AMR-050-0023: portable exact controls

These finite controls supplement the analytic proof; they are not a general
proof by sampling, a formal proof certificate, or an independent review.

Requirements: Python 3 and, for check_antipedal_centroid.py only, SymPy 1.13.1.
Install dependencies in an environment you control, for example:
  python3 -m pip install -r reproducibility/requirements.txt
Then, from the unpacked archive root:
  python3 reproducibility/check_antipedal_centroid.py
  python3 -I -B reproducibility/check_odd_repeat_centroid.py
The second command uses only Python's standard library.

The algebra entry point differs from the accepted research script only in its
dependency bootstrap: the repository lookup has been removed. Its symbolic
pair derivation and exact algebraic fixture calculations are byte-identical.
The geometry entry point is the unchanged accepted Fraction implementation.
Both parsed outputs were compared with the originating researcher's accepted
execution outputs. The local packaging harness supplies an already-installed
SymPy directory; it does not install a dependency or make a network request.
That local harness path is not required by either distributed entry point.

The algebra check verifies the paired-chord polynomial identity, support gap,
diagonal reduction, six genuine even controls at the center and both foci,
Euclidean equivariance, and an odd triangle repeated to a six-entry list.
The geometry check independently verifies a second repeated-triangle control
using rational arithmetic in a rescaled physical metric.

Scope: nondegenerate elliptic caustics with even least billiard period, admitted
coprime stars, and repetitions of even primitive cycles. The circle is separate.
The stronger unrestricted even-list interpretation is refuted by the explicit
twice-traversed triangles. No hyperbolic-caustic or unqualified whole-source-row
closure, absolute novelty, or independent/formal verification is claimed.

Source manifests retain bibliographic URLs and research-file hashes only.
Third-party papers, screenshots, raw web pages, and credentials are not included.
