# Verification report — AMR-044-0001 (Dragović–Radnović: periods of pseudo-integrable billiards, two concentric half-circles)

Verification date: 2026-10-09.

**Verdict.** The note gives a **partial answer**; every statement it makes is proved, and the scope is as follows.
- **Settled.** All regular trajectories with a given caustic are periodic if and only if ρ1 + ρ2 is rational
  (Theorem 1.2). There are at most two regions on the boundary, each mirror-symmetric; if ρ1 + ρ2 is irrational
  there is exactly one non-periodic region and at most one periodic region (Theorem 1.3).
- **Not new mathematics.** The criterion and the bound on the number of components are applications of published
  theorems: Boshernitzan (1988), Theorem 1.1, with his decomposition of interval exchanges; alternatively McMullen
  (2015), Corollary 7.4, and, for quadratic irrationals, McMullen (2003). What the note adds is the reduction that
  makes them applicable (Theorem 1.1: unfolding, chord map, the dictionary to McMullen's coupled rotations, and the
  period formula (2)), the statement about regions on the boundary, and explicit results.
- **Periods determined only in special cases.** ρ1 rational; ρ1 + qρ2 ∈ ½Z; pρ1 + ρ2 ∈ ½Z (Theorem 1.4(b)–(d));
  ρ1 + ρ2 = ½ (Theorem 1.5); rational pairs by a finite procedure (Proposition 8.1).
- **Open.** A closed formula for the periods when ρ1 + ρ2 is rational and different from ½; the existence of the
  periodic region when the generating relation pρ1 + qρ2 = n/2 has p, q ≥ 2; Question 3 of the source (more arcs)
  is not treated.

The note is unrefereed.

## Statement checked
- **Primary source.** V. Dragović, M. Radnović, "Periods of pseudo-integrable billiards", Arnold Math. J. 1 (2015)
  69–73, doi:10.1007/s40598-014-0004-0 (a problem contribution).
  - Read in the open HTML version on the journal's site,
    https://armj.math.stonybrook.edu/html-articles/Files-2015-2024/14-04 , with its three figures. The page was
    fetched on 2026-10-08 and again on 2026-10-09 (29,881 bytes, sha256
    `32e0ae872c65e6bf11ddbfd9ea3529fc10cdffd0cf72219141c3e940f56d74fd`), five times in all, with identical content.
  - The table: bounded by two concentric half-circles and two segments lying on the same diameter (Section 2,
    Fig. 1). Fig. 1, Remark 4 (two reflex angles) and the radii 2R, R√2 of the companion paper show that the two
    half-discs lie on opposite sides of the diameter.
  - Rotation numbers ρ_i = (1/π) arccos(r/R_i) for a caustic of radius r.
  - Question 1 asks to determine whether the trajectories are periodic, and for the periods. Question 2 asks for
    an arithmetic criterion for the number of non-periodic regions on the boundary, the number of periodic ones,
    and the corresponding periods. Question 3 asks the same for finitely many concentric arcs and radial segments.
  - Examples 1–3: (1/3, 1/4): periods 12 and 7; (1/4, 1/6): 5 and 6; (1/3, 1/5): 13 and 21; (1/4, 1/√30): period 6
    and non-periodic trajectories.
- **Companion paper.** V. Dragović, M. Radnović, "Pseudo-integrable billiards and arithmetic dynamics",
  J. Mod. Dyn. 8 (2014) 109–132, doi:10.3934/jmd.2014.8.109; read as arXiv:1206.0163v1 (the journal version was
  not compared). Used for the conventions: the period counts the reflections off the segments as well (Section
  4.1: four bounces on the smaller circle, six on the bigger one, one on each segment, for period 12); regular
  trajectories; regions (Theorem 6.1); the examples of Sections 4.3 and 8.2.
- **Corpus record.** ulamai/UnsolvedMath, AMR-044-0001 (dataset version 1.6.0; upstream status `open`). Its
  statement asks whether the trajectories with a fixed concentric caustic are periodic, and which periods are
  possible for given ρ1 and ρ2. This is Question 1 of the source. The record's summary gives the page range
  69–85; the journal and Crossref give 69–73.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Question 1, first part: are all regular trajectories with the given caustic periodic? | yes if and only if ρ1 + ρ2 ∈ Q | Theorem 1.2 |
| The same, read as: is there a periodic trajectory with the given caustic? | ρ1 + ρ2 ∈ Q: all are periodic. ρ1 + ρ2 ∉ Q: decided if 1, ρ1, ρ2 are independent or ρ2 ∈ Q (none), if ρ1 ∈ Q, if ρ1 + qρ2 ∈ ½Z, if pρ1 + ρ2 ∈ ½Z; open if the generating relation has p, q ≥ 2 | Theorem 1.4 |
| Question 1, second part: the periods | given by formula (2) from the reduced orbit; explicit in the cases of Theorems 1.4(b)–(d) and 1.5; a finite procedure for rational pairs; no closed formula in general | Theorem 1.1(b), Proposition 8.1 |
| Question 2: numbers of periodic and non-periodic regions | (1, 0) or (2, 0) if ρ1 + ρ2 ∈ Q, and (2, 0) if moreover ρ2 ∉ Q; (0, 1) or (1, 1) if ρ1 + ρ2 ∉ Q | Theorem 1.3 |
| Period = number of reflections off the whole boundary (the source's convention) | used throughout; the numbers of reflections off the large arc, the small arc and the segments are given separately | formula (2) |
| Table with both half-discs on the same side of the diameter (possible if only the corpus sentence is read) | not the source's table (its Fig. 1 and Remark 4); not treated | Section 1.1 |
| Question 3 (finitely many arcs and radial segments) | not treated | Section 9 |

## Results in the paper
- **Theorem 1.1 (reduction).** Unfolding in the real axis; the chord map (1) on states (u, b); the period formula
  (2), which counts reflections off the arcs and off the segments; the first-return map G of (3), an exchange of
  three arcs of a circle with discontinuities 0, δ, λ; the chord map is the coupled rotation of McMullen with
  (L1, τ1, L2, τ2; t) = (1, 2ρ1, 1, 2ρ2; 2ρ2), and the first-return map of the flow of slope 1 on an L-shaped
  surface of genus two with one cone point (Lemma 2.5).
- **Proposition 3.2, Lemma 3.3.** The decomposition into periodic and minimal components, derived from
  Boshernitzan's Proposition 2.9 and Theorem 2.16; a counting lemma for three discontinuities: at most one minimal
  component, at most two families, at most one family next to a minimal component.
- **Theorem 1.2.** "Only if": elementary (the mean of μ is 1). "If": the minimal component would be a minimal
  exchange of rank two, hence uniquely ergodic by Boshernitzan's Theorem 1.1, and a pigeonhole argument gives a
  contradiction. Remark 4.3 gives the second derivation from McMullen (2015): rank two, degree zero, single zero,
  Corollary 7.4; and from McMullen (2003) for quadratic irrationals.
- **Theorem 1.3.** Regions correspond to the components of G and are mirror-symmetric. The proof uses the time
  reversal ϑ(u) = α + λ − u, the five symmetric points c_1, …, c_5 and a count of connections (Lemmas 5.1–5.4).
- **Theorem 1.4.** (a) no periodic region if 1, ρ1, ρ2 are independent or ρ2 ∈ Q; (b) ρ1 = s/t: a region exists iff
  ρ2 < 1/(2t) (t odd) or ρ2 < 1/t (t even), period t + 2s; (c) ρ1 + qρ2 = n/2: iff q divides N or N + 1;
  (d) pρ1 + ρ2 = n/2: iff n = p and {1/(2ρ2)} > 1 − 2/p, period (2p + 1)N − 2, doubled if Np is odd;
  (e) in general q ≤ (N + 1)p is necessary. Here N = ⌊1/(2ρ2)⌋.
- **Theorem 1.5.** ρ1 + ρ2 = ½: periods 3s − 2 (s even) or 6s − 4 (s odd), s ∈ {N, N + 1}.
- **Propositions 8.1 and 8.2.** Rational pairs: an explicit permutation of M arcs; a finite certificate of
  non-periodicity.
- **Tables.** Table 1: all examples of the two source papers are recovered. Table 2: the 28 rational pairs with
  denominators ≤ 7. Tables 3 and 4: computed examples for the two open questions.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. The floating-point billiards used below simulate the table directly and do not
use the reduction.
- **Programs written with the note** (`writing_stage/check_paper.py`, standard library, about one minute; TOTAL failures: 0).
  They test the statements as printed: Theorem 1.1(a) and Lemma 2.3 chord by chord on 466,488 chords of 200 float
  trajectories; formula (2) against the period and the three hit counts counted in the table (600 starts, 60
  rational pairs); (3) against (1); Lemma 2.5 (12,000 points); Table 1; Theorems 1.2 and 1.3 (164 irrational pairs
  with rational sum: total measure exactly 1, two families; 171 with irrational sum: at most one family);
  Theorem 1.5 (436 values); Theorem 1.4(b), (c), (d) and Remark 6.2 (319, 165 and 147 pairs: family, data, width,
  arc, and a certificate for the complement in every case); Proposition 8.1 (231 pairs) and Table 2 as printed;
  Theorem 1.3(a) for 325 rational pairs in the full table (exact); Theorem 1.4(e) (112 pairs); Lemma 5.2(a) and
  (8) on 2,000 float states; Lemmas 5.2(b), 5.3 and 5.4 for families (325 + 76 pairs); Tables 3 and 4 and the long
  example of Section 9 as printed. A second program, `writing_stage/check_symmetric.py` (80 s; TOTAL failures: 0), tests
  Lemma 5.3(c), inequality (14) and the case distinctions in the proof of Lemma 5.4 on 325 rational pairs (exact)
  and 88 irrational pairs (infinite orbits up to 4000 steps).
- **Programs with which the results were first obtained** (`original/`): the exact chord map on the full table,
  three exact decomposition engines and a float billiard;
  1,011 irrational pairs with rational sum (all periodic, two families), 622 pairs for Theorem 1.5, 250 + 180 + 809
  pairs for Theorem 1.4(b)–(d) in its first form, 325 rational triples for the regions, 186 pairs of the open
  class.
- **Verification run A** (`verification_run_A/`): 838,333 chords in 300 float tables; 1,800 float runs and 1,600
  exact starts for formula (2); 1,286,300 irrational pairs with rational sum, all completely periodic with two
  families. Among them are 480,000 pairs with non-quadratic parameters (2^(1/3), 3^(1/3), π, e, ln 2, a Liouville
  number) and 120,000 pairs built on a quadratic irrational with very large partial quotients; these were treated
  with certified rational enclosures.
- **Verification run B** (`verification_run_B/`): 95 explicit and 4,829 scanned pairs; the period table for 231
  pairs by two programs; 864 float starts; 52,650 exchanges of three arcs for Lemma 3.3; regions for 231 rational
  and 12 irrational pairs; 1,155 pairs of the open class (546 with a family, 603 certified without, 13 not
  completely decided).
- **Verification run 3** (`independent_run_3/`, standard library, about 100 seconds; TOTAL failures: 0): two
  programs of its own, a float billiard in the real table which does not use the reduction and an exact engine for
  the maps (1) and (3) in the coordinate u, and their comparison.
  - Theorem 1.1(a), Lemma 2.3, (8) and Lemma 5.2(a) chord by chord: 410,479 chords of 150 float trajectories with
    random radii; 39,345 reflections off the segments; 30,017 reversed and mirrored states.
  - Period and the three numbers of reflections, start by start: 50 rational pairs (2,000 starts on the large arc,
    the small arc and the segments).
  - Regions from the trajectories in the table alone (classes of starts by continuity; time reversal and mirror
    as maps of starts): 12 rational and 9 irrational pairs; the number of regions equals the number of components
    of G, and no start lies in another region than its mirror image.
  - Explicit instances of every case of Theorems 1.4 and 1.5 with the predictions computed from the statements
    (63 pairs; exact widths; a certificate for the absence of further periodic points in all 45 instances of
    1.4(b)–(d)); 80 irrational pairs with rational sum; 58 pairs with irrational sum.
  - Proposition 8.1 on 231 pairs (30 with one region, 201 with two); Tables 2, 3, 4 and the long example as
    printed; the statements of Section 5 on 92 exact pairs; Lemma 2.5 (6,000 points); the dictionary (4) (4,000
    points); Theorem 1.3(a) and Lemma 5.1(a) exactly in the full table on 560 rational triples (M ≤ 16).
- **Re-runs.** On 2026-10-09, in the course of verification run 3, all programs of the package that have a
  recorded output were run again from an extracted copy of the archive. Their outputs are identical to the
  recorded ones, up to fields that record running times. The three period tables for denominators ≤ 12 are
  identical files, and run 3 computes the same lists a fourth time. Details are in `reproducibility/README.md` and
  `reproducibility/RERUN_LOG.txt`; `reproducibility/run_quick.sh` repeats the quick part (about eleven minutes).

## Independent verification runs
The results were first obtained in the coordinate x = u − λ/2. Three independent verification runs, all
AI-assisted, followed, each with its own programs. Runs A and B (2026-10-08) examined the first written version of
the results: run A the statement, the reduction and the criterion; run B the regions, the periodic family, the
case ρ1 + ρ2 = ½, the rational pairs, the certificate, the open part and the literature. Run 3 (2026-10-09)
examined the final text of the note.

| Item | Run A | Run B | Run 3 (final text) |
|---|---|---|---|
| Statement, table, conventions, examples against the source | CONFIRMED | CONFIRMED | CONFIRMED (source and figures read again; all examples recomputed) |
| Theorem 1.1 (reduction, period formula, first-return map, L-shaped surface) | CONFIRMED (re-derived) | used; its float billiard agrees in every test | CONFIRMED (re-derived in the coordinate u; dictionary compared with McMullen's text) |
| Lemma 3.3 (counting) | CONFIRMED | CONFIRMED | CONFIRMED |
| Theorem 1.2, "only if" | CONFIRMED | CONFIRMED | CONFIRMED |
| Theorem 1.2, "if" | CONFIRMED_WITH_FIXES (citations) | second, published route found (McMullen 2015) | CONFIRMED |
| Cited theorems of Boshernitzan and McMullen: statements and hypotheses | CONFIRMED_WITH_FIXES (Boshernitzan read in the paper) | not checked | CONFIRMED (statements compared with the texts) |
| Theorem 1.3 | not its part | CONFIRMED in its first form (families of G; regions) | CONFIRMED_WITH_FIXES in its final form (Section 5): one step written out |
| Theorem 1.4(a), (b), (e) | not its part | CONFIRMED | CONFIRMED |
| Theorem 1.4(c), (d) | not its part | CONFIRMED; (d) WITH_FIXES (closed form added) | CONFIRMED in the final form (half-open arcs) |
| Theorem 1.5, Propositions 8.1 and 8.2 | not its part | CONFIRMED | CONFIRMED |
| Description of the open part | not its part | ACCURATE; case (d) was moved from "criterion by return times" to "closed form" | ACCURATE |
| Novelty | the quadratic case follows from McMullen (2003) | the criterion and the bound on components follow from McMullen (2015); no prior source for this billiard | no prior source found (searches repeated); wording accurate after two corrections |

No run found a wrong theorem, proposition or lemma, or a gap that could not be closed.

**Corrections required by runs A and B**, all applied in the note:
1. (Run A) The decomposition is cited from Boshernitzan, Proposition 2.9 and Theorems 2.16–2.17, in his form
   (half-open intervals; forward orbit of every point), with a derivation: Proposition 3.2.
2. (Run A) Boshernitzan's theorem is cited as Theorem 1.1 with his definitions (rank of the lengths, minimal,
   uniquely ergodic, no irreducibility or Keane condition), and the hypotheses are checked: Theorem 4.2, the
   paragraph before it, and Step 2 of the proof.
3. (Run A) The note says that for quadratic irrationals the criterion follows from McMullen (2003), Theorems 6.1
   and 7.1: Remark 4.3(a), (c), Section 1.3.
4. (Run A, optional) The sheet function ε is defined for trajectories starting clockwise; the jumps of G are
   given with signs; the constant in the pigeonhole step is (T + 1)/(2g) − ½.
5. (Run B) The chord map is identified with McMullen's coupled rotation, with the dictionary (4), and the
   criterion is derived a second time from his Corollaries 3.2 and 7.4, Theorem 7.1 and Proposition 9.3:
   Theorem 1.1(d), Remark 4.3(b). The novelty wording was changed accordingly: Section 1.3 and "Scope and
   priority".
6. (Run B) Theorem 1.4(d) is stated in closed form (n = p and {1/(2ρ2)} > 1 − 2/p; data (Np, N, 1, Np)), with
   uniqueness of the family and l = N: Theorem 1.4(d) and its proof.
7. (Run B, optional) Theorem 1.4(c) in terms of n: Remark 6.2. "Exactly two regions if ρ1 + ρ2 ∈ Q and ρ2 ∉ Q":
   Theorem 1.3(b). The second open question is restricted to p, q ≥ 2: Section 9.

**Changes made when the note was written, after runs A and B.**
- All statements were transferred to the coordinate u = x + λ/2, in which the chord map is literally a coupled
  rotation, and all proofs were written out in this coordinate.
- The proof of Theorem 1.3 was reorganised around the reflection ϑ of the circle of G (Lemmas 5.2–5.4). Run B
  had checked the argument in its first form, for the map of two circles.
- In the proofs of Theorem 1.4(c) and (d) the index set is defined with the half-open arc [0, λ), so that the
  argument does not need that auxiliary points avoid the discontinuities.
- Lemma 3.3 is proved for half-open components; its Step 2 was shortened.

**Run 3: what it examined, and the result.**
- The three changes first. Every formula, interval and discontinuity in the coordinate u was compared with the
  first written version (chord map, first-return map with its three discontinuities 0, δ, λ and jumps λ − δ, −λ,
  δ, period data, the sets in Theorems 1.4 and 1.5, the M arcs of Proposition 8.1, the five symmetric points):
  correct.
- Section 5 was re-derived completely: Lemma 5.1; the time reversal u → α + λ − u (Lemma 5.2) and the identity
  ϑGϑ = G^{-1}, also on R/2Z; the symmetric elements (Lemma 5.3), counted as elements, since two of them can sit
  at one point; the reversal of connections; Lemma 5.4 in its three cases, including δ = 0; Steps 1–4 and parts
  (b), (c) of the proof of Theorem 1.3. Result: correct. In the proof of Lemma 5.1(b) the last step (the two
  minimal components over one minimal component of G are exchanged by the mirror) and the proof of (c) were not
  written out; they have been added.
- The proofs of Theorem 1.4(c), (d) with half-open arcs: correct; the identity "gap between consecutive indices
  = μ" holds for every index, also at the marked points, and regularity is used only where the family is
  identified.
- Then all other proofs, line by line, and the cited statements against the texts of Boshernitzan (1988),
  McMullen (2015) and McMullen (2003): correct.
- Its programs agree with the note in every comparison (see "Computations").
- It required ten corrections of the text and of the package; all were made:
  1. the Verification paragraph of the note describes the final state;
  2. the folder of the programs written with the note was renamed to `writing_stage/`, and the package was adapted;
  3. Lemma 5.1: the end of the proof of (b) and the proof of (c);
  4. abstract: the bound on the number of regions is presented, like the criterion, as an application of
     published theorems;
  5. Section 1.3: the companion paper contains four of the six examples of Table 1 (rows 1, 2, 5, 6), not all;
  6. Lemma 5.2(c): "orbits onto orbits" is stated for regular points;
  7. "Scope and priority": the searches of 9 October 2026;
  8. Section 10: the five parts of the package;
  9. the deposit metadata (description, identifier line, hashes);
  10. this report.
  None of them changes a theorem.

## Relation to the literature, novelty and scope
- **Searches (8 October 2026; repeated in part on 9 October 2026 by run 3).** arXiv (queries for
  pseudo-integrable billiards, for the authors of the problem, for billiards in concentric or half-circular
  tables, for coupled rotations, for interval exchanges of rank two), Crossref, zbMATH, the lists of citing works
  in OpenAlex (20 works citing the problem article, 29 citing the companion paper; titles and available abstracts
  read, texts not read), and six web searches.
  - No answer to the questions for this table was found.
  - The closest results concern almost every parameter: Frączek, Shi and Ulcigrai (2018) for ellipses with
    barriers, and Frączek (2021) for tables bounded by arcs of confocal conics (unique ergodicity on the invariant
    sets). They do not decide the arithmetic question.
  - McMullen's papers do not mention billiards of this kind; the papers of Dragović and Radnović do not mention
    coupled rotations.
- **What was read.** The problem article completely (HTML, three figures); the companion paper in the arXiv
  version v1; of Boshernitzan (1988), in the publisher's file: Sections 1–2 and the definitions of Section 3 at
  the writing, Sections 1–5 by run A, Sections 6–9 by nobody; of McMullen (2015), in the journal version: the
  definitions and statements of Sections 2, 3, 6, 7, 9; of McMullen (2003), in the author's version of 2007 in
  the repository of Harvard University, whose theorem numbers are used: the definitions and statements of Sections
  2, 4, 6, 7. The proofs of the cited theorems were not checked. Keane (1975) and Veech (1987) were not read; what
  the note says about them is taken from Boshernitzan. All bibliographic data were checked with Crossref (again by
  run 3: nine DOIs). Run 3 read the problem article and its figures again on the journal's site, and the companion
  paper, Boshernitzan (1988, Sections 1 to 3) and the two papers of McMullen in the text extractions made for the
  earlier runs.
- **Caveats.** In McMullen (2003) the proof of Theorem 7.1 is written for two distinct zeros, and the case of a
  double zero, which is the one needed, is said there to be similar; the note does not rely on it (the proof in
  the note goes through Boshernitzan's theorem). The journal version of the companion paper was not compared with
  the arXiv version. This negative search is not a proof of priority.
- **Scope.** The note answers the first part of Question 1 and the qualitative part of Question 2. The periods
  are determined only in the cases listed in the verdict. No priority is claimed for the criterion or for the
  bound on the number of components.

## 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/