The Golden Orbit Reversible Golden Scaling, Algebraic Orbits, and Coherence in a Graded Multilinear Algebra

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.