A Pentagonal Counterexample to a Poncelet Squared-Side Criterion

We construct an explicit fixed circle and strictly nested noncircular ellipse admitting a Poncelet family of primitive convex pentagons with constant sum of squared side lengths, even though the circle center is neither the ellipse center nor either focus. If t is the unique root in (1/10,1/5) of 25t^3-50t^2-40t+8=0, the invariant value is 175/18-(625/48)t. All tangencies occur on the actual edge segments. This gives an exact counterexample to the necessity direction of the fixed-family center-or-focus criterion proposed in Murad 2026, Section 6, Conjecture 1, without contradicting its proved triangle results. The construction starts from a bicentric circle pair. For the polygon obtained by unit inversion of the bicentric vertices about the inner center, we express the squared-side sum as a two-step normal trace. Jacobi's bicentric parametrization and the published MI-II cyclic identity of Khare, Lakshminarayan and Sukhatme show that this trace is constant. A nonsingular projectivity agrees with vertex inversion on the outer circle and carries the complete nested Poncelet configuration to the required circle/ellipse pair. An isolated cubic root then supplies a strictly convex primitive five-cycle, and Poncelet closure supplies the whole family. The investigation originated with frozen record AMR-050-0053, invariant k811. That source uses incompatible descriptions of its dual construction. We prove the squared-side invariance of two precise interpretations separately, crediting the published bicentric cosine theorem and MI-II identity, but do not silently repair or claim unqualified closure of k811. Two standard-library exact checkers certify the displayed algebraic pentagon and distinguish the reciprocal constructions on exact fixtures; the all-family theorem rests on the analytic proof, not finite sampling. This English preprint is AI-assisted, originating-researcher self-audited, and unrefereed. Novelty remains undetermined; no independent human review, proof-assistant verification, guaranteed indexing, or absolute-priority claim is made.