# Verification of focal inverted pedal area equality and common zeros

The result is supported by a displayed analytic proof and exact
computations. Classical effective-even-period antipodal symmetry exchanges
the two actual foci. Inversion and perpendicular projection commute with
the resulting half-turn, which identifies the two ordered pedal polygons
and their signed areas, including at zero.

complete_proof.md proves a genuine ten-periodic family of winding three
on ellipses with minor semiaxis one and major semiaxis a in [5/2,3].
A unique root of an explicit quartic determines its vertices. The proof
establishes ten distinct vertices, every reflection, tangency to one
nondegenerate confocal elliptic caustic and continuity of the derived
signed area. These geometric assertions are analytic, not conclusions
from small floating-point residuals.

The originating researcher actually reran
certify_n10_axis_endpoint_areas.py successfully on 1 October 2026.
n10-axis-endpoint-root-rerun-2.json records the result
EXACT_RATIONAL_INTERVAL_OPPOSITE_SIGNS_N10_TAU3_AXIS_FAMILY.
The checker uses 180 exact rational root bisections and enclosing interval
arithmetic with outward dyadic rounding at 224 bits. All required
divisions exclude zero; endpoint vertices are distinct and the inversion
and projection denominators have strictly positive lower bounds.
Its certified intervals imply

    6439/10000000 < A(5/2) < 161/250000,
    -2813/1000000 < A(3) < -703/250000.

The sign decisions use Fraction arithmetic only. Decimal displays do not
enter the certificate. The opposite signs and the analytic continuity
proof give an interior parameter at which both focal signed areas vanish.
No exact coordinate for that parameter or uniqueness is asserted.

check_inverted_pedal_pair.py was also actually rerun successfully.
receipt-root-rerun.json records
PASS_EXACT_CONSTRUCTION_REGRESSION_AND_SCOPE_CONTROLS: 400 exact
four-periodic source-family cases, including 399 noncircular cases and
one circle boundary case, and 63 arbitrary centrally paired polygon
controls. Arbitrary controls are not presented as source billiards.
Finite positive samples do not establish general nonvanishing.

four_periodic_area.md instead proves complete four-periodic nonvanishing
by a covering support-coordinate parameterization and an explicit
rational formula with positive factors. The retained symbolic computation
was actually rerun and found an identically zero difference. That
historical computation used SymPy; it is separate from the portable
standard-library checkers and their runtime requirements.

The scientific conclusion is equality for the stated effective even
elliptic-caustic scope, a source-valid common-zero existence theorem,
and the resulting quotient-domain qualification. Ratio one wherever
the common area is nonzero is not refuted. This self-audited, AI-assisted
preprint is unrefereed; no independent human review, formal
proof-assistant verification, exhaustive novelty search or absolute
priority is claimed.

The final English source compiled successfully with the desktop editor's
native compiler. The actual PDF was exported with the already installed
Tectonic using cached resources only. Its seven pages were all rendered
and visually inspected; the final export log contains no undefined
references, overfull boxes, underfull boxes or compiler errors. This
artifact review is separate from the mathematical argument.