylor (2010).
4. The version of the corpus record (1.7.0; in 1.6.0 the statement is the source's sentence).
5. Credit to Trutschnig and Fernández-Sánchez (JMAA 410 (2014)) for copulas with singular atomless conditional
   distributions (Section 1.2).
6. The citation of Dietrich–Trutschnig (2025) (their B-checkerboards are applied for B in a class C_Q; B = M is
   covered by Remark 3.6(c)) and of Kim (1968) (Theorem 1.1 and the remark after its proof).
7. Remark 3.6(d) explains why rational cell masses with the same margins exist (Birkhoff's theorem).

## Relation to the literature, novelty and scope
- **Searches (2026-10-14).** arXiv API (copulas with Baire, typical, meager, first category, generic; doubly
  stochastic measures and Markov operators with Baire category; the proposers' papers since 2015), zbMATH (copulas
  with Baire category; doubly stochastic operators with residual, category, generic, typical; the record of
  Mikusiński–Taylor 2010, whose review is not available), Crossref, OpenAlex (14 works citing the JMAA 2015 paper,
  39 citing the JTP 2015 paper, 48 citing Mikusiński–Taylor 2010), Semantic Scholar (citation contexts of the JMAA
  2015 paper and of the 62 works citing Mikusiński–Taylor 2010), nine web searches.
  - No text answering the question was found. The proposers' own 2026 paper on (C, D_1) (arXiv:2609.29275) cites
    the singular-copula result only for d_∞.
  - What Mikusiński–Taylor (2010) contains could not be read; their abstract (UCF repository record), their 2009
    paper (Ann. Polon. Math. 96, Definition 3 and Theorem 11), the closest source, Kasper–Fuchs–Trutschnig (2021,
    Section 2), Ansari–Wiesel (arXiv:2606.07354v1, Section 2.3) and Ansari–Fuchs (arXiv:2503.11390v2, Remark 3.7)
    indicate that it proves ∂-convergence of checkerboard, Bernstein and (very probably) check-min approximations
    of n-copulas, which is D_1-convergence for every conditioning coordinate (Remark 3.8).
  - Related Baire results in the strong operator topology concern other properties: ergodicity (Iwanik 1980,
    Proposition 4), mixing (Iwanik 1992) and zero entropy (Frej–Huczek, Ann. Funct. Anal. 10 (2019) 144–156); for
    subclasses of copulas with D_1: exchangeable/associative/Archimedean (Durante–Fernández-Sánchez–Trutschnig 2016),
    associative (Kasper–Fuchs–Trutschnig 2021).
- **What was read.** The source pp. 1124–1127; the closest source in full; Kim 1968, Brown 1966 and Iwanik 1980 in
  full; Mikusiński–Taylor 2009, Sections 1 and 3.4; Kasper–Fuchs–Trutschnig 2021 (arXiv v2) Sections 2–3;
  Dietrich–Trutschnig (arXiv v1) Sections 2–3; Mroz–Fuchs–Trutschnig (arXiv v2) pp. 3–5; arXiv:2609.29275 Sections
  1, 2, 4.1; arXiv:2606.07354 Section 2.3; arXiv:2503.11390 Section 3.3; Iwanik 1992 (zbMATH review only);
  Kasper–Fuchs–Trutschnig, Dependence Modeling 2021 and Frej–Huczek 2019 (abstracts only); Mikusiński–Taylor 2010
  (abstract only). Not read: Trutschnig 2011 and Fernández-Sánchez–Trutschnig 2015 (publisher sites refused
  automated access; their results are quoted from restatements), Trutschnig–Fernández-Sánchez 2014 (quoted from the
  closest source), FSS 284 (2016), EJOR 247 (2015), Metrika 72 (2010). All DOIs were checked with Crossref.
- **Caveats.** The search of the operator-theoretic literature on generic properties of doubly stochastic operators
  was limited; Theorem 1.2 for d = 2 might be known there in another language. The negative search is not a proof of
  priority.
- **Scope.** The note answers the question for every d >= 2 and adds Theorem 1.2 and Corollary 1.3. No priority is
  claimed for Theorem 1.1.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/