# Verification of the focal antipedal centroid theorem

The general result is accepted by the originating researcher after a
detailed self-audit. The mathematical argument is displayed in the
manuscript: fixed outer conic, finite consecutive antipedal intersections,
effective even-period antipodal pairing, opposite-edge formula, affine
normal reduction and compact-curve trace cancellation. Infinitely many
actual ordered pairs in the prescribed periodic family justify identity
extension; finite samples do not.

Root read all three accompanying standard-library checkers and actually
ran each successfully. Their exact stdout and hashes are supplied in
the reproducibility directory.

check_even_outer_antipedal.py returned PASS_EXACT_CONTROLS_AND_ALGEBRA.
It checks 24 generic opposite-edge identities on exact rational controls,
four discriminant/Vieta cases, five affine normal cases and the existing
four-period rectangle limit. Nonperiodic controls reject the stronger
claim that arbitrary central symmetry alone forces centroid O. They
are not counterexamples to the source's periodic-family assertion.

check_source_family.py returned PASS_EXACT_SOURCE_FAMILY_REGRESSION.
Five rational boundary starts in both tangent directions produce ten
primitive six-periodic billiards with a=5,b=3 and lambda=225/64. Reflection,
positive flight, confocal tangency, minimal closure, velocity return,
outer vertices and both-focus means are checked in exact quadratic fields.
The means are (-4/3,0) and (4/3,0). The exact eight-periodic example with
a squared=40,b squared=15,lambda=24/7 in Q(sqrt7) has means
(-35/18,0) and (35/18,0), with circle trace 4/5. No floating point is used.

symbolic_pair_certificate.py returned
PASS_GENERAL_FRACTION_POLYNOMIAL_IDENTITIES. Sparse polynomial coefficient
arithmetic over Q verifies nine general identities: both antipedal line
equations, the opposite-edge pair, its tangent component, the outer
focus-axis eigenvalue, the substituted billiard pair and the affine-normal
denominator and two components. These identities are general algebraic
certificates, not geometric sampling. The analytic proof separately
establishes domain inequalities, periodic pairing and global trace constancy.

The standalone English source compiled successfully in the built-in LaTeX
editor. A PDF was exported from that same source using the existing cached
Tectonic compiler, with no new installation. All six pages were rendered
and visually inspected; no clipping, malformed formula, missing reference
or layout defect was found. The final log has no warnings, overfull or
underfull boxes, or undefined references.

Source scope, argument correctness, current-literature search and actual
publication are separate facts. This self-audited preprint was prepared
with AI assistance; no independent human review, formal proof-assistant
verification, absolute priority or guaranteed search-engine indexing is
claimed. It proves k407's vertex-only constancy under its stated effective
period convention, not a general signed-area centroid or every source row.