# Verification report — AMR-049-0001 to AMR-049-0007 (Fuchs: billiard trajectories in regular polygons and geodesics on regular polyhedra; seven conjectures)

Verification date: 2026-10-09.

**Verdict.** The note decides all seven conjectures of the source. Every statement it makes is proved; the
outcome per record is as follows.
- **AMR-049-0001 (Conjecture 1.7): disproved as posed.** It is false for every odd n with at least two distinct
  prime factors. The smallest case is n = 15: the seven bands of trajectories parallel to a side have the factors
  15, 15, 5, 15, 3, 5, 15, with greatest common divisor 1. It is true for odd prime powers, and for even n when it
  is read for the trajectories which are not middle trajectories of bands (Theorem H).
  **The underlying description of the factors is not new**: it is due to Veech (1992), Proposition 8.11 with
  formulas (7.7), (8.16). The note adds the deduction of the answer from his formula (a few lines of arithmetic
  which are not in his paper), a proof of the counterexample which uses neither surfaces nor homology
  (Proposition 8.2), and a second derivation of the formula for odd n.
- **AMR-049-0002 (Conjecture 2.3): proved** for all n ≥ 5 (Theorem D).
- **AMR-049-0003 (Conjecture 2.4): disproved** for every n ≥ 5, with and without the words "infinitely many"
  (Theorem F). The vertex X_2 of the initial n-gon is a vertex of exactly one reachable n-gon; the reachable point
  w_n = 2 + 3ζ + ζ² is a vertex of none. True instead: a reachable point is a vertex of infinitely many reachable
  n-gons if and only if its type is A_0, and a point of another type is a vertex of at most one.
- **AMR-049-0004 (Conjecture 2.5): proved in corrected form** (Theorem E). As printed it is true for n = 5 and
  false for every n ≥ 6. It holds for all n ≥ 5 with λ² and λ² − 1 in place of λ + 1 and λ; these are the
  constants which the source's own argument for n = 6 gives.
- **AMR-049-0005 (Conjecture 2.6): proved** for all n ≥ 4 (Theorem B).
- **AMR-049-0006 (Conjecture 2.7): proved** for all n ≥ 4 (Theorem C).
- **AMR-049-0007 (Conjecture 3.2): proved**, with the definition of the type A_0 of the source (Theorem G). The
  argument with the half-turn and the statement that no short geodesic of type A_0 is closed are due to Athreya,
  Aulicino and Hooper; the same argument for the geodesics between vertices of the cube, the tetrahedron, the
  octahedron and the icosahedron, whose unfoldings are centrally symmetric, is in Troubetzkoy (Amer. Math. Monthly
  130 (2023)). The note adds the identification of the type A_0 with a centrally symmetric unfolding on the
  dodecahedron, and the statement on the distance 2. A public record (Zenodo, 10.5281/zenodo.21875076) presents a geodesic between
  two vertices at distance 2 as a counterexample: this geodesic exists, and with Definition 2.1 of the source its
  type is A_1 (Remark 7.4).
- **Not explained.** The census of the source for the dodecahedron has the total 3750; two independent exact
  enumerations give 3702. No statement of the note depends on the census.
- **Not treated.** Polyhedra other than the dodecahedron; anything beyond the seven conjectures.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Fuchs, "Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra",
  Arnold Math. J. 7 (2021) 493–517, doi:10.1007/s40598-020-00170-8 (a problem contribution).
  - Read completely in the public file on the journal's site,
    https://amj.math.stonybrook.edu/pdf-Springer-final/020-0170.pdf (fetched on 2026-10-09, and again by the third verification run; 3,572,784 bytes,
    sha256 `9550009b3a903727a89abba863c520def9bf900b6bec779b63ef6cc82d43f01d`): all 25 pages as text, and the
    pages with tables and figures as page images.
  - Short trajectories (from a vertex to a vertex, no vertex in between), developments, reachable points
    (§1.2, §2.1). The angles α and β of Fig. 8; the rule that α − β or α + β is a multiple of 2π/n according to
    the parity of the number N of segments (§2.2). Definition 2.1: n − 2 types A_k. The source's example: the
    diagonals X_0X_{k+1} have the type A_k. Reachable n-gons, unitary pairs (§2.4). Parallel trajectories
    (§1.1, §2.5; ℓ even for even n). Closed and preclosed trajectories, strongly parallel trajectories (§1.7).
    Short geodesics on the dodecahedron and their types (§3).
  - The seven conjectures: 1.7, 2.3, 2.4, 2.5, 2.6, 2.7, 3.2; they are restated in Section 1.2 of the note.
  - What the source proves: Conjecture 2.3, the four items of Conjecture 2.5 with the constants 3 and 2, and
    Conjecture 2.6, all for n = 6 in lattice coordinates; the types for n = 4 and n = 6 by explicit formulas.
    Its Proposition 1.8 (for prime n the factors of a class are all 1 or all n) is derived there from
    Conjecture 1.7, with a reference to Davis, Fuchs and Tabachnikov for n = 5 and n = 7.
- **Corpus records.** ulamai/UnsolvedMath, AMR-049-0001 to AMR-049-0007 (dataset version 1.6.0; upstream status
  `open` for all seven). Their statements are paraphrases of the conjectures. Record AMR-049-0004 has the printed
  constants; record AMR-049-0003 has "infinitely many"; record AMR-049-0007 defines the type A_0 only by
  β − α = 2π/5 for angles which it does not specify. With the angles of Fig. 8 and without the rule on the
  parity, that statement is false for a simple reason (a diagonal of a face). The note proves the statement of
  the source.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Definition 2.1 together with the rule on the parity of N (N = number of segments; β in the oriented polygon) | used throughout; it reproduces the source's formulas for n = 4 and n = 6 on all reachable points of length ≤ 120 (6875 and 7605 points), its examples and its two tables | Definition 1.1, Remark 1.2, Lemma 2.3 |
| Definition 2.1 without the rule on the parity | not the source's: every diagonal X_0X_{k+1} would have the type A_0, against the source's example | Remark 1.2(b) |
| Conjecture 2.3 as posed (n ≥ 5) | proved; it suffices that four of the vertices are reachable | Theorem D |
| Conjecture 2.4, with and without "infinitely many" | false for every n ≥ 5 in both readings | Theorem F |
| Conjecture 2.5 as printed (λ + 1, λ) | true for n = 5; false for every n ≥ 6 (item (b) is correct for m = −1) | Theorem E(c), Remark 5.1 |
| Conjecture 2.5 with λ², λ² − 1 | true for all n ≥ 5 | Theorem E(a), (b) |
| Conjecture 2.6 with the source's restriction (ℓ even for even n) | proved for n ≥ 4, including the existence of the parallel short trajectory | Theorem B, Lemma 2.13 |
| Conjecture 2.7 | proved for n ≥ 4, in a stronger form for all saddle connections of one direction | Theorem C |
| Conjecture 1.7 for the factors of bands (trajectories other than the middle trajectory) | false for odd n with at least two distinct prime factors; true for odd prime powers and for even n | Theorem H, Lemma 8.1 |
| Conjecture 1.7 with middle trajectories counted with their own factor | odd n: no change; even n: it would fail for n = 6 and n = 10, in contradiction with the table of the source, so this is not the source's reading | Remark 8.4 |
| Conjecture 3.2 with the type of the source (an edge, or N even and β − α = 2π/5 in the oriented pentagon) | proved | Theorem G, Section 7.1 |
| Record AMR-049-0007 read literally (β − α = 2π/5 without the rule on the parity) | false: a diagonal of a face | Section 1.4(5) |
| Conjecture 3.2 with the convention of the record 10.5281/zenodo.21875076 for β | not the source's convention: it gives another type than the source's formulas to every tested point with even N for n = 4 and n = 6 (280 of 280, 574 of 574); with it the geodesic of the record ends at distance 2 | Remark 7.4 |

## Results in the paper
- **Framework (Section 2).** The double n-gon D_n; the corners and rays at its vertices, the angular coordinate B
  and the invariant K of a saddle connection with values in Z/(n − 2); Lemma 2.3: the type of Definition 2.1 is K;
  the symmetries μ, ϱ and the reflections; the cylinders in the directions kπ/n (modulus 2 cot(π/n)) and the
  twists τ_k; finiteness and descent (n ≥ 4).
- **Theorem A.** The types are the orbits of saddle connections under the group G_0 generated by ϱ, μ and the
  twists.
- **Theorems B and C** (Conjectures 2.6, 2.7), n ≥ 4. **Theorem D** (Conjecture 2.3), n ≥ 5, with the
  characterization of unitary pairs. **Theorem E** (Conjecture 2.5): the reachable points on the two lines
  through a unitary pair are the points with the parameters in λ²Z ∪ (λ²Z − 1). **Theorem F** (Conjecture 2.4).
- **Theorem G** (Conjecture 3.2): the ends of a short geodesic of type A_0 are exchanged by a half-turn; the
  distance is 1, 3, 4 or 5 (Lemmas 7.1, 7.2).
- **Theorem H** (Conjecture 1.7). (a) Even n: every factor divides n/2. (b) Odd n = 2m + 1 ≥ 5: every class consists
  of m bands with the factors n′/gcd(n′, j), 1 ≤ j ≤ m, for a divisor n′ of n, and every divisor occurs. Parts (a)
  and (b) are due to Veech (Section 8.5 gives the dictionary to his paper); Section 8.6 is a second derivation
  for odd n. (c) Conjecture 1.7 holds for odd n if and only if n is a prime power (n = 3: directly).
- **Proposition 8.2, Corollary 8.3.** The trajectories parallel to a side for odd n, by a symmetry argument in the
  n-gon: factors n/gcd(n, j); the counterexample.
- **Tables.** Table 1: outcomes. Table 2: the census of the source and the exact enumeration. Table 3: n = 15.
  Table 4: factors for some odd n. Table 5: the table of experiments of the source and the computation.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. Lemma 7.2 is a finite statement which is proved in the text by a computation
with coordinates; `writing_stage/dodecahedron_halfturns.py` repeats it exactly and prints the 20 × 20 matrix of
the distances and the 15 half-turns.
- **Programs written with the note** (`writing_stage/`). `check_paper.py` (about ten seconds; TOTAL failures: 0)
  tests the statements as printed, parts A to L: formulas (1), (2), Lemmas 2.4, 2.8 for n ≤ 60; Definition 1.1,
  Lemma 2.3 and Lemma 7.1 on 1899 reachable points (n = 4..12, length ≤ 20); Theorems B and C on 1526 instances;
  Theorem D, Remark 4.2 and Theorem E on 377 pairs and 66 lines; the formulas of the proofs of Theorems D and F
  for n ≤ 60; Example 6.2 with the formulas of the source only; Lemma 7.2; Proposition 8.2 and Table 3 with a
  floating-point billiard in the n-gon which uses nothing of the note (504 trajectories, odd n ≤ 45); Lemmas 8.8
  to 8.10 for odd n ≤ 61; Theorem H(a) on 708 trajectories and Theorem H(b) for ten words in the twists
  (n = 9, 15) against this billiard; the tables and numbers of the note against the recorded outputs; Veech's
  formulas against the factors found by direct unfolding. `witness_check.py` (TOTAL failures: 0) contains the
  computations of Remark 7.4.
- **Programs with which the results were first obtained** (`original/`): exact arithmetic in Z[ζ]; all reachable
  points of length ≤ 80 for n = 5..12 (3289 to 3399 points per n): 30,654 instances of Theorems B and C; 8028
  unitary pairs with 60,360 points of Theorem E (none missing, none extra); 668 reachable n-gons inside the
  disc; the descent for all reachable points of length ≤ 60; all reachable n-gons through the points of type
  ≠ A_0 of length ≤ 9; the census of Table 2 and the half-turn on 2095 geodesics; closed trajectories by exact
  unfolding; the orbit of the rotation class for all odd n from 5 to 49.
- **Verification run A** (`verification_run_A/`): its own unfolding in Z[ζ_2n] and a billiard in the fixed n-gon
  with 60 digits. Lemma 2.3, the descent and Theorems B, C without exception for n = 4..12 up to the length 150
  (10,743 to 12,029 reachable points per n), for n = 13, 14, 15, 16, 18, 20 up to 60 and for n = 17, 24, 25, 30
  up to 40; Theorem D on all pairs with det = S of about half these lengths; for n = 5..12 all 8028 unitary
  pairs of length ≤ 80 span reachable n-gons with the types of Theorem D (vertices outside the disc decided by
  exact tracing), and no other pair does.
- **Verification run B** (`verification_run_B/`): its own arithmetic in Z[ζ_n] and its own unfolding of closed
  trajectories by reflections, without the double n-gon. Theorem E on 160 lines; Theorem F for n = 5..12 by an
  exhaustive search; Lemma 7.2; the census (the same twelve numbers) and the facts of Section 7.4; the half-turn
  on 2901 geodesics; the geodesic of the Zenodo record; the direction of a side for 19 odd n ≤ 105; 4776 classes
  of directions for 18 values of n (2407 for 14 odd n; 2369 for n = 6, 8, 10, 12), all as in Theorem H.
- **Verification run on the final text** (`independent_run_2/`): programs written from the definitions of the
  source alone (developments by literal reflection of the labelled polygon; the angles of Fig. 8 measured in the
  plane; the four lines of Definition 2.1 as printed; a billiard in the fixed n-gon in exact arithmetic in
  Q(ζ_2n); unfolding of closed trajectories without the double n-gon). The rule on the parity of N, Lemma 2.3,
  (3) and Lemma 7.1 for n = 4..12 up to the length 150 (the same 10,743 to 12,029 points per n as in run A) and
  for ten further n ≤ 30; the source's descriptions for n = 4 and n = 6 on 6875 and 7605 points. Theorems B and C
  on 175,162 instances. Theorem D, Remark 4.2 and Theorem E on 2798 pairs with det = S (938 unitary pairs) and
  1876 lines with 13,780 reachable points. Theorem F(b), (c) and Remark 6.3 for n = 5..12, by a search which is
  exhaustive for the vertices v_2, …, v_{n−2}. Proposition 8.2 and Table 3 with the exact billiard on 972
  trajectories for 17 odd n ≤ 63 (among them 15, 21, 33, 35, 45). Tables 4 and 5 with all numbers of classes
  (4776 classes; 114 of them compared with a floating-point billiard). Lemma 7.2; Table 2 and the numbers of
  Section 7.4; the half-turn on 2484 geodesics; the geodesic of the Zenodo record (16 segments, type A_1 by the
  printed definition). Lemma 8.1 on 867 pairs of trajectories, Lemmas 8.5 and 8.6 on 2557 unfolded bands, Lemma
  8.9 for odd n ≤ 61. Veech's matrices (his Lemma 4.3) and formula (17) against direct unfolding for 246 words
  and ten odd n ≤ 45, and the patterns of his formulas for n = 6, 8, 10, 12. The data of Figures 1, 3 and 4.
  About five minutes; details in `reproducibility/independent_run_2/README.md`.
- **Re-runs.** On 2026-10-09, when the note was written, all programs of the package which have a recorded
  output were run again, one process at a time. The three programs of `writing_stage/` passed, and every other
  output is identical to the recorded file, or identical up to the fields which record running times (136
  result lines, none failed). Details are in `reproducibility/README.md` and `reproducibility/RERUN_LOG.txt`;
  `reproducibility/run_quick.sh` (about 21 minutes) and `reproducibility/run_slow.sh` (about 30 minutes) repeat
  the re-runs. The verification run on the final text repeated both scripts, each from a freshly extracted copy
  of the archive as it was before that run: 117 and 19 comparisons, none failed (parts 3 and 4 of
  `RERUN_LOG.txt`).

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed on 2026-10-09, each with its own programs. Runs A and B examined the first
written version of the results: run A the conventions, the framework and Conjectures 2.6, 2.7, 2.3; run B
Conjectures 2.5, 2.4, 3.2, 1.7 and the literature. The third run examined the final text of the note; it is
described after the corrections of runs A and B.

| Item | Run A | Run B |
|---|---|---|
| Reading of Fig. 8 and Definition 2.1 with the rule on the parity; wording of 2.3, 2.6, 2.7 | CONFIRMED_WITH_FIXES (editorial) | re-derived for both parities; tested against the source's formulas for n = 4 and n = 6 |
| Lemma 2.3 (type = K) | CONFIRMED | re-derived |
| Theorem A with Lemmas 2.5–2.12 | CONFIRMED (state n ≥ 4 in Lemma 2.11) | the steps which it uses were re-derived; Lemma 2.11 read, not proved again |
| Theorem B (Conjecture 2.6) | CONFIRMED_WITH_FIXES: the existence of the parallel short trajectory was used and not proved; run A gave the lemma (now Lemma 2.13) | not its part |
| Theorem C (Conjecture 2.7) | CONFIRMED | not its part |
| Theorem D (Conjecture 2.3) | CONFIRMED | Steps 1 to 5 re-derived |
| Readings of Conjectures 1.7, 2.4, 2.5, 3.2 | not its part | CONFIRMED_WITH_FIXES |
| Theorem E (Conjecture 2.5) | not its part | CONFIRMED |
| Theorem F (Conjecture 2.4) | not its part | CONFIRMED (counterexamples recomputed; proof re-derived) |
| Theorem G (Conjecture 3.2) | not its part | CONFIRMED_WITH_FIXES: the record on Zenodo and the earlier work of Athreya, Aulicino and Hooper had to be cited |
| Counterexample to Conjecture 1.7 for n = 15 | not its part | CONFIRMED; run B gave the elementary proof (now Proposition 8.2) |
| Theorem H (Conjecture 1.7) | not its part | CONFIRMED_WITH_FIXES: Veech (1992) had to be read before any statement on novelty |
| Literature | not its part | three omissions found (the record on Zenodo; Athreya–Aulicino–Hooper, Section 5; Veech 1992 not read); all three are repaired |

Runs A and B found no wrong theorem, proposition or lemma, and no gap which could not be closed.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A) The existence of the parallel short trajectory: Lemma 2.13, used in the proof of Theorem B.
2. (Run A) n ≥ 4 in Lemma 2.11.
3. (Run A) The type A_0 is not characterized by β − α = 2π/n alone: Remark 1.2(b) and (3).
4. (Run A) ℓ even for even n in Theorem C; N always counts segments; the misprints of the source:
   Section 1.4(3). The sentence on the pairs inside the disc was restricted to what the first program decides,
   and the complete check is attributed to run A: Section 9.
5. (Run B) The full definition of the type A_0 of a geodesic, and the remark that the text of record
   AMR-049-0007, read without the rule on the parity, is false (a diagonal of a face): Section 7.1,
   Section 1.4(5).
6. (Run B) The record 10.5281/zenodo.21875076 is cited and the facts are stated: Section 1.5, Remark 7.4.
7. (Run B) Athreya–Aulicino–Hooper, Proposition 5.1, Theorem 5.2 and Section 10.2.1 are cited; the half-turn
   argument and the non-closedness are attributed to them: paragraph after Theorem G, Remark 7.3.
8. (Run B) What the table of the source shows and what is computed (factors (2, 1) for n = 8; classes with all
   factors 1 for n = 10 and n = 12; the reading for even n): Section 8.7, Section 1.4(4), Remark 8.4.
9. (Run B) The facts on the census: Section 7.4. The exception m = −1 of the printed item (b): Remark 5.1. The
   arXiv version of Davis–Fuchs–Tabachnikov proves the statement on lengths for n = 5 only: Section 1.5.
10. (Run B) Veech (1992) to be read in full: done, see the next item.

**After the two runs: the reading of Veech (1992).** The paper was read in full (scan of the Göttingen
digitisation centre, printed pages 341–379) by a separate AI-assisted reading run, which compared it with the
statements on Conjecture 1.7. It contains the description of the factors for all n > 4. The note was changed
accordingly: parts (a) and (b) of Theorem H are attributed to Veech (abstract, Theorem H, Table 1, Section 1.5);
Section 8.5 states his formula and the dictionary; the homology argument of Section 8.6 is presented as a second
derivation; Remark 3.2 mentions his statement on the circumferences of the cylinders. The two verification runs
did not examine the dictionary of Section 8.5; it is tested by part L of `check_paper.py` (odd n = 9, 15, 21,
25, 27, 45 and even n = 6, 8, 10, 12, against the factors found by direct unfolding).

**Changes made when the note was written, after the runs.** The proofs were written out again. The following
parts have a form which the two runs did not see: Lemmas 2.9 and 4.1; Lemma 6.1 as a separate lemma; the proof of
Lemma 7.2 with coordinates; Lemma 8.1, Lemmas 8.5 and 8.6 and Corollary 8.7 (the precise reading of
Conjecture 1.7 and the rotation class by labels); Theorem H(b) with the divisor n′. They are covered by
`check_paper.py`, parts B, D, G, H, I, J, and they were examined first by the third run.

**The verification run on the final text (third run).**

What it examined:
- The source again: all 25 pages as text; the pages with Conjecture 1.7 and the table of experiments, Fig. 8
  (also enlarged), Definition 2.1 with its two tables, Conjectures 2.3 to 2.7 with the proofs for n = 6, and the
  census with Conjecture 3.2 as page images. The restatements of Sections 1.1 and 1.2 of the note agree with
  them; the two source columns of Table 5 and the source rows of Table 2 are identical to the printed tables.
- All proofs of the note, line by line, the parts listed above first.
- Veech (1992): all 39 pages of the scan as page images. Every statement, number and page which the note
  attributes to this paper was compared with the page that carries it (n > 4; the surfaces X(n), Y(n) and the
  covering of degree N; Remark 1.11; the character χ; the classes ν_i, μ_j and the group E(n); Lemmas 4.3 and
  4.10; (7.5), (7.6), (7.7), (7.15); Lemmas 8.3 and 8.8, Proposition 8.11, (8.16); Theorems 1.3 and 9.25). All
  agree. Every page was also examined for a comparison of the factors of the cylinders of one direction: there
  is none; the numbers occur only as computed values and in sums. Veech's proofs were not checked.
- McMullen (2026, published version: Theorems 1.3, 2.2 to 2.4 and 7.1, Lemma 7.2 and the standing assumption of
  Section 7), Athreya–Aulicino–Hooper (arXiv version 2: Proposition 5.1 and Theorem 5.2 with their proofs,
  Section 10.2.1), Davis–Fuchs–Tabachnikov (arXiv version 1: Theorem 11, Corollary 12, the closing remarks on
  heptagons), Davis–Lelièvre (arXiv version 2: Theorems 3.30, 3.31): the statements quoted in the note agree
  with these texts.
- The record 10.5281/zenodo.21875076: its metadata from the Zenodo API (title, creator, date 2026-08-10,
  version 1.0.0, resource type software) and the text file with the data of its geodesic.
- The literature search, repeated (see below).
- Every page of the final PDF as an image.

What it found:
- No gap which could not be closed.
- One statement which was false as printed, in a case which is not used: Lemma 7.1 and the equivalence (3) were
  stated for n ≥ 3 and fail for n = 3 (there is only one type). No other wrong theorem, proposition, lemma or
  corollary.
- Theorem H(b) was stated for all odd n, while both derivations cover n ≥ 5.
- An earlier source for the argument of Theorem G which was not cited: S. Troubetzkoy, "Vertex to vertex
  geodesics on Platonic solids", Amer. Math. Monthly 130 (2023) 379–382. For the cube, the tetrahedron, the
  octahedron and the icosahedron it shows that the unfolding of a geodesic between two vertices is centrally
  symmetric and that the rotation by π about the axis through its midpoint exchanges the two ends; it notes
  that this proof does not carry over to the dodecahedron.

Corrections which it required, all applied in the note:
11. Troubetzkoy (2023) is cited, and the text says what Theorem G adds: abstract, the paragraph after
    Theorem G, Section 1.5, Remark 7.3, the Scope paragraph, the bibliography.
12. n ≥ 4 in (3) and in Lemma 7.1.
13. n ≥ 5 in Theorem H(b); the case n = 3 of part (c) is treated at the beginning of its proof.
14. Section 8.5: the formulas (7.5), (7.6) of Veech carry no attribution to his paper of 1989; the fact behind
    them is taken from it on his pages 349 and 363. The sentence was corrected.
15. Section 8.5, "Translation": for odd n, rot and χ are homomorphisms onto Z/n with the same kernel; the
    argument is now stated in this form.
16. Section 1.5: the cylinders of the double n-gon and the affine twists go back to Veech (1989), as recalled in
    Veech (1992), pages 348 and 351–353; this is now said.
17. Lemma 2.9: the proof is written out for (a), (b), (c) separately.
18. Remark 7.4: the angle which gives 2π/5 in the record is named in (c); the last sentence says that the paper
    and the programs of the record were not examined.
19. Section 1.4(3): "in two sentences" for the misprint in the proof of Conjecture 2.6 for n = 6.
20. Section 8.5, end: the classes for n = 15 are listed by the divisor n′; Theorem H: the list begins below the
    heading.
21. Section 9 and the Verification and Scope paragraphs describe the final state.

The corrections 11 to 21 were made by the third run and checked by it on the rebuilt text; no further
independent run followed.

## Relation to the literature, novelty and scope
- **Searches (9 October 2026).** The lists of works citing the source in OpenAlex and Crossref (one work, on
  another subject) and in Semantic Scholar (none); the review of the source in zbMATH, which presents the
  conjectures as open; arXiv (queries on billiards in regular polygons, on the pentagon and the double pentagon,
  on geodesics on the dodecahedron); Crossref and zbMATH; six web searches.
  - For Conjectures 2.3 to 2.7 and 3.2 no proof and no refutation was found in print; for Conjecture 1.7 no
    statement of its failure for composite n.
  - The description of the factors is in Veech (1992). The prime case of Conjecture 1.7 follows from theorems of
    McMullen (2026), and for n = 5 from Davis–Fuchs–Tabachnikov (2011) and Davis–Lelièvre; it also follows from
    Veech's formula.
  - The argument of Theorem G (a rotation by π about the midpoint of the geodesic exchanges its ends) is in
    Athreya–Aulicino–Hooper for closed saddle connections and in Troubetzkoy (2023) for the other four regular
    polyhedra.
  - The third run repeated the searches on the same day (OpenAlex, Crossref, Semantic Scholar, the zbMATH
    review of the source, twelve queries to the arXiv, queries to Crossref, OpenAlex and zbMATH, one web search)
    with the same result; Troubetzkoy's paper was found through the arXiv.
  - The only document found which claims to settle one of the seven conjectures is the record
    10.5281/zenodo.21875076 (see the verdict for AMR-049-0007).
- **What was read.** The source completely. Veech (1992) in full, by the reading run. McMullen (2026), published
  version: Sections 1, 2, 4–7. Davis–Fuchs–Tabachnikov, arXiv version 1: Sections 2.5–2.6 and the closing
  remarks. Davis–Lelièvre, arXiv version 2: Sections 1.2–1.5, Theorems 3.30, 3.31, the conjectures of Section 4.
  Athreya–Aulicino–Hooper, arXiv version 2: Sections 1, 5 and 10.2.1. Troubetzkoy, arXiv version 3: completely.
  Of the record on Zenodo: three text files and the metadata, not its paper or its programs. Not read: Veech (1989), whose facts are taken on the word of Veech (1992);
  Ward (1998) beyond the abstract; the journal versions of Davis–Fuchs–Tabachnikov, of
  Athreya–Aulicino–Hooper and of Troubetzkoy. All bibliographic data were checked with Crossref, zbMATH or the arXiv.
- **Caveats.** Veech's formulas are used as they are printed in his paper; the reading run and the third
  verification run compared the statements with the pages of the scan, and neither checked the proofs of his
  paper step by step; whether Veech (1989) already contains a part of the description of the factors was not
  checked. For odd n the note
  proves the formula a second time (Section 8.6); for even n it proves Theorem H(a) directly and tests the
  transcribed formulas by computation. Runs A and B examined the first written version of the results; the
  third run examined the final text, and its own corrections were not examined by a further run. The identification of the convention of the record on Zenodo
  rests on its stated edges and its relation between them, which two programs reproduce. A search which finds
  nothing is not a proof of novelty.
- **Scope.** The seven conjectures as listed in the verdict. No priority is claimed, in particular none for any
  statement on the factors.

## 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/