bekenstein_positive
plain-language theorem explainer
The declaration shows that the Bekenstein-Hawking entropy expression remains strictly positive for any positive boundary area A. Researchers closing the holographic gap in Recognition Science ledger models would cite it to confirm sign consistency after constant substitution. The argument reduces the inequality to a positivity check by rewriting via the central Bekenstein-Hawking identity derived from the ledger structure.
Claim. If $A > 0$, then $0 < A / (4 G_{rs} hbar_{rs})$, where $G_{rs} = phi^5$ and $hbar_{rs} = phi^{-5}$ are the Recognition Science values of Newton's gravitational constant and the reduced Planck constant.
background
The module derives the Bekenstein-Hawking bound from the RS ledger on the integer lattice Z^3 forced by T3 plus T8 of the unified forcing chain. Each voxel stores one unit of ledger information, so accessible information from a region is limited by its boundary, which scales as volume to the power 2/3 in three dimensions. In native units the constants satisfy G_rs = phi^5 and hbar_rs = phi^{-5}, forcing the product G hbar = 1 and yielding the simplified form S = A/4.
proof idea
The proof is a one-line wrapper. It rewrites the target inequality by applying the lemma bekenstein_hawking_from_rs, which supplies the explicit form A/4 after constant substitution, and then invokes the positivity tactic on the resulting positive term.
why it matters
This result supplies the sign verification step inside the derivation of the holographic bound from ledger capacity, supporting the later statements that information scales with boundary area rather than volume. It closes a consistency check required for the area_not_volume_certificate and aligns with the D=3 forcing and the Recognition Composition Law. The parent development appears in the module's treatment of Gap G3 in the brain holography argument.
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