What question did this study set out to answer?

The study aims to establish a comprehensive variational theory for optimal scaling in spectral-fractal systems.

March 27, 2026Open Access

Optimal Fractal Scaling in Spectral Systems: A Rigorous Variational Analysis

TMThierry MarechalF5 Networks (United States)

Key Points

The study aims to establish a comprehensive variational theory for optimal scaling in spectral-fractal systems.
Defined operators with fractal potentials scaling at a specific rate
Developed an energy functional to balance spectral energy and complexity
Proved existence and uniqueness of optimal scaling under precise conditions
Demonstrated that optimal scaling exists and can be calculated as α* = (γC_s/(βC_f))^{1/(β−γ)}
Identified that √2 is not universal for all cases, depending on scaling exponents
Provided complete proofs and quantitative assumptions for all findings

Abstract

Complete variational theory for optimal scaling in parametrized spectral-fractal systems. Given operators H_α with fractal potentials scaling at rate α, defines an energy functional balancing spectral energy against fractal complexity cost. Proves existence and uniqueness of optimal α* = (γCₛ/ (βCf) ) ^1/ (β−γ). Explicitly states that √2 is not universal — it emerges only when scaling exponents and constants satisfy specific relations. All assumptions quantitative, all proofs complete. Foundation for understanding optimal fractal scaling across physical and mathematical contexts.

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