Universal Classification of Primitive Scalar Witnesses in Fibonacci Dynamics Modulo Powers of Three
- 1ORCID
Abstract
We investigate the reconstruction of primitive time shifts in Fibonacci dynamics modulo powers of three. Using the Fibonacci transfer matrix and three scalar observables, we characterize the time positions at which the true primitive shift is uniquely determined by a single observation.
For every integer k >= 2, we establish an explicit classification of these positions in terms of residue classes modulo 12. Each scalar observable admits exactly one quarter of the full period as uniquely identifying positions, and the three corresponding witness sets are pairwise disjoint.
The proof combines a complete base classification modulo 9, 3-adic lifting arguments, Fibonacci reflection identities, and explicit constructions of competing primitive shifts.
The theoretical results are accompanied by reproducible computational verification using Python and its standard library. The computational audits independently test the underlying identities and reconstruction properties over finite precision levels.
This work contributes an explicit classification of scalar reconstruction uniqueness in modular Fibonacci dynamics, complementing classical results on Fibonacci periods and linear recurrences modulo prime powers.
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BibTeX
@misc{vajc2026universal,
title = {{Universal Classification of Primitive Scalar Witnesses in Fibonacci Dynamics Modulo Powers of Three}},
author = {Ivo Vajc},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23437165},
url = {https://zenodo.org/records/23437165},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Universal Classification of Primitive Scalar Witnesses in Fibonacci Dynamics Modulo Powers of Three
AU - Ivo Vajc
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23437165
UR - https://zenodo.org/records/23437165
AB - We investigate the reconstruction of primitive time shifts in Fibonacci dynamics modulo powers of three. Using the Fibonacci transfer matrix and three scalar observables, we characterize the time positions at which the true primitive shift is uniquely determined by a single observation. For every integer k >= 2, we establish an explicit classification of these positions in terms of residue classes modulo 12. Each scalar observable admits exactly one quarter of the full period as uniquely identifying positions, and the three corresponding witness sets are pairwise disjoint. The proof combines a complete base classification modulo 9, 3-adic lifting arguments, Fibonacci reflection identities, and explicit constructions of competing primitive shifts. The theoretical results are accompanied by reproducible computational verification using Python and its standard library. The computational audits independently test the underlying identities and reconstruction properties over finite precision levels. This work contributes an explicit classification of scalar reconstruction uniqueness in modular Fibonacci dynamics, complementing classical results on Fibonacci periods and linear recurrences modulo prime powers.
ER -