The Mirror Theorem: Central Incidence Saturation in Keith Recurrences
Abstract
Keith numbers are integers that reappear in sliding-window recurrences generated from their own digits. We introduce Keith mirrors, a subclass defined by saturation at the center of the recurrence's incidence profile. For an even index m, let C_d(m) denote the sum of the two central incidence counts. We prove that the central deficit m-C_d(m) is exactly the distance from m to the interval [2d,2d+2], which forces every Keith mirror of order d>=3 in base B>=3 to return at 2d+2. A digit identity and analytic capacity bounds then reduce the classification to finitely many cases. Exact enumeration, independently checked against the original recurrence and incidence definition, yields precisely 35 mirrors of order at least three, while order two consists of two explicit infinite families. We also classify exactly eight cross-base triple alignments satisfying R=C=x_1+x_d and record the conditional reconstruction property induced by incidence symmetry.
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BibTeX
@misc{zamponi2026mirror,
title = {{The Mirror Theorem: Central Incidence Saturation in Keith Recurrences}},
author = {Federico Ignacio Zamponi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10557},
eprint = {2610.10557},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10557},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Mirror Theorem: Central Incidence Saturation in Keith Recurrences
AU - Federico Ignacio Zamponi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.10557
UR - https://arxiv.org/abs/2610.10557
AB - Keith numbers are integers that reappear in sliding-window recurrences generated from their own digits. We introduce Keith mirrors, a subclass defined by saturation at the center of the recurrence's incidence profile. For an even index m, let C_d(m) denote the sum of the two central incidence counts. We prove that the central deficit m-C_d(m) is exactly the distance from m to the interval [2d,2d+2], which forces every Keith mirror of order d>=3 in base B>=3 to return at 2d+2. A digit identity and analytic capacity bounds then reduce the classification to finitely many cases. Exact enumeration, independently checked against the original recurrence and incidence definition, yields precisely 35 mirrors of order at least three, while order two consists of two explicit infinite families. We also classify exactly eight cross-base triple alignments satisfying R=C=x_1+x_d and record the conditional reconstruction property induced by incidence symmetry.
ER -