The odd-period focal-pedal sum invariant
- 1Independent researcher
Abstract
We verify the odd-period focal-pedal sum invariant k601 observed by Reznik, Garcia and Koiller: for a fixed proper nested confocal elliptic billiard pair, the product of the sums of ordinary distances from the two foci to the complete original side supporting lines is phase independent. All admitted coprime primitive star windings and both orientations are covered; repetitions of an odd primitive orbit scale by r². The proof applies Stachel's canonical coordinates and the classical generalized Landen formulas of Khare and Sukhatme, with a self-contained Fourier derivation of the shared trace normalization. Bialy–Tabachnikov's focal support identity and earlier cyclic identities are credited directly. No theorem, formula or method novelty, historical firstness, present-day openness, independent discovery or simultaneity is claimed. The bounded priority audit did not establish an earlier explicit full geometric application and does not exclude one elsewhere. An exact four-period pair has products 8 and 9, excluding an unrestricted even-period extension. The supplement includes portable physical calibration, required nonfinite/wrong-identity rejection controls, authored mathematical and priority audits, and clearly labelled immutable historical submission files. AI tools were used extensively in research, proof development, drafting, auditing, software and computation. Unrefereed preprint without conventional human peer review or proof-assistant certification.
Related
BibTeX
@misc{kriebel2026oddperiod,
title = {{The odd-period focal-pedal sum invariant}},
author = {Alec Kriebel},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23423290},
url = {https://zenodo.org/records/23423290},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The odd-period focal-pedal sum invariant
AU - Alec Kriebel
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23423290
UR - https://zenodo.org/records/23423290
AB - We verify the odd-period focal-pedal sum invariant k601 observed by Reznik, Garcia and Koiller: for a fixed proper nested confocal elliptic billiard pair, the product of the sums of ordinary distances from the two foci to the complete original side supporting lines is phase independent. All admitted coprime primitive star windings and both orientations are covered; repetitions of an odd primitive orbit scale by r². The proof applies Stachel's canonical coordinates and the classical generalized Landen formulas of Khare and Sukhatme, with a self-contained Fourier derivation of the shared trace normalization. Bialy–Tabachnikov's focal support identity and earlier cyclic identities are credited directly. No theorem, formula or method novelty, historical firstness, present-day openness, independent discovery or simultaneity is claimed. The bounded priority audit did not establish an earlier explicit full geometric application and does not exclude one elsewhere. An exact four-period pair has products 8 and 9, excluding an unrestricted even-period extension. The supplement includes portable physical calibration, required nonfinite/wrong-identity rejection controls, authored mathematical and priority audits, and clearly labelled immutable historical submission files. AI tools were used extensively in research, proof development, drafting, auditing, software and computation. Unrefereed preprint without conventional human peer review or proof-assistant certification.
ER -