# Verification of the focal inversion correction

The accepted result is a complete counterexample to a printed
constant, with a full proof of the replacement constant. On
x^2/25+y^2/16=1, the primitive axis diamond and its consecutive
boundary-tangent rectangle have focal-inverse signed areas 1/10
and 1/20 about (3,0). Thus the outer-over-original ratio is 1/2,
not the source's value 2.

The exact axis checker verifies boundary membership, all four
reflection equations, strict confocal caustic, all four tangencies,
the outer tangent intersections, nonzero inversion denominators and
both signed areas. The support-frame checker verifies 1000 rational
instances, of which 999 are noncircular; the one circle case is an
additional boundary regression rather than part of the source
theorem. It checks the explicit inverse formulas, the reflection
law, one confocal caustic, the signed areas and the outer squared
distance product 16(a^2 b^2)^2.

The full proof is algebraic: central symmetry and reflection yield
a rectangular tangent frame, the matrix b^2 I+F F^T gives the
support points, direct vertex inversion and shoelace give original
inverse area 2hk/(a^2 b^2), and a four-term cyclic identity gives
outer inverse area hk/(a^2 b^2). Their quotient is exactly 1/2
throughout the primitive four-periodic elliptic-caustic family.

The original PDF table and definitions were inspected to rule out
an OCR reversal or confusion with the outer polygon of an inverse
curve. The known original area-product-4 formula is credited, not
counted as another result. The complete proof and literal-source
coverage have been accepted by the originating researcher after
detailed self-audit. No independent human review, formalization or
absolute-priority certification is claimed.