A flag-curve proof of an outer-pedal area invariant
- 1Independent researcher
Abstract
For a noncircular ellipse and a proper nested nondegenerate confocal elliptical caustic with primitive least period N congruent to two modulo four, the signed area of the outer tangent polygon times its signed arbitrary-point pedal area is phase independent. The note presents an alternative flag-curve proof on the genus-one quotient: the original and pedal area traces have the same two possible simple poles; Riemann-Roch and reversal force proportionality. A self-contained verification of the credited Chavez-Caliz original/outer area product in this parity completes the proof. Primitive stars, reversal and signed zero or negative pedal areas are included. Alper Ferudun's contemporary Outer-Pedal Area Products in Elliptic Billiards, version1.0 DOI10.5281/zenodo.23087221 and version1.1 DOI10.5281/zenodo.23088516, already states the exact theorem, general six-period formula and repeated-triangle qualification, and is prominently credited. Steiner radiality, the Poncelet flag framework and meromorphic pole methods are prior mathematics. No theorem novelty, new general method, first disclosure or independently verified rediscovery is claimed. Observed October1 GitHub PR-creation and Zenodo/DataCite fields document contemporaneous separate records; they do not establish either proof's first public visibility or independence. The October16 note includes LaTeX source and a verification package with exact geometry, boundary controls, portable checkers, a finite noninterval physical-orbit sweep and bounded source/priority ledgers. AI tools were used extensively. This is an unrefereed preprint without conventional independent human peer review or proof-assistant certification.
Related
BibTeX
@misc{kriebel2026flagcurve,
title = {{A flag-curve proof of an outer-pedal area invariant}},
author = {Alec Kriebel},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23411482},
url = {https://zenodo.org/records/23411482},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A flag-curve proof of an outer-pedal area invariant
AU - Alec Kriebel
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23411482
UR - https://zenodo.org/records/23411482
AB - For a noncircular ellipse and a proper nested nondegenerate confocal elliptical caustic with primitive least period N congruent to two modulo four, the signed area of the outer tangent polygon times its signed arbitrary-point pedal area is phase independent. The note presents an alternative flag-curve proof on the genus-one quotient: the original and pedal area traces have the same two possible simple poles; Riemann-Roch and reversal force proportionality. A self-contained verification of the credited Chavez-Caliz original/outer area product in this parity completes the proof. Primitive stars, reversal and signed zero or negative pedal areas are included. Alper Ferudun's contemporary Outer-Pedal Area Products in Elliptic Billiards, version1.0 DOI10.5281/zenodo.23087221 and version1.1 DOI10.5281/zenodo.23088516, already states the exact theorem, general six-period formula and repeated-triangle qualification, and is prominently credited. Steiner radiality, the Poncelet flag framework and meromorphic pole methods are prior mathematics. No theorem novelty, new general method, first disclosure or independently verified rediscovery is claimed. Observed October1 GitHub PR-creation and Zenodo/DataCite fields document contemporaneous separate records; they do not establish either proof's first public visibility or independence. The October16 note includes LaTeX source and a verification package with exact geometry, boundary controls, portable checkers, a finite noninterval physical-orbit sweep and bounded source/priority ledgers. AI tools were used extensively. This is an unrefereed preprint without conventional independent human peer review or proof-assistant certification.
ER -