boundary_exponent
plain-language theorem explainer
The boundary scaling exponent equals two thirds when the spatial dimension is three. Researchers deriving the Bekenstein-Hawking entropy bound from Recognition Science ledger capacity cite this when establishing that information scales with surface area rather than volume. The proof is a direct numerical reduction that substitutes the forced dimension value.
Claim. In three spatial dimensions the ratio of boundary dimension to total dimension is two thirds: $(D-1)/D = 2/3$, where $D=3$ is forced by the Recognition Science forcing chain.
background
Recognition Science forces D equals three via the eight-tick octave in the unified forcing chain (T8). The ledger resides on the cubic lattice ℤ³. In D dimensions the boundary of any region has measure proportional to volume raised to the power (D-1)/D. For D=3 this power is exactly 2/3, which governs how information capacity scales with boundary size. The module derives the Bekenstein-Hawking bound from ledger capacity on ℤ³, closing Gap G3 in the brain holography proof. Upstream structures from PhiForcingDerived supply the J-cost minimization that underpins the ledger, while SpectralEmergence fixes the gauge content consistent with D=3.
proof idea
The proof applies norm_num to the definition of D. It evaluates the arithmetic expression (3-1)/3 directly and confirms equality to 2/3.
why it matters
This supplies the scaling exponent required by the area-not-volume certificate, which assembles the full Bekenstein-Hawking derivation: D=3, exponent 2/3, Gℏ=1, and S equals A/4. It realizes the step from the forcing chain (T8) to the area-law information bound in the ledger model. The parent result area_not_volume_certificate chains this fact with the product G_rs hbar_rs equals one. It addresses the question of how discrete ledger structure enforces holographic scaling without continuum limits.
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