bh_entropy_proportional_to_area
plain-language theorem explainer
Bekenstein-Hawking entropy scales linearly with horizon area. A physicist deriving holographic bounds from ledger projections would cite this when showing entropy follows area rather than volume. The proof is a direct algebraic reduction that unfolds the entropy definition, substitutes the doubled-area relation, and simplifies via ring arithmetic.
Claim. Let $S(a)$ denote Bekenstein-Hawking entropy for horizon area $a$. If $a_2 = 2 a_1$, then $S(a_2) = 2 S(a_1)$.
background
In Recognition Science the Bekenstein-Hawking entropy is obtained from the Ryu-Takayanagi formula $S_A = $ Area($γ_A$) / (4 $G_N$ ℏ), where $γ_A$ is the minimal bulk surface anchored to boundary region $A$. The module derives this from ledger projection: ledger entries are 2D surfaces, entanglement counts shared entries across a boundary, and the count is therefore proportional to area. Upstream, entropy of a configuration is defined as its total defect, with zero defect giving the minimum-entropy state.
proof idea
The proof is a one-line wrapper that unfolds the definition of Bekenstein-Hawking entropy, rewrites using the supplied area equality, and simplifies the resulting linear expression with the ring tactic.
why it matters
The result supplies the area-law step required for the Ryu-Takayanagi formula inside Recognition Science, confirming that entropy is set by boundary area rather than enclosed volume because ledger entries are fundamentally 2D. It directly supports the QG-008 derivation whose target is a PRL paper on the RT formula from the ledger structure. The proportionality is consistent with the framework constants $G = φ^5 / π$ and ℏ = $φ^{-5}$ already fixed in RS-native units.
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