entropy_linear_in_area
plain-language theorem explainer
Bekenstein-Hawking entropy scales linearly with horizon area for any nonnegative scale factor. Physicists deriving holographic bounds or black hole thermodynamics from J-cost minimization would cite this when establishing area laws. The proof reduces to unfolding the entropy definition and applying the ring tactic for algebraic verification.
Claim. Let $S(A)$ denote Bekenstein-Hawking entropy for horizon area $A$. For any scale factor $k$ with $k$ nonnegative, $S(k A) = k S(A)$.
background
In the Recognition Science module on black hole no-hair, the stationary state is the unique J-cost minimizer. For asymptotically flat spacetimes this leaves exactly three conserved charges (M, Q, J) corresponding to the symmetries of the RS voxel lattice. Entropy is defined as proportional to total defect of a configuration, with zero defect yielding the minimum entropy state, per the upstream InitialCondition.entropy definition. The Bekenstein-Hawking formula then sets entropy equal to one quarter of the horizon area in Planck units, as stated in the module key results.
proof idea
This is a term-mode proof. It unfolds the bekenstein_hawking_entropy definition then applies the ring tactic to confirm the linear scaling identity.
why it matters
The result supports the holographic principle by showing entropy depends on area rather than volume, consistent with T8 forcing D = 3 spatial dimensions. It fills the bekenstein_entropy_formula step listed in the module documentation and aligns with the broader no-hair claim that only three charges survive. No downstream uses are recorded yet, so it currently stands as a supporting identity rather than a parent theorem.
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