# Verification report — AMR-129-0019 (Tabachnikov: density of reflected wave fronts; balls and ellipses)

Verification date: 2026-10-18.

**Verdict.** The note gives a **partial answer**; every statement it makes is proved, and the scope is as follows.
- **Settled (balls).** In the open unit ball of R^d, d ≥ 2, the time-t front W_t(P) becomes dense in the sense of
  Kang–Knill (it meets every open ball for all sufficiently large t) for every source P other than the centre; for
  the centre it is a sphere of radius dist(t, 2Z) and does not become dense (Theorem 1.2).
- **Known in substance (disk).** For the disk, density for P ≠ centre follows from Colin de Verdière–Vicente,
  arXiv:2011.14886, Theorem 4.1 (Exp. Math. 33 (2024) 254–260): the length of the front inside every smooth subdomain
  grows linearly. Their proof is a sketch (the non-degeneracy of the phase for every Fourier mode is not written out)
  and they do not state the density corollary; Kang–Knill (2025/2026) and Tabachnikov (2026) regard the disk as
  open. The ball reduces to the disk by rotational symmetry (Corollary 3.5). The note gives a complete,
  self-contained proof.
- **Settled (ellipses), new as far as was found.** In an ellipse the front becomes dense exactly when P is not a
  focus (Theorem 1.3); from a focus all photons are at a focus at the times 2a, 4a, … (Proposition 4.5).
- **Known in substance (non-convex tables).** An explicit C^∞ non-convex table of Penrose type in which the fronts
  of all points of an open set miss a fixed disk forever (Proposition 5.1); Penrose 1958 and Rauch 1978 as reported
  by Chernov–Galperin 2003, and Kang–Knill Section 5 for the piecewise smooth Penrose table. Presented as a remark.
- **Open.** General convex and generic smooth domains (Kang–Knill, Conjectures 3–5), higher-dimensional ellipsoids,
  the Riemannian analogue.

The note is unrefereed.

## Statement checked
- **Primary source.** B. Albach, J.-F. Barraud, M. Bialy, …, S. Tabachnikov, "Open problems in billiards and
  quantitative symplectic geometry", arXiv:2602.12896v1 (13 February 2026), read in the LaTeX source, file
  `Contributions/Tabachnikov.tex`, paragraph "Density of reflected wave fronts": a domain with smooth boundary in
  Euclidean space, a generic interior point P as a light source, photons in all directions with optical reflection;
  the question is whether the time-t front becomes dense as t → ∞. The paragraph refers to Kang–Knill for the
  discussion, says that even the round circle is open, and cites Colin de Verdière–Vicente for the length growth.
- **Definition used.** E. Kang, O. Knill, "Density of wave fronts", arXiv:2501.14611v4 (LaTeX source), Definition 1
  (Section 3): W_t(P) becomes dense if for every open ball U there is τ(P, r) with W_t(P) ∩ U ≠ ∅ for all t > τ.
  Published in The Mathematical Intelligencer (online 10 June 2026), doi:10.1007/s00283-026-10528-z (Crossref; no
  volume or pages yet; the journal version was not compared).
- **Corpus record.** ulamai/UnsolvedMath, AMR-129-0019, status `open`; the statement reproduces the source
  paragraph.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Kang–Knill density ("for all t > τ") in a ball, P ≠ centre | yes | Theorem 1.2(a), Theorem 3.4, Corollary 3.5 |
| the same at the centre of a ball | no (front = sphere of radius dist(t, 2Z)) | Theorem 1.2(b) |
| the same in an ellipse, P not a focus | yes | Theorem 1.3(a), Section 4.4 |
| the same at a focus | no (front = a focus at t = 2an) | Theorem 1.3(b), Proposition 4.5 |
| "every smooth domain, generic P" (universal reading) | false without convexity (known in substance) | Remark 1.4, Proposition 5.1 |
| general convex / generic smooth domains (the intended setting of Kang–Knill's conjectures) | not treated; open | Scope and priority |

## Results in the paper
- **Proposition 2.2 (criterion).** If γ_θ(t) = X(κ(θ), Θ⁰(θ) + tω(κ(θ))) with κ and X continuous, Θ⁰ Borel, and for
  every k ≠ 0 the phase k·ω∘κ has a null critical set, then the law of γ_θ(t) (θ uniform) converges weakly as the real
  parameter t → ∞ (Riemann–Lebesgue, Stone–Weierstrass, continuous mapping); by the portmanteau theorem the front
  meets every open set of positive limit measure for all large t.
- **Disk.** Lemma 3.1 (explicit linear flow on a torus, frequencies (2)), Lemma 3.2 (k·ω non-constant on (−1, 1):
  limits 1/4 and 1/(2π), or a pole), Lemma 3.3 (the image of Haar measure has density
  1/(2π√(1−ρ²)√(|z|²−ρ²)) on the annulus ρ < |z| < 1), Theorem 3.4; Corollary 3.5 (ball = surface of revolution of
  the planar front).
- **Ellipse.** Elliptic coordinates and the separation (3); the integral κ, formula (4); invariant tori in both
  regimes with uniformising variables (5), (6); periods (7), (8); Lemma 4.1 (linearisation, Jacobian F/A > 0);
  Lemma 4.2 (joint continuity of (κ, Θ) ↦ X_κ(Θ)); Lemmas 4.3, 4.4 (twist for every Fourier mode, from the limits
  at the ends of the regimes: 1/perimeter vs 1/(4a), the 1/log expansion (11), and tan x < 4x/π); Section 4.4
  (the directions from P, continuity of Θ⁰, full support); Proposition 4.5 (foci).
- **Proposition 5.1.** C^∞ table of Penrose type; focal property; induction over the segments with the two cases
  (no segment on the axis; endpoints of the flat tops).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **`check_note.py`** (written with the note; standard library; about 0.5 s; 37 checks, all passed; its printed
  labels follow the numbering of the note since the run on the final text): Lemma 3.1
  against a direct simulation (1.1e-12); Lemma 3.2 limits; Lemma 3.3 identity exactly on a rational grid and the
  radial law by sampling; Corollary 3.5 in the 3-ball (1.7e-12); κ under 3,000 reflections (6e-15); the frequencies
  (9) against simulated rates in 8 trajectories (3e-4); the limits of Lemmas 4.3, 4.4 (7 digits, D_0 to 6 digits,
  first order of (11) to 6e-4); Proposition 4.5 exactly for 1,428 rational directions in the ellipse (5, 4, 3); the
  arithmetic of Section 5 exactly; the focal property (20,000 reflections).
- **Programs of the first version** (`original/`, NumPy/SymPy/mpmath): Lemma 3.2 symbolically, Lemma 3.1 against
  simulation (8.4e-13), disk fronts (coverage, Kolmogorov distance to the limit law, length), ellipse frequencies
  and limits, κ symbolically invariant, the focal property on 43,731 lines, ellipse fronts and the foci (2e-13), the
  arithmetic of Section 5.
- **Verification run A** (`verification_run_A/`): Lemma 3.1 (4.3e-12, 3,000 cases); coverage in the disk; the
  "all large t" probe (ball of radius 0.03 about (0.3, 0.2), 4,000 times in [2000, 2020], 2·10⁵ photons: met at every
  time for p = 0.5, missed 5 times for p = 0.1 — τ(P, r) can be large near the centre); κ (4e-15, 2e-15); frequencies
  (4–5 digits, 8 cases); all endpoint limits; the foci (2e-14); simulations of the table of Section 5 with sharp and
  rounded corners (no segment meets V).
- **Verification run B** (`verification_run_B/`): closed form vs simulation (3.7e-10 at t = 1000); largest empty
  ball of the sampled front up to t = 10⁶ (disk) and t = 10⁵ (3-ball), the centre never below 0.5; planar reduction
  of the 3-ball (2.4e-11); frequencies in 36 trajectories of three ellipses (1e-5) and the classical rotation-number
  formula (8 digits); deliberately wrong formulas rejected (25–81 % off); fronts in four ellipses; the table of
  Section 5 with 2.8·10⁵ photons and 4.9·10⁷ reflections (none meeting V; teeth moved off the foci leak); length of
  the disk front / t → 2 arcsin p; re-run of the seven programs of `original/` (byte-identical outputs).
- **Verification run on the final text** (`independent_run_2/`): see the section of that name below.

## Independent verification runs
The results were first obtained with written proofs and the programs in `original/`. Two independent verification
runs, both AI-assisted, followed (2026-10-18); each re-derived the proofs and wrote its own programs, and neither
read the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, Kang–Knill Definition 1, record text against the sources | CONFIRMED (LaTeX sources) | CONFIRMED (LaTeX sources) |
| Balls (Theorem 1.2), incl. "for all t > τ" via real t → ∞ | CONFIRMED (re-derived) | CONFIRMED (re-derived; numerics to t = 10⁶) |
| Disk implied by Colin de Verdière–Vicente Thm 4.1 (sketch, corollary not stated) | CONFIRMED (source read) | CONFIRMED (source read) |
| Ellipses (Theorem 1.3(a)) | CONFIRMED_WITH_FIXES: joint continuity of (κ, Θ) ↦ X_κ(Θ) and continuity of Θ⁰ to be proved (the first version only asserted them) | CONFIRMED (re-derived, no gap found) |
| Foci (Proposition 4.5) | CONFIRMED | CONFIRMED |
| Non-convex example (Proposition 5.1) | CONFIRMED_WITH_FIXES: no segment on the axis; endpoints of the flat tops | CONFIRMED; known in substance (Penrose 1958, Rauch 1978 via Chernov–Galperin) |
| Literature, novelty | disk implied by [dVV24]; novelty of balls/ellipses not checked | Chernov–Galperin 2003 found (illumination, not time-t density); nothing settling ellipses or balls found |

No run found a mathematical error.

**Corrections required by the runs**, all applied in the note:
1. (A1) Joint continuity of (κ, Θ) ↦ X_κ(Θ) on each regime interval × T², and continuity of θ ↦ Θ⁰(θ): Lemma 2.3
   (continuity of inverses), Lemma 4.2, Section 4.4; the "Lusin + Tietze" sentence of the first version is gone.
2. (A2) Proof of Proposition 5.1: no segment lies on the axis; the endpoints of the flat tops cannot be reached from
   below (infinite-order flatness).
3. (A3) Theorems 1.2 and 1.3 lead; the non-convex example is Remark 1.4 / Section 5; the disk is "settled by
   Theorem 1.2; implied by [dVV24, Theorem 4.1] (sketch proof, corollary not stated)".
4. (A4) One torus per librational κ (bands identified by w ∼ −w): Section 4.2.
5. (A5, B4) Kang–Knill Section 5 cited for the Penrose non-density, with the note in their Section 7 credited to
   Tabachnikov; [dV21] Corollary 5.1 numbered.
6. (A6) Novelty hedged; "a search that finds nothing is not a proof of novelty".
7. (A7) τ(P, r) can be large near the centre: Section 6.
8. (B1) Chernov–Galperin, Regul. Chaotic Dyn. 8 (2003) 225–241, cited in Section 1.3 and "Scope and priority"
   (illumination by a narrow beam in circles and ellipses, not time-t density).
9. (B2) The Penrose-type statement credited to Penrose 1958 and Rauch 1978 as reported by Chernov–Galperin.
10. (B3) The ball case presented as an immediate corollary of the disk case (Corollary 3.5).

**Written with the note, after the two runs.** Proposition 2.2 (the criterion in general form), Lemma 2.3, Lemma 4.2,
the uniformising variables of Section 4.2, Corollary 3.5 in its present form, and `check_note.py`. These parts, with
the whole text, were examined by the verification run on the final text (next section).

## Independent verification run on the final text (2026-10-18)

A further independent verification run, AI-assisted, examined the text of the note after it was written (the two
runs above had examined the results as first written), checking everything again without relying on the earlier
checks. Its programs and recorded outputs are in `reproducibility/independent_run_2/`.

| Item | Verdict |
|---|---|
| Question, Definition 1.1, Kang–Knill §3, §5 (Conjectures 3–5, the Penrose table), §7 note, against the LaTeX sources; corpus record | CONFIRMED |
| Lemma 2.1, Proposition 2.2, Lemma 2.3 | CONFIRMED; Proposition 2.2 needed the continuity of κ among its hypotheses (used for the continuity of G on J × T²) — added |
| Disk and ball: Lemmas 3.1–3.3, Theorem 3.4, Theorem 1.2(b), Corollary 3.5, Remark 3.6 | CONFIRMED (re-derived) |
| Ellipse: §4.1, §4.2, Lemma 4.1, Lemma 4.2, Lemmas 4.3, 4.4 (incl. (11) and tan x < 4x/π), §4.4, Proposition 4.5 | CONFIRMED (re-derived) |
| Proposition 5.1 | CONFIRMED (re-derived) |
| Cited results: Colin de Verdière–Vicente Theorems 3.1, 4.1 (arXiv v1) and the sketch of the proof of 4.1; Colin de Verdière 2021 Theorem 5.1, Corollary 5.1; Chernov–Galperin §1, Theorems 2.1, 2.3 | CONFIRMED; the description of Colin de Verdière 2021 corrected (Theorem 5.1 concerns integrable Hamiltonians on surfaces without boundary) |
| Bibliographic data (Crossref: all six DOIs; arXiv metadata of the four e-prints) | CONFIRMED |
| Novelty and credit | CONFIRMED: nothing found that settles balls or ellipses; disk credited to Colin de Verdière–Vicente Theorem 4.1; the example of §5 known in substance |

**Own programs** (`independent_run_2/`): Lemma 3.1 against a 40-digit simulation up to t = 3000 (3·10⁻³³) and
3,000 float cases (8·10⁻¹²); Lemmas 3.2, 3.3 symbolically; the Kolmogorov distance of the radial law of the disk
front (β uniform) to that of μ_p, from 0.71 and 0.14 at t = 10 to 0.0072 and 0.0011 at t = 10⁵ (p = 0.2, 0.6);
Theorem 1.2(b); Corollary 3.5 in balls of dimension 3–5 (5·10⁻¹²); ∫_D Ψ = π arcsin a and the front length
(2 arcsin p); Section 4.1 symbolically and numerically; Lemma 4.1 and (9) tested directly: the phases
Θ = Λ_κ(φ), computed from the states of 9 simulated trajectories in three ellipses (both regimes), satisfy
Θ(t) − Θ(0) − tω(κ) ∈ Z² to 3·10⁻¹¹ for t ≤ 1500; the limits of Lemmas 4.3, 4.4 and the coefficient of 1/L in
(11) with up to 160 digits (ε down to 10⁻⁶⁰), with the remainder decaying like 1/L; non-constancy of k·ω for
0 < |k_i| ≤ 6, including the exceptional direction k = (1, 1) of the ellipse (2, √3); Proposition 4.5 with 50
digits; a fixed ball met at 400 times in [800, 810] from sources on the focal segment, at the centre, on the axis
outside the foci and at a generic point; the table of §5 with polyline-rounded upper corners: 2.9·10⁶ segments
from sources in R₁, none meeting V, and the statements (i), (ii) of the induction hold for every segment (controls:
sources in Q_R reach V; teeth moved to ±0.6 leak). All checks pass.

**Re-run of the package** from an extracted copy of `source.zip`: all 27 recorded runs repeated (`independent_run_2/rerun_package_summary.txt`): 11 outputs byte-identical
(`check_note.py`, the seven programs of `original/`, three programs of run B); 11 identical apart from the line with
the elapsed time; `ellipse_check.py` identical apart from that line and a SciPy warning, which went to standard error
in the re-run; the four `vb_theoremA.py` runs, whose seeds were not recorded, give other counts with the default seed
and the same verdicts (teeth at ±1: no photon meets V, all axis crossings |x| ≤ 0.8657; teeth at ±0.6: 15,864 of
30,000 photons meet V, recorded 16,327).

**Corrections required by this run (all made):** (1) continuity of κ added to the hypotheses of Proposition 2.2;
(2) the abstract names the limit measure correctly (the distribution of the photons for uniformly distributed
directions, not the normalised length measure; Remark 3.6); (3) the description of Colin de Verdière 2021 in §1.3;
(4) "ω_ν = 1/|∂E| to 6 digits" in §6 (the recorded values for (1, 0.6) agree to 6 digits); (5) the labels printed
by `check_note.py` follow the numbering of the note ((4), (7)/(8), (9), (11), Proposition 4.5; it was printing
(4.2), (4.6)/(4.7), (4.9), (4.8), Proposition 4.6), and its header gives the running time; output re-recorded;
(6) the Verification paragraph and the description of the searches record this run; (7) this report and the
package README record this run; (8) §1.2 now states that the proofs give weak convergence for the directions of
an arc (in the disk, by symmetry, for all directions); for the ball in dimension d ≥ 3 the note proves density
via Corollary 3.5.

## Relation to the literature, novelty and scope
- **Searches (18 October 2026).** arXiv API (wave fronts with density, billiards, equidistribution, reflection,
  caustics, illumination, Wiedersehen, exponential map; recent work of Knill, Colin de Verdière, Vicente, Tabachnikov),
  Crossref, zbMATH, OpenAlex, Semantic Scholar (citing works of Kang–Knill, Colin de Verdière–Vicente, Colin de
  Verdière 2021 and of the problem collection: apart from these papers themselves, none), six web searches, Knill's
  blog post "Wave density conjectures" (20 December 2025; polygons). The run on the final text repeated the arXiv
  (15 queries), Crossref, zbMATH and OpenAlex searches and made one of the web searches; OpenAlex lists no work citing
  the journal versions of Kang–Knill and Colin de Verdière–Vicente.
  - No printed answer to the question was found, apart from the consequence of [dVV24, Theorem 4.1] for the disk.
  - Colin de Verdière, arXiv:2101.01919: surfaces without boundary with an integrable Hamiltonian (Theorem 5.1);
    geodesic flows on closed surfaces, length inside a domain (Corollary 5.1),
    no billiards.
  - Chernov–Galperin 2003: illuminated set of a narrow beam in circles and ellipses (Theorems 2.1, 2.3) — union over
    time; the same twist mechanism.
- **What was read.** Tabachnikov's paragraph and Kang–Knill (arXiv v4) in LaTeX; Colin de Verdière–Vicente in the
  LaTeX source of arXiv v1 (the journal version was not accessible to us; whether Theorem 4.1 is in the published
  version is not known); the relevant parts of
  Colin de Verdière 2021; Chernov–Galperin in the publisher's file (Sections 1–2). Tabachnikov, Geometry and
  Billiards: only the Crossref record of Chapter 4 ("Billiards inside conics and quadrics", pp. 51–71) — the
  note derives what it uses. Penrose 1958 and Rauch 1978 not seen (cited after Chernov–Galperin).
- **Side remark.** Kang–Knill, Section 5, quote the length of the disk front as arcsin(|P|) t + o(t); the formula in
  Colin de Verdière–Vicente (Theorem 3.1) and the computations give 2 arcsin(|P|) t + o(t). Stated neutrally in the
  note.
- **Scope.** Balls in every dimension and planar ellipses. No priority is claimed. This negative search is not a
  proof of novelty.

## 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/