Finite-Sample Rigidity for Left-Translated Similarity Orbits and Off-Diagonal Quadratic Realizations
Abstract
We study finite-sample rigidity for left translates of similarity orbits over commutative rings. Let R be a commutative ring with 2∈ R×, and let C∈Mat2(R) be cyclic. Given matrices Gi=FKi, where F∈GL2(R) is common and the underlying points Ki lie in Orb(C), we determine when a finite translated sample has exactly the same multiplier ambiguity as the complete translated orbit. The key structural result is that the orbit differences span sl2(R), which yields a scalar description of the left stabilizer. We then give a determinantal saturation criterion, expressed directly in the matrices Gi, under which finite-sample consistency reduces to trace equations and one determinant condition. Off-diagonal quadratic operators on pairs of free rank-two modules realize the translated-orbit model exactly. Over fields we obtain sharp sample thresholds under the saturation criterion, and over finite fields we derive exact saturation probabilities for regular semisimple and nonzero nilpotent orbits. Torsion restrictions of elliptic isogenies provide a natural arithmetic realization of the rank-two module framework.
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BibTeX
@misc{minamide2026finitesample,
title = {{Finite-Sample Rigidity for Left-Translated Similarity Orbits and Off-Diagonal Quadratic Realizations}},
author = {Hiroki Minamide},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05932},
eprint = {2609.05932},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05932},
}
RIS
TY - UNPB
TI - Finite-Sample Rigidity for Left-Translated Similarity Orbits and Off-Diagonal Quadratic Realizations
AU - Hiroki Minamide
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05932
UR - https://arxiv.org/abs/2609.05932
AB - We study finite-sample rigidity for left translates of similarity orbits over commutative rings. Let R be a commutative ring with 2∈ R^×, and let C∈Mat_2(R) be cyclic. Given matrices G_i=FK_i, where F∈GL_2(R) is common and the underlying points K_i lie in Orb(C), we determine when a finite translated sample has exactly the same multiplier ambiguity as the complete translated orbit. The key structural result is that the orbit differences span sl_2(R), which yields a scalar description of the left stabilizer. We then give a determinantal saturation criterion, expressed directly in the matrices G_i, under which finite-sample consistency reduces to trace equations and one determinant condition. Off-diagonal quadratic operators on pairs of free rank-two modules realize the translated-orbit model exactly. Over fields we obtain sharp sample thresholds under the saturation criterion, and over finite fields we derive exact saturation probabilities for regular semisimple and nonzero nilpotent orbits. Torsion restrictions of elliptic isogenies provide a natural arithmetic realization of the rank-two module framework.
ER -