Published March 5, 2026 | Version 1.0
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Lucian Law Determines Feigenbaum Universality - A Synthesis of Three Results

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  • 1. The Emergence

Description

Description/Abstract: We show that Feigenbaum universality—the appearance of the constants δ = 4.669... and α = 2.502... across all period-doubling cascades—is not a foundational principle but a consequence of a deeper geometric law. The Lucian Law states that bounded nonlinear systems with coupling necessarily produce fractal geometry. Three papers, taken together, establish a closed logical chain: (1) the Lucian Law generates the Feigenbaum constants through the geometry of the cascade, (2) the Feigenbaum constants structure the Lucian Law’s own prerequisites through the decay bounce mechanism, and (3) the loop closes—the Law is self-grounding. This note synthesizes the argument and establishes the four-layer universality hierarchy.

Keywords: Feigenbaum universality, period-doubling cascade, renormalization, self-grounding, fractal geometry, Lucian Law

 

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Additional details

Is part of
Preprint: 10.5281/zenodo.18865365 (DOI)
Is supplemented by
Preprint: 10.5281/zenodo.18818006 (DOI)
Preprint: 10.5281/zenodo.18818008 (DOI)
Preprint: 10.5281/zenodo.18868816 (DOI)
Preprint: 10.5281/zenodo.18764176 (DOI)
Preprint: 10.5281/zenodo.18764623 (DOI)
Preprint: 10.5281/zenodo.18764576 (DOI)
Preprint: 10.5281/zenodo.18772441 (DOI)

Dates

Created
2026-03-05
Restricted (Not for Public Distribution)

References

  • [1] M. J. Feigenbaum, "Quantitative universality for a class of nonlinear transformations," J. Stat. Phys. 19, 25–52 (1978).
  • [2] M. J. Feigenbaum, "The universal metric properties of nonlinear transformations," J. Stat. Phys. 21, 669–706 (1979).
  • [3] O. E. Lanford III, "A computer-assisted proof of the Feigenbaum conjectures," Bull. Amer. Math. Soc. 6, 427–434 (1982).
  • [4] D. Sullivan, "Bounds, quadratic differentials, and renormalization conjectures," AMS Centennial Publications, vol. II, 417–466 (1992).
  • [5] L. Randolph, "The Lucian Law," Zenodo, DOI: 10.5281/zenodo.18818006 (2026).
  • [6] L. Randolph, "The Geometric Necessity of Feigenbaum's Constant," Zenodo, DOI: 10.5281/zenodo.18818008 (2026).
  • [7] L. Randolph, "The Decay Bounce: Reflection Geometry of the Feigenbaum Stable Manifold," Zenodo, DOI: 10.5281/zenodo.18868816 (2026).