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ZENODO
Preprint . 2026
License: CC BY
Data sources: ZENODO
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
ZENODO
Preprint . 2026
License: CC BY
Data sources: Datacite
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Deriving the Golden Ratio as a Fractal Attractor in Fractal Theory

Authors: Morgan, Mark;

Deriving the Golden Ratio as a Fractal Attractor in Fractal Theory

Abstract

The Golden Ratio Φ has long appeared across natural systems, yet most explanations treat it as a geometric constant or an imposed tuning parameter. In this paper, Φ is derived directly from the generative operators of Fractal Theory (FT) Morgan, M. (2025), without invoking geometry, harmonic rules, or external proportional assumptions. FT models all recursive structures through the interaction of Unity (coherence), Division (differentiation), Scale (recursive elevation), Drift (operator tuning), and Memory (cumulative structural history). We show that under Drift, Unity grows by integrating Division and Division grows by differentiating Unity, producing the operator transformations U’ ∝ U+D and D’ ∝ U. A recursion shell is stable only when the operator ratio U/D is preserved across a Scale step, which imposes the self-similarity condition U’/D’ = U/D. Substituting the FT operator relations yields the functional equation r = 1 + 1/r, whose unique positive solution is the Golden Ratio r = Φ = (1+√5)/2. Thus, Φ emerges as the sole recursion-stable attractor of the FT kernel, arising naturally from the interplay of Unity, Division, Scale, Drift, and Memory. This revised edition strengthens the original derivation by: (i) deriving the linear recursion form from Scale-invariance rather than asserting it; (ii) connecting the general Drift matrix to the proportional Drift regime used in the derivation; (iii) proving Memory accumulation at Φ; and (iv) establishing that equal proportionality coefficients follow from Scale symmetry.

Keywords

Drift, Unity, Recursive, Fractal Theory, Golden Ratio, Fixed Points, FOS: Mathematics, Recursion, Attractors, Operator Theory, Mathematical Physics, Division

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
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