The Golden Orbit Reversible Golden Scaling, Algebraic Orbits, and Coherence in a Graded Multilinear Algebra
Abstract
The Golden Orbit
Reversible Golden Scaling, Algebraic Orbits, and Coherence in a Graded Multilinear Algebra
C. J. Tully
Independent Researcher and Engineer
ORCID: 0009-0007-5661-7332
October 2026
Abstract
We study the discrete dynamics generated by repeated application of a reversible golden-scaling transformation on a graded multilinear algebra. Let K = ℚ(√5) and let
φ = (1 + √5)/2
be the fundamental positive unit of the golden field. For an integer-graded algebra
L = ⊕_{k ∈ ℤ} L_k,
the golden-scaling operator S_φ is defined on homogeneous elements by S_φ(x_k) = φ^k x_k. Under the assumption that the multilinear operation is homogeneous of degree zero, S_φ is an invertible algebra automorphism. We define the bilateral golden orbit
O_φ(x) = { S_φ^n(x) : n ∈ ℤ }
and derive its explicit componentwise form. The orbit is algebraically reversible. Bounded bilateral orbits occur only for states supported in degree zero. Any coherence predicate that is invariant under a single application of S_φ remains constant along the entire orbit. These results separate algebraic symmetry, coherence preservation, and metric stability for the iterated golden scaling introduced in earlier work.
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BibTeX
@misc{tully2026golden,
title = {{The Golden Orbit Reversible Golden Scaling, Algebraic Orbits, and Coherence in a Graded Multilinear Algebra}},
author = {Chloe Tully},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23407270},
url = {https://zenodo.org/records/23407270},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Golden Orbit Reversible Golden Scaling, Algebraic Orbits, and Coherence in a Graded Multilinear Algebra
AU - Chloe Tully
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23407270
UR - https://zenodo.org/records/23407270
AB - The Golden Orbit Reversible Golden Scaling, Algebraic Orbits, and Coherence in a Graded Multilinear Algebra C. J. Tully Independent Researcher and Engineer ORCID: 0009-0007-5661-7332 October 2026 Abstract We study the discrete dynamics generated by repeated application of a reversible golden-scaling transformation on a graded multilinear algebra. Let K = ℚ(√5) and let φ = (1 + √5)/2 be the fundamental positive unit of the golden field. For an integer-graded algebra L = ⊕_{k ∈ ℤ} L_k, the golden-scaling operator S_φ is defined on homogeneous elements by S_φ(x_k) = φ^k x_k. Under the assumption that the multilinear operation is homogeneous of degree zero, S_φ is an invertible algebra automorphism. We define the bilateral golden orbit O_φ(x) = { S_φ^n(x) : n ∈ ℤ } and derive its explicit componentwise form. The orbit is algebraically reversible. Bounded bilateral orbits occur only for states supported in degree zero. Any coherence predicate that is invariant under a single application of S_φ remains constant along the entire orbit. These results separate algebraic symmetry, coherence preservation, and metric stability for the iterated golden scaling introduced in earlier work.
ER -