A telescoping verification of a focal-inversion area invariant

Abstract

For a proper elliptical billiard with a fixed strictly nested nondegenerate confocal elliptical caustic, the original signed shoelace area times the signed area of the straight polygon obtained by unit inversion of its vertices about either original focus is a positive phase-independent constant when the genuine least collision period is divisible by four. This attributed verification note presents an explicit local Jacobi decomposition, a derivative telescope and a classical Fourier/Landen grid reduction. It covers every admitted coprime star winding, either focus, reversal and repeated traversals of admitted primitive cycles. The theorem and the failure of an unrestricted repeated-list interpretation already appear in Alper Ferudun, DOI 10.5281/zenodo.23093706, and are credited. Classical local and cyclic Jacobi master identities and generalized Landen transformations are also credited, along with the prior four-periodic value and canonical billiard coordinates. An exact triangular family gives a check of the repeated-list boundary. The supporting source package includes explicit exact algebra certificates, independent Euclidean ray diagnostics and failure controls that remain active under optimized Python. Finite floating-point diagnostics do not replace the analytical proof. GitHub dates PR207 to 1 October 2026; matching Zenodo file metadata dates Ferudun's deposit to 2 October 2026. The present telescoping reconstruction and boundary fixture were recorded on 16 October 2026. These records do not establish first discovery, causal independence or simultaneous discovery. No theorem or method novelty or current-open status is asserted. AI tools were used extensively in development, verification, source checking and drafting. This is an unrefereed preprint without conventional human peer review or proof-assistant formalization.