The least-period criterion for outer focal products in elliptic billiards
- 1Independent researcher
Abstract
For a noncircular ellipse and a fixed proper nested nondegenerate confocal elliptical caustic with primitive least billiard period N at least three, the ordinary product of distances from consecutive boundary-tangent intersections to either original focus is phase independent if and only if four divides N. All admissible coprime star windings are covered; the two constants then agree. A paired-distance factorization and cyclic divisor prove both directions, with an explicit odd-period argument and the exceptional four-period degree drop. Repeated traversals cannot restore constancy. The positive direction is the primitive interpretation of k115 and is also proved by Alper Ferudun, Focal Distance Products of Outer Elliptic Billiard Polygons, DOI 10.5281/zenodo.23093258; its closely overlapping mechanism and explicit constant are prominently credited. Canonical billiard parametrization, polarity and elliptic-function norm methods are established background. The observed October 1 GitHub PR and October 2 competing deposit/DOI registration document contemporaneous separate treatments, not certified first disclosure or independent discovery. The converse was added during the October 16 audit; its historical novelty is unestablished. The deposit contains the six-page research note, LaTeX source and portable verification package with exact reflection/tangency certificates, divisor checks, direct tangent-intersection controls and bounded priority/source ledgers. Numerical tests supplement the analytic proof and do not certify it. AI tools were used extensively. This is an unrefereed preprint without conventional independent human peer review or formal proof-assistant certification.
Related
- Period Relations for Theta Products and Elliptic Integral Moments
- Reflective projective billiards in the plane: odd periods, period five, and metric rigidity
- The outer center-pedal area product in elliptic billiards
- A Mixed Elliptic--Hyperbolic Monge--Ampère Problem: Modeling and a Least-Squares Solver
- A Criterion for classifying Elliptic curve that are unsolvable in the set integers
BibTeX
@misc{kriebel2026leastperiod,
title = {{The least-period criterion for outer focal products in elliptic billiards}},
author = {Alec Kriebel},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23409879},
url = {https://zenodo.org/records/23409879},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The least-period criterion for outer focal products in elliptic billiards
AU - Alec Kriebel
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23409879
UR - https://zenodo.org/records/23409879
AB - For a noncircular ellipse and a fixed proper nested nondegenerate confocal elliptical caustic with primitive least billiard period N at least three, the ordinary product of distances from consecutive boundary-tangent intersections to either original focus is phase independent if and only if four divides N. All admissible coprime star windings are covered; the two constants then agree. A paired-distance factorization and cyclic divisor prove both directions, with an explicit odd-period argument and the exceptional four-period degree drop. Repeated traversals cannot restore constancy. The positive direction is the primitive interpretation of k115 and is also proved by Alper Ferudun, Focal Distance Products of Outer Elliptic Billiard Polygons, DOI 10.5281/zenodo.23093258; its closely overlapping mechanism and explicit constant are prominently credited. Canonical billiard parametrization, polarity and elliptic-function norm methods are established background. The observed October 1 GitHub PR and October 2 competing deposit/DOI registration document contemporaneous separate treatments, not certified first disclosure or independent discovery. The converse was added during the October 16 audit; its historical novelty is unestablished. The deposit contains the six-page research note, LaTeX source and portable verification package with exact reflection/tangency certificates, divisor checks, direct tangent-intersection controls and bounded priority/source ledgers. Numerical tests supplement the analytic proof and do not certify it. AI tools were used extensively. This is an unrefereed preprint without conventional independent human peer review or formal proof-assistant certification.
ER -