The Universal Scale Lemma: Scale-Invariant Energy Distribution Beyond Frequency and Time

This paper identifies Lemma 1 of Beyer 2026h as a universal principle — valid for any scale variable under a 1/f-type distribution, not specific to temporal frequency. The Universal Scale Lemma (Lemma USL) proves that for any positive scale variable x with S(x) ∝ x⁻¹, the energy per unit log-interval is constant. Two corollaries follow: (i) generalisation of C*(r) = 1/(D+1) to continuous D ∈ ℝ⁺, with E_residual = {D}·ln(b) where {D} is the fractional part; (ii) the spatial 1/f theorem — for receivers without Axiom 3 (no time arrow), the optimal signal is S(k) ∝ |k|⁻¹ and C*(r) = 1/(D_k+1). This closes Thesis C of Beyer 2026g. Reviewed: Mistral (June 2026). You're now past your plan's included usage. Your session limit resets at 5:0