bhEntropy_eq
plain-language theorem explainer
Black hole entropy in the recognition model equals four pi M squared for mass parameter M. Workers on holographic bounds and maximal saturation in gravity cite this when closing the bandwidth ledger. The proof reduces directly by unfolding the entropy as area over four and the area expression, followed by ring simplification.
Claim. For mass $M$, the black hole entropy $S(M)$ satisfies $S(M) = 4 pi M^2$, where $S$ is the Bekenstein-Hawking entropy obtained from horizon area divided by four in units with Planck length one.
background
In the Black Hole Bandwidth module, a black hole represents the case of complete recognition bandwidth saturation at the horizon. The horizon area is given by sixteen pi M squared, and entropy is defined as that area divided by four. This builds on horizonArea definitions from the Gravity and Quantum modules that encode the Schwarzschild geometry, together with the local bhEntropy definition as area over four. The module documentation states that the Bekenstein-Hawking entropy S_BH equals four pi M squared, realizing the area-equals-entropy relation under the recognition operator.
proof idea
The proof unfolds the definitions of bhEntropy and horizonArea, then applies the ring tactic to equate the expression to four pi M squared.
why it matters
This result supplies the explicit entropy formula that the positivity theorem bhEntropy_pos relies upon. It fills the entropy_is_bandwidth step in the module's saturation argument, connecting to the eight-tick octave through the Hawking temperature derivation. It closes a link between geometric area and recognition cost in the unification framework.
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