# Verification report — AMR-050-SYM7 (Reznik–Garcia–Koiller rows k406,a, k607, k609, k610, k905, k906, k907,b: central symmetry and seven area identities for even N-periodics in the elliptic billiard)

Verification date: 2026-10-15.

**Verdict.** The note proves all seven rows **in the equality form**, with the hypotheses and the exceptional sets
stated.
- **Settled.** For every billiard trajectory in an ellipse with a confocal *elliptic* caustic and *even least
  period* (convex or star, either orientation, every member of the family): the two signed areas of each of the
  rows k607, k609, k610, k905, k906, k907,b are equal (Theorems 2–7), and for k406,a the vertex centroid of the
  antipedal polygon R* of the outer polygon with respect to O is O and its area centroid is O wherever S(R*) ≠ 0
  (Theorem 1). Hence each ratio of the source equals 1 wherever its denominator is nonzero. The restriction
  N ≡ 0 (mod 4) of k607 is not needed. Rows k607, k609, k905, k906, k907,b are defined at every member; k610 exactly
  at the members with no vertex on a directrix x = ±a″²/c of the caustic, and such members exist iff a″² ≤ ac
  (Theorem 4, Proposition 6.1).
- **Where the ratios are undefined (numerical, Section 6).** k607: the common area vanishes along whole families,
  N = 6 at b/a = 1/2 (exact zero at one member), N = 8 at 0.5394187523…, N = 10 at 0.5543486017…, N = 12 at
  0.5618074057…, N = 16 at 0.5688313760… (N = 8, 12, 16 inside the row's class N ≡ 0 mod 4); numerically identified
  algebraic equations for N = 8 and N = 10. k905: isolated zeros (e.g. N = 4, b/a = 1/2). k610: poles (degenerate
  members) and isolated zeros. k906: isolated zeros in star families. k406,a: isolated zeros of S(R*) (N = 12,
  b/a = 0.1). No zero found for k609 and k907,b. Nonvanishing elsewhere is not proved for any row.
- **k907,b.** The symbol A^†_{j,dual} is not defined in the source; the theorem holds for every reading that applies
  one and the same isometry-natural recipe at f1 and f2 (Theorem 7(c), Remark 5.1); three readings checked.
- **Not new mathematics.** Central symmetry of even periodic trajectories is classical (Stachel, J. Geom. 112 (2021)
  art. 40, Corollary 4.2(i)); the source marks its sibling rows k604,b, k605,b, k903,b, k904,b as proved by this
  symmetry. The note makes the one-line argument complete and checkable and records the exceptional sets.
- **Checked three times.** Three independent AI-assisted verification runs: two examined the results before the note
  was written, the third examined the final text and the package (2026-10-15). No run found a wrong lemma, theorem
  or proposition; all corrections they required were made.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Reznik, R. Garcia, J. Koiller, arXiv:2004.12497v11 (29 October 2020), "Eighty New Invariants
  of N-Periodics in the Elliptic Billiard"; 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 (not read; its table numbering may
  differ, so the rows are cited by their v11 codes).
  - Read in the tex source of v11 (by the first runs and again, from a fresh download, by the third run):
    preliminaries (signed areas, area centroid, J), antipedal polygons (Table with k406,a, l.27), pairs of pedal
    polygons (Table with k607, k609, k610, l.18–21, and the note after it, l.28, which names k604,a and k604,b),
    inversive objects (Table with k905, k906, k907,b, l.78–81; figure caption defining the polar and the dual polygons
    and O_p), symbol table (A_{j,dual} defined, A^†_{j,dual} not).
- **Corpus records.** ulamai/UnsolvedMath 1.6.0, AMR-050-0024 (k406,a), -0035 (k607), -0037 (k609), -0038 (k610),
  -0064 (k905), -0065 (k906), -0067 (k907,b); upstream status "open". Each asks to prove that the row's quantity,
  "equal to 1" (k406,a: the two centroids, "both equal O"), is constant along the family. The k406,a text is
  garbled; its meaning is fixed by the table row. The records say "a fixed confocal caustic"; the source's setting
  is the confocal ellipse pair.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Ratio = 1 at every member (literal) | true wherever the denominator is nonzero; the denominators vanish on the sets of Section 6 (whole families for k607, including N ≡ 0 mod 4) | Theorems 1–7, Section 6 |
| Equality of the two signed areas (and centroids at O where defined) | true at every member, for every even least period, convex and star, both orientations | Theorems 1–7 |
| "N-periodic" = least period N (source) | covered | Lemma A, Section 5 |
| "N-periodic" = closes after N reflections | true if the least period is even (repetitions of even orbits); false (computed) for repetitions of odd-period orbits, e.g. the 3-periodic listed 2 or 4 times | Section 7, check_note.py (ii) |
| Hyperbolic caustic | not treated; the symmetry is axial for N ≡ 0 mod 4 (ratio −1 found by run A) | Section 7 |
| Unsigned areas | equalities hold for absolute values too | Section 7 |
| Radius of the ‡-inversion (unspecified) | any common radius r; areas scale by r⁴ | Theorem 6(c) |
| A^†_{j,dual} (undefined symbol) | covered for every reading using one natural recipe at f1 and f2 (D_j, D̃_j, Inv(D_j; f_j), …) | Theorem 7(c), Remark 5.1 |

## Results in the paper
- **Lemma 3.1, Corollary 3.2.** Joachimsthal: a chord (X, v) touches the caustic iff |⟨AX, v⟩| = √(a²−a″²)/(ab), and
  ⟨AX, v⟩ is invariant under the billiard map; hence the reflection in the normal at any point of E swaps the two
  tangent lines from that point to the caustic, and neither is the normal.
- **Lemma A (central symmetry).** Even least period m = 2h ⇒ P_{k+h} = −P_k, and the outer and inner polygons are
  centrally symmetric with the same shift. Self-contained proof: invariant circle Γ of chords with the caustic on
  the left, T(Γ) = Γ (using Corollary 3.2), invariant measure from the invariant area form (proved by the generating
  function) and Joachimsthal's integral, T and the half-turn are rotations of Γ, least period ⇒ odd turning number
  ⇒ T^h = half-turn; clockwise trajectories by time reversal. Matches Stachel, Corollary 4.2(i) (arXiv v2 numbering;
  derived there from Theorem 3.6).
- **Lemma B (naturality), Lemma 4.2.** Constructions natural under the half-turn and cyclic relabelling map
  σR = R^[k] to equal signed areas at σ-related points; centroids at O for σ-symmetric polygons.
- **Theorems 1–7** as in the verdict; Theorem 6 also gives E′ (semi-axes a²/a″, b²/b″) and its foci (x-axis iff
  a″ > a²/√(a²+b²); circle for the 4-periodic caustic); Theorem 7 the circles of the polar and dual polygons.
- **Proposition 6.1.** The k610 degenerate members are the members whose polygon passes through a point where a
  directrix of the caustic meets E (at most two polygons, at most 2N parameters); such members exist iff
  a″² ≤ ac.
- **Tables.** Table 1 (the rows), Table 2 (k607 zero families), Table 3 (isolated zeros).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The exact program of the note** (`check_note.py`, mpmath at 50 digits, 16.2 s; 64 checks, TOTAL failures: 0):
  (i) Lemma A and Theorems 1–7 on 25 families (four aspect ratios × N = 4, …, 12; three star families with
  τ = 3, 3, 5; the 4-periodic listed twice and three times), five members, both orientations (250 polygons):
  largest symmetry defect 1.1e-45, largest relative difference of the two areas 1.1e-43; three readings of k907,b;
  siblings k604,b, k605,b, k903,b, k904,b; E′ and its foci; r⁴ scaling; polar/dual circles. (ii) Controls: odd
  N = 3, 5, 7, the 3-periodic listed twice and four times, non-antipodal partner points: every row fails at every
  member (relative deviations from 4.15e-5 to order 1; for the repeated 3-periodic 0.026 to 1.26, and the vertex
  centroid of R* at distance 0.946 to 1.45 from O). Part (ii) was revised by the third run (see below). (iii) Section 6: the k607 families (root-finding in b; areas below 1e-48 at 12
  members; exact N = 6 member in Q(√3, √5); numerical identifications for N = 8, 10; k403,b and k404 constancy),
  the zeros of Table 3, the k610 degenerate members of Proposition 6.1.
- **The program with which the results were first obtained** (`original/sym7_check.py`, 60 digits, 22 even
  families with nine members each and controls). Re-run on 2026-10-15: output identical apart from timing; the
  minima extraction identical.
- **Verification run A** (`verification_run_A/`): caustics by Cayley's criterion, trajectories by the reflection law,
  340 polygons (34 even families, both orientations, τ up to 5) — largest relative area difference 2.9e-42 (star
  families) and 2.4e-45 (convex families); controls and repetitions; exact sympy check of the N = 6 member; sign scans and bisection: k607
  families N = 8, 12 (24 members, |Ā*_j| ≤ 2.3e-57 at 60 digits), k905 and S(R*) zeros, k610 pole; hyperbolic
  caustics.
- **Verification run B** (`verification_run_B/`): two independent billiard maps, 46 family runs (N up to 16, τ up to
  5), 360-member scans with bisection of every sign change, aspect-ratio scans, wrong-input detection; k607 families
  N = 6, 8, 10, 12, 16 with integer relations for N = 8, 10; zeros of k905, k610, k906 (stars); poles of k610.
- **Third verification run** (`independent_run_2/`, about 40 s): caustic from the elliptic-integral formula for the
  rotation number, trajectories by the reflection law alone, E′ fitted to outer vertices. Lemma A and Theorems 1–7 on
  400 labelled polygons in 40 families (N ≤ 18, τ ≤ 7, both orientations, repetitions; largest relative difference
  2.9e-42); Lemma 3.1, Corollary 3.2, T*Ω = Ω (Jacobian, 1e-38), the invariant measure of Step 3 (vertices equally
  spaced in u, 3e-29); controls per member; Table 2 by root-finding at t = 1.1 (agreement to the 20 printed
  decimals; vanishing at 7 further members; polynomials; a sign scan with exactly one change for N = 6, 8, 10, 12,
  16 and none for N = 4); Table 3 (all 12 zeros to the 12 printed decimals); Proposition 6.1 (degenerate members,
  720-member scans, the boundary case a″² = ac); the N = 6 member in exact arithmetic (sympy); 110 checks, all
  passed. It also re-ran the package from the archive (`independent_run_2/outputs/package_rerun_summary.txt`).

## Independent verification runs
The results were first obtained with proofs of Lemma A, Lemma B and the seven applications and with the program in
`original/`. Three independent verification runs, all AI-assisted, followed (2026-10-15); each wrote its own
programs. Runs A and B examined the results before the note was written and re-ran the original program (output
reproduced); the third run examined the final text and the package.

| Item | Run A | Run B | Run 3 (final text) |
|---|---|---|---|
| Rows and definitions against the source (v11 tex) | CONFIRMED | CONFIRMED (read independently) | CONFIRMED (fresh download of v11; records compared) |
| Lemma A | CONFIRMED_WITH_FIXES (Step 2: reflected chord is the other tangent; clockwise case; "odd τ ⇔ least period" one direction only) | CONFIRMED_WITH_FIXES (Step 2 gap) | CONFIRMED, with the writing-stage additions (Lemma 3.1, Corollary 3.2, invariance of Ω); one sentence added on the coordinates of the phase space |
| Lemma B, centroid lemma | CONFIRMED (also with any shift k: repetitions of even orbits) | CONFIRMED (label-dependent construction breaks it: hypothesis needed) | CONFIRMED |
| Theorems 1–7 (equality form) | CONFIRMED; k610 degeneracy criterion found; E′-foci remark corrected | CONFIRMED; E′ equation and polar/dual circles verified | CONFIRMED; one sentence added to Theorem 6(b) (every 4-periodic family has the caustic of the rhombus) |
| Proposition 6.1, exact N = 6 member | (not yet written) | (not yet written) | CONFIRMED (one reference added to the proof); exact check repeated with sympy |
| Numerical scope statements of the first write-up | WRONG: k607 zero families in N ≡ 0 mod 4; k905 and S(R*) zeros; k610 poles | WRONG: k607 families N = 6–16; k905, k610, k906 zeros/poles | Tables 2, 3 and Section 6 reproduced; Section 7 numbers for the repeated 3-periodic WRONG (corrected) |
| Literature, novelty | no printed proof of the seven rows found; Stachel Theorem 3.6 (not 3.5); dating inference unsupported; source note names k604,a/b only | no printed proof found; cite arXiv:2011.06640v3 (video 03) and the DOI of Bialy–Tabachnikov | no printed proof found; the Garcia–Reznik survey (IMPA 2021) read: 3-periodics only |
| Package | — | — | re-run from the archive: reproduced, except run B's table output in the four N = 4 families (recorded before a program change; disclosed) |

No run found a false lemma, theorem or proposition.

**Corrections required by runs A and B**, all applied in the note: the exceptional sets (Section 6, Tables 2, 3); the claim that N ≡ 0 (mod 4) avoids the zero is dropped;
every sentence on vanishing or size corrected; Step 2 of Lemma A (Corollary 3.2) and the clockwise case; even least
period suffices, least period N ⇒ odd τ but not conversely; the E′-foci statement (x-axis iff a″ > a²/√(a²+b²),
equivalently τ/N < 1/4); Stachel's Theorem 3.6 (arXiv v2); no journal-dating inference; the source's note after the
pairs table names k604,a and k604,b; the k608 Zenodo quote dropped and paraphrased from the abstract; no numerical
"exactly 0"; the k907,b reading condition; citations of arXiv:2011.06640v3 and of Bialy–Tabachnikov with its DOI.

**Changes made when the note was written, after runs A and B** (examined by the third run): the caustic property
derived from Lemma 3.1 instead of quoted; the invariance of the area form proved; Proposition 6.1; the exact
computation in Q(√3, √5) in check_note.py (run A had made it with sympy).

**Corrections required by the third run**, all applied: Section 7, the numbers for the 3-periodic listed twice and
four times (the vertex centroid of R* lies at distance 0.95 to 1.45 from O, not 0.37, which was a normalised value;
the relative differences at the members range from 0.026 to 1.3, the earlier range 0.66–1.3 being the maxima per
row); check_note.py part (ii) now prints the actual distance and the per-member minimum and requires every row to
fail at every member (Section 8 now says 4·10⁻⁵ instead of 5·10⁻⁵ for the smallest deviation); one sentence each in
the proofs of Lemma A (coordinates on the phase space), Theorem 6(b) (uniqueness of the 4-periodic caustic) and
Proposition 6.1 (uniqueness of the polygon through a point); Section 6.1, "the proposers appear to have known" the
N = 6 zero (the video title does not name the point; the antipedal area with respect to O does not vanish there);
Section 6.3, the sentence on further k905 zeros rewritten, with the reason why a zero of a convex member needs a
focus outside the inner polygon; the survey and the searches recorded; the package README discloses the run B
table discrepancy and lists the programs re-run.

## Relation to the literature, novelty and scope
- **Searches (2026-10-15, first write-up and the three runs).** arXiv API (antipedal; inversive and pedal polygons
  in elliptic billiards; centrally symmetric billiards; works of Reznik, Garcia, Koiller, Stachel), OpenAlex (the 19
  works citing the source; full-text searches), Semantic Scholar (31 citers of the arXiv version), zbMATH Open,
  Crossref, the Zenodo API and DataCite (about twenty 2026 deposits on sibling rows, by A. Ferudun and A. Kriebel);
  one general web search (third run). No printed or preprint statement of any of the seven rows was found.
- **What was read.** The source (v11 tex, sections listed above); Stachel, arXiv:2105.03362v2 (Section 4, statement
  of Theorem 3.6); Bialy–Tabachnikov, arXiv:2001.08469v2 (proof of the theorem on k202, which uses the symmetry as a
  well known fact); Garcia–Reznik, arXiv:2011.06640v3 (abstract, video table: entry 03 is titled "N=6 zero-area
  antipedal @ a/b=2"); abstracts of the Zenodo deposits and the prior-observation section of the k608 note
  (doi:10.5281/zenodo.23091871); the survey R. A. Garcia, D. S. Reznik, "Discovering Poncelet Invariants in the
  Plane" (IMPA, 2021, Zbl 1511.14001) in its open-access edition, text searched by the third run: it treats
  3-periodics and contains none of the seven rows. Not read: the journal version of the source; the videos.
- **Related unrefereed notes.** doi:10.5281/zenodo.23079798 (k403,b; explains the k607 zero families for
  N ≡ 0 mod 4), doi:10.5281/zenodo.23149774 (A. Kriebel, k606), doi:10.5281/zenodo.23091871 (k608, same symmetry
  for the outer polygon, zero at a/b = 1 + √3).
- **Caveats.** The exceptional sets are numerical (except Proposition 6.1 and one exact member); the k607 families
  for N ≡ 0 (mod 4) vanish identically if the k403,b law holds (proved only in an unrefereed note), for N = 6, 10
  if k404 holds (not proved). The τ/N < 1/4 form of the E′-foci statement uses the standard monotonicity of the
  rotation number, not proved in the note. A negative search is not a proof of priority.
- **Scope.** Complete for the seven rows in the equality form under the stated hypotheses; no priority 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/