# Verification report — AMR-020-0402 (Bolsinov–Matveev–Miranda–Tabachnikov: the algebraic Birkhoff conjecture for outer billiards on the sphere and the hyperbolic plane)

Verification date: 2026-10-17.

**Verdict.** The note gives a **complete answer** to the question in the reading of its source (a polynomial first
integral forces a conic), and more generally for rational first integrals, for closed C⁴ curves with nowhere
vanishing geodesic curvature (C³ suffices on the sphere for polynomial integrals). Every statement it makes is
proved. The answer is a **corollary of published theorems of A. Glutsyuk** and is presented as such.
- **Settled.** Theorem 1.2: if the outer billiard about an oval γ on S² (inside an open hemisphere) or on H² has a
  first integral which is the restriction of a polynomial, or of a rational function, of the coordinates of R³,
  then γ is a conic Σ ∩ {⟨Bx,x⟩ = 0} (rank B = 3, B not proportional to A). Proposition 1.3: conversely every such
  conic has the quadratic integral ⟨Bx,x⟩, and its invariant curves are the conics of the pencil spanned by γ and
  the absolute.
- **Not new mathematics.** The theorems which carry the proof are Glutsyuk's: J. Eur. Math. Soc. 23 (2021),
  Theorem 1.25 (rationally integrable I-angular billiards; it completes work of Bialy and Mironov) and Theorem 1.6
  (polynomially integrable billiards, used on the sphere through the spherical duality), and Invent. Math. 245
  (2026), Theorem 1.16 (germs with a rationally integrable dual billiard structure; numbering of
  arXiv:2112.07056v4). What the note adds is the identification of the outer billiard on S² and H² with an
  I-angular billiard of the absolute (the point reflection is the projectivized pseudo-symmetry), the reduction of
  an integral in the coordinates of R³ to the settings of these theorems (Lemmas 3.1, 3.2, 4.1), and the write-up.
  On the sphere the result was within reach through the spherical duality recalled by the source.
- **Not covered.** Algebraic, non-rational first integrals; ovals of class C² or C³ on H²; the smooth version
  (foliation by closed invariant curves without an algebraic integral).

The note is unrefereed.

## Statement checked
- **Primary source.** A. Bolsinov, V. S. Matveev, E. Miranda, S. Tabachnikov, "Open problems, questions and
  challenges in finite-dimensional integrable systems", Phil. Trans. R. Soc. A 376 (2018), no. 2131, 20170430,
  doi:10.1098/rsta.2017.0430; read as arXiv:1804.03737v2 (TeX source and PDF), Section 4.2 and 4.2.1.
  - In v2, conjectures and problems share one counter: Conjecture 4.1, Problem 4.2, Conjecture 4.3,
    **Problem 4.4** (the statement of the record), Problem 4.5. The journal version was not compared.
  - The problem asks to "prove the algebraic version of Birkhoff conjecture for outer billiards" on a
    non-Euclidean surface of constant curvature.
  - The preceding paragraph fixes the meaning: the conjectures "have algebraic versions"; the polynomial Birkhoff
    conjecture was proved for inner billiards, including non-zero constant curvature (Bialy–Mironov, Glutsyuk), and
    for outer billiards in the affine plane (Glutsyuk–Shustin). Problem 4.5 (rational, resp. algebraic, first
    integral for dual billiards) is called a generalization.
  - Section 4.2 says that outer billiards can be defined in spherical and hyperbolic geometry and that in the
    spherical case inner and outer billiards are conjugate under the spherical duality.
- **Corpus record.** ulamai/UnsolvedMath, AMR-020-0402 (dataset version 1.6.0; upstream status `open`; title
  "Open Problems in Integrable Systems — Around the Birkhoff conjecture"). The statement is the source text word
  for word. The record's background calls it "Problem 4.2" because it counts only the problems; it is Problem 4.4
  of v2.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Polynomial first integral in the coordinates of R³ (the source's "algebraic version") | γ is a conic; conversely conics have quadratic integrals | Theorem 1.2, Proposition 1.3 |
| Rational first integral (denominator without zeros near γ) | γ is a conic | Theorem 1.2, Section 4.2 |
| Polynomial integral in the coordinates of the gnomonic, Klein, Poincaré or half-plane model | a rational function of x₁, x₂, x₃ without poles near γ: covered | Definition 1.1(c) |
| Tabachnikov's "algebraically integrable" (invariant curves are components of level curves of a polynomial) | gives a polynomial first integral: covered | Definition 1.1(c) |
| Algebraic, non-rational first integral | not covered | Section 6 |
| Smoothness | C⁴ (C³ on S² for polynomial integrals); C² or C³ on H² not covered | Section 6, Proposition 4.2 |
| Curve on S² not in a hemisphere | the outer billiard is not globally defined; excluded | Definition 1.1(a) |
| Euclidean plane | not part of the question; it is the theorem of Glutsyuk–Shustin (Tabachnikov under a non-degeneracy assumption) | Section 6 |

## Results in the paper
- **Lemma 2.1.** ρ_P(x) = 2ε⟨x,P⟩_A P − x is an involutive A-isometry preserving Σ, the outer billiard map is
  T(X) = ρ_P(X) for the selected tangency point P, and the projectivization of ρ_P is the I-angular symmetry with
  centre π(P) (Glutsyuk, JEMS 2021, Definition 2.4; this is his Proposition 2.5 for the pseudo-symmetry ρ_P). In
  the chart: σ(z₀ + tw) = z₀ + μ(t)w with μ(t) = −ct/(c + bt), c = |z₀|² + ε, b = 2z₀·w.
- **Lemma 3.1.** Invariance on an arc of the tangent geodesic propagates (Bézout) to the whole conic
  Q_P = W_P ∩ {⟨x,x⟩_A = ε}, including the second branch −L_P on the lower sheet for H².
- **Lemma 3.2.** Even/odd splitting and homogenization by s = ε⟨x,x⟩_A give rational functions R₊, R₋ on RP² with
  poles on the absolute, invariant under the I-angular symmetries on the tangent lines, at least one non-constant,
  and constant on π(γ).
- **Lemma 4.1.** Extension of the invariance from a real arc to every point of the irreducible component of the
  Zariski closure (the coefficients of a polynomial identity vanish on infinitely many points of the curve).
- **Proof for polynomial integrals (Section 4.1).** Components of the curve {R = r₀}, Glutsyuk's JEMS Theorem
  1.25, pasting of conical arcs with C⁴ (intersection multiplicity ≥ 5 > 4), globalization.
- **Proof for rational integrals (Section 4.2).** Same reduction with N = F₁(x)F₂(−x), D = F₂(x)F₂(−x); finite
  exceptional set; Glutsyuk's Invent. Math. Theorem 1.16 for germs; pasting. For polynomial integrals this is a
  second proof.
- **Proposition 4.2.** On S², for C³ ovals (Definition 1.1(b),(c) and Lemmas 3.1, 3.2 need only C²): the outer
  billiard about γ is the pole picture of the inner billiard about the dual curve (Q × ϱ(v) = ρ_P(Q × v));
  G(q × v) is a polynomial integral of that inner billiard, and JEMS Theorem 1.6 applies.
- **Proposition 1.3.** ⟨Bx,x⟩ = b²⟨BV,V⟩ on W_P; the involution is the Desargues involution of the pencil spanned
  by π(γ) and the absolute (Tabachnikov's observation, Invent. Math. Example 1.14); geodesic circles also have a
  linear integral.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The program of the note** (`check_note.py`, SymPy, about 5 s; 71 checks, all PASS): Lemma 2.1 (abstract P, V
  and three explicit rational frames for S² and H²), the chart formula and the swap of the absolute points, the
  I-angular symmetry (every line through the centre fixed, polar line fixed pointwise), equal distance in the
  chart; Proposition 1.3 for a generic symmetric B and for a general chart conic through a symbolic point with a
  symbolic tangent, for the whole pencil; geodesic circles and their linear integrals; formula (3) and the descent
  on a test polynomial (both sheets for H²) and on two integrable non-homogeneous examples; the duality identities
  of Proposition 4.2; three controls that must fail and do.
- **Program of the first computation** (`original/verify_outer_billiard.py`, SymPy, under 1 s): 44 exact
  identities (Lemma 2.1, the identity behind Proposition 1.3, the descent on a degree-5 polynomial, a circle
  example).
- **Verification run A** (`verification_run_A/`): 24 exact identities and exact null spaces over Q of the
  invariant homogeneous polynomials of degree ≤ 6 along germs (parabola: dimensions 0, 2, 0, 3, 0, 4; two non-conic
  germs: 0, 1, 0, 1, 0, 1, i.e. only powers of ⟨x,x⟩_A; a geodesic circle: also the odd integral x₃), identical for
  S² and H²; T computed from its definition in double precision for an ellipse and a non-conic oval
  (|T − ρ_P| ≤ 2.5·10⁻¹⁵) and numerical dimensions of invariant polynomials (evidence only).
- **Verification run B** (`verification_run_B/`): T from its definition at 50 digits (|T − ρ_P| ≤ 6·10⁻⁵⁰); the
  quadratic integral of conics along orbits; the spherical duality; exact linear algebra up to degree 14: the
  invariant integrals of conic germs are exactly the pencil polynomials, three non-conic germs on S² and H² have
  none besides powers of ⟨x,x⟩_A; Euclidean and exotic controls; deliberately wrong inputs detected.
- **Third verification run, on the text of the note** (`independent_run_2/`): `run2_exact.py` (SymPy, 114 exact
  checks, 34.5 s): Lemma 2.1 with exact rational frames (Cayley transforms), Lemma 2.1(c) over C, the chart identity
  with the harmonic position of the two points of a line on I, the algebra of Lemma 4.1 (on a conic the invariance
  holds at all complex points, by division by the equation of the conic; a cubic control fails), Proposition 1.3 for
  a general conic, the descent of an integral with even and odd parts, the duality identities, controls;
  `run2_numeric.py` (mpmath, 40 digits, 16 checks, 25.4 s): T from its definition on S² and H² for an ellipse and a
  non-conic oval; in the chart T coincides, within 2·10⁻⁴¹, with the involution constructed directly from Glutsyuk's
  Definition 2.4 (fix the point of tangency, interchange the two points of the line on I); the quadratic integral
  along 300 steps; a control.
- **Re-runs.** On 2026-10-17 every program of the package was run again from a copy of the folder; all outputs
  agree with the recorded ones, up to timing fields (`reproducibility/RERUN_LOG.txt`). The third verification run
  repeated this from an extracted copy of `source.zip`, with the same result. Both earlier verification runs had
  re-run the first program with byte-identical output.

## Independent verification runs
The result was first obtained with the proof through Invent. Math. Theorem 1.16 (now Section 4.2). Two
independent verification runs, both AI-assisted, followed, each with its own programs; neither read the other's
report. A third independent verification run examined the final text (see the subsection after the table).

| Item | Run A | Run B |
|---|---|---|
| Statement and source (Problem 4.4 of v2; reading "polynomial integral") | CONFIRMED | CONFIRMED |
| Lemma 2.1 (T = ρ_P) | CONFIRMED (re-derived; numerical construction of T) | CONFIRMED (re-derived; 50-digit construction of T) |
| Lemma 3.1 (Bézout on Q_P, second branch on H²) | CONFIRMED | CONFIRMED |
| Lemma 3.2 (descent, non-constancy) | CONFIRMED | CONFIRMED |
| Use of Invent. Math. Theorem 1.16 (hypotheses: C⁴ non-linear germ, Definitions 1.6 and 1.12) | CONFIRMED against arXiv v4 | CONFIRMED against arXiv v4 (TeX source) |
| Globalization, rational case, pasting with C⁴ | CONFIRMED | CONFIRMED |
| Proposition 1.3 | CONFIRMED (exact, symbolic ellipse and pencil) | CONFIRMED (numerical and exact) |
| I-angular identification; JEMS Theorem 1.25 route | found by this run | — |
| S² case from JEMS Theorem 1.6 by the spherical duality | consistent | stressed by this run (novelty) |
| Novelty | not stale; corollary level | not stale; S² half not new in substance |

Neither run found a mathematical gap. **Corrections required by the runs**, all applied:
1. (A1) The identification of the outer billiard with an I-angular billiard (JEMS Definition 2.10, Theorem 1.25) is
   stated and used: Lemma 2.1(c), Section 4.1 with Lemma 4.1; the claim that the H² case needs the dual-billiard
   theorem was dropped, and the weight of the proof is attributed to both theorems of Glutsyuk.
2. (A2, B1) The source's sentence on the spherical duality is recalled, and the S² case is derived from JEMS
   Theorem 1.6 (Proposition 4.2); the note says that on the sphere the result was within reach through this duality.
3. (A3, B2) The Semantic Scholar entry left unresolved in the first account is Glutsyuk's two-page abstract of a
   minicourse at the Simons Center (Stony Brook) on planar, dual and projective billiards; it says nothing about
   outer billiards on S² or H² and is cited as such.
4. (A4, B4) The descriptions of arXiv:2605.24434 and arXiv:2301.00464 were corrected (both mention outer billiards,
   only in the plane), and arXiv:2110.10750 is cited with its six authors as Arnold Math. J. 8 (2022) 411–422.
5. (A5) Conics are defined with rank B = 3 (Definition 1.1(d)).
6. (A6, B3) Theorem numbers of the Invent. Math. paper are those of arXiv:2112.07056v4 (the journal version was not
   compared); those of the JEMS paper were read in the published version and agree with arXiv:1706.04030v7.
7. (A7) The result is presented as a short corollary of Glutsyuk's published theorems, in the title, the abstract,
   the introduction and "Scope and priority".
8. (B4) Wording: strict convexity is derived from non-vanishing geodesic curvature (not called equivalent to the
   support property); the running time of the first program is under one second.

### Third independent verification run (on the text of the note, 17 October 2026)
A third independent verification run, also AI-assisted, examined the final text of the note (the first two runs had
examined the first written account, before Lemma 4.1 and the proof through JEMS Theorem 1.25 were written). It
checked:
- **Statement.** The TeX source of arXiv:1804.03737v2: numbering (Conjecture 4.1, Problem 4.2, Conjecture 4.3,
  **Problem 4.4**, Problem 4.5, Problem 4.6 in Section 4.2.1), the quotation, the reading "polynomial first integral"
  (the source's paragraph on the polynomial version, "one modifies these conjectures in an obvious way" for outer
  billiards, and the paragraph on "algebraic versions"), the spherical-duality sentence of Section 4.2.
- **Cited results, against the texts.** JEMS 23 (2021), published version: Definitions 1.1, 2.2, 2.4, 2.10,
  Proposition 2.5, Corollary 2.11, Proposition 2.13, Theorems 1.6 and 1.25 and the proof in Section 6.1; Invent.
  Math. 245 (2026) as arXiv:2112.07056v4: Definitions 1.6, 1.12, Examples 1.13, 1.14, Theorems 1.11 and 1.16
  (numbering confirmed for v4); arXiv:1901.01881v2 (Remark 1.12 in v2 numbering); Bialy–Mironov, J. Geom. Phys.
  (arXiv:1602.05698v1: the dual of the inner billiard is the outer billiard on S², on the de Sitter surface for H²);
  arXiv:2110.10750 (Problem 2). All hypotheses of JEMS Theorems 1.25 and 1.6 and of Invent. Math. Theorem 1.16 are
  met where they are used.
- **Proofs, line by line**, with Lemma 4.1 (added at the writing stage) first: the complex involution ρ̂_C, the
  identity ⟨ρ̂y,ρ̂y⟩ = ⟨C,C⟩²⟨y,y⟩, the cofinite set Γ°, the tangent vector v(C), the homogeneity of the
  coefficients of H, the agreement of the tangent lines of Γ and γ̄ on J, Bézout, and the limit argument at the
  remaining points of Γ \ I (all local branches). Then Lemmas 2.1, 3.1, 3.2, Steps 1–4, the rational case,
  Propositions 4.2 and 1.3, and the remarks. No mathematical error and no gap.
- **Computations**: its own programs (above) and a re-run of the whole package from `source.zip`.
- **Literature**: searches repeated (below); the result was not found in print.

It required seven corrections, all applied:
1. Date of the note: 17 October 2026.
2. This run recorded in the Verification paragraph, in this report and in `reproducibility/independent_run_2/`.
3. The number "Remark 1.12" of [Glutsyuk, J. Fixed Point Theory Appl. 2022] is that of arXiv:1901.01881v2; the note
   now says which version was read, and the bibliography gives the arXiv number.
4. Abstract and "Scope and priority": on the sphere the duality route gives the **polynomial** case for C³ curves
   (the abstract had "the result").
5. "Outer billiards on S² and H² appear ... for other questions, in [Tab02, Glu22]" → "for example in" (papers on
   polygonal outer billiards in H² exist).
6. The search account: searches repeated on 17 October 2026, OpenAlex lists of works citing JEMS 2021 and Invent.
   Math. 2026 obtained, four web searches in all.
7. Proposition 4.2: Definition 1.1(b),(c) and Lemmas 3.1, 3.2 are used for a C³ curve; the proof now says that they
   need only class C².

## Relation to the literature, novelty and scope
- **Searches (16–17 October 2026).** arXiv (all papers of Glutsyuk; outer and dual billiards with sphere,
  hyperbolic plane, constant curvature, polynomial, rational or algebraic integrability; papers of Bialy and
  Mironov; the most recent outer and dual billiard papers), Crossref, zbMATH Open, Semantic Scholar (citations of
  the source and of the Invent. Math. paper; partly rate-limited), OpenAlex, and four web searches. The third
  verification run repeated the arXiv, Crossref and zbMATH searches on 17 October 2026 (latest outer and dual
  billiard papers up to arXiv:2610.06418, planar; nothing new from Glutsyuk after arXiv:2607.26158, which is on
  Josephson junctions), obtained the OpenAlex lists of works citing JEMS 2021 (9 works) and Invent. Math. 2026
  (1 work), none of them on outer billiards on S² or H², read the 2025 survey of Bialy and Mironov
  (arXiv:2510.03790; outer billiards only in the plane), and made one web search.
- **Read for the staleness check.** Glutsyuk, JEMS 23 (2021), published version (CC BY): Sections 1.1, 1.2, 2.1 and
  6 (Definitions 1.1, 2.2, 2.4, 2.10, Proposition 2.5, Corollary 2.11, Theorems 1.6 and 1.25 with their proofs'
  structure), and a full-text search: outer billiards appear only in the plane (historical remarks, Remark 3.2, the
  reference to Glutsyuk–Shustin). Glutsyuk, Invent. Math. 245 (2026), arXiv v4: Sections 1.1, 1.2, the start of 8.1;
  outer billiards only in the plane (Remark 1.10, Example 1.13, history). Glutsyuk, arXiv:2301.00464,
  arXiv:2509.11257 (Regul. Chaotic Dyn. 31 (2026)), arXiv:2605.24434, Glutsyuk–Shustin (arXiv:1607.07593): planar
  outer billiards only; arXiv:2509.11257 describes the dual of inner billiards on S² and H² as I-angular billiards.
  Glutsyuk, "On curves with the Poritsky property" (J. Fixed Point Theory Appl. 24 (2022), arXiv:1901.01881): outer
  billiards on S² and H² for the area Poritsky property, with the spherical duality recalled in its Remark 1.12
  (numbering of arXiv v2, the version read); no polynomial integrals. Glutsyuk–Izmestiev–Tabachnikov (arXiv:1909.09028): Poritsky and Liouville properties. The
  minicourse abstract (above). Bialy–Mironov, J. Geom. Phys. 115 (2017): the dual of the inner billiard on S² is the
  outer billiard on S², on H² the outer billiard on the de Sitter surface; no theorem on outer billiards on S² or H².
  The surveys of Bialy and Mironov (Phil. Trans. R. Soc. A 2018; arXiv:2510.03790) and the open problems list
  arXiv:2110.10750: outer billiards only in the plane, apart from the duality.
- **Result of the search.** No source states that the outer billiard on S² or H² is an I-angular billiard, and none
  states the conclusion of Problem 4.4 for these surfaces. Outer billiards in H² are studied for other questions
  (Tabachnikov 2002; papers on polygonal outer billiards and tilings). The record is not stale. A search that finds
  nothing is not a proof of novelty.
- **Caveats.** The proofs of the cited theorems (55 pages, resp. 98 pages in arXiv v4) were not checked; they are
  refereed. The Inventiones version was not compared with arXiv v4 (the publisher's page refused automated access,
  and this was not worked around), and the journal version of the Poritsky paper was not compared with arXiv v2.
- **Scope.** Complete for polynomial and rational first integrals and C⁴ ovals (C³ on S² for polynomial
  integrals). 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/