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.
Thierry Marechal (Wed,) studied this question.