# Verification report — AMR-050-0011, AMR-050-0013, AMR-050-0016 (Reznik–Garcia–Koiller: pedal areas of N-periodics with respect to an arbitrary point)

Verification date: 2026-10-15.

**Verdict.** The note proves the three rows in a precise form; every statement it makes is proved, and the scope is as
follows.
- **Settled.** Row k_{203,a} (AMR-050-0011): for every ellipse x^2/a^2 + y^2/b^2 = 1 (a > b > 0), every confocal
  elliptic caustic whose billiard trajectories have least period N ≡ 0 (mod 4), every winding number and every point
  M (real or complex), A·A_M = C0 + C2 M·M is constant along the family (Theorem A).
- **Settled on the domain where the ratio is defined.** Row k_{204} (AMR-050-0013), N ≡ 2 (mod 4): A_M = κ(M)·A with
  κ(M) = κ0 + κ2 M·M, κ0 > 0 (Theorem B). Row k_{304} (AMR-050-0016), N ≡ 0 (mod 4): A'_M = κ'(M)·A' with
  κ'(M) = κ'0 + κ'2 M·M, κ'0 > 0 (Theorem C). Hence A/A_M and A'/A'_M are constant for every M with κ(M) ≠ 0,
  resp. κ'(M) ≠ 0; on the exceptional set the pedal area vanishes along the whole family and the ratio is undefined.
  The exceptional set can be a real circle in star families: exactly for k_{304} (N = 8, τ = 3, a^2 = 32/27, b = 1,
  λ = 32/35: A'_M = 0 on |M|^2 = 85/27), numerically for k_{204} (N = 10, τ = 3). For winding number 1 it has no
  real points for k_{304} (Kriebel's δ ≥ 0); for k_{204} this was only tested.
- **Not new.** The rows are Reznik–Garcia–Koiller's. The Jacobi parametrisation is Stachel's canonical
  parametrisation. The case M = O of k_{203,a} is row k_{203,b}, proved in the author's earlier note
  (doi:10.5281/zenodo.23085903, concept DOI). For k_{304}, the dependence of A'_M on M for every even N,
  A'_M = A'_O (2F + δ|M|^2)/(2F), with A'_O > 0 and δ ≥ 0 for winding number 1, was proved by A. Kriebel
  (doi:10.5281/zenodo.23331331, 13 October 2026, row k_{307}); only the constancy of A'/A'_O for N ≡ 0 (mod 4) is
  new in Theorem C.
- **By-product.** For even N the part of A_M (and of A'_M) linear in M vanishes (central symmetry; Proposition 3.2).
- **Not treated.** Hyperbolic caustics; closed walks that repeat a trajectory of smaller least period (the statements
  are then false in general); the positivity of κ2 for winding number 1 in Theorem B.

The note is unrefereed. Three independent AI-assisted verification runs examined it: runs A and B the first written
version of the results, and run 2, the third run, the text of the note (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), journal version "Fifty new invariants of N-periodics in the
  elliptic billiard", Arnold Math. J. 7 (2021) 341–355, doi:10.1007/s40598-021-00174-y (Crossref-checked).
  - Read in the tex source of v11: preliminaries ("In this paper all polygon areas are signed", shoelace formula),
    the subsection "Pedals with respect to N-periodic" (feet of the perpendiculars from M onto the sides of the
    N-periodic; A_m the area of the polygon of the feet), the subsection on the outer polygon, and the tables
    `tab:inv-ped` and `tab:inv-ped-tang`.
  - Rows: k_{203,a}: A A_m, N ≡ 0 (mod 4), M all, proven "?"; k_{204}: A/A_m, N ≡ 2 (mod 4), M all, proven "?";
    k_{304}: A'/A'_m, N ≡ 0 (mod 4), M all, proven "?". The same rows and marks are in Tables 3 and 4 of the journal
    version (text extraction of the journal's file).
  - Neighbouring rows: k_{203,b} (A A_m, N ≢ 2 (mod 4), M = O); k_{202,a}, k_{202,b} (credited by the source to
    Bialy–Tabachnikov); k_{201}, k_{205} (credited by the source to private communications of A. Akopyan);
    k_{303,a}, k_{303,b} (A' A'_m); k_{307} (area centroid of the outer pedal polygon, even N). The two tables have
    further rows with M = "all" (k_{205}, k_{302}, k_{303,a}, k_{305}, k_{306}, k_{307}); the note treats three of
    them.
- **Corpus records.** ulamai/UnsolvedMath v1.6.0: AMR-050-0011 (record 5100011, k_{203,a}), AMR-050-0013 (5100013,
  k_{204}), AMR-050-0016 (5100016, k_{304}); upstream status `open`. Each asks to prove that, for the stated class of
  N and every M, the quantity is constant as P varies through the family. The texts agree with the source rows. The
  records say "a fixed confocal caustic" without saying that it is an ellipse (the source's setting is the confocal
  ellipse pair, as in the note), and cite the "Eighty New Invariants" title with the year 2021 of the journal
  version, whose title is "Fifty New Invariants".

## Readings
| Reading | Answer | Where |
|---|---|---|
| N-periodic = least period N (the source's experiments: one caustic per N) | proved, every winding number coprime to N | Theorems A–C, Lemma 2.2 |
| N-periodic = closed walk of length N (repeated traversals allowed) | false in general: a 3-periodic listed four times (N = 12) has non-constant A·A_M; products scale by r^2 and ratios are unchanged under r-fold repetition | Section 1.4, Section 6 |
| Signed areas (the source's convention) | as proved; reversing the orientation changes the signs of all areas and not the statements | Section 1.4 |
| Every real point M | proved | Theorems A–C |
| Complex points M (feet by the bilinear formula, M·M for \|M\|^2) | proved | Theorems A–C |
| Ratio rows read literally ("A/A_M constant for every M") | true for every M with κ(M) ≠ 0 (resp. κ'(M) ≠ 0); on the exceptional circle the ratio is undefined | Theorems B, C |
| Caustic a confocal ellipse ("confocal ellipse pair") | the setting of the note | Section 1.1 |
| Hyperbolic caustic | not treated | Scope and priority |

## Results in the paper
- **Lemmas 2.1–2.5.** Stachel's canonical parametrisation P(u) = (−a sn u, b cn u) with modulus
  k^2 = (a^2−b^2)/(a^2−λ), vertex step h = 2Δ, sn^2 Δ = λ/b^2; the side lines m(t)·X = 1; the law of reflection
  (the two chords at a vertex are mirror images in the normal) verified directly; closure Nh = 4τK with
  gcd(τ, N) = 1; m·m = dn^2 t/(b^2 cn^2 Δ) (side normals isotropic exactly at the double zeros of dn); n·n with
  simple zeros at Π' ± Δ; parities about the poles and the zeros of dn.
- **Lemma 3.1.** A_M = A_O + c1·M + c2 M·M with explicit formulas (c1 = −½ Σ w_i (m_i + m_{i+1}),
  c2 = ¼ Σ w_i (m_i·m_{i+1})); the isotropy of the quadratic part (Steiner) for complex normals.
- **Proposition 3.2.** For even N, A_{−M} = A_M and A'_{−M} = A'_M; so c1 = c'1 = 0.
- **Lemmas 3.3–3.4.** Behaviour under u → u + 2K, u + 2iK'; positivity of A, A', A_O, A'_O on the real family.
- **Section 4 (Theorem A).** A has at most simple poles, at the vertex poles 𝒫_A; A_M has at most simple poles, at
  the isotropic phases 𝒫_iso (the double pole of a foot collapses because the neighbouring feet coincide);
  𝒫_A and 𝒫_iso are cosets of one group, disjoint exactly for N odd or N ≡ 0 (mod 4) (Lemma 4.4); A_M vanishes on
  𝒫_A for N ≡ 0 (mod 4) (Lemma 4.5, an order-reversing involution composed with the half-turn); A vanishes on
  𝒫_iso (Lemma 4.6); Liouville. N = 4: C2 = 0 and C0 = 8a^4b^4/(a^2+b^2)^2 (from the rhombus).
- **Section 5 (Theorems B, C).** Nonzero residues of A (Lemma 5.1; the sign factor between residues is ±1);
  for N ≡ 2 (mod 4) the pole sets of A and A_M coincide and the residues are matched; the outer polygon is
  Γ·P(u + Δ) (Lemma 5.2) and its pole set coincides with the isotropic set exactly for N ≡ 0 (mod 4).
  Remark 5.3 identifies κ'2/κ'0 = δ/(2F) with Kriebel's formula.
- **Table 2.** Exact values: N = 4 (a^2 = 16/9): A·A_M = 2048/625, A'_M = A'/2; N = 6 (a^2 = 4, λ = 4/9):
  A_M = (7 + M·M) A/9; N = 8, τ = 1 (a^2 = 8/3, λ = 8/35): A·A_M = 256(13 + 3M·M)/175, A'_M = 9(5 + M·M)A'/50;
  N = 8, τ = 3 (a^2 = 32/27, λ = 32/35): A·A_M = −512(27M·M − 13)/4725, A'_M = (85 − 27M·M)A'/600 (all b = 1).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **Program of this note** (`check_note.py`, about 10 seconds, TOTAL failures: 0, 118 checks).
  - (i) Trajectories built by the reflection law alone, caustic by bisection on the eccentric-angle advance,
    50 digits: 24 families ((a,b) ∈ {(1.3,1), (2,1), (3.5,1), (2.6,1.7)}, (N,τ) ∈ {(4,1), (8,1), (8,3), (12,1),
    (12,5), (16,3)}), 8 members, 5 points M (one complex): closure and tangency ≤ 5·10^-42; relative spread of
    A·A_M ≤ 4·10^-42 (≤ 3·10^-44 unless b^2 − λ < 2·10^-8); A_M/A not constant. Controls: N = 6, 10, 14 (product
    spread ≥ 8·10^-6, ratio constant to 5·10^-48); N = 3, 5, 7 (A·A_O constant to 6·10^-46, A·A_M for M ≠ O varies
    by ≥ 7·10^-3); 3-periodic × 4 and 6-periodic × 2 (spread ≥ 0.02); 4-periodic × 2 (constant).
  - (ii) Theorem B on 12 families: A_M/A constant to 9·10^-48; κ0 + κ2 M·M fits to 3·10^-46; κ0 > 0; κ2 < 0 exactly
    for (N, τ) = (10, 3), with A_M = 0 (to 6·10^-48 relative) on the circle |M|^2 = −κ0/κ2 (3.0378…, 5.8229…,
    15.4158… for a = 1.3, 2, 3.5). Theorem C on 24 families: A'_M/A' constant to 2·10^-42, κ'0 > 0,
    κ'2/κ'0 = δ/(2F) to 7·10^-44.
  - (iii) Exact N = 4 certificate (sympy; symbolic a, b, phase, M), with three rejected false identities.
  - (iv) Exact trajectories in quadratic number fields for the four families of Table 2 (3–4 members each) and a
    50-digit check of A'_M = 0 on |M|^2 = 85/27 at ten members of the star family.
- **Programs with which the results were first obtained** (`original/`): `check_k203a.py` (60 digits; identities,
  geometric cross-check, Theorem A on 18 families, controls, pole and zero orders, involutions), `exact_N4.py`
  (N = 4 exactly), `check_siblings.py` (c1 = 0, Theorems B and C, residue formula).
- **Verification run A** (`verification_run_A/`): Jacobi-free geometry (`geo_check.py`, 54 families;
  `geo_check_extra.py`, 14 harder ones, N up to 32, a/b from 1.01 to 30), the reflection law at every vertex
  (`reflect_check.py`), the Jacobi shift formulas (`jacobi_shift_check.py`), exact certificates symbolic in the phase
  for the four families of Table 2 (`exact_cert.py`).
- **Verification run B** (`verification_run_B/`): reflection-law geometry with controls and wrong inputs
  (`vb_geom.py`), every lemma at random complex phases, pole orders, involutions, residues, the arithmetic of
  Lemma 4.4 for N ≤ 200 (`vb_jacobi.py`), residue quotients at 80 digits (`vb_resB.py`), extreme families
  (`vb_extreme.py`), non-confocal Poncelet controls (`vb_nonconfocal.py`), exact N = 4 certificates with its own
  parametrisation (`vb_exact_N4.py`, symbolic in a, b, phase, M; `vb_exact_N4_num.py`). It re-ran the three original
  programs: outputs byte-identical.
- **Verification run 2** (`independent_run_2/`, the third run, on the text of the note): `r2_algebra.py` (exact symbolic identities,
  49 checks), `r2_numeric.py` (reflection-law trajectories for four further shapes and 20 pairs (N, τ) each, complex
  phases, 51 checks), `r2_exact.py` (Table 2 exactly at two members per family in Q(√2, √3, √5, √7), 31 checks);
  all pass. It re-ran every program of the package from an extracted copy of `source.zip`
  (`independent_run_2/package_rerun_comparison.txt`): all recorded outputs were reproduced byte for byte or up to
  timing lines, except (a) part A of `vb_geom.py`, which with the program as included reproduces 22 of the 37
  recorded families (same printed values) and does not construct the other 15 (the bisection interval was enlarged
  after the recorded run; these families are covered by `check_note.py`, run A's `geo_check.py` and
  `r2_numeric.py`), and (b) the last two lines of the recorded output of `reflect_check.py` (the reference value
  at two phases), which were printed by a call that is not part of the program. The README of the package
  describes both.

## Independent verification runs
The results were first obtained in a written version containing Theorems A–C with the proofs of the note. Two
independent verification runs, both AI-assisted, followed (2026-10-15), each with its own programs; the second wrote
its programs before reading those of the first write-up.

| Item | Run A | Run B |
|---|---|---|
| Statement, rows, definitions against the source | CONFIRMED (tex v11 and journal Tables 3–4) | CONFIRMED (record and transcription; arXiv/journal metadata by API) |
| Parametrisation, chords, closure (Lemmas 2.1–2.2) | CONFIRMED (re-derived; reflection law checked numerically) | CONFIRMED (λ from the reflection-law construction equals the Jacobi closure value) |
| m·m, parities, poles and feet (Lemmas 2.3–2.4, 4.1–4.3) | CONFIRMED | CONFIRMED (numerically at random complex points; simple, not double, poles of A_M) |
| Lemma 4.4 (residue class mod 4) | CONFIRMED | CONFIRMED (also brute force for N ≤ 200) |
| Lemmas 4.5–4.6 and Theorem A | CONFIRMED | CONFIRMED |
| Odd-N remark of the first write-up | gap (one foot at a pole of m at u0); fix required | same observation; fix required |
| Proposition 3.2, Lemma 3.3 | CONFIRMED | CONFIRMED |
| Lemma 5.1 (residues) | CONFIRMED_WITH_FIXES (sign factor ±1, not ±i) | CONFIRMED_WITH_FIXES (same) |
| Theorem B | CONFIRMED in the precise form; domain statement required | CONFIRMED (residue quotients to 25 digits) |
| Theorem C | CONFIRMED in the precise form; literature fix required | CONFIRMED |
| Novelty | Kriebel's note (doi:10.5281/zenodo.23331331) contains the M-dependence of k_{304}; not cited by the write-up | no prior proof of the three rows found |

No run found a wrong theorem or lemma, or a gap in the proofs of Theorems A–C.

**Corrections required by runs A and B**, all applied in the note:
1. (A1) Kriebel's note (doi:10.5281/zenodo.23331331) is cited as the closest prior source for k_{304}; it was read in
   full; the novelty of Theorem C is restricted to the constancy of A'/A'_O for N ≡ 0 (mod 4); the statement that
   the only arbitrary-M pedal-area result was the author's k_{303,a} note was replaced (Section 1.3, Remark 5.3,
   Scope and priority).
2. (A2) Domains in Theorems B and C: the ratio is constant for κ(M) ≠ 0 (resp. κ'(M) ≠ 0); the pedal area vanishes
   identically on the exceptional set; it can be a real circle (exact N = 8 star for k_{304}; numerical N = 10,
   τ = 3 for k_{204}); winding number 1: none for k_{304} (Kriebel), only tested for k_{204}.
3. (A3, B3) Lemma 5.1: residues r_A on iK' + G' and −r_A on 3iK' + G' (sign ±1).
4. (A4, B1) One DOI for the k_{203,b} note: the concept DOI 10.5281/zenodo.23085903, labelled as such.
5. (A5) k_{201} and k_{205}: credited by the source to private communications of A. Akopyan.
6. (A6, B2) The odd-N remark of the first write-up was removed; Remark 4.7 only says that the argument does not
   apply for odd N and what the computations show.
7. (A7) Minor: the explicit formula for c1 (Lemma 3.1); M·M for complex M; the check of the journal Tables 3–4
   recorded; the theorem number "Tabachnikov, Thm 4.4" dropped (the book is cited only for the classical fact, and
   the reflection law is verified directly in Lemma 2.2).

When the note was written, the positivity lemma (Lemma 3.4) and its consequences κ0 > 0, κ'0 > 0, C0 > 0, the
formula for c2 for complex normals in Lemma 3.1, the direct verification of the reflection law in Lemma 2.2, the
value of C0 for N = 4 from the rhombus, the description of 𝒫_A and 𝒫_iso as cosets in Lemma 4.4 and the program
`check_note.py` were added. Run 2 examined them.

### Verification run 2 (the third run, on the text of the note)
A third independent verification run, also AI-assisted (2026-10-15), examined the text of the note line by line. It wrote its own programs before reading or re-running the other programs of the package.

| Item | Run 2 |
|---|---|
| Statement fidelity | CONFIRMED: the rows and definitions re-read in the tex source of v11 and in Tables 3–4 of the journal's file; Stachel's Theorem 2 (arXiv v2) matches Section 2; Kriebel's note read in full (file checksum equal to the Zenodo record), every statement about it accurate; the statement of the k_{203,b} note matches. Wording fix: the abstract and Section 1.1 suggested that only three rows of the pedal tables concern an arbitrary point |
| Lemma 2.2(a), the algebraic check of the reflection law | CORRECT (all identities verified symbolically); the definition of the law of reflection had to exclude the reversal of a chord |
| Lemma 3.1, the formula for c2 (complex vectors) | CORRECT (identity verified for N = 3, 4, 5 with generic symbolic vectors) |
| Lemma 3.4 (positivity) and C0, κ0, κ'0 > 0 | CORRECT (also at every member of 80 numerical families and exactly at two members of each Table 2 family) |
| N = 4: C2 = 0, C0 = 8a^4b^4/(a^2+b^2)^2 from the rhombus | CORRECT (symbolic in a, b, M, with the feet computed from the vertices) |
| Lemma 4.4 in coset form | CORRECT (brute force over all 24338 pairs (N, τ) with N ≤ 400) |
| Other lemmas, Theorems A–C, cited results (DLMF §§22.4, 22.6, 22.8, 22.16, Table 22.4.3) | CORRECT; one side remark wrong: after the proof of Theorem C, A'A'_M was said to be constant for odd N, which holds only for M = O |
| Own programs | `r2_algebra.py` 49, `r2_numeric.py` 51, `r2_exact.py` 31 checks, all pass |
| Re-run of the package | results reproduced; two documentation defects (part A of `vb_geom.py`, two recorded lines of `reflect_check.py`) |
| Novelty and credit | no prior proof of k_{203,a} for arbitrary M or of k_{204} found; Kriebel credited correctly; the attribution of k_{202} to Bialy–Tabachnikov had to follow the source |
| References, house style, rendering | DOIs verified (Crossref, DataCite); all pages inspected |

**Corrections required by run 2**, all applied:
1. The abstract and Section 1.1 say that the note treats three rows on pedal areas for an arbitrary point (the
   tables have further such rows, among them k_{303,a}); the abstract says that the ratios are proved wherever
   defined.
2. After the proof of Theorem C: for N ≡ 2 (mod 4) A'A'_M is constant for every M [Fer26b]; for odd N only A'A'_O is
   constant [Fer26b], and A'A'_M is not constant for the other points tested (run 2: relative spread ≥ 6·10^-6).
3. The law of reflection is defined by "mirror images of each other in the normal", which excludes the reversal of a
   chord used implicitly in Lemma 2.2(d).
4. The source's credit of k_{202,a}, k_{202,b} to Bialy–Tabachnikov is stated as a credit of the source; three
   references that were cited only as "not read" were removed.
5. The README of the package describes the re-run of part A of `vb_geom.py` and the recorded output of
   `reflect_check.py` correctly.
6. The Verification and Scope paragraphs, the list of sources read and the caption of Table 2 record run 2 and its
   searches.
7. Two numbers of Section 6: 3.0378… (not 3.0379…) and 3·10^-46 (not 2·10^-46).
8. Notation: φ in Lemma 2.4, Ψ in Lemma 4.5, ρ' in Lemma 4.6 and Γ in (4) and Lemma 5.2 (the symbols F, D, σ had
   two or three meanings).
9. This report, the README and the deposit description were brought in line with the revised note.

No run found a gap in the proofs of Theorems A–C.

## Relation to the literature, novelty and scope
- **Searches (15 October 2026)** by the first write-up, the runs A and B, and run 2: arXiv API (pedal with billiard or
  Poncelet, invariants of the elliptic billiard, N-periodic billiards, the proposers and Stachel), OpenAlex (19 works
  citing the source; pedal and elliptic-billiard searches), Semantic Scholar (31 citations; rate-limited for one
  run), zbMATH Open, Crossref, the Zenodo search for elliptic-billiard deposits (repeated when the note was written
  and by run 2; the deposits of 15 October 2026 treat other rows), five web searches.
  - No printed or deposited proof of k_{203,a} for arbitrary M or of k_{204} was found.
  - For k_{304}: Kriebel's note proves the M-dependence for every even N (see above).
  - Related: the author's notes on k_{203,b} (M = O) and k_{303,a}, k_{303,b} (outer polygon, the complementary
    class); Kriebel's notes on k_{606} (foci) and k_{403,a}; Bialy–Tabachnikov on k_{202} (credited by the source).
- **What was read.** The source in the tex of arXiv v11 (preliminaries, tables of pedal polygons) and Tables 3–4 of
  the journal version; Stachel, Eur. J. Math. 8 (2022), Theorem 2 in arXiv v2 (the canonical parametrisation);
  Kriebel's note doi:10.5281/zenodo.23331331 in full (version 1.0; the file's md5 agrees with the Zenodo record);
  the abstracts of the other Zenodo notes cited and the statement of the k_{203,b} note; DLMF §§22.4, 22.6, 22.8,
  22.16 and Table 22.4.3. Run 2 read these sources again in the same versions. The text of Bialy–Tabachnikov and
  Tabachnikov's book were not read for the note; their bibliographic data were checked with Crossref (Zenodo DOIs
  with DataCite).
- **Caveats.** A search that finds nothing is not a proof of novelty. The positivity of κ2 for winding number 1
  (Theorem B) is not proved. Part A of the recorded output of `vb_geom.py` cannot be reproduced with the program as
  included (see Computations).
- **Scope.** The three 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/