# Verification report — AMR-050-0056, AMR-050-0057, AMR-050-0058, AMR-050-0059, AMR-050-0061 (Reznik–Garcia–Koiller: rows k813, k814, k815, k816, k818 of the table of inversive objects)

Verification date: 2026-10-15.

**Verdict.** Standing hypotheses of every statement: a > b > 0, a confocal elliptic caustic, closed billiard
trajectories of least period N with any winding number coprime to N, signed areas; each statement holds for both
foci.
- **AMR-050-0056 (k813, A_{j,pol}/A_j^†, all N): proved** (Theorem 1), for every N ≥ 3 and every winding number;
  for N = 4 the constant is 2a^4/b^4.
- **AMR-050-0057 (k814, A_{j,pol}/A_{j,dual}, all N): proved** (Theorem 2) with "dual" read as the polar reciprocal
  of the polar polygon with respect to a circle centred at O_p, the centre of the circle touched by its sides (the
  reading fixed by the source's caption and figure; the radius only scales the constant); for N = 4 the constant is
  2a^6/b^10 (unit circle). It also holds for the point-wise reading (inversion of the polar vertices in a circle at
  O_p): that polygon is the midpoint polygon of the reciprocal dual, and its area is a positive constant multiple of
  A^†.
- **AMR-050-0058 (k815, A^†_{j,ped}·A_{j,dual}, odd N): answered in corrected form.** The symbol A^†_{j,ped} is not
  defined in the source. Reading (i) (pedal polygon of the inversive polygon; the reading of the source's row
  k908,b and of the 2026 deposits on k908): false, exact certificate for N = 3; numerically false for every odd N
  tested. Reading (ii) (inversion of the focal pedal polygon, which is the polar polygon): false for every N, exact
  values for N = 3. Reading (iii) (the focal pedal area Ā_j): true for every odd N (Theorem 3(B)).
- **AMR-050-0059 (k816, A^†_{j,ped}/A_{j,dual}, even N): true for every even N ≥ 6 under every reading; N = 4
  depends on the reading.** Reading (i): false for N = 4 (exact certificate; the inversive pedal area has a pole of
  order exactly 3 there), true for every even N ≥ 6 (Theorem 4(B); the constant can be 0). Reading (ii): true for all
  even N (Theorem 2). Reading (iii): true for all even N (Theorem 4(C)).
- **AMR-050-0061 (k818, A_j^†/A_{j,ant}, N ≡ 2 mod 4): a scoped corollary of two earlier deposits of the same
  author**, doi:10.5281/zenodo.23094118 (A/A^† constant for N ≡ 2 mod 4) and doi:10.5281/zenodo.23079799 (A_ant = R·A
  for every even N, with R explicit); the note gives an independent proof. The quotient is constant on every family
  with R ≠ 0, and undefined at every member of the families with R = 0, i.e. λ(α−β)^2 = αβ(α−3β) (α = a^2, β = b^2):
  for N = 6 exactly b/a = 1/2; numerically one aspect ratio each for N = 10, 14, 18 (winding 1). For N = 6 the
  constant is (4a^2 − b^2)/(4a^3b^2(2b − a)) (b ≠ a/2), from the value of A/A^† given by Garcia and Reznik.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Reznik, R. Garcia, J. Koiller, "Eighty new invariants of N-periodics in the elliptic
  billiard", arXiv:2004.12497v11 (29 October 2020), read in its tex source: the section on inversive objects
  (`090_inversive.tex`, both tables and the figure of the polar and dual polygons), the definitions of signed area,
  pedal and antipedal polygons (`020`, `050`, `055`), the table of symbols (`210`) and the list of animations
  (`110`). **Published 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, examined by the third verification run in the publisher's PDF on the
  journal's website: it lists 56 invariants, and its table of inversive objects (Table 9 there) has only the rows
  k801–k804 (sums of inverse focal distances, perimeter and sum of cosines of the inversive polygon). None of the five
  rows, and no polar or dual polygon, occurs there; the rows exist only in the arXiv version, whose codes the corpus
  uses.
  - Rows (table `tab:inversive`, all "value ?", "proven ?"): k813 `A_{j,pol}/A_{j,inv}` (all N), k814
    `A_{j,pol}/A_{j,dual}` (all), k815 `A^†_{j,ped}·A_{j,dual}` (odd), k816 `A^†_{j,ped}/A_{j,dual}` (even), k818
    `A_j^†/A_{j,ant}` (≡ 2 mod 4). The caption credits k814 to a private communication of A. Akopyan; the mark is
    printed at k811 in the table, and the communication's title, "Sum of squared sidelengths of focus-polar polygon
    is invariant", concerns a sum of squared side lengths, the quantity of row k811.
  - Definitions: inversion in the unit circle at f_j; polar polygon = "antipedal of the inversive"; dual =
    "inversion of the polar with respect to O_p", said to be inscribed in a circle centred at O_p; A_{j,ant} = area of
    the antipedal polygon of P with respect to f_j; A_{j,ped} is in the table of symbols, A^†_{j,ped} and A_{j,inv}
    are not.
- **Corpus records.** ulamai/UnsolvedMath 1.6.0, records 5100056, 5100057, 5100058, 5100059, 5100061 (upstream
  status `open`). They transcribe the rows with the source's definitions; k813 writes A_j^† for the source's
  A_{j,inv}.

## Readings
| Reading | Answer | Where |
|---|---|---|
| A_{j,inv} = A_j^† (the corpus record's transcription) | k813 holds for every N | Theorem 1 |
| "dual" = polar reciprocal about O_p (caption, figure) | k814 holds for every N | Theorem 2(a) |
| "dual" = point-wise inversion of the polar vertices at O_p | k814 holds for every N | Theorem 2(b) |
| A^†_{j,ped} reading (i): pedal of the inversive polygon | k815 false (N = 3 exact; numerically every odd N tested); k816 false for N = 4 (exact), true for even N ≥ 6 | Theorems 3(A), 4(A), 4(B) |
| A^†_{j,ped} reading (ii): inversion of the focal pedal polygon (= polar polygon) | k815 false for every N; k816 true for every even N | Theorems 3(A), 2 |
| A^†_{j,ped} reading (iii): focal pedal area Ā_j (dagger a slip) | k815 true for every odd N; k816 true for every even N | Theorems 3(B), 4(C) |
| Radius of the circle at O_p | irrelevant (multiplies A_dual by r^4) | Section 1.1 |
| Least period N (not a repeated walk) | used throughout; a 3-periodic listed twice fails the even-N rows | Section 1.4, Section 8 |
| Hyperbolic caustics | not treated | Section 1.4 |

## Results in the paper
- **Method (Sections 2–4).** Stachel's parametrisation P(u) = (−a sn u, b cn u); every area is an elliptic-type
  function of the phase (in the class F: h- and 4K-periodic, 2iK′-antiperiodic). Its singular phases are the cosets
  I_1 (a side of P is an isotropic line through f), I_2 (the same for the other focus) and J (a vertex at infinity):
  I_1 = I_2 iff N is even, J = I_1 iff N ≡ 2 mod 4 (Lemma 4.1). Folding (Lemma 3.4): at I_1 ∪ I_2 the areas are odd
  in the local variable, so poles of order ≤ 2 are simple and holomorphic areas vanish. Comparison (Lemma 4.3):
  functions of F with simple poles on one coset are proportional; products that are entire are constant.
- **Theorem 1** (k813): A_pol = κ_1 A^†; N = 4: 2a^4/b^4 (Proposition 5.1, at the rhombus).
- **Theorem 2** (k814): A_dual = κ_2 A^†; A_dual^pt = κ_3 A^† with κ_3 > 0 (positivity from the formula
  A_dual^pt = (b^4/2a^2) Σ cos(θ_l/2) cos(θ_{l+1}/2) sin((θ_l+θ_{l+1})/2), θ_l the angle P_l f P_{l+1});
  A_dual^pt = A_dual/4 (N = 3), A_dual/2 (N = 4); N = 4: A_pol/A_dual = 2a^6/b^10.
- **Theorem 3** (k815): (A) certificate at a^2 = 8, b^2 = 3, λ = 72/25, members M_1, M_2: reading (i)
  243√10/256000 + 243/32000 vs 6561/1024000; reading (ii) 27√10/5 + 999/40 vs 729/40, and A_pol A_dual = κ_1κ_2 (A^†)^2
  is never constant. (B) Ā·A_dual constant for odd N (= 6561/25000 at M_1, M_2). Remark 6.3: for odd N the product
  under reading (i) is constant iff the residue of A^†_ped on I_1 vanishes; for N = 3 it does not (double poles on
  I_1; A_dual vanishes on I_2).
- **Theorem 4** (k816): (A) certificate at a = 5, b = 4, λ = 400/41, rhombus and rectangle: 2/1681 vs 1/800 (reading
  (i)); Proposition 6.4: pole of order exactly 3 of A^†_ped on I_1 for N = 4. (B) A^†_ped = κ_5 A_dual for even
  N ≥ 6 (Lemma 6.2: simple poles); κ_5 = 0 on the N = 10, τ = 3 family of doi:10.5281/zenodo.23073120 (Remark 6.5).
  (C) Ā = κ_6 A_dual for even N; N = 4: 2a^6/(b^2(a^2+b^2)^2).
- **Theorem 5** (k818): A_ant = R A and A^† = μA for N ≡ 2 mod 4 (corollary of the two deposits; independent
  proof); R = 0 iff λ(α−β)^2 = αβ(α−3β), which needs b/a < 1/√3; for N = 6, λ = a^2b^2/(a+b)^2 (the caustic given by
  Garcia and Reznik) and R = (2b−a)(a+b)^2/(b^2(2a+b)), zero exactly at b/a = 1/2 (exact hexagon with focal antipedal
  area 0); with their value A/A^† = 4a^3b^4/((2a−b)(a+b)^2) for N = 6, the constant of k818 is
  A^†/A_ant = (4a^2 − b^2)/(4a^3b^2(2b − a)) (35/16 at (a, b) = (1, 0.8)).

## Computations (sanity checks; programs and outputs in reproducibility/)
Apart from the exact certificates (Theorems 3(A), 4(A), the hexagon H), no proof depends on a computation.
- **Program of the note** (`check_note.py`, mpmath 50 digits and sympy, about 15 seconds; 171 checks, TOTAL
  failures: 0): 33 reflection-law families (three shapes, N = 3 to 12, windings up to 5): k813 and k814 (both duals)
  constant to 7e-36 (1e-42 unless k^2 > 0.9999), the identifications of Lemma 3.1 to 2e-43, positivity, the N = 4
  constants; the exact certificates for N = 3 and N = 4 under all readings; the symbolic rhombus; k816 constant for
  N = 6, 8 under all readings and both duals; k818: A_ant/A = R and A^†/A constant (N = 6, 10, 14), the exact
  hexagon H, the symbolic hexagon and R for N = 6, the roots of R = 0 for N = 10, 14, 18; controls outside the
  classes of N, unclosed orbits, a repeated 3-periodic.
- **Programs with which the results were first obtained** (`original/`): a 57-family screen at 50 digits under all
  readings, Laurent coefficients and vertex pole orders at the singular phases, closed forms at complex arguments,
  exact certificates, the exceptional hexagon, scans of A_ant/A. One symbolic program (N = 4) did not finish; it is
  not used.
- **Verification run A** (`verification_run_A/`): 84 families at 50 digits with geometric certification of every
  member; exact certificates; Laurent coefficients in 12 families (60 digits); A_ant/A = R in 66 even families; the
  symbolic rhombus; the midpoint relation of the two duals; the κ_5 = 0 family; isotropic directions at N = 4.
- **Verification run B** (`verification_run_B/`): 87 reflection-law families at 60 digits (110 for ill-conditioned
  stars) with three negative controls; exact certificates with the reflection law verified exactly; complex-phase
  parity, pole orders and residue quotients; pole-location grids; the exceptional aspect ratios for N = 6, 10, 14, 18;
  re-run of the original programs (identical outputs, except the unfinished symbolic program).
- **Third verification run** (`independent_run_2/`): exact arithmetic in multiquadratic number fields (standard
  library, no floating point) for M_1, M_2, R_1, R_2, H: certification, all 28 entries of Table 3, the products and
  quotients of Theorems 3(A), 4(A), the identifications of Lemma 3.1, the formula of doi:10.5281/zenodo.23073120;
  sympy for the rhombus and the 6-periodic through (a, 0) with general a, b (λ, R, A_ant/A = R, A/A^†, the k818
  constant for N = 6); 48 families at 50 digits with trajectories built from tangent lines of the caustic, λ by
  closure, O_p fitted from three sides (worst closure 1.4e-40; k813, k814 both duals ≤ 1.4e-34; k815 (iii) ≤ 8e-36;
  k816 (i) for N ≥ 6 ≤ 8e-44, for N = 4 spreads 0.10–0.55; A_ant/A = R ≤ 1.6e-35; Lemma 3.2 formulas ≤ 4.2e-41,
  A_dual^pt > 0); Laurent coefficients at I_1 (and I_2 for odd N) in 11 families (orders, residue quotients equal to
  the invariants to ≥ 15 digits, the order-3 pole for N = 4, nonzero residues of A^†_ped on I_1 for N = 3, 5, 7); the
  roots of R = 0 for N = 10, 14, 18; and a re-run of the whole package from the archive: all recorded outputs
  reproduced byte for byte (26 programs), except the running-time line of `check_note.py`.

## Independent verification runs
The results were first obtained with the proofs of Theorems 1, 2(a), 3(B), 4(B), 4(C), the proportionalities of
Theorem 5 and the certificates. Two independent verification runs, both AI-assisted, followed (2026-10-15); each
wrote its own programs from the source's definitions.

| Item | Run A | Run B |
|---|---|---|
| Rows, definitions, readings against the source's tex and figure | CONFIRMED (readings of "dual" and A^†_{j,ped} analysed; k908,b decides reading (i)) | CONFIRMED |
| Lemmas of Sections 2–4 (closed forms, cosets, folding, vertex singularities) | CONFIRMED (re-derived) | CONFIRMED (argument checked; parity, orders, residues computed) |
| Theorem 1, Theorem 2(a) | CONFIRMED | CONFIRMED |
| Point-wise dual | PROVED proportional to A^† (new); multiple positive numerically | constant ratio numerically |
| Theorem 3(A), 4(A) (certificates) | CONFIRMED (own exact program) | CONFIRMED (own exact program, reflection law exact) |
| Theorem 3(B), 4(B), 4(C) | CONFIRMED (4(B) re-derived, including N = 6) | CONFIRMED (4(B) checked against Laurent data) |
| Theorem 5 | CONFIRMED; found the deposits 23094118, 23079799 (corollary) | CONFIRMED; found 23396998, 23091871 (exceptional families) |
| N = 4 constants | PROVED at the symbolic rhombus | numerically and exactly at two members |
| Literature / deposits | 23094118, 23079799, 23099540 added; journal version not examined | all 2026 deposits of the two authors listed; 23396998, 23091871 added |

Neither run found a wrong theorem or a gap in a proof.

**Corrections required by the two runs**, all applied in the note:
1. Row k818 presented as a scoped corollary of doi:10.5281/zenodo.23094118 and doi:10.5281/zenodo.23079799, with R
   and the exceptional set explicit (abstract, Section 1.2, Section 1.3, Theorem 5, Section 7).
2. The exceptional families credited to doi:10.5281/zenodo.23396998 and doi:10.5281/zenodo.23091871 and to the
   proposers' animations; no claim that their vanishing along whole families is new; no "exact value not
   determined" (the closed equation R = 0 is given, and for N = 6 the exact value).
3. The reading of A^†_{j,ped} named in every statement on k815 and k816; reading (ii) treated; the k816 summary
   "true for every even N ≥ 6 under every reading; N = 4 depends on the reading".
4. The explanation of the failure of k815 under reading (i) corrected: the obstruction is a double pole on the
   f-isotropic set I_1; A_dual vanishes on I_2 (Remark 6.3).
5. A statement on the journal version corrected (it had not been examined then; the third run examined it).
6. The N = 4 constants proved at the rhombus (Proposition 5.1) instead of by the unfinished symbolic program.
7. The numerical separation of true and false candidates stated correctly (false candidates down to 2.9e-10 and
   7.8e-12).
8. Typographical error: for N = 4, A^†·A_ant = 8 and A·A^† = 4.
9. The point-wise reading of "dual" (midpoint polygon, constant multiple of A^†) and the possible vanishing of κ_5
   added.
10. Wording on row k908 (only k908,b is the N = 3 row); the deposit list completed (23073120, 23073495, 23099540,
    23094118, 23400541); the data of Reznik–Garcia–Helman (Eur. J. Math. 8 (2022) 1550–1565) and of
    Roitman–Garcia–Reznik (Arnold Math. J. 7 (2021) 619–637), checked with Crossref; Akopyan's credit for k814; the
    remark on parity at phases with a vertex at infinity restricted to odd N (Remark 3.5).

**Added when the note was written** (examined by the third run below): the positivity of A_dual^pt (Lemma 3.2),
which completes Theorem 2(b); the failure of reading (ii) of k815 for every N; the caustic λ = a^2b^2/(a+b)^2 and the
formula for R for N = 6 (Theorem 5(c)); Proposition 6.4 (order three, from the certificate); the treatment of odd N
in Lemma 6.2 and the conclusion of Remark 6.3 for N = 3; the comparison with the N = 4 formula of
doi:10.5281/zenodo.23073120.

### Third independent verification run (on the text of the note, 2026-10-15)
An AI-assisted run examined the text of the note before its last revision, first the additions just listed, and
wrote its own programs before opening those of the package (`reproducibility/independent_run_2/`).

| Item | Result |
|---|---|
| Deposits of A. Kriebel (Zenodo, all 131 records, newest first) | none treats k813, k814, k815, k816 or k818 (the newest, of 15 October 2026, treat k605,a and k904,a) |
| Rows and definitions (arXiv v11 tex and figure); corpus records | CONFIRMED; readings as in the note |
| Published version of the source | examined: contains only k801–k804 of the inversive table; none of the five rows |
| Positivity of A_dual^pt (Lemma 3.2) and Theorem 2(b) | CONFIRMED (re-derived: midpoint relation, angle formulas, identity sin x + sin y + sin(x+y) = 4 cos(x/2) cos(y/2) sin((x+y)/2)) |
| Failure of reading (ii) of k815 for every N | CONFIRMED |
| N = 6: λ = a^2b^2/(a+b)^2, R = (2b−a)(a+b)^2/(b^2(2a+b)) | CONFIRMED (by hand and symbolically); the caustic and the member are in Garcia–Reznik (credit added) |
| Proposition 6.4 | CONFIRMED (order 3 observed: c_-3 = 0.08i, c_-2 ≈ 0 at b/a = 1/2) |
| Lemma 6.2, all cases (odd N at I_1 and I_2, even N ≥ 8, N = 6, N = 4) | CONFIRMED (re-derived, including the index arithmetic) |
| Lemma 4.1(c), coset argument | CONFIRMED |
| Theorems 1–5, all other lemmas | CONFIRMED (re-derived; residue quotients and pole orders computed) |
| Exact certificates and Table 3 | CONFIRMED (own exact arithmetic, no floating point) |
| Cited results (Stachel Thm 2 and Cor 4.2(i); RGH Section 3; RGR; GR22; the deposits) | CONFIRMED in the versions cited; 23094118 and 23079799 read in full |
| Bibliography | DOIs resolved (Crossref, DataCite); data agree |
| Package | re-run from the archive: all outputs reproduced |

It found no error in the mathematics. **Corrections required by the third run**, all applied in the note:
1. The published version of the source described (Section 1.4, Verification, Scope).
2. Garcia–Reznik (Ann. Math. Inform., arXiv:2011.06640v3, Section 3 and Appendix B) credited for the caustics and
   members for N = 4 and N = 6 and the values A·A^† = 4 (N = 4) and A/A^† (N = 6).
3. The constant of k818 for N = 6 stated in closed form (Section 7).
4. Row k908,b attributed to the source; Reznik–Garcia–Helman state the same observation.
5. The abstract's description of the dual made precise for star polygons ("the circle touched by its sides").
6. The title of Akopyan's communication quoted.
7. This run recorded (Verification, Section 8, this report, `independent_run_2/`).
8. The searches extended (Scope and priority).

## Relation to the literature, novelty and scope
- **Credits.** The rows: Reznik, Garcia and Koiller (k814 credited by them to A. Akopyan). The parametrisation:
  Stachel, Eur. J. Math. 8 (2022), Theorem 2; central symmetry of even-N periodics: Stachel, J. Geom. 112 (2021),
  Corollary 4.2(i). The caustics and members for N = 4 and N = 6 and the values A·A^† = 4 (N = 4) and
  A/A^† = 4a^3b^4/((2a−b)(a+b)^2) (N = 6): Garcia and Reznik, Ann. Math. Inform. (2022), arXiv:2011.06640v3. The
  bicentric polar family and the Liouville method for it: Roitman, Garcia and Reznik, Arnold Math. J. 7 (2021). Row
  k908,b is the source's; Reznik, Garcia and Helman, Eur. J. Math. 8 (2022), state the same observation. Row k818:
  the author's deposits 23094118 and 23079799. Exceptional antipedal-zero families: the author's deposits 23396998
  and 23091871, and the proposers' animations (arXiv v11 of the source; Garcia–Reznik, arXiv:2011.06640v3). Related
  deposits: 23073120, 23073495 (k908,a/b), 23099540 (k811; incompatible descriptions of the dual; midpoint relation),
  23400541 (the Jacobi method for pedal areas).
- **New, to our knowledge:** the proofs of k813 and k814 for all N (including the point-wise reading), the analysis
  of k815 and k816 under the three readings with exact certificates, and the independent Jacobi proof of the
  proportionalities behind k818.
- **Searches (15 October 2026, by the first write-up and the three runs).** The source (arXiv v11 and the published
  version); the book of Garcia and Reznik, "Discovering Poncelet Invariants in the Plane" (IMPA, 2021; full text
  searched); the 19 works citing the source in OpenAlex (titles); arXiv listings for nine queries; zbMATH Open;
  Crossref; the 2026 Zenodo deposits of A. Ferudun and A. Kriebel (all records listed, the billiard abstracts read)
  and the 2026 OpenAlex works for "elliptic billiard"; five web searches; Semantic Scholar answered with a rate
  limit. No printed or deposited proof of any of the five rows was found. A search that finds nothing is not a
  proof of novelty; no priority claim.
- **Scope.** Elliptic caustics only; least period N; for reading (i) of k815 the failure is proved for N = 3 and
  observed for every odd N tested; the exceptional aspect ratios of k818 for N ≥ 10 are numerical (winding 1).

## 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/