# Verification report: four-periodic counterexamples

This report records the analytic argument, exact executed checks and final
manuscript verification. It is not a claim of peer review or formal
proof-assistant verification.

## Literal scope and analytic argument

The two universal assertions are refuted at primitive period four:

- AMR-050-0001 / raw ID 5100001: constancy of the angle-area quantity
  (A'/A) product_i sin(theta_i/2) for N congruent to 0 modulo 4, source k107.
- AMR-050-0005 / raw ID 5100005: constancy of A'A'' for even N,
  old k111 in arXiv:2004.12497v11. This old row is absent from the
  final journal table; its renumbered k111 is not the target.

The angles belong to the original billiard polygon, not the outer polygon.
All areas follow the source's ordered signed shoelace convention.

Put M=diag(a^2,b^2), s=a^2+b^2, D=a^2 b^2, with a>b>0.
For positively oriented orthonormal vectors n,m, set
h^2=n^T M n, k^2=m^T M m, d=n^T M m, P=M n/h, Q=M m/k.
Then h^2+k^2=s and h^2 k^2-d^2=D. The vertices P,Q,-P,-Q
give a continuous family of convex primitive four-periodic billiards
with fixed positive caustic matrix C=M-(D/s)I. Its semiaxes are
a^2/sqrt(s), b^2/sqrt(s), and it is confocal with the outer ellipse.

The explicit support and velocity calculations establish reflection,
finite tangent intersections and strictly interior caustic contacts.
The positive determinant det(P,Q)=D/(hk) gives four distinct convexly
ordered vertices, not a repeated two-periodic path. The area and
original-angle formulas are

    A = 2D/(hk), A' = 4hk, A'' = 4hk D/s^2,
    product_i sin(theta_i/2) = h^2 k^2/s^2.

For d(t)=(b^2-a^2) sin(t) cos(t), it follows that

    A'A'' = 16D(D+d(t)^2)/s^2,
    (A'/A) product_i sin(theta_i/2) = 2(D+d(t)^2)^2/(D s^2).

The attained ranges on this displayed connected family are

    [16D^2/s^2, 4D],
    [2D/s^2, s^2/(8D)].

Both have distinct endpoints for a>b>0. No classification of other periods
is inferred. Both derived areas are positive. Reversal, cyclic relabeling
and taking absolute areas do not remove these contradictions.

For a=4,b=3, the same caustic has semiaxes 16/5,9/5:

| Quantity | Axis diamond | Axis-aligned rectangle |
| --- | ---: | ---: |
| A | 24 | 576/25 |
| A' | 48 | 50 |
| A'' | 6912/625 | 288/25 |
| Original half-angle sine product | 144/625 | 1/4 |
| A'A'' | 331776/625 | 576 |
| Angle-area quantity | 288/625 | 625/1152 |

The ordered 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).

## Actual execution and manuscript verification

Both standalone Python 3 checkers were actually executed successfully
using standard-library Fraction arithmetic. The retained outputs are
reproducibility/k107-root-rerun.json and
reproducibility/old-k111-root-rerun.json. Each checker verifies eight
reflection equations and eight strictly interior caustic tangencies
across the two rational orbits, with exact boundary, ordering and area
checks. check_k107.py computes interior cosines from outgoing and
negative incoming unit vectors, multiplies the four half-angle sine
squares, and takes the positive exact square root only after multiplication.
No floating-point angle approximation or third-party library is used.
The analytic docstrings and accompanying proofs establish the connected
family and sharp ranges; finite fixture execution does not replace them.

The final English main.tex compiled successfully with the native document
editor. Its separately exported PDF has four pages, all rendered and
actually visually inspected with no located layout defects. The retained
source SHA-256 is
707fc32991840548033dca7cd2b3c0dcc1adc8927bf710103bb48d6afdc2aace;
the PDF SHA-256 is
df4f6b6ad1afd46ae274db52eb75bee63a722322642c72e6f829b3785f708d46.
The native compilation and visual inspection are manuscript checks, not
independent mathematical review. No warning-free export claim is made.

To reproduce the exact checks after extracting arxiv_source.zip, run:

```sh
python3 -I -B reproducibility/check_k107.py
python3 -I -B reproducibility/check_four_periodic.py
```

Each prints a JSON result with status PASS. Compare the parsed JSON to
the corresponding retained root-rerun file; key ordering and whitespace
are immaterial. The archive includes both complete proofs, frozen source
records and authored provenance audits, but no third-party source PDFs.
Frozen source records preserve historical upstream metadata verbatim;
their provisional open-status text is superseded by this version-specific
source review, not treated as a current-openness certificate. Original
research checkers and outputs are preserved byte-for-byte, including
historical internal labels that confer no publication authority.

Garcia and Reznik's Proposition 4.9 identities are prior mathematics.
Bounded source checks do not certify novelty or priority. This
AI-assisted, unrefereed manuscript claims neither independent human
review nor formal proof-assistant certification.