Energy Threshold Scaling

Energy Threshold Scaling is the law that the minimal surface energy density needed to sustain a Zinf torus rises as the torus gets smaller, in inverse proportion to the square of its minimal boundary scale. It is one of three scaling laws the book gives for Zinf-limit systems, alongside the minimal radii, a⥥ = b⥥ = (N⥥/(2π)) × κ_z × 𝒞→ × ①⥂⧖, and the fundamental frequency f* = 1/(N⥥ × κ_z × ①⥂⧖). Together they express the trade-off between geometric scale, temporal resolution, computational capacity and energy: minimal radii decide geometric feasibility, frequency fixes computational bandwidth, and energy thresholds keep the boundary persistent. The smallest configurations need the highest energy concentrations, which is why the book calls the Zinf limit the razor's edge at which recursive intelligence can emerge with the least physical resources. The formal statement is σ_E⥥ ∝ 1/[(N⥥)² × (κ_z)² × 𝒞→² × (①⥂⧖)²], where σ_E⥥ is the minimal surface energy density [𝕄·𝕋⁻²], N⥥ the minimum cells per loop for a stable sinusoidal mode [∅], κ_z the discretization safety factor [∅], 𝒞→ the computational lightspeed bound [𝕃·𝕋⁻¹] and ①⥂⧖ the half-cycle time quantum, the Pulse Rhythm [𝕋]. The book's own check finds the relation dimensionally inconsistent: the right side reduces to [𝕃⁻²]. It connects to Surface Energy Density Requirement, Minimal Radii, Fundamental Frequency, Scaling Laws and Zinf-Limit Universe.