cube_sv_ratio
plain-language theorem explainer
The cube_sv_ratio theorem states that for positive real L the ratio of a cube's surface area 6L² to its volume L³ simplifies exactly to 6/L. Researchers deriving holographic entropy bounds from ledger structures on ℤ³ cite this to confirm the surface scaling that makes boundary information dominate over volume. The proof is a one-line term proof applying field_simp to cancel powers of L under the positivity hypothesis.
Claim. For all real $L > 0$, $6L^2 / L^3 = 6/L$.
background
The module derives the Bekenstein-Hawking bound from ledger capacity on the ℤ³ lattice forced by the T0-T8 chain. Volume of a region is defined as the cardinality of its vertex set, while the ledger L records one unit of information per voxel. Boundary information is required to scale as volume to the power (D-1)/D; for D=3 this yields the 2/3 exponent that makes accessible information proportional to surface area rather than bulk volume. Upstream results include the volume definition and the ledger instances from Recognition and Cycle3.
proof idea
The proof is a one-line term proof that applies the field_simp tactic to the rational expression 6 * L^2 / L^3 under the hypothesis 0 < L, which directly cancels the L^3 denominator against the numerator to obtain 6/L.
why it matters
This identity supplies the concrete scaling relation used in the module's derivation of S_BH = A/4 in RS units where Gℏ = 1. It supports the boundary_dimension and info_scales_subvolume results that close Gap G3 in the brain holography argument, confirming that information flows through the (D-1)-dimensional boundary as required by the Recognition Composition Law and the eight-tick octave structure. The result is cited by the area_not_volume_certificate that completes the ledger-to-entropy step.
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