Elliptic-billiard invariant k_{907,a}

Definitions and mathematical statement

Fix the billiard ellipse $E$, its center $O$, its foci $f_1,f_2$, an admissible confocal caustic $C$, and an odd period $N$. Let $\mathcal P_N(E,C)$ be the corresponding Poncelet family of nondegenerate closed billiard polygons $P=(P_1,\ldots,P_N)$: vertices lie on $E$, consecutive sides are tangent to $C$, and the billiard reflection law holds at each vertex. The outer polygon $P'$ is formed by intersections of consecutive tangents to $E$, and the inner polygon $P''$ by caustic tangency points, as in the source.

Unit-circle inversion about the focus $f_j$ is \[ \iota_j(z)=f_j+\frac{z-f_j}{\|z-f_j\|^2}\quad(z\ne f_j), \] and $P_j^\dagger$ is obtained by applying it to the vertices of $P$. The source also names focal polar, dual, and antipedal areas $A_{j,pol},A_{j,dual},A_{j,ant}$, but supplies no construction of the focal dual or the composition indicated by $A_{1,dual}^\dagger$.

With those source constructions fixed and defined, invariant $k_{907,a}$ asserts \[ \exists K\quad\forall P\in\mathcal P_N(E,C)\text{ admissible for the constructions}, \qquad A_{1,dual}^\dagger(P)\,A_{2,dual}(P)=K. \] Here $K$ may depend on the fixed ellipse, caustic, odd period, inversion radius, and construction conventions, but not on the orbit parameter. The exact asymmetric product is retained: only the first factor bears a dagger. The statement must not be replaced by the product with both factors inverted, or by a ratio. The source's other auxiliary areas, lengths, and angles do not appear in this target.

A complete formulation still requires the focal-dual construction, whether the dagger precedes or follows duality, area conventions for self-intersecting polygons, the admitted periods and orbit types, and the treatment of singular configurations. Invariance on nonsingular members does not assert an extension through undefined configurations. The supplied research summary says the table's proven column is "?" and that the invariant was experimentally detected in April--May 2020; it reports no general-period proof located in its later survey, not a verified impossibility or a new proof.

Short English statement

For fixed ellipse and confocal caustic, prove that $A_{1,dual}^\dagger A_{2,dual}$ is independent of the chosen odd-period orbit. The source-specific dual/inversion composition and area conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID $5100066$, exact source code catalogue entry, "Elliptic-billiard invariant $k_{907,a}$."
• Dan Reznik, Ronaldo Garcia, Jair Koiller, Eighty New Invariants of N-Periodics in the Elliptic Billiard (2021), table row $k_{907,a}$, cluster "Pairs of inversive objects"; https://arxiv.org/abs/2004.12497. Source-TeX extraction; recorded access 2026-07-29.
• The supplied research survey mentions Roitman--Garcia--Reznik's bicentric paper, arXiv:2103.11260; Reznik--Garcia--Helman's inversive-triangle paper, arXiv:2012.03020; and Garcia--Reznik's self-intersected paper, arXiv:2011.06640. These are described only as surveyed follow-ups, not proofs of this invariant.
• The stored research summary then ends with an incomplete Garcia--Koiller--Reznik "New$\ldots$" reference; its missing continuation is not reconstructed.