entropy_is_bandwidth_capacity
plain-language theorem explainer
Black hole entropy equals horizon recognition bandwidth multiplied by the recognition Boltzmann constant and the eight-tick cadence. Holographic gravity researchers would cite this to equate horizon area with information capacity under full recognition saturation. The proof unfolds the bandwidth definition, asserts positivity of the constant product, and cancels the division via a standard cancellation lemma.
Claim. For mass parameter $M > 0$, the Bekenstein-Hawking entropy $S_{BH}(M)$ equals the horizon recognition bandwidth $R(M)$ multiplied by the recognition Boltzmann constant $k_R$ and the eight-tick cadence: $S_{BH}(M) = R(M) · k_R · C_8$.
background
The module treats a Schwarzschild black hole as the limiting case of maximal recognition saturation, where every holographic bit on the horizon is consumed by gravitational dynamics. Horizon area is $A = 16πM²$, Bekenstein-Hawking entropy is $S_{BH} = 4πM²$, and recognition bandwidth at the horizon is defined as $R_{max} = S_{BH} / (k_R · 8τ_0)$. The recognition Boltzmann constant is given by $k_R = ln(φ)$, the fundamental cost per ledger bit. The eight-tick cadence arises from the forcing chain's T7 octave period. Upstream results establish $k_R > 0$ because $φ > 1$, via the lemma that applies the logarithm positivity property to the golden-ratio constant.
proof idea
The proof unfolds the horizonBandwidth definition to expose the quotient form, introduces the auxiliary fact that $k_R · C_8 > 0$ by multiplying the positivity lemmas for $k_R$ and the cadence, then rewrites via the division-multiplication cancellation rule to obtain equality.
why it matters
The declaration confirms the module claim that area equals information equals bandwidth in the black-hole limit, directly supporting the listed key results on maximal saturation, Hawking temperature from bandwidth, and the no-hair statement from bandwidth exhaustion. It instantiates the eight-tick cadence (T7) inside the unification setting and closes one link in the recognition composition law application to gravitational horizons. No open scaffolding remains for this specific equality.
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