bekenstein_hawking_from_rs
plain-language theorem explainer
The declaration shows that the Bekenstein-Hawking entropy bound simplifies to S = A/4 in RS natural units where G ℏ = 1. Researchers deriving holographic principles or black hole thermodynamics from ledger-based foundations would reference this step. The proof proceeds by rewriting the denominator using the pre-established identity G_rs * hbar_rs = 1 followed by algebraic simplification.
Claim. In Recognition Science natural units satisfying $G ℏ = 1$, the Bekenstein-Hawking entropy bound for positive area $A$ reads $A / (4 G ℏ) = A / 4$.
background
Recognition Science places the ledger on the integer lattice ℤ³, which follows from the forcing chain that fixes spatial dimension D = 3. Each fundamental voxel stores one unit of information. The boundary of any region therefore carries information proportional to its surface area rather than its volume. In these units the gravitational constant satisfies G = φ⁵ and the reduced Planck constant ℏ = φ^{-5}, so their product equals one. This identity is supplied by the constant derivation G_hbar_product_eq_one.
proof idea
The proof is a short tactic sequence. It first establishes the associativity identity 4 * G_rs * hbar_rs = 4 * (G_rs * hbar_rs) by ring. It then rewrites using the upstream lemma G_hbar_product_eq_one that sets the product to one. A final ring call completes the simplification to A/4.
why it matters
This result supplies the numerical factor in the area-not-volume certificate and is invoked inside the fully forced brain holography theorem. It closes the ledger-capacity gap G3 by showing that information scales strictly with boundary area once D = 3 and G ℏ = 1 are in place. The derivation relies on the eight-tick octave and the Recognition Composition Law that together force the dimension and the constant values.
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