{ "schema_version": 1, "problem_number": "AMR-050-0053", "title": "A Pentagonal Counterexample to a Poncelet Squared-Side Criterion", "author": { "name": "Alper Ferudun", "affiliation": "Mercury Software GmbH", "url": "https://github.com/AlperTheKing" }, "abstract": "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.", "result_type": "COMPLETE_COUNTEREXAMPLE_TO_ADJACENT_CONJECTURE", "categories": [ "math.DS", "math.MG" ], "keywords": [ "Poncelet porism", "bicentric polygons", "squared side lengths", "convex pentagon", "central conic", "Jacobi elliptic functions", "cyclic identities", "projective geometry", "exact computation", "counterexample" ], "manuscript_version_date": "2026-10-02", "publication_date": "2026-10-02", "publication_date_kind": "first public online release", "version": "1.0", "date_modified": "2026-10-02", "presentation_revision_only": false, "doi_archived_file_version": "1.0", "status": "unrefereed preprint", "canonical_url": "https://eulersolve.org/papers/amr-050-0053/", "pdf_url": "https://eulersolve.org/papers/amr-050-0053/paper.pdf?v=0afb30180869", "doi": "10.5281/zenodo.23099540", "zenodo_record_url": "https://zenodo.org/records/23099540", "license": "https://creativecommons.org/licenses/by/4.0/", "scope_caveat": "The two explicitly distinguished reciprocal constructions D and I of a strictly nested bicentric Poncelet polygon have phase-independent squared-side sums for all admissible periods, by credited published-theorem transfer. An exact strictly nested circle/ellipse family of primitive convex pentagons has constant squared-side sum 175/18-625t/48, with 25t^3-50t^2-40t+8=0 and 1/10