schwarzschild_entropy_eq
plain-language theorem explainer
The theorem establishes that the entropy of a Schwarzschild black hole of mass M equals 4πM² in Planck units. Physicists deriving black hole thermodynamics from J-cost minimization in Recognition Science would cite this equality when specializing the area law. The proof is a direct algebraic reduction obtained by unfolding the definitions of Schwarzschild entropy and Bekenstein-Hawking entropy followed by ring simplification.
Claim. For a Schwarzschild black hole of mass $M$ in Planck units, the entropy $S$ satisfies $S = 4πM^2$.
background
In Recognition Science the stationary state of any system is the unique J-cost minimizer. For asymptotically flat spacetimes exactly three conserved charges survive (M, Q, J) corresponding to the symmetries forced by the voxel lattice. All other classical information decays because it carries positive J-cost. The module develops the no-hair theorem from this principle and recovers the standard thermodynamic relations as consequences.
proof idea
The proof unfolds the definition of Schwarzschild entropy, which applies the Bekenstein-Hawking entropy (area divided by 4) to the horizon area 16πM², then simplifies the resulting expression with the ring tactic.
why it matters
This equality confirms the Bekenstein-Hawking area law for the Schwarzschild case inside the Recognition Science derivation of the no-hair theorem. It supports the interpretation that entropy counts the number of ledger J-bits crossing the horizon per unit area. The result sits inside the module whose paper reference is RS_No_Hair_Theorem.tex and connects to the forcing chain that produces three spatial dimensions and the three surviving charges.
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