Information Density Ratio

The Information Density Ratio is the check of a quantum substrate design against the Planck limit on information density: stored information divided by the substrate volume, compared with one bit per Planck volume. The book uses it as a physical realizability test. Any substrate must stay within the maximum storage per unit volume that spacetime allows, ρ⥣ = 1/l_P³. Applied to the book's reference design, 6.4 × 10¹⁰ bits in a cube of (100 voxels)³ each one Planck length on a side, the ratio exceeds the limit by a factor of 6.4 × 10⁴, a clear violation. The book concludes that voxel size must be increased, or stored information reduced, for the design to be physically realizable, so that ρ ≤ ρ⥣. The formal statement is ρ = I/V = 6.4 × 10⁴ · ℨᵇ/l_P³, where ρ is the information density, I the information content (6.4 × 10¹⁰ bits), V the substrate volume ((100 voxels)³ × l_P³ = 10⁶ l_P³), l_P the Planck length and ρ⥣ = 1/l_P³ the Planck limit, giving ρ/ρ⥣ = 6.4 × 10⁴ ≫ 1. Dimensions as the book gives them: ρ [𝕃⁻³·1ᵇ], ρ⥣ [𝕃⁻³], I [1ᵇ], V [𝕃³], l_P [𝕃]. It connects to Maximum Information Density, Information Density per Voxel, Causal Resolution, Voxel Lattice Structure and Lloyd's Computational Bound.