# Verification report — AMR-050-0022, AMR-050-0033, AMR-050-0047, AMR-050-0063, AMR-050-0066 (Reznik–Garcia–Koiller: five focal area invariants of N-periodics in the elliptic billiard)

Verification date: 2026-10-15.

**Verdict.** The note proves the five rows; every statement it makes is proved, and the scope is as follows.
- **Settled (standing hypotheses: a > b > 0; confocal elliptic caustic, 0 < λ < b²; trajectories of least period N,
  any winding number τ coprime to N; signed areas).**
  - Row k806,a (AMR-050-0047): for every N ≥ 3, A'†_j = κ·A†_j with one constant κ > 0 for both foci, and A†_j > 0;
    so A'†_j/A†_j = κ along the family (Theorem A). For N = 4, κ = 1/2; the value 2 printed in row k806,b is the
    reciprocal.
  - Row k904,a (AMR-050-0063): for odd N, A'†_1·A'†_2 is a positive constant, equal to κ²·A†_1·A†_2 (Theorem B).
    First proved by A. Kriebel (doi:10.5281/zenodo.23394004, 15 October 2026); Theorem B is an independent proof,
    and no novelty is claimed.
  - Row k605,a (AMR-050-0033): for odd N, Ā'_1·Ā'_2 is a positive constant (Theorem C). First proved by A. Kriebel
    (doi:10.5281/zenodo.23398175, 15 October 2026); Theorem C is an independent proof, and no novelty is claimed.
  - Row k907,a (AMR-050-0066): for odd N, A_{1,dual}·A_{2,dual} is constant (Theorem D), in two readings of "dual"
    (polar reciprocal of the polar polygon w.r.t. a circle centred at O_p; point-wise inversion of its vertices);
    the dagger printed on the first factor is not defined in the source and is read as absent (stated in the
    theorem). In the reciprocal reading the constant is positive and the statement is equivalent to the constancy of
    (Σ sin∠P_i f_1 P_{i+1})(Σ sin∠P_i f_2 P_{i+1}).
  - Row k404 (AMR-050-0022): for even N, A*_{f_j} = κ*·A; for N ≡ 2 (mod 4), Ā_j = κ_p·A with κ_p > 0, so
    A*_M/A_M = κ*/κ_p for M = f_1, f_2 (Theorem E). The focal pedal area Ā_j is positive at every trajectory of every
    family (Proposition 6.3), so the ratio is defined at every trajectory: there is no exceptional set. κ* can vanish
    (N = 6, τ = 1, a = 2b: the ratio is identically 0, still constant).
- **Additional statements.** For N ≡ 0 (mod 4) the k404 ratio is defined everywhere and is constant if and only if
  κ* = 0 (then identically 0); κ* = 2 for every 4-periodic family; numerically κ* = 0 for N = 8 and 12 (τ = 1) at
  a/b = 1.8538473043818… and 1.7799694162448…. For odd N also Ā_1Ā_2 (row k604,a, not a record) and the product of
  the two polar areas are constant (Remark 6.6).
- **Not new.** The rows are Reznik–Garcia–Koiller's. The parametrisation is Stachel's (Eur. J. Math. 8 (2022),
  Theorem 2); the central symmetry of even periodics is Stachel's (J. Geom. 112 (2021), Corollary 4.2(i)); A·A' is
  constant for even N by Chavez-Caliz (Arnold Math. J. 7 (2021), Theorem 6). The case N = 4 of Theorem A (value 1/2
  for every 4-periodic) is the theorem of the author's note doi:10.5281/zenodo.23065067. Rows k904,a and k605,a were
  first proved by A. Kriebel in deposits of 15 October 2026 (doi:10.5281/zenodo.23394004, created 11:39 UTC, concept
  DOI 23394003; doi:10.5281/zenodo.23398175, created 14:26 UTC, concept DOI 23398174), before the present note was
  written. Kriebel proves, for odd least period, an elliptic caustic and every coprime winding (stars included),
  that A'†_1A'†_2 is a positive constant (23394004, Theorem 1) and that Ā'_1Ā'_2 is constant (23398175, Theorem 1);
  his corollaries add repeated traversals and the circular billiard. Both proofs locate the possible poles on a torus
  of reduced periods from explicit formulas (the inverse-edge terms of the outer polygon, resp. the focal feet; the
  double principal parts cancel in the cyclic sum) and conclude with complementary zeros from a reflection and an
  imaginary antiperiod; 23398175 also derives k605,a from the author's k903,a note (23096993) through the
  proportionality of Ā'_j and A†_{3−j}, and describes a route through a Landen-type identity of Khare and Sukhatme.
  Theorems B and C are independent proofs of the same statements by an argument of the same kind, organised
  differently (one list of possible poles for all area functions); the note does not use these deposits, which were
  read in full (version 1.0 PDFs) by the verification run on the final text. For k404: part (i) of Theorem E is Lemma 4 of the author's note
  doi:10.5281/zenodo.23079798 (with the closed form κ* = (a²b²(3b²−a²) + λ(a²−b²)²)/(2a²b²(b²−λ))), also the
  corollary of doi:10.5281/zenodo.23403150 for winding number 1; part (ii) is the case M = f_j of Theorem B of
  doi:10.5281/zenodo.23400541; the positivity of the focal pedal areas of P and P' (every term of the centred
  shoelace sums, every winding) is proved in Kriebel's doi:10.5281/zenodo.23149774 (Theorem 1 and Section 4), and that
  of the terms of A'†_j in 23394004 (eq. (12)); the families with κ* = 0 were found in doi:10.5281/zenodo.23396998, and an animation of
  the case N = 6, a/b = 2 is listed by Garcia and Reznik (arXiv:2011.06640v3). So the k404 ratio follows from these
  notes; the note gives a self-contained proof.
- **New, to our knowledge (rows k806,a, k907,a, k404 only).** Theorem A for N ≠ 4 (row k806,a), Theorem D (row
  k907,a), the ratio statement of row k404 as such, and Theorem E(iii) for N ≥ 8. Theorem E(iii) also follows at
  once from Theorem 1 of 23079798 and Chavez-Caliz's Theorem 6; Theorems C and D (and the polar product) also follow
  from row k903,a (23096993), because the list of poles makes Ā'_1 a constant multiple of A†_2 and the dual and polar
  areas constant multiples of A†_1 (Remark 6.6). No novelty is claimed for rows k904,a and k605,a, and no priority
  claim is made.
- **Not treated.** Hyperbolic caustics; closed walks that repeat a trajectory of smaller least period (for them the
  rows can fail: the 3-periodic listed twice does not satisfy k404); literal daggered readings of k907,a (three such
  readings were tested and are not constant).

The note is unrefereed. Two independent AI-assisted verification runs examined the first written version of the
results, and a further one examined the final text (below).

## 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; arXiv API: journal_ref "Arnold Mathematical Journal volume 7,
  pages 341-355, 2021"). Journal version: "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 (Crossref-checked).
  - Read in the tex source of v11: signed areas ("signed-a"); pedal polygons (feet on the side lines, A_m); the
    antipedal polygon (intersections of the perpendiculars at P_i to P_i − M; the running text once writes A_m for
    its area by a slip, the table and the table of symbols write A*_m); focus inversion in the unit circles at
    f_1, f_2 (P†_j, A†_j; primes for the outer polygon); the caption of the figure on polar and dual polygons
    ("polar: antipedal of the inversive wrt f_1 … dual: inversion of the polar with respect to O_p", a family
    "inscribed in a circle centered on O_p"); the sentence "the dual is the inverse of the pedal"; the table of
    symbols (A_{j,pol}, A_{j,dual}; no daggered dual).
  - Rows: k404 (Table 5): A*_m/A_m, N ≡ 2 (mod 4), M = f_1, f_2, proven "?". k605,a (Table 7): Ā'_1·Ā'_2, odd, "?".
    k806,a (Table 9): A'†_j/A†_j, all N, "?"; k806,b: value 2 for N = 4, "?". k904,a (Table 10): A'†_1·A'†_2, odd,
    "?". k907,a (Table 10): "A^†_{1,dual}·A_{2,dual}", odd, "?"; k907,b: A^†_{1,dual}/A^†_{2,dual} = 1, even.
  - Journal version (text extraction of the journal's file): the numbering scheme lists inversive objects only up
    to k804 and no rows k9xx; Table 5 contains k404; Table 7 lists k601–k607. So k806,a, k904,a, k907,a are in the
    arXiv version only, and all row codes are cited from v11.
- **Corpus records.** ulamai/UnsolvedMath v1.6.0: AMR-050-0022 (record 5100022, k404), AMR-050-0033 (5100033,
  k605,a), AMR-050-0047 (5100047, k806,a), AMR-050-0063 (5100063, k904,a), AMR-050-0066 (5100066, k907,a); upstream
  status `open`. The record texts are faithful copies of the rows, including the undefined dagger of k907,a. The
  records say "a fixed confocal caustic" without saying that it is an ellipse; the source's setting is the confocal
  ellipse pair.

## Readings
| Reading | Answer | Where |
|---|---|---|
| N-periodic = least period N (the source's experiments: one caustic per N), any winding coprime to N | proved | Theorems A–E, Lemma 2.2 |
| N-periodic = closed walk of length N (repeated traversals) | not N-periodics in the note's sense; products scale by r², ratios unchanged; the 3-periodic listed twice fails k404 | Section 1.4, Section 7 |
| k404: A_M = area of the pedal polygon of P (table and table of symbols) | proved for N ≡ 2 (mod 4), defined at every trajectory | Theorem E |
| k404: A_M = antipedal area (the slip in the running text) | the ratio would be 1 trivially; excluded by the table | Section 1.4 |
| k404 for N ≡ 0 (mod 4) (not the record) | constant iff κ* = 0 | Theorem E(iii) |
| k907,a dagger read as absent: A_{1,dual}·A_{2,dual} | proved, both readings of "dual" | Theorem D |
| "dual" = polar reciprocal w.r.t. a circle at O_p (the only reading under which the duals are inscribed in a circle centred at O_p, as the source says) | proved; positive constant | Theorem D(a), Remark 3.1 |
| "dual" = point-wise inversion of the polar vertices in a circle at O_p | proved | Theorem D(b) |
| "dual" = polar polygon ("the dual is the inverse of the pedal") | product constant for odd N | Remark 6.6 |
| literal daggers: inverted dual × dual; both duals inverted; pedal of P†_1 × dual | not constant | Remark 6.5, Section 7 |
| Radius of the circle at O_p, radius 1 at the foci | immaterial for ratios; products scale by a constant | Section 1.4, Remark 3.1 |
| Hyperbolic caustic | not treated | Scope and priority |

## Results in the paper
- **Section 2.** Stachel's parametrisation P(u) = (−a sn u, b cn u), modulus k² = (a²−b²)/(a²−λ), step h = 2Δ,
  sn²Δ = λ/b²; chords (Lemma 2.1); the reflection law verified directly and closure Nh = 4τK (Lemma 2.2); identities
  (Lemma 2.3): focal distances a ∓ c sn u, m·m, n·n, P(t−Δ) × P(t+Δ) = 2ab s₀c₀ dn t/(1 − k²s₀² sn²t), and
  (V − f_1)² with roots 1/k and r₁ = dn h/(k cn h) (one root for N = 4); parities (Lemma 2.4); orientation and
  turning (Lemma 2.5); N = 4 ⟺ λ = a²b²/(a²+b²), rhombus member (Lemma 2.6).
- **Section 3.** All vertices as natural formulas in the bilinear products (Φ, Ψ, ι, α) and their explicit forms
  (Lemma 3.2); Remark 3.1 (polar = inverse of the focal pedal = antipedal of the inversive polygon; the circle of
  radius a/b² about O_p = (1 + 1/b²) f_j touched by the polar sides; reciprocal duals on the circle of radius b²/a;
  A_{1,dual} = (b⁴/(2a²)) Σ sin∠P_l f_1 P_{l+1}); symmetries under h, 2K, 4K, 2iK' (Lemma 3.3).
- **Section 4.** The phase sets P_A, S₁, S₂ as cosets and their intersections by N mod 4 (Lemma 4.1); configuration
  at special phases, including the N = 6 adjacency (Lemma 4.2); the isotropic line through f_1 at S₁ containing
  V_{−1}, V_0, V_1 (Lemma 4.3); the cross product d_l × d_{l+1} with zeros of order 3 on Π'₊ and 1 on Π'₋
  (Lemma 4.4); folding and reversal (Lemmas 4.5, 4.6).
- **Section 5.** The pole map (Table 2): A (simple poles exactly on P_A, residues ±ρ_A), A' (exactly on P_A − Δ:
  a pole at S₁ iff N ≡ 0 (mod 4)), inversive and dual areas (poles on S₁, vanishing on S₂ for odd N), focal pedal
  areas of P and P' (poles on S₂), polar areas (S₁), focal antipedal area (poles on P_A, order ≤ 2, ≤ 1 for even N).
- **Section 6.** Liouville and proportionality lemmas; positivity of A, A', Ā_j, Ā'_j, A†_j, A'†_j, A_{j,dual} for
  every winding (Proposition 6.3); the rhombus values A = 2ab, A†_j = 2/(ab), A'†_j = 1/(ab), A*_j = 4ab
  (Proposition 6.4); proofs of Theorems A–E; Remarks 6.5–6.7 (dagger readings; further products; the split k403,b /
  k404 with Chavez-Caliz's theorem).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of this note** (`check_note.py`, 18 s, 56 checks, TOTAL failures: 0).
  - (i) Reflection-law trajectories (no Jacobi functions), 50 digits, 48 families ((a,b) ∈ {(1.5,1), (2.2,1),
    (2.6,1.7)} × 16 pairs (N, τ), N ≤ 14, stars), 6 members each; self-checks ≤ 1.1·10⁻³¹. Largest in-class
    relative spreads: k404 7·10⁻⁴⁹, k605,a 3·10⁻⁴⁴, k806,a 1·10⁻⁴¹ (same constant for both foci to 9·10⁻⁴²),
    k904,a 1.2·10⁻⁴², k907,a 1.3·10⁻⁴¹ (R), 1.2·10⁻³⁴ (P), 4.5·10⁻³³ (polar product), the largest values in the
    star family a = 2.2, (9,4) with b² − λ < 5·10⁻¹⁰. Controls outside the classes: ≥ 3.5·10⁻⁷ (products, even N),
    ≥ 7.9·10⁻⁵ (k404). N = 4: κ = 1/2 and κ* = 2 at all members (three shapes).
  - (ii) Exact rhombus (sympy, symbolic a, b): tangency, reflection law, A'†/A† = 1/2, A*/A = 2, both foci.
  - (iii) Exact hexagon for a = 2, b = 1, λ = 4/9 in Q(√3, √5): focal antipedal areas exactly 0, A = 20√5/9,
    Ā_j = 200√5/81; 50-digit members |A*_j/A| < 10⁻⁴⁵; κ* for N ∈ {4, 6, 8, 10, 12} (τ = 1) and (8,3), (10,3),
    (12,5) at 8 aspect ratios agrees with the closed form of Lemma 4 of doi:10.5281/zenodo.23079798 to 6·10⁻³⁴;
    sign changes for τ = 1; roots a/b = 1.85384730438182101462… (N = 8), 1.77996941624483732972… (N = 12), with
    |A*_j/A| < 5·10⁻⁴⁰ resp. 2·10⁻³³ at six members while A varies.
  - (iv) Positivity of every term of the focal pedal shoelace sums on 16 star families (smallest Ā_j/A = 2.3·10⁻¹²).
  - (v) The three daggered alternatives of k907,a vary (relative spread ≥ 3.7·10⁻⁹).
  - (vi) Jacobi form, 60 digits: identities of Lemmas 2.1, 2.3, 3.2, 4.4 at random complex points (residual
    4.1·10⁻⁴⁸), Theorems A–E at a complex phase, and orders at S₁, S₂, P_A in seven families within the bounds of
    Table 2 (A*_1 has double poles on P_A for odd N; A' has a pole at S₁ exactly for N ≡ 0 (mod 4)).
- **Programs with which the results were first obtained** (`original/`): `ebtools.py`, `scan_rows.py` (32 families,
  50 digits), `local_checks.py` (orders and residue quotients), `kappa_scan.py` (κ_p and κ* for N ≡ 2 (mod 4)),
  `probe_k806.py` (the root r₁, isotropic outer vertices).
- **Verification run A** (`verification_run_A/`): tangent-stepping geometry (`vA_geom.py`), 59 families
  (`vA_invariance.py`), κ* signs on 297 even-N families (`vA_kappa_scan.py`), the κ* = 0 families for N = 8, 12
  (`vA_zero_k404.py`), dagger readings (`vA_dagger_readings.py`), 110-digit re-run of three stars
  (`vA_highprec_star.py`); recorded output of an orders program that was not preserved (`vA_local_output.txt`).
- **Verification run B** (`verification_run_B/`): reflection-law geometry (`vb_geom.py`), 112 families
  (`vb_rows.py`), wrong inputs (`vb_wrong.py`), κ_p search on 208 families (`vb_kappa.py`), algebraic lemmas at
  random parameters (`vb_lemmas_numeric.py`), the proof mechanism at complex phases (`vb_jacobi.py`); `vb_sympy.py`
  was stopped (too slow) and superseded.
- **Re-run of the package.** When the package was assembled, every program except `vb_sympy.py` was re-run from a
  copy of `reproducibility/`; see `reproducibility/RERUN_LOG.txt`.
- **Verification run on the final text** (`independent_run_2/`): tangent-stepping geometry with every polygon by its
  elementary construction and O_p located geometrically (`r2lib.py`); 80 families at 90 digits (`r2_rows.py`); exact
  checks including the worked examples of Kriebel's deposits (`r2_exact.py`); Lemma 4.4, Lemma 3.2 and Table 2 at
  complex phases (`r2_complex.py`); the families with κ* = 0 and other constants of Theorem E (`r2_kstar.py`).

## Independent verification runs
The results were first obtained in a written version containing Theorems A–E with proofs by the method of the
note (2026-10-15). Two independent verification runs, both AI-assisted, followed on the same day, each with its own
programs, written without using those of the first write-up.

| Item | Run A | Run B |
|---|---|---|
| Rows and definitions against the source | CONFIRMED (tex v11 read; record texts faithful) | CONFIRMED |
| Parametrisation, chords, closure, identities (Lemmas 2.1–2.3) | CONFIRMED (re-derived; λ from tangent stepping equals the Jacobi value) | CONFIRMED (Jacobi orbit = reflection-law orbit to 10⁻⁷⁶) |
| Vertex formulas, special phases, index arithmetic | CONFIRMED | CONFIRMED (brute force N ≤ 40) |
| Folding, reversal, symmetries | CONFIRMED | CONFIRMED (complex phases) |
| Pole map (all area functions) | CONFIRMED (90-digit orders, 8 families); the description of A' at S₁ in the write-up was wrong (fix) | CONFIRMED (80-digit orders, 11 families) |
| Theorem A (k806,a), all N | CONFIRMED; N = 4 value 1/2 at the rhombus | CONFIRMED (residue quotients) |
| Theorems B, C, D (odd N products) | CONFIRMED; D for the stated reading only | CONFIRMED; third reading (polar product) also constant |
| Theorem E (k404) | CONFIRMED; exceptional set empty (positivity proof); the claim "fails for N ≡ 0 (mod 4)" holds only when κ* ≠ 0 (families with κ* = 0 for N = 8, 12) | CONFIRMED; κ_p > 0 on 208 families; example N = 6, a/b = 2 |
| Literature | no proof of the five rows found; four deposits to add; journal title | no proof found; deposit 23094117 to add; RGH volume/year |

No run found a gap in the proofs.

**Corrections required by runs A and B**, all applied in the note:
1. (A1) k404 for N ≡ 0 (mod 4): non-constant exactly when κ* ≠ 0, identically 0 on the families with κ* = 0
   (N = 8, 12 numerically); the κ* scan was extended to N ≡ 0 (mod 4) (Theorem E(iii), `check_note.py` (iii)).
2. (A2, B3) The possible exceptional set of k404 was replaced by the positivity of the focal pedal areas
   (Proposition 6.3), credited to the abstract of doi:10.5281/zenodo.23149774; the example N = 6, a = 2b
   (κ* = 0) is stated.
3. (A3) Poles of A' at S₁: a pole for N ≡ 0 (mod 4), regular for N ≡ 2 (mod 4) (Lemma 5.1, Table 2).
4. (A4, B1) The journal version's title "Fifty New Invariants…" and the fact that rows k8xx, k9xx are in the arXiv
   version only (checked in the journal's numbering scheme); all row codes cited from v11.
5. (B1) Reznik–Garcia–Helman cited as Eur. J. Math. 8 (2022) 1550–1565 (Crossref).
6. (A5, B2) Deposits 23094117 (k805), 23312252 (cosine sums), 23271171 (k108), 23331330 (k307) added to the
   related-work discussion with what each does; none covers the five rows.
7. (A6) Theorem D states the reading of the dagger; the literal daggered readings are recorded as not constant.
8. (A7) Full zero set of d_l × d_{l+1}: order 3 on Π'₊ and order 1 on Π'₋ (Lemma 4.4, with a closed form).
9. (A7) N = 4 in Lemma 5.2: V(iK' + w) is odd in w, which gives d'·d' = −2(r'·f_1)/w + O(1).
10. (B4) Elliptic caustic and least period stated in every theorem (standing hypotheses) and in Scope and priority.
11. (B3) Theorem E keeps the explicit domain statement; with positivity the ratio is defined everywhere.

When the note was written, the closed form of the cross product (Lemma 4.4), the positivity of A†_j, A'†_j, Ā'_j for
every winding (giving κ > 0 and positive products), the polar product (Remark 6.6), Proposition 6.4 and
`check_note.py` were added, and the search of the Zenodo deposits was repeated through DataCite. That search found
Kriebel's deposits on k904,a and k605,a and the author's k401 note, and the reading of the deposited sources showed
that 23065067, 23079798 and 23396998 contain the case N = 4 of Theorem A, part (i) of Theorem E and the families
with κ* = 0; the novelty statements of the note were restricted accordingly.

## Verification run on the final text (second verification round)
A further independent verification run, also AI-assisted, examined the final text of the note on 15 October 2026
(19:42–20:30 UTC), after the note had been written; it worked with its own programs (`reproducibility/independent_run_2/`)
and did not use those of the note or of runs A and B.

| Item | Result |
|---|---|
| Rows, definitions, records (tex of arXiv:2004.12497v11 read again: §3.9, Tables 5, 7, 9, 10, the polar/dual caption, the table of symbols) | CONFIRMED; record texts faithful (AMR-050-0022 does not define A_M; the records do not say that the caustic is an ellipse; AMR-050-0066 copies the undefined dagger) |
| Kriebel's deposits 23394004 (k904,a) and 23398175 (k605,a), read in full | Theorem 1 of each has the statement of the note's Theorem B (including positivity) resp. Theorem C (constancy), for every coprime winding; corollaries on repeated traversals and the circular billiard; method of the same kind (poles on a reduced torus from explicit formulas, complementary zeros); 23398175 also reduces k605,a to k903,a (Ā'_j ∝ A†_{3−j}) and gives a Landen-identity route |
| Kriebel's 23149774 (k606), read in full | Theorem 1 and Section 4 prove the positivity of Ā_j and Ā'_j term by term for every winding |
| Parts added when the note was written: Lemma 2.5, Proposition 6.3, closed form of Lemma 4.4, polar product, Proposition 6.4, Theorem E(iii) | re-derived; CONFIRMED |
| Lemmas 2.1–6.2, Table 2, Theorems A–E, Remarks 6.5–6.7, cited results (Stachel Theorem 2 and Corollary 4.2(i), Chavez-Caliz Theorem 6, DLMF numbers) | re-derived line by line or checked in the sources; CONFIRMED, no gap |
| Own computations | 80 families at 90 digits by tangent stepping, O_p located geometrically: largest in-class relative spread 6·10⁻⁴⁷ (k806,a 2.2·10⁻⁶⁸, k904,a 1.8·10⁻⁷², k605,a 1.9·10⁻⁷⁴, k907,a 1.6·10⁻⁶⁹ (R), 1.1·10⁻⁵¹ (P), polar product 5.3·10⁻⁴⁷, k404 2.8·10⁻⁸⁸), controls ≥ 3.5·10⁻⁷; all shoelace terms of Proposition 6.3 positive; exact rhombus, hexagon and the worked examples of 23394004, 23398175, 23149774, 23065067 (Kriebel's B_± are the note's A'†_j and Ā'_j); Lemma 4.4, Lemma 3.2 and Table 2 at complex phases in 10 families (no violation); κ* = 0 at a/b = 2, 1.8538473043818…, 1.8039190445851…, 1.7799694162448… (N = 6, 8, 10, 12, τ = 1) |
| Package | re-run from an extracted copy of the archive: all recorded outputs reproduced (see `RERUN_LOG.txt`) |
| Literature (searches repeated 19:59–20:02 UTC) | no proof of rows k806,a (N ≠ 4), k907,a or of the k404 ratio statement; DOIs checked (Crossref, DataCite) |

It found no mathematical error. **Corrections required by this run**, all applied:
1. The content and the method of Kriebel's two deposits stated exactly (§1.3, Scope and priority, What was read),
   replacing the statements that only their abstracts could be read; Theorem B's parenthesis records that the
   positivity is part of Kriebel's theorem; Kriebel's deposits and Chavez-Caliz's Theorem 6 added to the credit
   for the method.
2. The positivity of Ā_j, Ā'_j (23149774, Theorem 1 and Section 4) and of the terms of A'†_j (23394004, (12))
   credited after Proposition 6.3, in §1.3, Theorem C and Scope and priority; abstract: "which Kriebel proved".
3. Theorem E(iii) for N ≥ 8: noted that it also follows from Theorem 1 of 23079798 and Chavez-Caliz's Theorem 6.
4. Remark 6.6: Ā'_1 = γA†_2 (Kriebel's comparison) and the polar and dual areas are constant multiples of A†_1, so
   Theorems C and D and the polar product also follow from row k903,a; recorded in §1.3 and in the novelty statement.
5. The Verification paragraph describes this run; package part `independent_run_2/`.
6. Four numerical bounds of Section 7 rounded the wrong way corrected (3.5·10⁻⁷, 7.9·10⁻⁵, 3.7·10⁻⁹, 4.1·10⁻⁴⁸).
7. Bibliography: the journal title of Garcia–Reznik ("Exploring self-intersected N-periodics in the elliptic
   billiard").

## Relation to the literature, novelty and scope
- **Searches (15 October 2026)** by the first write-up, the runs A and B, and when the note was written: arXiv API
  (elliptic billiards; antipedal, inversive, polar, dual polygons; the proposers; Stachel; recent "elliptic billiard"
  listings), OpenAlex (19 works citing the source; 2025–2026 billiard searches), Semantic Scholar (31 citations),
  zbMATH Open, Crossref, and the Zenodo deposits that DataCite lists for "billiard" and for the five record ids or
  row codes (last search 19:33 UTC; no deposit after 18:39 UTC on the subject). Zenodo's own API refused automated
  access at that time (HTTP 403); it was not worked around. The verification run on the final text repeated the
  searches at 19:59–20:02 UTC (DataCite; Zenodo's own search API, which then answered normally; arXiv API; OpenAlex;
  Crossref; zbMATH Open): no new deposit or paper on the five rows; the newest billiard deposit is 23403150.
  - No printed or deposited proof of row k806,a for N ≠ 4, of row k907,a or of the k404 ratio statement was found.
  - Rows k904,a and k605,a: first proved in Kriebel's deposits of 15 October 2026 (above), made before the note was
    written; the note's proofs are independent.
  - Related: 23065067 (k806,b, N = 4), 23096993 (k903,a), 23094117 (k805), 23079798 (k403,b; Lemma 4),
    23403150 (k401), 23400541 (k203,a, k204, k304), 23396998 (seven rows by central symmetry; κ* = 0 families),
    23091870 (k608 zero, credits the N = 6, a/b = 2 observation to Garcia–Reznik), 23149774 (Kriebel, k606),
    23334000 (Kriebel, k403,a), 23312252 (focal-inversive cosine sums), 23271171 (Kriebel, k108), 23331330
    (Kriebel, k307); Reznik–Garcia–Helman, Eur. J. Math. 8 (2022) 1550–1565 (N = 3 inversive triangle);
    Roitman–Garcia–Reznik, Arnold Math. J. 7 (2021) 619–637 (bicentric polygons; invariants of the focus-inversive
    polygons via elliptic functions and Liouville's theorem); Garcia–Helman–Reznik, arXiv:2604.26035 (2026,
    triangles only).
- **What was read.** The source in the tex of v11 (definitions, the tables containing the five rows, the polar/dual
  caption, the table of symbols) and the journal's numbering scheme and Tables 5, 7 (text extraction); Theorem 6 of
  Chavez-Caliz in arXiv:2004.05404v1; the deposited sources of 23065067 (in full) and 23079798 (Theorem 1, Lemma 4),
  the relevant sections of 23091870, 23396998, 23400541, 23403150; the video list of arXiv:2011.06640v3; the
  abstracts of Roitman–Garcia–Reznik and Garcia–Helman–Reznik (arXiv API) and of all other deposits (DataCite).
  Stachel's Theorem 2 and Corollary 4.2(i) and the Reznik–Garcia–Helman paper were read by run A (arXiv v2), and
  again by the verification run on the final text, which also read Kriebel's deposits 23394004, 23398175 and
  23149774 in full (version 1.0 PDFs, downloaded anonymously from Zenodo), re-read 23065067 (Theorem 1), 23079798
  (Theorem 1, Lemma 4), 23400541 (Theorem B), 23396998 (Section 6.1), 23403150 (Theorem 2, Corollary 3) and checked
  the cited section and table numbers of the DLMF.
- **Caveats.** A search that finds nothing is not a proof of novelty. Deposits made after the last search
  are not covered. The families with κ* = 0 for N = 8, 12 are located numerically (with the closed form of κ* they
  are the families with λ = a²b²(a²−3b²)/(a²−b²)²).
- **Scope.** The five rows for least period and confocal elliptic caustics; 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/