# Verification report for the antipedal area ratio

This report states the mathematical scope and the exact reproducibility checks accompanying *Nonconstant Antipedal Area Ratios in Four-Periodic Elliptic Billiards*. It is not a peer-review or proof-assistant certificate.

## Literal source scope

The frozen target is AMR-050-0019, raw ID 5100019, from ulamai/UnsolvedMath v1.6.0. Section 3.5 and Table 5, printed pages 347–348, of Reznik, Garcia and Koiller's final 2021 article define the original-orbit antipedal and list k402 as A'/A*_M, with N congruent to zero modulo four and M arbitrary. The unprimed denominator is not the antipedal of the outer polygon. The same row is in arXiv:2004.12497v11. A fixed center M=O and primitive period N=4 lie in the printed domain. This note refutes that universal constancy assertion; it makes no claim about other rows or a differently restricted conjecture.

## Analytic family and antipedal formula

Fix a>b>0 and write B=diag(a^2,b^2), s=a^2+b^2 and D=a^2 b^2. For a positively oriented orthonormal pair n(t),m(t), set h^2=n^T B n, k^2=m^T B m, d=n^T B m, P=B n/h and Q=B m/k. In the rotated coordinates P=(h,d/h), Q=(d/k,k), h^2+k^2=s and h^2 k^2-d^2=D>0.

The polygon P,Q,-P,-Q is a continuous, strictly convex primitive four-periodic billiard family. Direct reflected-direction calculations prove the billiard law. Its common confocal elliptic caustic is B-(D/s)I, with semiaxes a^2/sqrt(s),b^2/sqrt(s). Support equations and the parameters k^2/s and h^2/s place the caustic contacts strictly inside the actual segments. Consecutive outer tangents form a rectangle of area A'=4hk.

At O the antipedal lines are P dot X=|P|^2, Q dot X=|Q|^2 and their opposite lines. The linear map X to (P dot X,Q dot X) has positive determinant Delta=det(P,Q)=D/(hk). It sends their consecutive intersections to the corners of a counterclockwise rectangle with area 4|P|^2|Q|^2. The antipedal is therefore finite, strictly convex, counterclockwise and has positive area

    A*_O=4|P|^2|Q|^2/Delta.

Using |P|^2=s-D/h^2 and |Q|^2=s-D/k^2 gives

    R(t)=A'/A*_O=D(D+d(t)^2)/(D^2+s^2 d(t)^2).

Its derivative with respect to z=d(t)^2 is D^2(D-s^2)/(D^2+s^2 z)^2<0. As t ranges from zero to pi/4, z ranges continuously over [0,(s^2-4D)/4]. Thus the sharp attained range on this explicit family is

    D s^2/(s^2-2D)^2 <= R <= 1.

The endpoints differ for a>b because (s^2-2D)^2-D s^2=(s^2-D)(s^2-4D)>0. The circle boundary has the constant value one. No classification of all higher-period orbits is used.

## Exact witnesses

For a=4,b=3, both witnesses have the caustic with semiaxes 16/5 and 9/5 and lambda=144/25.

| Quantity | Axis diamond | Axis-aligned rectangle |
| --- | ---: | ---: |
| Outer tangent area A' | 48 | 50 |
| Original-orbit antipedal area at O | 48 | 113569/1800 |
| Ratio A'/A*_O | 1 | 90000/113569 |

The orbit vertices are respectively (4,0),(0,3),(-4,0),(0,-3), and (16/5,9/5),(-16/5,9/5),(-16/5,-9/5),(16/5,-9/5). The positive ratio difference is 23569/113569. Both originals and derived polygons are convex and positively oriented. Reversing traversal reverses both areas, leaving the ratio unchanged; absolute areas give the same two ratios. The center is one fixed point, not chosen separately for each orbit. Nonzero defining determinants and areas also make the discrepancy stable under sufficiently small perturbations of that fixed point.

## Executed rational checks and reproduction

The two originating checkers were actually executed with exact standard-library Fraction arithmetic. The root checker reconstructs defining line intersections and verifies boundary membership, eight reflection equations, eight strictly interior caustic contacts, convex order, exact areas, reversal and a further fixed point (1/10,1/10). Its retained JSON has status PASS_EXACT_WITNESSES. The separately authored checker has status PASS and additionally checks cyclic shifts and 35 rational support-matrix geometries, including a circular boundary control. Those additional finite cases test the implementation; the analytic proof establishes the continuous-family result.

After extracting arxiv_source.zip, reproduce the two originating results with:

```sh
python3 -I -B reproducibility/check_k402.py
python3 -I -B reproducibility/check_antipedal.py
```

Compare the parsed JSON output with reproducibility/root-exact-result.json and reproducibility/independent-exact-result.json, respectively. JSON spacing and key order are immaterial. Neither checker requires repository imports or third-party packages. Do not use the root checker's optional --report argument for this comparison; it is unnecessary and would create a new file.

The package builder executes both checkers again from a fresh isolated ZIP extraction and requires exact parsed equality with those originating results before installing the package. Its actual portable receipt records the script, archive and output hashes. It also requires successful final native compilation, the actual exported PDF, and root-recorded inspection of every rendered page, with exact source/PDF/log/image bindings. These manuscript and packaging checks are not mathematical peer review. No warning-free export is claimed.

## Credit and limitations

The standard four-periodic family, caustic and outer-rectangle geometry are prior mathematics, including Garcia and Reznik's Propositions 4.7–4.9 and Appendix B. The author's earlier note DOI 10.5281/zenodo.23075934 uses that family for different source expressions. Here the distinct original-orbit antipedal ratio is the target. The bounded primary-source audit does not certify absolute originality, author intent, or absence of an unindexed correction.

The archive includes authored proofs, frozen source metadata and provenance audits, but no third-party source PDFs. Historical upstream open-status text in a frozen source record is not a current-openness certificate. Retained research records describe their actual time and scope; they confer no external publication authority. This AI-assisted preprint is unrefereed and has not undergone independent human review or proof-assistant verification.