A two-pole proof of the centered pedal-area invariant
- 1Independent researcher
Abstract
For a noncircular ellipse and a proper nested confocal elliptical caustic, the signed original orbit area times the signed area of its origin-pedal polygon is phase independent when the least physical billiard period is odd or divisible by four. This attributed alternative research note gives a direct-foot Laurent cancellation and reduced two-pole Jacobi trace proof, including primitive stars and both orientations. The exact theorem is already proved in Alper Ferudun's contemporaneous Center-Pedal Area Products in Elliptic Billiards, version1.0, DOI10.5281/zenodo.23085904. Our explicit trace coefficient follows from that paper and standard addition identities; its later arbitrary-point companion DOI10.5281/zenodo.23400541 also contains adjacent-foot pole cancellation. No theorem, formula or method novelty, historical firstness, independent discovery or literal simultaneity is claimed. GitHub's original PR204 creation and the centered Zenodo record creation are both dated October1,2026, with distinct server times; these records do not by themselves establish discovery chronology or historical visibility. The source table does not literally state least period; an exact doubled-six twelve-entry counterexample makes this qualification necessary. Established experimental, canonical-coordinate and complex-method contributions are credited. LaTeX source and verification materials preserve all20 original submission files, independent geometric derivations, source and priority ledgers, explicit-exception corrected historical controls, portable scientific verification and noninterval physical diagnostics. AI tools were used extensively in derivation, auditing, computation, software and drafting. Unrefereed preprint without conventional human peer review or proof-assistant certification.
Related
BibTeX
@misc{kriebel2026twopole,
title = {{A two-pole proof of the centered pedal-area invariant}},
author = {Alec Kriebel},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23421134},
url = {https://zenodo.org/records/23421134},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A two-pole proof of the centered pedal-area invariant
AU - Alec Kriebel
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23421134
UR - https://zenodo.org/records/23421134
AB - For a noncircular ellipse and a proper nested confocal elliptical caustic, the signed original orbit area times the signed area of its origin-pedal polygon is phase independent when the least physical billiard period is odd or divisible by four. This attributed alternative research note gives a direct-foot Laurent cancellation and reduced two-pole Jacobi trace proof, including primitive stars and both orientations. The exact theorem is already proved in Alper Ferudun's contemporaneous Center-Pedal Area Products in Elliptic Billiards, version1.0, DOI10.5281/zenodo.23085904. Our explicit trace coefficient follows from that paper and standard addition identities; its later arbitrary-point companion DOI10.5281/zenodo.23400541 also contains adjacent-foot pole cancellation. No theorem, formula or method novelty, historical firstness, independent discovery or literal simultaneity is claimed. GitHub's original PR204 creation and the centered Zenodo record creation are both dated October1,2026, with distinct server times; these records do not by themselves establish discovery chronology or historical visibility. The source table does not literally state least period; an exact doubled-six twelve-entry counterexample makes this qualification necessary. Established experimental, canonical-coordinate and complex-method contributions are credited. LaTeX source and verification materials preserve all20 original submission files, independent geometric derivations, source and priority ledgers, explicit-exception corrected historical controls, portable scientific verification and noninterval physical diagnostics. AI tools were used extensively in derivation, auditing, computation, software and drafting. Unrefereed preprint without conventional human peer review or proof-assistant certification.
ER -