Published July 16, 2026 | Version 3.1
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Formalizing the Domain of Feigenbaum Universality: Class Membership Across All Dynamical Systems

Authors/Creators

  • 1. Independent Researcher

Description

Feigenbaum’s discovery of universal constants δ = 4.669201... and α = 2.502907... identified the governing numbers of period-doubling cascades. Lanford’s subsequent proof established that these constants hold universally within the domain of smooth unimodal maps. Both results operate within a single specific domain: they characterize the behavior of smooth unimodal maps with a nondegenerate quadratic maximum, systems already known to exhibit the cascade. But in the years since the universality was proven for that one system, numerous unrelated systems and equations have been discovered to exhibit the cascade that follows from Feigenbaum constants. It is well established that the full domain of which equations qualify for Feigenbaum universality is larger than just smooth unimodal maps with a nondegenerate quadratic maximum. The question of which dynamical systems belong to this overall Feigenbaum universality domain — and why — has not been answered by pure mathematics. This paper formalizes the class membership requirements.

We prove the Feigenbaum Domain Theorem (FDT): three structural conditions C₁ (analytic dissipative boundedness), C₂ (nondegenerate parametric fold), and C₃ (transversal spectral crossing with infinite accumulating cascade) are necessary and sufficient for membership in the Feigenbaum universality class. The proof is purely analytic. The result applies to all dynamical system types without restriction to unimodal maps and provides the mathematical tool to determine membership qualification of any particular system from formula analysis alone.

The FDT identifies what qualifies a formula to exhibit the universality that Feigenbaum and Lanford proved: it formally establishes the full universality domain that prior results operated within. Companion results establish that no scalar quantity survives a cascade transition (the No-Scalar Theorem [33]) and that C₁+C₂+C₃ membership constitutes the fundamental structural divide in mathematics between equations capable of producing novelty and equations that cannot. Feigenbaum’s constants govern this divide.

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Additional details

Dates

Created
2026-04-14
Updated
2026-04-15
Updated
2026-04-30
Updated
2026-05-21
Updated
2026-07-13
Updated
2026-07-14
Updated
2026-07-16

Software

Repository URL
https://github.com/lucian-png/resonance-theory-code
Programming language
Python
Development Status
Active

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