# Verification report — AMR-050-0002, AMR-050-0028, AMR-050-0039, AMR-050-0032 (Reznik–Garcia–Koiller rows k108, k502, k701, k603: half-angle sines, Steiner pedal areas, evolute areas, focal antipedal sums)

Verification dates: 2026-10-15 (two verification runs, then the writing of the note); 2026-10-16 (a third verification
run, which examined the text of the note).

**Verdict.** The note gives complete answers to the four records, with two stated gaps on definedness.
- **Row k108 (AMR-050-0002): false as printed; corrected form proved.** The quotient (A'/A)/∏ sin(θ_i/2) is constant on
  no family with N ≡ 2 (mod 4) (Theorem 1(b)); the product (A'/A)·∏ sin(θ_i/2) is constant on every family with
  N ≡ 2 (mod 4), N ≥ 6, every ellipse and every winding number (Theorem 1(a)). Exact certificate: a = 1, b = 4/5,
  λ = 16/81, two members with quotients 6908733/2151296 and 29760696/9282325 and the common product 91/162
  (Proposition 3.2). The falsity of the printed row was shown first by A. Kriebel (Zenodo,
  doi:10.5281/zenodo.23271172, 8 October 2026; quotients 11664/3125 and 3645/1024 on x²/4 + y² = 1); no novelty is
  claimed for the disproof.
- **Row k502 (AMR-050-0028): proved for every odd N on its domain.** A'/A'_K' = c²σ/(2(π_g − π_f)), where σ = A'·Σ sin 2θ'_i,
  π_f and π_g (products of the pedal areas of the outer polygon at the real and at the imaginary foci) are constant
  on the family (Theorem 2). The ratio is defined at every member of a family or at none; it is defined iff σ ≠ 0
  and π_g ≠ π_f. Neither condition is proved for every family; no family violating them was found (Remark 4.4).
  Value 4 for N = 3.
- **Row k701 (AMR-050-0039): proved for every N ≥ 5 in the form A_ev = νA.** ν = −c⁴κ/(16a²b²) with κ in closed form
  (Theorem 3, Proposition 5.2); κ = 4 for N = 6, where A_ev = −c⁴A/(4a²b²) holds for every centrally symmetric
  hexagon inscribed in the ellipse (Proposition 5.3); κ → 6 as N → ∞ with τ fixed. A > 0 at every member, so the
  ratio is constant at every member if ν ≠ 0; ν ≠ 0 is not proved (observed: κ > 0 for τ/N < 1/3, κ < 0 for
  τ/N > 1/3; ν = 0 would make the row vacuous, not false). For N = 4 the ratio is not constant (Proposition 5.5:
  A_ev/A = −(c⁴/(4a²b²)) sin²(φ1 + φ2) for every inscribed parallelogram; −c⁴/(4a²b²) at the rhombus member, 0 at
  the rectangular member); for N = 3, A_ev = 0.
- **Row k603 (AMR-050-0032): Kriebel's theorem** (Zenodo, doi:10.5281/zenodo.23191247, 6 October 2026, Theorem 1).
  The note gives only a short confirmation (Proposition 6.2: q*_{1,i} − q*_{2,i} = C(y_{i+1} − y_i) with C in
  closed form) and claims nothing new for this row.
- **Not new.** The rows and conventions (Reznik–Garcia–Koiller); the parametrisation by Jacobi functions (Stachel,
  Eur. J. Math. 2022, Theorem 2 of arXiv:2105.03624v2; a short proof is included); the reflection property, the
  angle formula and the central symmetry of even trajectories (Stachel, J. Geom. 2021, Lemmas 2.1, 2.2 and
  Corollary 4.2(i) of arXiv:2105.03362v2); Steiner's theorem on pedal areas (as recalled in the source); the
  constancy of π_f (row k605,a; Kriebel, doi:10.5281/zenodo.23398175, and independently the author,
  doi:10.5281/zenodo.23404834); the quadratic form of the pedal area with Steiner's centroid as its stationary point,
  for real polygons (Lemma 4.1(b), (d); classical, and in the author's doi:10.5281/zenodo.23089778, eq. (8), and
  doi:10.5281/zenodo.23090907, Lemma 2); the constancy of σ is the analogue for the outer polygon of
  doi:10.5281/zenodo.23089778, Lemma 3; the circumcentre formula behind Lemma 5.1 is used on the caustic in the
  author's doi:10.5281/zenodo.23061834, Lemma 3.
- **New, to our knowledge:** the product identity of Theorem 1(a) with the non-constancy of the printed quotient on
  every family; Theorem 2 for every odd N on its domain; Theorem 3 for every N ≥ 5, with Propositions 5.3 and 5.5.
- **Open:** σ ≠ 0 and π_g ≠ π_f for every odd family (k502); ν ≠ 0 for every family with N ≥ 5 (k701). Hyperbolic
  caustics are not treated.
- **Checked three times.** Two independent verification runs examined the first write-up; a third examined the text
  of the note, including every part added at the writing stage, and found no error in a theorem, proposition or
  lemma (section "Third verification run" below).

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 of N-periodics in the elliptic billiard"). The v11 e-print (tex) was read at
  the writing and by both verification runs; the journal version (PDF on the journal's host,
  armj.math.stonybrook.edu) was read at the writing and by the second run. Bibliographic data checked with
  Crossref and the arXiv interface on 2026-10-15.
  - Row k108: Table 2 (v11; Table 2, p. 345 of the journal version), invariant k103/k105 with k103 = A'/A and
    k105 = ∏ sin(θ_i/2), value "?", which N "≡ 2 (mod 4)", date 1/20, proven "?". Neighbouring row k107 = k103·k105 for
    N ≡ 0 (mod 4).
  - Row k502: Table 6 (both versions; p. 349), invariant A'/A'_k, odd N; A'_k is the area of the pedal polygon of the
    outer polygon P' with respect to its Steiner curvature centroid K' (v11 Section 3.4, equation (4)
    K = Σ sin(2θ_i)R_i/Σ sin(2θ_i), the R_i being the vertices). The source adds that for even N the centroids are at
    the origin.
  - Row k603: Table 7 (both versions; p. 349), Σ q*_{1,i}/Σ q*_{2,i} = 1 for all N; q*_{j,i} = |Q̄*_{j,i} − f_j| with
    Q̄*_{j,i} the vertices of the antipedal polygon of the trajectory with respect to f_j (v11 Section 3.5).
  - Row k701: Table 8 (both versions; p. 350), A/A_ev for N > 4; the evolute polygon has its vertices at the
    intersections of successive perpendicular bisectors of the sides (v11 Section 3.6, after Arnold, Fuchs,
    Izmestiev, Tabachnikov and Tsukerman). The source notes that the evolutes vanish for N = 3 and that A'_ev = 0 for
    N = 4 (P' a rectangle); it makes no statement on A/A_ev for N = 4.
  - Conventions: signed areas by the shoelace formula (v11 equation (1)); the active caption of Figure 1 describes
    the outer polygon as tangent to the billiard ellipse at the vertices of the trajectory; the foci are written
    (±c, 0) without naming f_1; the experimental method starts from vertices spread evenly over the ellipse (simple
    trajectories).
- **Corpus records.** ulamai/UnsolvedMath, dataset version 1.6.0: AMR-050-0002 ("Elliptic-billiard invariant
  k_{108}", status open), AMR-050-0028 (k_{502}, partially_solved upstream), AMR-050-0032 (k_{603}, open),
  AMR-050-0039 (k_{701}, partially_solved upstream). Each asks to prove that the quantity is constant as P varies
  through the Poncelet family, for the N of the row. The records are faithful to the rows; AMR-050-0028 writes A'_{K'}
  for the source's A'_k.

## Readings
| Reading | Effect | Where |
|---|---|---|
| θ_i = interior angle of the trajectory at P_i, in (0, π) | with it the source's row k101 (Σ cos θ_i = JL − N) holds; turning angles would change its sign | Section 2.1 |
| Signed areas (source's convention) | for counterclockwise members A > 0 and A' > 0 (Lemma 2.2); no statement depends on the orientation | Sections 2.1, 2.2 |
| Winding numbers | the source's families are simple; all theorems hold for every winding number coprime to N, which includes τ = 1 | Section 1.1 |
| "N-periodic" = least period N | the families are those of Lemma 2.1(iii) | Section 2.1 |
| Pedal areas at the imaginary foci (0, ±ic) | used only as values of the quadratic polynomial M ↦ A'_M (Lemma 4.1(a)); at real members A'_{g2} is the conjugate of A'_{g1} | Section 4 |
| The source's K' (Steiner centroid of P') | identified with the stationary point of M ↦ A'_M, also for star outer polygons (all turns of P' are left turns) | Lemma 4.1(b), (d), Lemma 2.2 |
| Definedness of the k502 and k701 ratios | k502: defined at every member iff σ ≠ 0 and π_g ≠ π_f, otherwise at none; k701: A > 0, so defined at every member iff ν ≠ 0 | Theorems 2, 3 |
| Which focus is f_1 | irrelevant: the four rows are symmetric in f_1 ↔ f_2 | Section 1.1 |

## Results in the paper
- **Lemma 2.1 (Stachel's parametrisation)** with proof: P(u) = (−a sn u, b cn u), modulus m = c/a'', cn Δ = b''/b,
  vertices P(u0 + 2kΔ), N-periodic iff Δ = 2Kτ/N; outer vertices V(u + (2j+1)Δ). **Lemma 2.2**: positivity of A, A'
  and left turns of P and P'. Section 2.3: local expansions at iK', K + iK', residues, isotropic phases.
- **Theorem 1 (k108)**: Lemma 3.1 (sin²(θ/2) = a''²dn²v/(a² − c²sn²v), proof via focal distances; Stachel's Lemma
  2.2); divisor of S on C/L (−2 at X = iK', +2 at Y = K + iK'); A has simple poles at X, X + 2iK' with
  Res_X A = −2iab(1 − dn 2Δ)/(m² sn 2Δ) and fold zeros at Y, Y + 2iK'; A' has simple poles at Y, Y + 2iK' with
  Res_Y A' = −2ia²b²(1 − dn 2Δ)/(a''b''m² sn 2Δ) and fold zeros at X, X + 2iK'; hence (A'/A)S is constant and (A'/A)/S
  has orders +4, −4. **Proposition 3.2**: the exact certificate with a table of values. **Remark 3.3**: N ≡ 0 (mod 4)
  (observed reversal; numerical tables of Garcia–Reznik).
- **Theorem 2 (k502)**: Lemma 4.1 (pedal area of a polygon given by its side lines: quadratic in M, Steiner's centre
  = the source's centroid, four-foci identity A_K = 2(A_{g1}A_{g2} − A_{f1}A_{f2})/(c²Σ)); Lemma 4.2 (feet of a point on
  the tangents of the ellipse: symmetries, poles only at isotropic phases with residues along the isotropic normal,
  exactly one real and one imaginary focus on each isotropic tangent); Lemma 4.3 (σ, π_f, π_g constant for odd N);
  Remark 4.4 (domain; numerical scans).
- **Theorem 3 (k701)**: Lemma 5.1 (centre of a circle through four points of the ellipse), the area formulas (12);
  pole argument for odd N ≥ 5 and even N ≥ 8; **Proposition 5.2** (closed form of κ; κ = 6 − (2 − m²)h² + O(h⁴);
  κ → 6 for fixed τ); **Proposition 5.3** (F = 4Φ for centrally symmetric hexagons); **Remark 5.4** (the cross terms
  for N = 6 written out); **Proposition 5.5** (N = 4: parallelogram formula, rhombus and rectangle members);
  **Remark 5.6** (sign of κ, observed).
- **Section 6 (k603)**: Lemma 6.1 and Proposition 6.2 (Kriebel's edge identity in the notation of the note).

## 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 mpmath and sympy, about 100 s; 150 checks, TOTAL
  failures: 0). (i) Trajectories from the reflection law alone, 50 digits, six members per family on two ellipses:
  Theorem 1 for (N, τ) = (6,1), (10,1), (10,3), (14,3), with 91/162 and λ = 16/81; the reversal of Remark 3.3 for
  (4,1), (8,1), (8,3), (12,5); Theorem 2 with its closed form for six odd families and the stationarity of the pedal
  area at the source's centroid; Theorem 3 for eight families with 5 ≤ N ≤ 8 (κ = 4 for N = 6), Proposition 5.5 on the
  4-periodic family, A_ev = 0 for N = 3; row k603 and Proposition 6.2. (ii) Exact (sympy): every entry of
  Proposition 3.2 and the cosines in its proof; Kriebel's two members (11664/3125, 3645/1024, product 5/9); the
  rhombus and rectangle of Proposition 5.5 (A_ev/A = −81/1600 and 0; equal quotients 1681/200, different products);
  two wrong inputs rejected. (iii) Propositions 5.3, 5.5 and Remark 5.4 symbolically; Proposition 5.3 on random
  hexagons with a non-symmetric control. (iv) The sign rule of Remark 5.6 on 24 pairs of a family and an ellipse.
  (v) With Jacobi functions: Lemmas 2.1, 3.1, the two residues of Section 3.2 by contour integrals, (A'/A)S at complex
  phases, Proposition 5.2 against direct values (seven families) and its expansion, the identity of Proposition 6.2,
  Lemma 4.1(b), (c) at complex points.
- **Programs with which the results were first obtained** (`original/`, 40 digits): the toolkit with an independent
  check of the reflection law at every vertex; a screen of the four rows over 18 families; numerical checks of the
  divisor table of Theorem 1 (local orders by logarithmic slopes), of Lemmas 4.1–4.3 and Theorem 2, of the residues,
  of Theorem 3 and κ, and of a second proof of k603; the exact certificate.
- **Verification run A** (`verification_run_A/`, own toolkit, 40 digits; 90 digits for one pole order): Theorem 1 and
  Remark 3.3 on three ellipses with local orders and complex phases; Theorem 2 with the source's centroid, the
  four-foci identity on random polygons; k603 on 30 families; Theorem 3 for N = 5–12 and κ → 6 up to N = 192;
  Proposition 5.3 on random hexagons; scans of σ, π_g − π_f (odd N ≤ 15) and κ (N ≤ 16) over b/a = 0.02, …, 0.98.
- **Verification run B** (`verification_run_B/`, trajectories from the reflection law only, 50 digits): the four rows
  on 60 pairs of a family and an ellipse; exact reconstruction of Proposition 3.2 and of Kriebel's members from their
  first vertices, with negative controls; complex-phase local orders; κ against its closed form and as a function of
  a continuous rotation number; scans of κ, A'_K'/A', σ/A'² and (π_g − π_f)/A'² over b/a = 0.06, …, 0.98.
- **Verification run 3** (`independent_run_2/`, 2026-10-16, written independently of `check_note.py` and of the
  earlier programs): `r2_exact.py` (126 exact checks), `r2_geom.py` (215 checks; plane geometry at 60 digits on
  trajectories built by the tangent construction), `r2_complex.py` (130 checks; Jacobi functions at complex phases).
  Details in the section "Third verification run" and in the folder's README.
- **Re-runs.** On 2026-10-15 all 29 programs of `original/`, `verification_run_A/` and `verification_run_B/` were run
  again from a copy of the package with the recorded arguments: every output is identical to the recorded one.
  `check_note.py` was run from the final folder and again from an extracted copy of `source.zip` (identical apart
  from the line with the running time). On 2026-10-16 the third run re-ran the 29 programs and `check_note.py` from an
  extracted copy of the archive of the previous version: all 29 outputs byte-identical, `check_note.py` identical
  apart from the running time; the three programs of `independent_run_2/` were run again from an extracted copy of
  the new archive and reproduced their outputs apart from the running time. `main.tex` compiles without warnings and
  without overfull boxes (21 pages).

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Two independent verification
runs, both AI-assisted, followed (2026-10-15); each wrote its own programs, and neither read the other's report.
Run A examined the statements against the source and every proof line by line; run B examined the computations,
the sources, the literature, novelty and the bibliography.

| Item | Run A | Run B |
|---|---|---|
| Rows and records against the source (v11 tex; journal version) | CONFIRMED (tex read) | CONFIRMED (tex and journal PDF read) |
| Parametrisation | PROVED without the literature | CONFIRMED against reflection-law orbits; Stachel's Theorem 2 read in arXiv v2 |
| Theorem 1 (divisors, fold zeros, residues) | PROVED (every count re-derived) | PROVED; residue of A' at Y confirmed numerically only (now derived in the note) |
| Exact certificate (Proposition 3.2), Kriebel's members | CONFIRMED (own sympy code) | CONFIRMED (built exactly from the first vertices; controls rejected) |
| Remark on N ≡ 0 (mod 4) | CONFIRMED numerically | CONFIRMED numerically and exactly for N = 4 |
| Lemma 4.1 (pedal quadratic, four-foci identity) | PROVED; supplied the proof that the stationary point is the source's centroid | CONFIRMED (stationarity checked for convex and star polygons) |
| Lemmas 4.2, 4.3, Theorem 2 | PROVED | PROVED |
| Domain of k502 | σ ≠ 0 missing in the first write-up; no violation in scans | the same; no violation in scans; sign of τ/N − 1/4 observed |
| Lemma 5.1, area formulas, pole argument, κ | PROVED; closed form of κ re-derived; supplied F = 4Φ for N = 6 | PROVED; N = 6 cross terms confirmed numerically only (now written out) |
| Sign of κ, ν ≠ 0 | κ < 0 for stars with τ/N > 1/3; the stated range of κ to be corrected | the same; single zero of the closed form at rotation number 1/3 |
| N = 4, N = 3 | CONFIRMED; the source does not claim non-invariance for N = 4 | CONFIRMED (pole of order 3; ratio varies) |
| Row k603 | covered by Kriebel's deposit (PDF read); supplied the edge identity of Proposition 6.2 | covered by Kriebel's deposit (PDF read); edge identity checked |
| Literature and novelty | Zenodo (all 131 records of A. Kriebel), OpenAlex, arXiv | Zenodo, OpenAlex, Semantic Scholar, arXiv, Crossref, zbMATH Open, one web search |
| Bibliography | Stachel's theorem numbering to be checked | Crossref checked; [GR22] and the title and pages of [CC21] to be added |

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 cosine with the normal at (±5/9, ±8√14/45) is √2/3, not 5/9 (5/9 belongs to (±1, 0)): proof of
   Proposition 3.2, with all four cosines listed.
2. (A2, B2) The domain of k502: σ ≠ 0 and π_g ≠ π_f, each holding for a whole family or failing for it, neither
   proved; scans cited; the exceptional family of row k503 mentioned: Theorem 2, Remark 4.4.
3. (A3) The source's families are simple; all theorems cover every coprime winding number: Section 1.1, abstract.
4. (A4) Proof that Steiner's stationary point is the source's K': Lemma 4.1(b), (d) (with Lemma 2.2 for the sign of
   the turning angles).
5. (A5, B1) The incorrect range of κ removed; the observed sign rule (κ > 0 for τ/N < 1/3, κ < 0 for τ/N > 1/3) and
   "for fixed τ" in the limit κ → 6: Theorem 3, Proposition 5.2, Remark 5.6.
6. (A6) A > 0 at every member (Lemma 2.2), so the k701 ratio is defined at every member iff ν ≠ 0; the N = 4 case
   attributed correctly (the source only restricts the row): Theorem 3, Section 5.3.
7. (A7) The N = 6 case settled by the identity F = 4Φ: Proposition 5.3.
8. (A8, B3) Stachel's theorem cited with the numbering of arXiv:2105.03624v2; the journal numbering could not be
   checked (publisher's page refused automated access), and the deposits of Kriebel cite it as Theorem 4.3.
9. (A9) The active caption of the source's Figure 1 is cited, not a commented-out one: Section 1.1.
10. (B3) Garcia–Reznik, Ann. Math. Inform. 55 (2022) 49–75, doi:10.33039/ami.2022.02.001, cited for the numerical
    k107/k108 tables: Remark 3.3 (these tables are in an appendix file of its arXiv source that is not compiled; see
    the third run).
11. (B3) Chavez-Caliz cited with title and pages.
12. (B4) Both parts of the k108 status kept visible: disproof by Kriebel, product form proved here.
13. (Run B, numerical-only items) The residue of A' at Y and the cancellation of the cross terms for N = 6 are derived
    in the text: Section 3.2 and Remark 5.4.

**Changes made when the note was written, after runs A and B.** Lemma 3.1 is proved from two classical facts on the
foci (the first write-up used Joachimsthal's integral); Proposition 5.5 (N = 4) and the proof of the limit κ → 6 in
Proposition 5.2 were added; the constant C of Proposition 6.2 is given in closed form; the deposits of 15 October
2026 on row k605,a (Kriebel; the author) are credited for the constancy of π_f; the program `check_note.py` was
written. These additions were not examined by the two runs; they are tested by `check_note.py`, and the third
verification run examined them (below).

## Third verification run (2026-10-16)
A third verification run, also AI-assisted, examined the text of the note, first the parts added or rewritten at the
writing stage (the proof of Lemma 3.1, Proposition 5.5, the proof of the limit κ → 6, the closed form of C, the residue of
A' at Y, the cross terms for N = 6, the identification of Steiner's stationary point with the source's centroid), then
everything else.
- **First check.** A. Kriebel's newest Zenodo records listed again (131 records, newest 23398175 of 2026-10-15, k605,a):
  none treats the corrected k108, k502 or k701; Zenodo queries for k108, k502, k701 and the four record numbers found
  nothing new. Kriebel's deposits 23271172 (k108) and 23191247 (k603) were read in full.
- **Proofs.** Every proof re-derived line by line: no error in a theorem, proposition or lemma, no gap that matters.
  Among the parts added at the writing stage, the residue −2ia²b²(1 − dn 2Δ)/(a''b''m² sn 2Δ), the formula for T₁ when
  N = 6, the telescoping identity behind the stationary point, the focal-distance proof of Lemma 3.1, the parallelogram
  formula A_ev = −(c⁴/(4a²b²)) sin²(φ1 + φ2)A, the expansion κ = 6 − (2 − m²)h² + (5m⁴ − 2m² + 2)h⁴/12 + O(h⁶) and the
  argument for fixed τ (via dn(K/2) = √m'), and C = c(a²b² − λ(a² + b²))/(ab√λ(b² − λ)) = −Γ (Kriebel) are all correct.
- **Sources.** The source (v11 tex and the journal tables), Stachel's Theorem 2 and Lemmas 2.1, 2.2, Corollary 4.2, the
  journal version of Garcia–Reznik and the sources and compiled PDFs of its arXiv versions v1–v3, and the deposits
  cited were compared with the note. The numerical tables of k107/k108 that the note attributed to an appendix of
  arXiv v3 are in a file of the arXiv source that is not included in any compiled version.
- **Programs** (`independent_run_2/`): 471 checks, 0 failures; the package re-run as described under "Re-runs".
- **Corrections required (10), all applied:** (1) the provenance of Kriebel's example (he takes the member with a vertex
  at (2, 0), with its caustic, from Garcia–Reznik, Appendix B.5.1, and verifies the other himself); (2) the description
  of the Garcia–Reznik tables (Remark 3.3, bibliography, "What was read"); (3) credit to the author's earlier deposits
  23089778, 23090907 and 23061834 for three ingredients (Section 1.3); (4) the reason why Q_f(u_{−j}) = Q_f(u_j) at Y in
  the proof of Lemma 4.3, which must also cover the feet on the isotropic lines; (5) the Verification paragraph; (6)
  Section 7 (the folder of this run); (7) the zero of κ at rotation number 1/3 is exact for every modulus (Remark 5.6);
  (8) "What was read" and the searches; (9) the punctuation of (2); (10) this report, the package README, the archive
  and the deposit metadata.

## Relation to the literature, novelty and scope
- **Searches (2026-10-15, 2026-10-16).** By the runs, in part again at the writing, and except Semantic Scholar by the
  third run: arXiv (elliptic billiards and their
  invariants, evolute polygons, Steiner curvature centroids, antipedal polygons), the 19 works citing the journal
  version of the source in OpenAlex and the 31 citing records in Semantic Scholar, Crossref, zbMATH Open, Zenodo
  (all 131 records of A. Kriebel, the newest of them listed again at the end of the writing and on 2026-10-16; the records of the
  author; queries for
  elliptic billiards, k108, k502, k701, evolutes and curvature centroids), the journal version of the source, the
  paper of Garcia and Reznik with its arXiv versions, and one web search in run B and one in the third run.
  - k108: the falsity of the printed row is in Kriebel's deposit 23271172 (one N = 6 family; it notes the common
    product 5/9 and asserts no repaired identity). The corrected identity was found nowhere. The N = 4 deposits on k107
    (the author, 23075934; Kriebel, 23231145) assert no corrected invariant.
  - k502: not treated anywhere; the deposits 23089778 (k501) and 23090907 (k503) of the author exclude it; the
    constancy of π_f used in the proof is row k605,a (Kriebel 23398175; the author 23404834).
  - k701: not treated anywhere; the deposit 23061834 of the author treats k702 and k703 only.
  - k603: Kriebel's deposit 23191247 (Theorem 1, every closed trajectory).
  - Garcia–Reznik: the arXiv source of their paper (v1–v3) contains an appendix file, not included in any compiled
    version or in the journal version, with values of k107 and k108 at one member per aspect ratio; no correction or
    proof.
- **What was read.** The source in the v11 tex (tables of the four rows and of k107, definitions, symbols, method,
  captions) and Tables 2, 6, 7, 8 of the journal version; Stachel's Theorem 2 with its proof and his Lemmas 2.1,
  2.2 and Corollary 4.2 (arXiv v2 versions); the statements of Theorems 1–3 and Corollary 6.2 of Akopyan–Schwartz–
  Tabachnikov (arXiv v2); the journal version of Garcia–Reznik (appendices, Table 1) and the sources and compiled
  PDFs of its arXiv versions v1–v3; Kriebel's deposits 23271172 and 23191247 in full (third run) and 23398175
  (statements and the parts cited); the statements and scope of the author's deposits 23089778, 23090907, 23061834
  and 23404834 (third run); the Zenodo descriptions of the other deposits cited.
  Chavez-Caliz (2021) and Arnold–Fuchs–Izmestiev–Tabachnikov–Tsukerman (2017) were not read; what the note says
  about them is taken from the source. The proofs of the cited theorems were not checked; the parametrisation, the
  angle formula and the central symmetry are proved in the note, the reflection property and two classical facts on
  the foci are quoted.
- **Caveats.** The journal numbering of Stachel's theorem was not checked. The k502 and k701 ratios may be undefined
  on whole families if σ = 0, π_g = π_f or ν = 0; no such family is known, none is excluded by proof. This negative
  search is not a proof of priority.
- **Scope.** The note settles rows k108 (false as printed; product form proved), k502 (on its domain) and k701
  (N ≥ 5, up to ν ≠ 0; N = 4 fails; N = 3 vacuous), and confirms Kriebel's proof of k603. No priority is claimed for
  the disproof of k108, for k603 or for the constancy of π_f.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
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