# Verification report — AMR-050-0018 (Reznik–Garcia–Koiller row k401: outer area times antipedal area, N ≡ 2 (mod 4), every point M)

Verification date: 2026-10-15.

**Verdict.** The row is **false as printed**; this is a complete negative answer to the record. The corrected focal
statement is proved.
- **Disproof (Theorem 1, Proposition 3.1).** For a = 8/5, b = 1 the billiard trajectories with the caustic
  x²/(96/65)² + y²/(105/169) = 1 (caustic parameter λ = 64/169) form a convex 6-periodic family. Its member with a vertex
  at (a, 0) has A'·A*_O = 801792/21875, its member with a vertex at (0, b) has A'·A*_O = 3241/100; difference
  371293/87500 ≠ 0. For every ellipse a > b > 0 the two 6-periodic members symmetric in both axes differ at M = O by
  4(a−b)³(a+b)⁵/(ab(2a+b)(a+2b)) > 0, with all vertices, outer and antipedal vertices written out.
- **Theorem 2 (every N ≥ 3).** For every member of a convex N-periodic family with confocal elliptic caustic
  parameter λ, A*_{f1} + A*_{f2} = 2κA with κ = [a²b²(3b² − a²) + λ(a² − b²)²]/(2a²b²(b² − λ)).
- **Corollary 3 (even N).** A*_{f1} = A*_{f2} = κA, and A'·A*_{fj} = κ·A'A is constant; so row k401 holds at the foci
  for every N ≡ 2 (mod 4). N = 4: 16a²b²; N = 6: 4a²(4b² − a²). The focal antipedal area vanishes along a whole
  family exactly when κ = 0, i.e. λ = a²b²(a² − 3b²)/(a² − b²)²; for N = 6 exactly when a = 2b.
- **Not new.** The row and its conventions are Reznik–Garcia–Koiller's; the parametrisation by Jacobi functions is
  Stachel's (Eur. J. Math. 2022, Theorem 2; a short proof of the form used is included); the central symmetry of
  even periodic trajectories is Stachel's (J. Geom. 2021, Corollary 4.2(i) of arXiv v2); the constancy of A'A for
  even N is Chavez-Caliz's (Arnold Math. J. 2021, Theorem 6); the even-N identity A*_f = κA is Lemma 4 of the
  author's unrefereed Zenodo note on k403,b (doi:10.5281/zenodo.23079799); the equality A*_{f1} = A*_{f2} for every
  even N (row k607) and the zero families for N = 6, 8, 10, 12, 16 are in the author's unrefereed Zenodo note
  doi:10.5281/zenodo.23396998; a zero-area antipedal for N = 6, a/b = 2 was noticed by Garcia and Reznik
  (arXiv:2011.06640, table of videos); for N = 4 the values κ = 2 and A'A = 8a²b², hence A'A*_f = 16a²b², are also
  in Section 6 of the k403,b note.
- **New, to our knowledge:** the disproof of k401 as printed, Theorem 2 for every N (odd N included), and the
  explicit focal product for N = 6.
- **Only tested numerically:** non-constancy of A'A*_M at points M other than O and the foci; the identity for star
  families (not stated). For odd N the observed constancy of A*_O/A follows from rows k203,b and k403,a (AA_O and
  A_OA*_O constant for odd N), proved in the unrefereed Zenodo notes doi:10.5281/zenodo.23085904 (Ferudun) and
  doi:10.5281/zenodo.23334001 (Kriebel); this is credited in Remark 3.2.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Reznik, R. Garcia, J. Koiller, "Fifty new invariants of N-periodics in the elliptic
  billiard", Arnold Math. J. 7(3) (2021) 341–355, doi:10.1007/s40598-021-00174-y; arXiv:2004.12497v11 (29 October
  2020, titled "Eighty new invariants ..."). Bibliographic data checked with Crossref and the arXiv interface on
  2026-10-15.
  - Row k401 is the first row of the table "Invariants of antipedal polygons" (arXiv v11, file `055_antipedals.tex`;
    Table 5 on p. 348 of the journal version): invariant A'A*_m, value "?", which N "≡ 2 (mod 4)", M "all", date 4/20,
    proven "?".
  - Definitions (arXiv v11 tex): the antipedal polygon w.r.t. M is formed by "the intersections of rays shot from
    every P_i along (P_i − M)^⊥"; the list of symbols gives A_m^* as the area of the antipedal polygon of the
    N-periodic (the running text writes A_m); all areas are signed (shoelace formula); A' is the area of the outer
    polygon (tangents to the ellipse at consecutive vertices).
  - The source explains the table columns invariant, value, which N, date and proven, but not the column M; the
    entries of that column are O, "f1,f2", "O,f1,f2" and "all".
- **Corpus record.** UnsolvedMath 1.6.0, AMR-050-0018, "Elliptic-billiard invariant k_{401}": "Prove that, for
  N ≡ 2 (mod 4), every M, the quantity A'A_M^* is constant as P varies through the family." Faithful to the row.

## Readings
- "all" in the column M is read as "every point of the plane" (the record's reading; the reading is ours). The
  counterexample is at M = O, covered by every reading that includes the centre.
- Signed versus unsigned areas: at M = O both antipedal hexagons of the counterexample are convex and counterclockwise,
  so both readings give the same values.
- Antipedal of P' instead of P: also not constant at O (271752/6125 versus 264992/5625), Remark 3.2.
- "N-periodic" = least period N (our convention; the source does not spell it out). It plays no role in the
  counterexample (the family is fixed by the caustic; both members have six distinct vertices).
- Orientation: reversing the labels changes the sign of every signed area; no statement is affected.

## Results in the paper
- Theorem 1 and Proposition 3.1: exact counterexample; every-ellipse closed forms (vertices of P, P', Ant(P;O),
  areas, products, difference), proved by direct computation that is written out.
- Theorem 2 (Section 4): Lemma 4.1 (antipedal area side by side), Lemma 4.2 and Proposition 4.3 (two foci on one
  side), Lemma 4.4 (parametrisation, with proof; credited to Stachel), Lemmas 4.5–4.6 (one elliptic function φ),
  Lemma 4.7 and Proposition 4.8 (poles, principal parts, residue condition μ = μ*, S ≡ 0), and the identification
  2 − μ* = κ in closed form.
- Corollary 3 (Section 5): central symmetry, half-turn argument, general-position hypothesis of Chavez-Caliz's
  theorem verified, closed forms for N = 4, 6, zeros of κ.
- Remarks 3.2, 4.9, 5.1: numerical observations (other points M, odd N, star families, zero families); for odd N,
  A*_O/A is constant by the two results credited in Remark 3.2; for N = 3, a = 1.6, b = 1, A*_f1/A ranges from
  9.670… to 21.347… along the family (Remark 4.9).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The exact program of the note** (`check_note.py`, Python with sympy and mpmath, about 6 s; 116 checks, TOTAL
  failures: 0). (i) Both members of Theorem 1 generated from their first vertex by the reflection law in exact
  arithmetic in Q(√105, √39) (implemented in the program, exact sign tests): vertices on the ellipse, tangency with
  contact strictly inside every side, reflection law, return after 6 reflections and not earlier, convexity; every
  entry of Table 1; rows k106 and k113 at both members; the reading with Ant(P'; O). (ii) sympy: Proposition 3.1 for
  every ellipse; A*_f = κ6·A at both members for every ellipse (a = u²+1, b = 2u); κ4 = 2, κ6, the closed forms of
  Corollary 3; κ = 2 − μ* and the closed form of μ*; the general-position determinants. (iii) mpmath, 50 digits:
  Theorem 2 for N = 3..10 on a/b = 1.6, 2, 3, 1.15 (six members each; largest relative deviation 4.0e-47);
  Corollary 3 for even N (defects ≤ 1.6e-46); controls: A'A*_O and A'A*_G (G = (0.31, 0.17)) not constant for every
  N, single focal ratios not constant for odd N. (iv) The displayed formulas of Lemmas 4.1, 4.2, 4.4, 4.5, 4.6 and
  Proposition 4.3 side by side for N = 3..7 (errors ≤ 2.8e-47); S(v) at complex points ≤ 2.7e-48 for μ = μ*, and
  ≥ 3.2e-3 for μ = μ* + 0.1. (v) κ_N = 0 for N = 8, 10, 12, 16 at the aspect ratios printed (20 digits) in
  doi:10.5281/zenodo.23396998.
- **Programs with which the results were first obtained** (`original/`): exact certificate in sympy and with fractions
  only; the whole 4- and 6-periodic families for every ellipse in exact rational-function arithmetic over a quadratic
  extension; κ = 2 − μ* symbolically; numerical tests of every lemma; a 50-digit sweep (five ellipses, N = 3..18,
  five points M). The recorded output of `k401_numeric.py` was produced with the argument list
  `6 10 14 18 4 8 12 3 5 7` (the default list of the script omits 18, 12 and 7).
- **Verification run A** (`verification_run_A/`): exact counterexample with the standard library only in
  Q(√105)(√39); sympy checks of the every-ellipse difference, κ4, κ6 and 2 − μ* = κ; a direct billiard harness
  (40 digits) for Theorem 2 (N = 3..10, 14; three ellipses), Corollary 3, the controls, Lemma 1 on 200 random
  polygons, the parametrisation, and five star families; residues and principal parts of φ by contour integrals;
  the zero families of the SYM7 note.
- **Verification run B** (`verification_run_B/`): two exact constructions of the counterexample from the reflection
  law (standard library; sympy number field), with rejection of perturbed inputs; symbolic construction of the two
  symmetric 6-periodic members for every ellipse; own residue computation; a 50-digit sweep (five ellipses,
  N = 3..18, five points M); each identity of the proof against reflection-law orbits.
- **Third verification run** (`independent_run_2/`, own README): exact counterexample from the first vertices by the
  reflection law with the standard library only (own fields Q(√105), Q(√105)(√39)); Proposition 3.1 and Corollary 3
  for every ellipse by the reflection law over Q(a,b)(s) and Q(a,b)(t) with the foci adjoined (46 checks); residue of
  Lemma 4.7 by series expansion, 2 − μ* = κ, principal parts and S(v) with complex Jacobi functions (45 checks);
  reflection-law families in 60 digits: Theorem 2, Corollary 3, controls, the numbers of Remarks 3.2 and 4.9, odd N at
  the centre, the lemma chain of Section 4 (51 checks); the aspect ratios of Remark 5.1 by two routes, all twenty
  digits (9 checks); the range of A*_f1/A for N = 3 (Remark 4.9).
- **Re-runs.** On 2026-10-15 every program of the package was run again from a copy of the package; the outputs are
  identical to the recorded ones up to lines that record running times and two shell status lines of the recorded
  runs (details in `reproducibility/README.md`). The third verification run repeated this from an extracted copy of
  `source.zip` (all 23 runs of `original/`, `verification_run_A/`, `verification_run_B/` and `check_note.py`: 116
  checks, 0 failures; all outputs reproduced), and ran its own programs again from the extracted final archive.
  `main.tex` compiles from the archive without warnings and without overfull boxes.

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed (2026-10-15); each wrote its own programs. The first two (runs A and B) examined the
first write-up before the note was written and did not read each other's report: run A examined the statement and
the proofs line by line; run B examined the computations, the sources, the literature and the bibliography. The
third run examined the final text of the note and the package (below).

| Item | Run A | Run B |
|---|---|---|
| Row, record, definitions against the source (tex v11, journal Table 5) | CONFIRMED; M column and least period to be presented as our reading | CONFIRMED (tex, figure, journal PDF) |
| Counterexample (Theorem 1) | CONFIRMED by own exact computation in Q(√105)(√39), standard library | CONFIRMED by two own exact implementations from the reflection law; perturbed inputs rejected |
| Every-ellipse difference (Proposition 3.1) | CONFIRMED (sympy) | CONFIRMED (own symbolic construction) |
| Lemma 4.1 (antipedal area side by side) | PROVED (re-derived; 200 random polygons) | CONFIRMED (algebra re-derived; random polygons) |
| Lemma 4.2, Proposition 4.3 | PROVED | CONFIRMED (tests to 1e-29 or better) |
| Lemma 4.4 (parametrisation) | PROVED without the literature (cn identity); the necessity direction to be written out | CONFIRMED against Stachel's Theorem 2 (tex read) |
| Lemmas 4.5, 4.6 | PROVED | CONFIRMED (key steps re-derived; tests) |
| Lemma 4.7, Proposition 4.8 | PROVED; principal parts and residues checked by contour integrals | CONFIRMED; own symbolic residue gives the same μ*; S ≡ 0 numerically |
| 2 − μ* = κ | CHECKED (sympy and a hand-derived intermediate form) | CHECKED (sympy) |
| Corollary 3 | PROVED given Chavez-Caliz's theorem; its general-position hypothesis to be verified (verified by run A) | PROVED given Chavez-Caliz's theorem (statement checked in the tex) |
| Zero families | κ_N = 0 reproduces the aspect ratios of the SYM7 note | credit for the N = 6, a/b = 2 zero to Garcia–Reznik required |
| Literature and novelty | spot checks; the SYM7 note (2026-10-15) to be cited | full search; no statement, proof or refutation of k401 found |
| Bibliography | CHECKED (Crossref, arXiv) | CHECKED (Crossref, Zenodo); Stachel's corollary to be cited with the arXiv numbering |

Neither run found a mathematical error.

**Corrections required by runs A and B**, all applied in the note (details in `audit/fixes_applied.md` of the
working folder):
1. (A1) The general-position hypothesis of Chavez-Caliz's Theorem 6 is verified for the confocal pair: four distinct
   common points (±x0, ±y0), nonzero gradient determinant, no common point at infinity (Section 5).
2. (A2) The SYM7 note doi:10.5281/zenodo.23396998 is cited for k607 and its zero families, and the zero families
   are explained by κ = 0 (Sections 1.3, 5, Remark 5.1); the Zenodo count is 11 records.
3. (A3) Explicit caustic parameters are credited to Ramírez-Ros (Nonlinearity 27 (2014) 1003–1028), single author,
   as cited by the source (Section 3).
4. (A4) The source's row k607 is described as stated for N ≡ 0 (mod 4); Corollary 3 and the SYM7 note give it for
   every even N (Section 1.3, Remark 5.1).
5. (A5) The reading of the column M and the least-period convention are presented as ours (Sections 1.1, 2); the
   source's A_m / A_m^* notation is mentioned (Section 2).
6. (A6) Lemma 4.4 is proved in full, including the direction "convex N-periodic ⇒ Δ = 2K/N", and the relation to
   Stachel's Theorem 2(1)–(2) is stated (Section 4.3).
7. (A7) Novelty statements say "to our knowledge" and "not found in the sources searched"; no priority is claimed.
8. (B-F1) Garcia and Reznik (arXiv:2011.06640) are credited for the zero-area antipedal at N = 6, a/b = 2
   (abstract, Sections 1.3, 5, "Scope and priority").
9. (B-F2) The SYM7 note is cited by its DOI wherever its central-symmetry argument is used, and the overlap with the
   first half of Corollary 3 (row k607 for every even N) is stated (Sections 1.3, 5, "Scope and priority").
10. (B-F3) The argument list of the original sweep is recorded (`reproducibility/README.md`).
11. (B-F4) Stachel's central-symmetry corollary is cited with the numbering of arXiv:2105.03362v2, where it is printed
    as Corollary 4.2, item (i) (theorem counters per section; checked in the arXiv text). The journal numbering was not
    verified, and the bibliography entry says that statement numbers refer to the arXiv version.

**Changes made when the note was written, after runs A and B.**
- Lemma 4.4 has a self-contained proof (addition theorems, the identity cn x cn y + sn x sn y dn(x − y) = cn(x − y),
  and the reflection property of confocal conics, Stachel J. Geom. 2021, Lemma 2.1 of arXiv v2), following run A.
- The identification 2 − μ* = κ is written as a short closed-form computation that needs no computer algebra:
  μ* = (a² + b²)(a²b² − λ(a² + b²))/(2a²b²(b² − λ)).
- Proposition 3.1 lists the vertices of P, P' and Ant(P; O) of both symmetric 6-periodic members for every ellipse,
  and the degree-8 identity behind the difference is expanded in the text.
- The per-side expression of antipedal areas used in Lemma 4.1 is also in the k403,b note; this is credited.
- Theorem 2 and Corollary 3 are stated for "a convex N-periodic family with caustic parameter λ", without using the
  uniqueness of that caustic.
- New program `check_note.py` tests the statements as printed. Numbering: Theorem A of the first write-up is
  Theorem 2, Corollaries B/C are Corollary 3 and Section 5, Lemmas 1–7 are Lemmas 4.1–4.7 (Proposition 3 is 4.3),
  Theorem 8 is Proposition 4.8.
### Third run: examination of the final text
The third run examined the note and the package line by line as if nothing had been checked, starting with the parts
added when the note was written (the vertex lists and the proof of Proposition 3.1, the full proof of Lemma 4.4 with
identity (3) and the direction statement of (ii), the closed form of 2 − μ*, the second form of μ*, the derivation of
α_{i+1} in Lemma 4.1, the general-position computation as written, and Remarks 3.2, 4.9, 5.1 and Section 6 with their
numbers). It read again, in fresh anonymous downloads, the source (arXiv v11 tex: the row, the definitions, the list
of symbols, the column explanations, rows k106, k113, k607 and the citation of Ramírez-Ros; the journal's Table 5 on
p. 348), Stachel's Theorem 2 with its proof (arXiv:2105.03624v2) and his Lemma 2.1 and Corollary 4.2(i)
(arXiv:2105.03362v2, PDF), Chavez-Caliz's definition of general position and Theorem 6 (arXiv:2004.05404v1), the k403,b
note completely, the SYM7 note (abstract, central symmetry, Table 2 with the twenty digits), the main theorems of the
notes doi:10.5281/zenodo.23085904 and doi:10.5281/zenodo.23334001, and the video table of Garcia–Reznik. It
re-derived every proof by hand and recomputed the statements with its own programs (`independent_run_2/`).

| Item | Third run |
|---|---|
| Row, record, definitions, readings, scope | CONFIRMED |
| Theorem 1, Table 1, Proposition 3.1 | CONFIRMED (own exact construction from the first vertices by the reflection law; own symbolic construction for every ellipse; hand re-derivation of every vertex, area and product) |
| Lemmas 4.1–4.7, Propositions 4.3 and 4.8, Theorem 2 | PROVED (re-derived; residue by an independent series expansion; principal parts by contour integrals; lemma chain on reflection-law trajectories to 1e-55) |
| Corollary 3 | PROVED given Chavez-Caliz's theorem, whose hypotheses were verified again |
| Cited statements and numbering | CORRECT (Stachel Theorem 2, Lemma 2.1, Corollary 4.2(i); Chavez-Caliz Theorem 6; k403,b note Lemma 4, eq. (16), Section 6; SYM7 Table 2) |
| Remarks 3.2, 4.9, 5.1 | CORRECT except the "9.8" of Remark 4.9 (a six-member spread) and the "not proved" of Remark 3.2 (corrected) |
| Package | REPRODUCED from the archive |
| Novelty | not already published; the claim for N = 4 restricted (corrected) |

It found no mathematical error. **Corrections required by the third run**, all applied:
1. Remark 4.9: along the 3-periodic family of a = 1.6, b = 1 the ratio A*_f1/A ranges from 9.670… (member with a
   vertex at (−a, 0)) to 21.347… (vertex at (a, 0)); the earlier "varies by 9.8" was the spread over six members.
2. Remark 3.2: the constancy of A*_O/A for odd N follows from rows k203,b and k403,a, proved in
   doi:10.5281/zenodo.23085904 and doi:10.5281/zenodo.23334001; both are cited, and "not proved" now refers only to
   the other observations.
3. Novelty: only the focal product for N = 6 is claimed as new; κ = 2 and A'A = 8a²b² for N = 4 are credited to
   Section 6 of the k403,b note (Section 1.3, "Scope and priority").
4. The Verification paragraph records three runs (the first two on the first write-up, the third on the final
   text) and what was added when the note was written; Section 6 lists the fourth part of the package.
5. Wording: after Lemma 4.4, Part (i), the tangency statement of (ii) and the vertex parameters are attributed to
   Stachel (the orientation statement of (ii) is proved in the note).
6. Wording: the general-position determinant in Section 5 is written at a common point (x, y).
The date of the note stays October 15, 2026 (the day of writing, of all searches and of the cited SYM7 deposit).

## Relation to the literature, novelty and scope
- **Searches (2026-10-15).** arXiv interface (antipedal in abstracts and full records: 0 relevant; elliptic billiard
  invariants; the authors of the source; Stachel), the 19 works citing the journal version in OpenAlex (titles and
  abstracts) and OpenAlex full-text searches (antipedal billiard / antipedal polygon Poncelet / elliptic billiard
  invariants pedal), Crossref, zbMATH Open, Zenodo (all 11 records mentioning "antipedal" on that day, and the
  records on elliptic billiards), the authors' IMPA survey (2021), and web searches; Semantic Scholar returned HTTP 429
  and was not used. On the day the note was written the Zenodo, arXiv and Crossref/DataCite checks were repeated
  (log `queries.log` of the working folder): the "antipedal" Zenodo set is unchanged (11 records), a Zenodo query for
  "k401" returns nothing relevant, and a web search returns only the source and its known follow-ups. The third
  verification run repeated the queries to arXiv (antipedal; elliptic billiard, newest; Poncelet and area; Reznik,
  newest), OpenAlex (citing works; antipedal), Crossref, zbMATH Open (antipedal; elliptic billiard invariants) and
  Zenodo (antipedal: 11 records; k401; elliptic billiard, newest 25) and made one web search, all on 2026-10-15; the
  new records of that day concern pedal areas (rows k203,a, k204, k304) and focal inversions (k904,a).
- **Result.** No printed statement, proof or refutation of k401 was found. The closest works are the author's
  unrefereed Zenodo notes on k403,b (doi:10.5281/zenodo.23079799; Lemma 4 is the even-N factor) and on seven
  even-N rows (doi:10.5281/zenodo.23396998; k607 for every even N and the zero families), and other unrefereed 2026
  Zenodo notes on sibling rows (k402, k403,a, k405, k407, k603, k607 domain, k608), none of which states, proves or
  refutes k401. For odd N, the constancy of A*_O/A observed in Remark 3.2 follows from the unrefereed notes
  doi:10.5281/zenodo.23085904 (row k203,b) and doi:10.5281/zenodo.23334001 (row k403,a). A search that finds nothing
  is not a proof of novelty.
- **What was read.** The source (arXiv v11 tex: tables, definitions, list of symbols, figure of antipedal polygons;
  journal Table 5); Stachel's Theorem 2 with its proof (arXiv:2105.03624v2) and his Lemma 2.1 and Section 4
  (arXiv:2105.03362v2); in Chavez-Caliz (arXiv:2004.05404v1) the definition of general position and Theorem 6; the
  k403,b note completely; the sections of the SYM7 note on central symmetry and on zero families; the table of videos
  of Garcia–Reznik (arXiv:2011.06640v3; the videos were not watched); the main theorems, with their conventions, of
  the notes doi:10.5281/zenodo.23085904 and doi:10.5281/zenodo.23334001. Ramírez-Ros (2014) was not read. The proofs
  of the cited theorems were not checked; the parametrisation is proved in the note.
- **Bibliography.** DOIs checked with Crossref (RGK21, Stachel 2021 and 2022, Chavez-Caliz 2021, Ramírez-Ros 2014,
  Garcia–Reznik 2022) and DataCite (the four Zenodo notes), again by the third run; arXiv versions with the arXiv
  interface; the IMPA survey with zbMATH Open (Zbl 1511.14001) and its URL; DLMF version 1.2.8 (2026-09-15) for the
  Jacobi-function facts (§§22.4, 22.8).
- **Scope.** The row is settled as printed (false). Points other than O and the foci are only tested numerically;
  star families and hyperbolic caustics are not treated.

## 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/