# Verification report — OWR-16415-018 (Gekhtman: preimages of four-holed and three-holed spheres under finite covers of surfaces)

Verification date: 2026-10-09.

**Verdict.** The note gives a **partial answer**; every statement it makes is proved or is an exhaustive finite
computation, and the scope is as follows.
- **Settled, in genus 2 only.** The four-holed question has a negative answer in genus 2: there is a cover
  S_10 → S_2 of degree 9, with monodromy group M_9 = 3^2:Q_8 of order 72, for which the preimage of every
  four-holed sphere with essential boundary components is connected (Theorem 1.1). In genus 2 every such four-holed
  sphere is the complement of two disjoint curves, so its boundary curves are isotopic in pairs (Lemma 3.1). The
  statement is therefore one about the literal reading of the question. If the four boundary curves are required to
  be pairwise non-isotopic, genus 2 contains no such subsurface and the question starts at genus 3.
- **A family.** The same holds for covers of S_2 of degree p^2 with group F_p^2 ⋊ Q_8, p ≡ 3 (mod 4) prime, whose
  monodromy does not lift to a central extension by Z/p (Theorem 1.2). For p = 5 and p = 13 the conclusion fails.
- **Not a counterexample elsewhere.** The same groups give no example in genus ≥ 3, under either reading
  (Proposition 1.3). The example is not a regular cover, and its Galois closure is not a counterexample.
- **Positive results.** In every genus there are three-holed and four-holed spheres with disconnected preimage for
  covers which factor through a non-trivial abelian cover, for abelian-by-cyclic monodromy groups, for regular covers
  with characteristic kernel, and when the genus is at least three times the order of the monodromy group
  (Theorem 1.4).
- **Computations in genus 2.** 9 is the smallest degree of an example; in degrees 9 and 10 the only examples are the
  covers with group M_9 whose monodromy does not lift; none in degree 11 unless the group contains A_11; none among
  regular covers with deck group of order < 12180 (Theorems 1.5 to 1.7).
- **Open.** The three-holed question (the covers of the note do have pairs of pants with disconnected preimage); the
  four-holed question in genus ≥ 3 under both readings; both questions for regular covers.
- **Not new mathematics.** The obstruction is the classical invariant of a surface group homomorphism in H_2 of the
  target group (Livingston 1985; Zimmermann 1987; Edmonds; Dunfield–Thurston 2006, Section 6); its use follows the
  model of the lifting invariant of Hurwitz theory (Fried–Serre; Conway–Parker, Fried–Völklein). No priority is
  claimed.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Gekhtman, Problem 12 of the problem session (compiled by R. Penner, pp. 2525–2529) in:
  "New Trends in Teichmüller Theory and Mapping Class Groups", Oberwolfach Reports 15 (2018), no. 3, 2475–2534,
  Report No. 40/2018, doi:10.4171/OWR/2018/40; the problem is on p. 2528.
  - Read in the open file of the Oberwolfach repository,
    https://publications.mfo.de/bitstream/handle/mfo/3662/OWR_2018_40.pdf (60 pages; sha256
    `b4fce9564e67a49dd92f61341f04f53a65c434647b1f3545c3c48b53e3e54b9c`), fetched five times on 2026-10-09 (first
    stage, runs A and B, at the writing, and by run 2, which read p. 2528 as a rendered image); the checksum
    recorded by run A agrees with the one above.
  - The problem fixes a finite cover f : S_h → S_g with h > g ≥ 2 and asks whether there is always a sphere minus
    four discs H with essential boundary components in S_g such that the interior of f^{-1}(H) is disconnected; then
    the same for a sphere minus three discs; it adds that one may consider only regular covers.
  - Motivation given there: for the induced map of Teichmüller spaces the proposer has a result (the smaller space
    is not a holomorphic retract of the bigger one) which needs such an H; the requirement may be unnecessary.
  - The name printed with the problem is Dmitri Gekhtman.
- **Corpus record.** ulamai/UnsolvedMath, OWR-16415-018 (dataset version 1.6.0; upstream status `open`). Its
  statement gives the two questions; it omits the remark on regular covers and the motivation.
- **The proposer's theorem.** Its statement and exact hypothesis were not found: not in his thesis (Caltech 2019,
  doi:10.7907/XKMM-8591; abstract read, text searched by both verification runs), not in the abstracts of his
  papers that were found. So the motivation cannot be used to decide between the readings.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (G) literal: each boundary curve of H is essential; two of them may be isotopic. Genus 2 | **no**: the cover of Theorem 1.1 | Theorem 1.1, Lemma 3.1 |
| (S) strict: moreover the four boundary curves are pairwise non-isotopic. Genus 2 | void: no such subsurface exists in S_2 | Lemma 3.1(a) |
| (G) or (S), genus ≥ 3 | open; yes for the classes of Theorem 1.4 and for the groups F_p^2 ⋊ Q_8 | Theorem 1.4, Proposition 1.3 |
| Three-holed spheres, any genus | open; yes for the classes of Theorem 1.4; in genus 2 yes for degree ≤ 10 and further classes | Proposition 1.8, Section 10 |
| Regular covers only | open; yes for non-perfect deck groups, for characteristic kernels, and in genus 2 for deck groups of order < 12180 | Theorems 1.4, 1.7 |
| "Interior of f^{-1}(H) disconnected" | equivalent to f^{-1}(H) disconnected and to: the image of the group of H is intransitive | Lemma 2.1 |
| Why (G) is taken as the main reading | the source allows g = 2 and speaks of the interior of the preimage; this is an inference, the source also makes sense under (S) | Section 1.1 |

## Results in the paper
- **Lemma 3.1.** In genus 2 a four-holed sphere with essential boundary components is the complement of two
  disjoint annuli whose cores are non-isotopic, nonseparating and jointly nonseparating; homeomorphisms act
  transitively on these subsurfaces; for a suitable standard generating system the image of the fundamental group
  is J = < a1, [a1,b1], a2 > = < a1, b1 a1 b1^-1, a2, b2 a2 b2^-1 >.
- **Lemma 4.1.** For a central extension G~ → Γ and a homomorphism φ : π1(S_g) → Γ, the element
  κ = ∏ [x~_i, u~_i] does not depend on the lifts, vanishes exactly if φ lifts, and its vanishing does not depend
  on the standard generating system.
- **Lemma 5.1 (main lemma).** p odd, Q < SL(2,p) non-abelian of order prime to p in which only ±1 have an
  eigenvalue in F_p; Γ = F_p^2 ⋊ Q with its central extension by Z/p. If x, u, v, w generate Γ, [x,u][v,w] = 1 and
  J is intransitive, then κ = 1.
- **Theorem 1.1.** φ0(a1) = φ0(a2) = (z → z + (1,0)), φ0(b1) = (z → −Iz), φ0(b2) = (z → IJz + (1,0)) on F_3^2;
  as permutations (composition of maps, [p,q] = p q p^-1 q^-1): (0 1 2)(3 4 5)(6 7 8), (1 6 2 3)(4 7 8 5),
  (0 1 3 5)(2 8 4 7). φ0 is onto M_9, κ(φ0) = [0,2;1] ≠ 1, and every four-holed sphere with essential boundary
  components has connected preimage. The same for every epimorphism onto M_9 that does not lift.
- **Theorem 1.2.** The family for p ≡ 3 (mod 4), with explicit epimorphisms and κ = −c1 − (1+α)/2; the general
  hypothesis on Q. **Proposition 6.1:** explicit non-liftable epimorphisms with J intransitive for p = 5 and 13.
- **Proposition 1.3.** For g ≥ 3 and every homomorphism π1(S_g) → F_p^2 ⋊ Q_8 with transitive image: (a) a four-holed
  sphere with four pairwise non-isotopic nonseparating boundary curves, (b) one with two isotopic boundary curves,
  with disconnected preimage.
- **Theorem 1.4.** (C1) abelian factor, (C2) abelian-by-cyclic, (C3) characteristic kernel, (C4) large genus;
  Table 2 of the note says which reading each case serves.
- **Theorems 1.5 to 1.7 (computations).** Degree ≤ 8; degrees 9, 10 (no restriction on the monodromy group) and 11
  (group not containing A_11); regular covers with deck group of order < 12180.
- **Proposition 1.8, Section 10.** The three-holed question: what is known; it remains open.

## Computations (programs and outputs in reproducibility/)
Theorems 1.1, 1.2, 1.4 and Propositions 1.3, 1.8(a), (b), 10.1 are proved without computation. Theorems 1.5 to 1.7,
Proposition 1.8(c), Proposition 6.1 for p = 13 and Remarks 5.2(b), 6.2, 6.3 are exhaustive computations.
- **Program of the note** (`writing_stage/check_note.py`, standard library, about 6 seconds; TOTAL failures: 0).
  It checks the statements as printed: the affine maps and the permutations; the relation in the four conventions
  (it holds for maps with p q p^-1 q^-1 and for left-to-right products with p^-1 q^-1 p q; it fails in the two mixed
  conventions); the product rule of the central extension for p = 3 (associative on all 216^3 triples; centre of
  order 3; projection onto M_9) and formulas (7), (8); κ(φ0) for all 81 lifts; all 1,592,136 homomorphisms
  π1(S_2) → M_9 (1,036,800 epimorphisms; 414,720 liftable, of which 25,920 with J intransitive; 622,080 non-liftable,
  all with J transitive); the 68 subgroups of M_9; the orbit of φ0 (1440 classes, for both sets of moves); pairs of
  pants; Theorem 1.2(b) for 13 primes; Proposition 6.1 with its printed intermediate values; hypothesis (E) for the
  groups of Remark 6.3; the cyclic order (4), the position of the diagonals and the boundary curves of Lemma 2.3(a)–(d)
  by face tracing of the ribbon graphs (78 checks in all).
- **Programs with which the results were first obtained** (`original/`): `verify_m9.py`, `verify_family.py`,
  `verify_m9_polygon.py`, `validate_twists.py`; `g2search.c` (n ≤ 7: all tuples, 1,188,326,736 for n = 7; n = 8:
  813,196,586 normalised tuples, 12 survivors, all settled by `g2orbit.py`); `regfull.c` (16 simple groups in the
  regular action: no exhausted orbit; natural actions of AGL(2,3), PΓL(2,8), PΓL(2,9), S_5 on pairs, AGL(1,11), M_11,
  F_p^2 ⋊ Q_8 for p = 3, 5, 7, 11: exhausted orbits only for p = 3, 7, 11, exactly p − 1 of them, and one in
  AGL(2,3)).
- **Verification run A** (`verification_run_A/`): M_9 only. Independent model of the central extension (3×3
  matrices over F_4, order 216); all 1,592,136 homomorphisms; orbit of φ0 (1440 classes modulo AGL(2,3), 4320 modulo
  M_9); three orbits on epimorphisms (4320, 4320, 5760 classes modulo M_9); cut-and-glue model: one component for
  φ0 and for all 1440 classes; the formula for J on 10,500 random homomorphisms (0 mismatches; the three sign
  variants of the formula fail); genus-3 samples.
- **Verification run B** (`verification_run_B/`): `hb.c`, all homomorphisms to Sym(n), n ≤ 10 (for n = 9:
  96,124,376,168,956,800 transitive homomorphisms, one exceptional orbit of 1440 classes with group of order 72;
  for n = 10: 95,938,557,393,889,324,800 transitive homomorphisms, no exceptional block); `orb.c` with `kappa.py`:
  F_p^2 ⋊ Q_8 for p = 3, 5, 7, 11, 13, the groups 7^2:Q_16, 11^2:Dic_3, 11^2:SL(2,3), AGL(2,3), M_11 on 11 points,
  the 16 simple groups (9 in the regular action, 7 in a faithful action), Sym(4) to Sym(7) as a cross-check.
- **Verification run 2** (`independent_run_2/`, the final verification run; programs written from the text of the
  note, without reading those of the other parts): `r2_moves.py` (its own substitutions: relator, Sp(4,Z/2),
  Sp(4,Z/3), braid and commutation relations; the ten substitutions (12) of the note); `r2_m9.py` (55 checks, 0
  failed: the maps, the permutations, the four conventions; M_9; the central extension; κ(φ0) for all 81 lifts;
  all 1,592,136 homomorphisms; three orbits on epimorphisms of 4320, 4320, 5760 classes modulo M_9 and one orbit
  of 1440 classes modulo AGL(2,3), J transitive on it; a cut-and-glue model of its own: genus 10, one component
  for φ0, agreement with the formula for J on 46,638 homomorphisms; pants; Galois closure); `r2_ribbon.py` (34
  checks: the cyclic order (4) from the relator, Lemma 2.3(a)–(e) for g = 2, 3, 4); `r2_family.py` (35 checks:
  Theorem 1.2(b), Proposition 6.1 with every printed value); `r2_fam.c` (all homomorphisms to F_p^2 ⋊ Q_8,
  p = 3, 5, 7, 11, 13: the numbers of Remark 6.2 for each value of κ); `r2_blk.c` (all homomorphisms to Sym(n),
  n ≤ 10: every entry of Table 3; for n = 10 about two minutes and 1.8 GB); `r2_grp.c` (M_11 on 11 points: all
  22,145 transitive blocks contain or lead to a failing tuple; the 16 simple groups in faithful actions of degree
  5 to 28: no exceptional block); `r2_lists.py` (the two lists of groups against Butler–McKay, Table 11A, and
  OEIS A001034, A109379).
- **Re-runs.** On 2026-10-09, when the note was written, the quick programs of the first four parts were run again
  from an extracted copy of the archive (`run_quick.sh`, one process, about 7 minutes): all outputs are identical
  to the recorded ones, or identical up to fields recording running times. Four slow programs were re-run
  separately with identical outputs (`hb 10`, `orb` for M_11, `vA_step6_genus3.py`, `genus3_test.py` for p = 7).
  Run 2 repeated `run_quick.sh` from the archive (0 differences) and `hb 10` and `orb` for M_11 (identical
  outputs), and ran `independent_run_2/run_all.sh` from an extracted copy of the final archive (0 differences).
  The long runs of `g2search` (n = 7 mode 0, n = 8 mode 1) and of `orb` for the other large groups were not
  repeated; their recorded outputs are in the package. Details: `reproducibility/README.md`,
  `reproducibility/RERUN_LOG.txt`.

## Independent verification runs
The results were first obtained with the programs in `original/`. Three independent verification runs, all
AI-assisted, followed on 2026-10-09; each wrote its own programs. Run A examined the statement and the main
example, run B the family, the positive results, the computations, the three-holed question and the literature;
both examined the first written version of the results. Run 2 (second round; "the final verification run" in the
note) examined the text of the note, and first the parts that were reorganised when it was written.

| Item | Run A | Run B | Run 2 (text of the note) |
|---|---|---|---|
| Statement, proposer, page, readings against the source | CONFIRMED_WITH_FIXES (wording of the choice of reading; the qualification must accompany every summary) | source read again; proposer and wording confirmed | CONFIRMED (p. 2528 read as an image); two wording fixes; the scope completed in the abstract and in the first paragraph |
| Four-holed spheres in genus 2 (Lemma 3.1): classification, transitivity, the group J | CONFIRMED (re-derived; the group in two ways; signs confirmed by the cut-and-glue model) | used; the group re-derived | CONFIRMED in the present form (Lemma 2.2, Lemma 2.3, H_0 = N(a1 ∨ δ1 ∨ a2)): cyclic order recomputed from the relator, boundary curves by tracing ribbon graphs, cut-and-glue model of its own |
| Obstruction (Lemma 4.1); φ0 does not lift | CONFIRMED (own model of the extension) | CONFIRMED (κ(φ0) = 2 with its own program) | CONFIRMED (all 81 lifts; associativity of the extension on all triples) |
| Main lemma | CONFIRMED for M_9 (re-proved; every step checked on all 1,036,800 generating tuples) | CONFIRMED in the general form of Lemma 5.1 (own proof) | CONFIRMED (all cases read; all homomorphisms to F_p^2 ⋊ Q_8 for p = 3, 7, 11: no epimorphism with κ ≠ 0 and J intransitive) |
| Theorem 1.1 | CONFIRMED; no gap | not its part | CONFIRMED |
| Theorem 1.2 (family); p ≡ 1 (mod 4) | not its part (read: no obstacle) | CONFIRMED; explicit systems for p = 5, 13 found; orbits for p = 3, 5, 7, 11, 13 | CONFIRMED (the closed form of κ; every printed value of Proposition 6.1; the numbers for each value of κ, p ≤ 13) |
| Proposition 1.3 | (b) CONFIRMED; (a) proved by run A | (b) CONFIRMED; (a) proved independently | CONFIRMED (Lemma 7.1 with the integral lifting step; both parts) |
| Theorem 1.4 (C1)–(C4) | not its part | CONFIRMED (own proofs); credits to Nielsen/Edmonds/Livingston, Dunfield–Thurston, Funar–Lochak required | CONFIRMED ((C4) in the present form through Lemma 2.3(e)); the quoted statements compared with the arXiv versions |
| Theorems 1.5 to 1.7 | the part on M_9 CONFIRMED | CONFIRMED with an independent method; extended to all monodromy groups in degrees 9 and 10 | CONFIRMED with its own programs: Table 3 entry by entry; M_11; the 16 simple groups |
| Three-holed question | pairs of pants with disconnected preimage for the example CONFIRMED | CONFIRMED as open; no example of degree ≤ 10 in genus 2 | CONFIRMED as open (190 and 240 of the 1440 classes recomputed) |
| Novelty | not its part | no answer, example or this use of the obstruction found in print; the invariant is classical | search repeated in a smaller form: nothing found |

No run found a wrong theorem, proposition or lemma, or a gap in the mathematics.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A, fix 1; run B, fix 1) The statement for genus ≥ 3 in both readings: Proposition 1.3(a) (pairwise
   non-isotopic boundary curves) and (b) (two isotopic boundary curves); Table 2.
2. (Run A, fix 2) The cover is defined by the affine maps (1); the permutations (2) are printed with the convention,
   and the paragraph "The convention matters" says what changes in the other conventions.
3. (Run A, fix 3) The qualification (genus 2; boundary curves isotopic in pairs; the question with pairwise
   non-isotopic boundary curves starts at genus 3 and is open; three-holed spheres and regular covers open; the
   Galois closure is not a counterexample) is in the title, the abstract, the first paragraph, after Theorem 1.1,
   in Table 1 and in "Scope and priority".
4. (Run A, fix 4) "is most naturally read" instead of "must be read"; the observation on the word "interior"; the
   statement that the proposer's theorem and its hypothesis were not found: Section 1.1.
5. (Run A, fix 5; run B, fix 6) Generation of the mapping class group of S_2 by five chain twists: cited from
   Lickorish (1964), Humphries (1979), Birman–Hilden (1971) and Farb–Margalit, Section 4.4, without theorem
   numbers; the Dehn–Nielsen–Baer theorem is stated in Section 2.2 as the reason why a substitution mapping the
   relator to a conjugate of itself is induced by a homeomorphism.
6. (Run B, fixes 2, 3, 4) Livingston (arXiv:math/0002162), Lemma 4.1 (Nielsen) and Theorem 4.2 (Edmonds' argument);
   Funar–Lochak, Theorem 1.4, for characteristic quotients; the pigeonhole argument of (C4) credited to the proof of
   Dunfield–Thurston, Proposition 6.16 (bound g > |Q| there), with the remark that Livingston's theorem of 1985
   contains no bound: Section 1.3, Remark 8.1.
7. (Run B, fix 5) κ is presented as the classical invariant in H_2 and the lifting invariant of Hurwitz theory is
   named as the model: Section 1.3, Remark 4.2.
8. (Run B, optional fixes 7, 8) Theorem 1.6 is stated without restriction on the monodromy group in degrees 9 and
   10; the first open cases are listed as run B lists them; Lemma 5.1 is stated in the general form, with the
   examples of Remark 6.3 and the systems of Proposition 6.1.
Optional points of run A that were taken up: an example showing that the generation hypothesis of the main lemma
is needed, and the sharpness of the criterion for M_9 (Remark 5.2).

**Changes made when the note was written, after runs A and B.** They were examined by run 2 (next paragraph).
- All subsurfaces and their groups are derived from one lemma on regular neighbourhoods of wedges of loops
  (Lemma 2.2); H_0 is defined as N(a1 ∨ δ1 ∨ a2). Run A had checked the group of H_0 by two other derivations.
- Standard systems are defined through automorphisms of the surface group, with the Dehn–Nielsen–Baer theorem;
  in Lemma 7.1 the isotropic plane is lifted to an integral symplectic basis, so that only the surjectivity of the
  mapping class group onto Sp(2g, Z) is used.
- (C4) is proved through Lemma 2.3(e).
- In Theorem 1.2(b) the term ω(b, Jb) is evaluated (it equals α), which gives κ = −c1 − (1+α)/2; in the proof of
  Proposition 6.1 the case p = 5 is written out.
- The remark that a non-liftable epimorphism onto M_9 has no essential simple closed curve in its kernel
  (Remark 5.2(c)) was found by run B and is included with its proof.

**Run 2: what it examined, and the result.** It was made on the text of the note, before any further edit.
- The four reorganised parts, line by line. Lemma 2.2: the bands are attached compatibly with the orientation
  (one long side joins x_2 to y_1, the other y_2 to x_1), so that every region of the chord diagram gives one
  boundary curve, read as stated; N is an embedded subsurface of genus 0. The cyclic order (4) was recomputed from
  the relator: it is the stated one, read clockwise if the sides of the polygon are numbered counterclockwise;
  the sense plays no role (boundary words are replaced by their inverses), and the note now says so. The
  boundary curves of Lemma 2.3(a)–(e) were recomputed by tracing ribbon graphs for g = 2, 3, 4. Lemma 3.1 (every
  four-holed sphere with essential boundary in genus 2 is the complement of two annuli; transitivity; the group),
  Lemma 4.1 (liftability is invariant under all automorphisms of the surface group), Lemma 7.1, (C4),
  Theorem 1.2(b) and Proposition 6.1 were re-derived. The group of H_0 was confirmed with a cut-and-glue model
  that does not use Lemma 2.2.
- All other proofs, line by line, among them Lemma 5.1 with its cases (W = 0; |W| = p with A = 1, which uses that
  Q is non-abelian; A = −1), Proposition 1.3, (C1)–(C3), Lemmas 9.1 and 9.2 and the reductions in the proofs of
  Theorems 1.5 to 1.7.
- The quoted statements against the TeX sources of the arXiv versions: Livingston, math/0002162v1, Section 4
  (the lemma attributed to Nielsen with a proof in Edmonds; Theorem 4.2, whose proof the acknowledgements
  attribute to Edmonds; order 32 in the abstract); Dunfield–Thurston, math/0502567v3, §§6.12–6.18 (the class c_f;
  the recipe c_f = ∏[s_i, t_i] with lifts to a Schur cover in the proof of Lemma 6.13; Proposition 6.16 with
  g > |Q| and the pigeonhole principle; Theorem 6.18 attributed to Livingston); Funar–Lochak, 1702.07866v2,
  Theorem 1.4. All attributions in the note are exact.
- The two lists of groups: Butler and McKay (1983), Table 11A (eight transitive groups of degree 11: 11, 11.2,
  11.5, 11.10, L(2,11), M_11, A_11, Σ_11), read in the scanned copy linked from the OEIS; OEIS A001034 and A109379
  (the sixteen orders below 12180, none repeated).
- All finite claims, with programs of its own (see "Computations"). Every number agrees with the earlier programs.
- All 23 DOIs of the bibliography through Crossref (the thesis through DataCite); every page of the PDF rendered
  and looked at, before and after the corrections.
- Result: no wrong theorem, proposition or lemma and no gap. Twelve corrections of the text and of the package
  were required and made; none changes a statement: (1) the Verification paragraph in its final form; (2) the two
  lists of groups compared with sources and cited; (3) the scope completed in the abstract and in the first
  paragraph (the example is not a regular cover, its Galois closure is not an example; with pairwise non-isotopic
  boundary curves the question begins in genus 3); (4) the sentence on the word "interior" made precise; (5) "an
  essential simple closed curve" in the description of Livingston's theorem; (6) the sense of the cyclic order (4);
  (7) the programs of run 2 named in Sections 9.2, 9.3 and Table 3; (8) Section 11 (five parts, re-runs); (9)
  "What was read" and the searches; (10) the reference to the classification of surfaces in Remark 5.2(c);
  (11) this report and the README; (12) the Zenodo metadata.

**What rests on what.** Theorem 1.6(a), (b) for monodromy groups containing A_9 or A_10 rested on one program
(`hb.c` of run B) when the note was written; it is now confirmed by `r2_blk.c` of run 2 (the same decomposition
into blocks, written independently, the same numbers in every column of Table 3). The orbit counts of Remark 6.3
rest on the programs of run B alone. The list of primitive groups of degree 9 and 10 mentioned in the proof of
Theorem 1.6 was not compared with a table; it only describes what the first programs did and is not needed.

## Relation to the literature, novelty and scope
- **Searches (9 October 2026).** arXiv (about forty queries: the proposer; holomorphic retracts of Teichmüller
  spaces; preimages and lifts of subsurfaces, pairs of pants and simple closed curves in finite covers; lifting
  invariants; orbits of mapping class groups on homomorphisms to finite groups), Crossref, OpenAlex, Semantic
  Scholar and zbMATH, including the works citing the report (one citing work was listed, unrelated), and six web
  searches. Run 2 repeated the search on the same day in a smaller form (14 search queries to the arXiv, 4 to
  zbMATH, OpenAlex, Crossref, one web search). No work stating, answering or citing Problem 12 was found. A search
  that finds nothing is not a proof of novelty.
- **What was read.** The problem session of the report (pp. 2525–2529 by run A, p. 2528 by run B and again at the
  writing); the proposer's thesis (abstract; text searched by both runs); Livingston, arXiv:math/0002162, in full
  (arXiv version; the journal version was not compared); Dunfield–Thurston, arXiv:math/0502567v3, Section 6
  (run B) and §§6.12–6.18 again at the writing; the first page of Livingston (1985); Funar–Lochak,
  arXiv:1702.07866v2, the introduction with Theorem 1.4; Boggi–Putman–Salter (introduction) and Malestein–Putman,
  Koberda–Santharoubane, Klukowski, Putman–Wieland (abstracts), each with a keyword search of the arXiv text, by
  run B; the abstract of Fried (2010). By run 2: p. 2528 of the report; Section 4 of Livingston (2000),
  §§6.12–6.18 of Dunfield–Thurston and the introduction of Funar–Lochak in the TeX sources of the arXiv versions;
  Tables 11A to 11C of Butler–McKay (1983) in the scanned copy linked from the OEIS; the OEIS entries A001034,
  A109379, A002106. Not read: Edmonds (1982, 1983), Zimmermann (1987), Serre (1990),
  Fried–Völklein (1991), Lickorish (1964), Humphries (1979), Birman–Hilden (1971), and the books cited for
  standard theorems. The proofs of the cited theorems were not checked. All bibliographic data were checked with
  Crossref, zbMATH or the arXiv.
- **Credits.** D. Gekhtman (the problem). Livingston (1985), Zimmermann, Edmonds, Dunfield–Thurston (the invariant
  in H_2, stable equivalence, the pigeonhole argument). Fried–Serre, Conway–Parker, Fried–Völklein (the lifting
  invariant as the model). Nielsen, Edmonds, Livingston (2000) (adapted generating systems, abelian-by-cyclic
  groups, kernels without simple loops). Funar–Lochak (characteristic finite quotients). Boggi–Putman–Salter,
  Malestein–Putman, Koberda–Santharoubane, Klukowski, Putman–Wieland (neighbouring questions on simple closed curves
  and finite covers; none states or answers the question, as far as read).
- **Scope.** A partial answer: the four-holed question in genus 2 under the literal reading. No statement is made
  about the proposer's application to Teichmüller spaces. No priority is claimed.

## 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/