# Verification report — AMR-050-0049 (Reznik–Garcia–Koiller rows k807, k808, k809, k810 and k110: midpoint polygons and four inversive area invariants of N-periodics in the elliptic billiard)

Verification dates: 2026-10-15 (runs A and B, on the first write-up) and 2026-10-15/16 (third run, on the text of the
note).

**Verdict.** The note proves the four inversive rows completely, and it shows that the fifth row (k110) is true
but not new.
- **Settled (new, to our knowledge).** For every ellipse with a > b > 0, every confocal *elliptic* caustic, every
  *least* period N of the stated parity, every turning number τ (convex and star trajectories) and signed areas:
  - k807 (AMR-050-0049) and k809 (AMR-050-0051), even N: A·A^⊗ = (c1/2 + c2/4)/κ and A′·A′^⊖ = κ(c1/2 + c2/4) at
    every member, where κ = ab/(a″b″) and c1 = F(u)F(u+Δ), c2 = F(u)G(u+Δ) are the constants of Theorem A;
    for N = 4 the values are 4a⁴b⁴/(a²+b²)² and 4a²b² (Theorem 7.1(a), Corollary 7.2(a)).
  - k808 (AMR-050-0050) and k810 (AMR-050-0052), odd N: A′^⊖ = (2+γ)A/4 and A^⊗ = (2+γ)A/(4κ) at every member,
    with γ = 2a²b²(a²b² − (a²+b²)λ)/(a⁴b⁴ − c⁴λ²), λ = a² − a″², and 2 + γ > 0 for every caustic (Lemma 6.1);
    hence A/A^⊗ = A′/A′^⊖ = 4κ/(2+γ) at every member with A ≠ 0, which is every member for τ = 1
    (Theorem 7.1(b),(c)). For N = 3: γ = −1 and the value is 4κ.
- **k110 (AMR-050-0004): true, not new.** A·A″ = κ⁻²·A·A′ is constant for every even N. This follows at once from
  Chavez-Caliz, Arnold Math. J. 7 (2021) 91–106, Theorem 3 (p. 93) = Theorem 6 (p. 104) (A·A′ constant for even
  n) together with Stachel's relation A′ = κ²A″ (J. Geom. 112 (2021) Art. 40, Subsection 3.2, Definition 3.10,
  Lemma 3.11), or from Chavez-Caliz's theorem applied to the inner polygons by pole–polar duality
  (Proposition 7.4). Theorem 7.1(a) re-proves it; for N = 4 the value is 8a⁴b⁴/(a²+b²)².
- **Not new mathematics.** The Jacobi parametrisation is Stachel's canonical parametrisation (Eur. J. Math. 8
  (2022), Theorem 2); the inner and outer polygons as axial images of the member with phase u + Δ are Stachel's
  conjugate billiard; the method (poles at degenerate polygons, cancellation, Liouville) is Chavez-Caliz's. The
  cases N = 3 of k808, k810 follow from Akopyan–Schwartz–Tabachnikov (Eur. J. Math. 8 (2022), Theorem 3) with
  Stachel's relation, and the cases N = 4, 6 of k807, k809 from Chavez-Caliz's theorem and the midpoint lemma.
- **Checked three times.** Two independent AI-assisted verification runs examined the results before the note was
  written (2026-10-15); a third, also AI-assisted, examined the text of the note (2026-10-15/16), first the parts
  added when the note was written: Lemma 6.1 (the algebraic form of γ and 2 + γ > 0), the identity
  sn(K − Δ) = a″/a in the proof of Lemma 3.3(a), and the program `check_note.py`. No run found an error in a lemma
  or a theorem; all corrections they required were made.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Reznik, R. Garcia, J. Koiller, arXiv:2004.12497v11 (29 October 2020), "Eighty New
  Invariants of N-Periodics in the Elliptic Billiard", tex source: rows k807 = A·A^⊗ (even), k808 = A/A^⊗ (odd),
  k809 = A′·A′^⊖ (even), k810 = A′/A′^⊖ (odd) in the table of invariants of inversive objects; row k110 = A·A″
  (even) in the table of distance, area and angle invariants; all with value "?" and proof "?". Definitions:
  signed areas (preliminaries); outer polygon (tangent intersections) and inner polygon (points of contact with
  the caustic); P^⊗ = inversions of the vertices of P in the caustic, P′^⊖ = inversions of the vertices of P′ in
  the billiard ellipse (list of symbols; footnote to the inversive table: the inverted vertices lie at the
  midpoints of the sides of the inner polygon and of the trajectory); the inversion of a point in an ellipse is
  described as the midpoint of the chord joining the points of contact of the tangents from the point. One
  sentence of the inversive section attaches the two symbols the other way round; the list of symbols, the
  footnote and the corpus records agree with the reading used.
- **Journal version.** "Fifty New Invariants of N-Periodics in the Elliptic Billiard", Arnold Math. J. 7 (2021)
  341–355, doi:10.1007/s40598-021-00174-y: k110 is listed without value or proof (Table 2); the inversive table
  (Table 9) has only k801–k804, so k807–k810 are rows of arXiv v11 only. The journal renumbers v11's k113
  (A′/A″) as k112; the note uses the v11 codes throughout.
- **Corpus records.** ulamai/UnsolvedMath 1.6.0, AMR-050-0049 (k807), -0050 (k808), -0051 (k809), -0052 (k810),
  -0004 (k110): "Prove that, for even N [odd N], the quantity … is constant as P varies through the family";
  the records define A^⊗ and A′^⊖ as "the areas after elliptic inversion of P in its caustic and of P′ in the
  billiard ellipse"; upstream status open.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Inversion in an ellipse: radial (|OY||OY*| = |OQ|²) or midpoint of the chord of contact | the same point Y/(Y·MY) | Lemma 2.1 |
| Which polygon is inverted in which conic | P in the caustic, P′ in the ellipse (list of symbols, footnote, records); inverting P in the ellipse would return P | Section 2, Section 8 |
| "N-periodic" = least period N | covered; all trajectories with a given caustic have the same least period | Lemma 3.3 |
| Turning number τ ≥ 2 (star trajectories) | covered: products at every member, ratios at every member with A ≠ 0 | Theorem 7.1 |
| Unsigned areas | same answer for τ = 1 (all five areas positive) | Theorem 7.1(c) |
| Hyperbolic caustics | not treated (the source's caustic is an ellipse); tested numerically by both runs only | Section 8, "The hypotheses" |
| Value of the constant | not asked; closed forms for N = 4, explicit 4κ/(2+γ) for odd N | Corollary 7.2, Lemma 6.1 |

## Results in the paper
- **Lemma 2.1.** Inversion in a central conic {X·MX = 1}: Y ↦ Y/(Y·MY) is the midpoint of the chord of contact and
  the radial inversion.
- **Lemmas 3.1–3.3, Proposition 3.4.** sn²Δ = λ/b², b″ = b cn Δ, a″ = a cn Δ/dn Δ, κ = dn Δ/cn²Δ,
  sn(K − Δ) = a″/a; the chord P(u−Δ)P(u+Δ) touches the caustic at Q(u) = σ⁻¹P(u), and the tangents at its ends
  meet at R(u) = σP(u); existence of the caustic for every (N, τ) (with K → ∞ and K − Δ → arsinh(c/b) as
  λ → b²), closure, least period, turning number, and the converse; P″ = σ⁻¹P^[u+Δ], P′ = σP^[u+Δ].
- **Lemmas 4.1–4.3, Corollary 4.4.** Midpoint polygon area S/2 + S₂/4; P′^⊖ and P^⊗ are the midpoint polygons of
  P and P″; A = F(u), A′ = κF(u+Δ), A″ = F(u+Δ)/κ, A′^⊖ = F/2 + G/4, A^⊗ = (F/2 + G/4)(u+Δ)/κ;
  A′/A″ = κ², k809(u) = κ²k807(u+Δ), k810(u) = k808(u+Δ).
- **Lemmas 5.1–5.3.** Periods h, 2K, anti-period 2iK′, evenness; singular phases S₀ = iK′ + Λ₀ and S₁ = S₀ − Δ
  (equal for odd N, disjoint for even N); at most simple poles; G ≡ 0 for N = 4; folding: X(p − ε) = −X(p + ε)
  at every singular phase.
- **Theorem A (even N).** F(u)F(u+Δ) and F(u)G(u+Δ) are constant (Liouville).
- **Theorem B (odd N), Lemma 6.1.** F, G have the period Δ, G = γF with γ = sn 2h(1 + dn h)/(sn h(1 + dn 2h));
  residue of F at iK′: ρ = iab(dn h − 1)/(k² sn h) ≠ 0; γ = 2a²b²(a²b² − (a²+b²)λ)/(a⁴b⁴ − c⁴λ²), 2 + γ > 0.
- **Theorem 7.1, Corollary 7.2, Remark 7.3, Proposition 7.4.** The rows (as in the verdict); N = 4, 3, 6; what
  follows from print; k110 from print in two ways, with the general-position check of Chavez-Caliz's hypothesis
  for the pairs (E, C) and (C, E*).
- **Proposition 8.1 (outside the records).** k113 = κ² for all N; for odd N, A′ = κA and A = κA″, so k103 = k109 = κ
  and k112 = 1 wherever A ≠ 0; A′A″ (k111 of v11) is constant for no N (residue ρ for odd N, 2ρ for even N).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The program of the note** (`check_note.py`, mpmath at 50 digits and sympy, about 13 s; 194 checks, all
  passed). (i) 56 families (four ellipses; N = 3, …, 10 with τ = 1; six star families per ellipse with N ≤ 12),
  seven members each, trajectories by the reflection law alone, derived polygons from their definitions: the five
  rows constant in the stated parity (largest relative spread 3.1e-42; largest closure defect 3.4e-42) and varying
  in the other (smallest relative spread 3.0e-13); side rows as in Proposition 8.1; all areas positive for τ = 1.
  (ii) Constants against the closed forms (c1, c2 from Jacobi functions at complex phases; G = γF; three forms of
  γ; 4κ/(2+γ)): largest relative deviation 2.2e-39. (iii) Exact N = 4 (sympy): the rhombus member and the whole
  family as identities modulo the relations of sn, cn, dn: tangency, reflection law and the four values of
  Corollary 7.2(a); the algebra of Lemma 6.1 and of the proof of Proposition 7.4 as identities in (a, b, λ).
  (iv) Lemma 3.1, Proposition 3.4 and Lemma 4.3 on the reflection-law orbits: 2.0e-38.
  (v) Residues at iK′ (ρ for odd N, 2ρ for even N), oddness, sn(K − Δ) = a″/a and the limit of Lemma 3.3(a),
  positivity in Lemma 6.1, Corollary 7.2(b),(c), the N = 6 closed forms of k106 and k110 printed by Garcia and
  Reznik.
- **The program with which the results were first obtained** (`original/verify_rows.py`, 40 digits, 50 in one
  block; 669 checks, all passed, about 5 s).
- **Verification run A** (`verification_run_A/`): geometric construction without Jacobi functions at 60 digits
  (95 families, 14 of them star families, 8 members each; radial inversion and chord-of-contact midpoints;
  relative spread of the constant rows ≤ 4e-58); Jacobi checks at complex phases (42 families); exact sympy
  checks (midpoint lemma N = 3..9, chord lemma, caustic axes, inversion lemma, N = 4 rhombus); N = 4 and N = 6
  closed forms; side rows; families with hyperbolic caustics (outside the setting).
- **Verification run B** (`verification_run_B/`): reflection-law construction at 70 digits (245 families: seven
  ellipses, all turning numbers for N ≤ 14, and N = 16, 20), Jacobi cross-check (15 families), the whole N = 4
  family in exact arithmetic (sympy, Gröbner reduction, about 10 minutes), 2 + γ for all odd N ≤ 51 and all τ on
  a grid of moduli (always positive), values of the side rows and of γ, wrong-input detection and the other
  readings of the definitions (no invariants), the inner family as a Poncelet family for (C, E*), families with
  hyperbolic caustics.
- **Verification run 3** (`independent_run_2/`, written independently of the programs above): plane geometry at
  60 digits with trajectories built from the tangents to the caustic (reflection law checked, inversions by the
  radial definition and as midpoints of chords of contact; 140 families: five ellipses, all turning numbers for
  N ≤ 13, seven members each; the sign of A on 48 members of each of the 85 star families: always positive); Jacobi
  functions at complex phases (Theorems A and B, residues by contour integrals, Lemma 3.3(a) at 250 digits, γ for
  890 odd families with N ≤ 41 and five moduli); exact algebra (every step of Lemma 6.1, Lemmas 2.1, 3.1, 3.2, 4.1,
  Proposition 7.4, Corollary 7.2(a) at the rhombus and at two members with irrational vertices). 236 checks, all
  passed.
- **Re-runs (2026-10-15, from a copy of the package).** `check_note.py` twice (identical apart from the timing
  lines); `original/verify_rows.py` and every program of both verification runs: all recorded outputs reproduced
  byte for byte (list with running times in `reproducibility/README.md`). The third run repeated all re-runs from an
  extracted copy of the archive: every recorded output reproduced byte for byte.

## Independent verification runs
The results were first obtained with proofs of all lemmas and of Theorems A and B, and with the program in
`original/`. Two independent verification runs, both AI-assisted, followed (2026-10-15); each re-derived every
proof, wrote its own programs, and re-ran the original program (output reproduced byte for byte).

| Item | Run A | Run B |
|---|---|---|
| Rows and definitions against the source | CONFIRMED (v11 tex and journal version; the symbol slip noted) | CONFIRMED (read independently; other readings give no invariants) |
| Lemmas 2.1, 3.1–3.4, 4.1–4.4 | CONFIRMED; the limit λ → b² in Lemma 3.3(a) needed an argument (K diverges) | CONFIRMED |
| Lemmas 5.1–5.3, Theorems A and B | CONFIRMED (both index maps of the folding lemma redone) | CONFIRMED |
| Theorem 7.1 and the N = 4 values | CONFIRMED (exact rhombus member) | CONFIRMED (whole N = 4 family exact) |
| k110 | true; follows from Chavez-Caliz + Stachel (not new) | true; follows from Chavez-Caliz by pole–polar duality, or + Stachel (not new) |
| k807–k810, novelty | no print or deposit found (bounded search) | no print or deposit found (bounded search) |
| Side statements | k111: even-N nonconstancy argument missing | k111: even-N residue missing; ratio rows for τ ≥ 2 need 2 + γ ≠ 0 |
| Citations | journal numbering of Chavez-Caliz; journal version of Garcia–Reznik; journal version of the source lacks k807–k810 | journal data of Akopyan–Schwartz–Tabachnikov and Bialy–Tabachnikov; journal version of Garcia–Reznik |

No run found a false lemma or theorem.

**Corrections required by runs A and B**, all applied in the note: k110 reported as a consequence of printed
results, with exact theorem numbers (Chavez-Caliz Theorem 3 p. 93 = Theorem 6 p. 104; Stachel Subsection 3.2,
Definition 3.10, Lemma 3.11), both routes (Proposition 7.4); the limit as λ → b² in Lemma 3.3(a) (K diverges and
K − Δ → arsinh(c/b)); the even-N residue 2ρ in Proposition 8.1(c); journal citations verified with Crossref
(Chavez-Caliz; Garcia–Reznik, Annales Math. Inform. 55 (2022) 49–75; Akopyan–Schwartz–Tabachnikov, Eur. J. Math.
8 (2022) 1313–1327; Bialy–Tabachnikov, Eur. J. Math. 8 (2022) 1341–1354); k807–k810 as rows of arXiv v11 only;
bounded novelty wording, with the unread chapter "On the Diagonals of Billiards" (doi:10.1007/978-3-031-13588-0_2)
and Stachel's KoG 26 paper (abstract only) named; the scope remark on hyperbolic caustics; the condition
2 + γ ≠ 0 for τ ≥ 2 (now proved, Lemma 6.1).

**Additions made when the note was written** (examined by the third run; checked by `check_note.py` and by the
programs of the third run): Lemma 6.1 (γ as an algebraic function of a, b, λ, and 2 + γ > 0 for every caustic); the
identity sn(K − Δ) = a″/a in the proof of Lemma 3.3(a); the general-position check in Proposition 7.4 written out;
`check_note.py`.

**Third run (2026-10-15/16, on the text of the note).** It re-derived the additions first — Lemma 6.1 from the
quotient of the residues through every displayed step to the λ-form, with 2 + γ > 0 (the denominator is positive
since c²λ < c²b² < a²b²; q is concave with q(0) = 2a⁴b⁴ and q(b²) = b⁶c²), and the identity sn(K − Δ) = a″/a
(from sn(K − u) = cd u) with the limit in Lemma 3.3(a) — and then every other proof line by line. It read
`check_note.py` and compared it with Section 9 of the note; it compared the definitions and statements with the
arXiv v11 tex of the source, the journal PDF of Chavez-Caliz (Theorem 3 on p. 93, Theorem 6 on p. 104), the arXiv
tex of Stachel's two papers (Subsection 3.2, Definition 3.10, Lemma 3.11; Theorem 2), of Akopyan–Schwartz–Tabachnikov
(Theorem 3) and of Garcia–Reznik, and the journal PDF of Garcia–Reznik (Proposition 4.9); it repeated the literature
and Zenodo searches (the newest records of A. Kriebel and of the author included: none treats k807–k810 or k110) and
read the author's preprint of Stachel's chapter on diagonals (no statement on areas). Its programs and the re-run of
the package agree with the note. It found no error in a lemma or a theorem and required ten corrections, all made:
the abstract also credits the cases N = 4, 6 of k807, k809 to Chavez-Caliz's theorem; Proposition 8.1(b) is stated
by the identities A′ = κA, A = κA″; two sentences of Section 2 were made precise (the lines fixed by the reflection
in the normal; if one trajectory with a given caustic is periodic, then all are); the Verification paragraph, the
searches and the list of what was read record the third run; the running time of `check_note.py` is given
consistently (13 s); a comment of `check_note.py` named a non-existent Lemma 6.3 (corrected to 6.1; the output is
unchanged); this report and the package record the run (`independent_run_2/`).

## Relation to the literature, novelty and scope
- **Searches (2026-10-15, first write-up and both runs; Zenodo repeated when the note was written).** arXiv API
  (elliptic billiards; areas, midpoints and inversions of Poncelet polygons; works of Reznik, Garcia, Stachel,
  Tabachnikov, Chavez-Caliz), OpenAlex (19 works citing the source, 23 citing Chavez-Caliz, 13 citing Stachel's
  J. Geom. paper), Semantic Scholar (31 works citing the arXiv version of the source), zbMATH Open, Crossref,
  Zenodo (the labels k807–k810 give no record; all records of the two authors of the 2026 notes on sibling rows
  were screened), one general web search. Repeated by the third run on 15–16 October 2026 (arXiv API, OpenAlex,
  zbMATH Open, Crossref, Zenodo including the newest records of A. Kriebel and of the author, one more web search):
  nothing new. No printed or deposited proof of k807–k810 for general N was found.
- **What was read.** The source (arXiv v11 tex; journal version, Tables 2 and 9); Chavez-Caliz (arXiv v1 tex and
  the journal PDF: Theorems 3 and 6 and the proof of Theorem 6); Stachel, J. Geom. 2021 (arXiv v2 tex, Section 3;
  the journal numbering was not compared); Stachel, Eur. J. Math. 2022 (arXiv v2, Theorem 2 and its proof);
  Akopyan–Schwartz–Tabachnikov and Bialy–Tabachnikov (arXiv v2 tex); Garcia–Reznik (arXiv v1–v3 tex and the
  journal PDF); the IMPA survey of Garcia and Reznik (text searched). Not read: the published chapter "On the
  Diagonals of Billiards" (an author's preprint with this title was read by two runs: envelopes of diagonals and
  hyperboloids, no statement on areas); Stachel's "A Triple of Projective Billiards", KoG 26 (2022), abstract
  only. The proofs of the cited theorems were not checked.
- **Related unrefereed notes of the author.** doi:10.5281/zenodo.23065067 (k806,b) and
  doi:10.5281/zenodo.23094117 (k805), rows of the same inversive table; doi:10.5281/zenodo.23075934 (k107 and
  k111 for N = 4); doi:10.5281/zenodo.23396998 (central symmetry, seven rows); doi:10.5281/zenodo.23400541 and
  doi:10.5281/zenodo.23403151 (the same method for other rows).
- **Caveats.** The even-N constants c1, c2 are not given in closed form beyond N = 4 (and N = 6 with Garcia and
  Reznik's printed closed forms, confirmed only numerically); hyperbolic caustics are not treated; for odd N and
  τ ≥ 2 the note does not exclude members with A = 0 (where the ratio rows are 0/0; none was found in the tested
  families); a negative search is not a proof of priority.
- **Scope.** Complete for k807–k810 under the stated hypotheses; k110 follows from printed results; no priority
  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/