cube_isoperimetric
plain-language theorem explainer
The identity shows that a cube's surface area equals six times its volume raised to the two-thirds power. Recognition Science uses this scaling to relate boundary ledger capacity to enclosed volume in three-dimensional space when deriving the Bekenstein-Hawking entropy bound. The proof is a short sequence of real-number rewrites that reduce the exponent equality via multiplication rules for powers.
Claim. For any real number $L > 0$, $6L^{2} = 6(L^{3})^{2/3}$.
background
In the Recognition Science framework the ledger lives on the integer lattice in $D=3$ dimensions, where $D=3$ is forced by the unified forcing chain (T8). The boundary of any region is $(D-1)$-dimensional and therefore scales as volume to the power $(D-1)/D = 2/3$. This supplies the explicit equality for the cube case: surface area $6L^{2}$ equals six times volume to the two-thirds power. The module derives the Bekenstein-Hawking bound $S = A/(4Gℏ)$ by noting that information flows through the boundary and that in RS units $Gℏ=1$, so entropy equals boundary area over 4.
proof idea
The tactic proof first obtains positivity of $L^3$ by positivity. It rewrites the exponent $2/3$ as an explicit division of reals, converts the powers to natural-number casts, reduces the goal by congruence to the core equality $L^2 = (L^3)^{2/3}$, applies the real-power multiplication rule under the positivity hypothesis, and finishes with norm_num.
why it matters
This supplies the concrete scaling relation needed for the boundary information count in the derivation of the Bekenstein-Hawking entropy from the RS ledger on ℤ³. It closes the step that information accessible from a region is proportional to its surface rather than its volume, as required for the holographic bound. The result sits inside the chain from the forcing of D=3 to the entropy formula $S_{BH}=A/4$ in natural units.
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