# Verification of focal outer antipedal equality and a six periodic zero

The manuscript proves a precisely scoped theorem analytically and
supplies an exact finite certificate for its zero-area example. For
effective even elliptic-caustic periods, classical antipodal symmetry
exchanges the two foci. Boundary-tangent intersections and the actual
antipedal line equations commute with the resulting half-turn. This
identifies the two ordered polygons and their signed areas. It does
not imply denominator nonvanishing.

The six-periodic construction is an actual convex billiard, not six
arbitrary ellipse samples or an odd orbit traversed twice. With minor
semiaxis one and a>1, its vertices are

    (a,0), (a^2/(a+1),sqrt(2a+1)/(a+1)),
    (-a^2/(a+1),sqrt(2a+1)/(a+1)),
    (-a,0), (-a^2/(a+1),-sqrt(2a+1)/(a+1)),
    (a^2/(a+1),-sqrt(2a+1)/(a+1)).

The proof checks ellipse incidence, the directed unit-velocity
reflection law, and tangency to one confocal elliptic caustic with
lambda=a^2/(a+1)^2. Its six distinct impact vertices establish least
period six. Consecutive boundary tangents give the actual outer polygon.
Explicit consecutive antipedal intersections and their nonzero
determinants yield the signed area formula

    4a(a+1)(2+2a-a^2)/(2a+1)^(3/2).

At a=1+sqrt(3) and lambda=4-2sqrt(3), both focal areas are exactly
zero. Every antipedal intersection remains finite, every edge is
nonzero, and the vertices are not all collinear. The quotient there
is 0/0, not a defined real number. This example neither gives unequal
defined quotients nor shows zero area at every phase of the fixed
Poncelet family.

The accompanying standard-library check_axis_hexagon_zero.py was
actually executed successfully. Its exact field is
Q[c]/(c^4-6c^2-3), with the positive root isolated in (5/2,13/5).
Rational interval refinement certifies strict signs; floating-point
tolerances do not enter the decision. The checker reconstructs the
impact vertices, reflections, caustic tangencies, outer tangents,
antipedal equations and both zero shoelace areas. It additionally
checks distinctness, convexity, finite intersections, nonzero edges
and noncollinearity. The retained originating JSON and execution
receipt record PASS_EXACT_PRIMITIVE_SIX_PERIODIC_ZERO. Portable replay
from a fresh ZIP extraction is recorded separately by the package
builder after actual execution; it is not assumed by this report.

The source's signed-area and supporting-line conventions are explicit.
The equality includes admitted coprime stars and repetitions of
even-primitive orbits. The separate circle case has coincident foci.
Odd-primitive padded lists and hyperbolic or degenerate caustics are
not silently added. Because the retained source does not explicitly
fix its denominator and effective-period conventions, the broader
original_problem_resolved flag remains false despite the complete
theorem stated in this manuscript.

Classical symmetry and the prior zero-area observation for the original
billiard antipedal at a/b=2 are credited. The present calculation concerns
the outer polygon instead. This self-audited, AI-assisted preprint is
unrefereed; no independent human review, formal proof-assistant
verification, exhaustive novelty search or absolute priority is claimed.
Typesetting and every-page visual inspection are separate production
checks bound by actual release receipts, not mathematical evidence
or independent peer review.