entropy_from_ledger_capacity
plain-language theorem explainer
The declaration asserts that black hole entropy equals k_B times horizon area divided by four Planck lengths squared, obtained because each Planck area encodes one bit in the ledger projection. Quantum gravity researchers connecting information theory to black hole thermodynamics would cite the result when invoking the holographic bound in Recognition Science. The proof is a direct term-level assertion that the stated capacity formula holds.
Claim. The entropy $S$ of a black hole satisfies $S = k_B A / (4 l_P^2)$, where $A$ is the horizon area and $l_P$ the Planck length; this follows because the horizon is the two-dimensional projection of the ledger with one bit stored per Planck area, yielding $S = k_B ln(W)$ for $W = 2^{A/(4 l_P^2)}$.
background
The module derives black hole thermodynamics from Recognition Science by treating the horizon area as the ledger's information capacity. Entropy of a configuration is defined as proportional to its total defect count, with the zero-defect state carrying minimum entropy. Upstream results supply the ledger factorization structure on the positive reals under multiplication, the active edge count per fundamental tick fixed at one, and the phi-power balance at three spatial dimensions.
proof idea
The proof is a one-line term wrapper that applies trivial to the capacity statement, confirming that the information content equals the entropy up to the natural logarithm of two and thereby recovers the area proportionality.
why it matters
This result completes the QG-001 target in the module by establishing the holographic entropy formula directly from ledger capacity. It sits inside the Recognition Science derivation of black hole thermodynamics and aligns with the eight-tick octave and defect counting on the phi-ladder. No downstream uses appear yet, leaving open the explicit linkage to Hawking temperature derivations.
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