bh_entropy_from_ledger
plain-language theorem explainer
The theorem equates the Bekenstein-Hawking entropy of a Schwarzschild black hole to one-fourth the ledger capacity limit at the horizon area. Researchers deriving black-hole thermodynamics from information or recognition principles would cite the equivalence. The proof is a short tactic sequence that introduces the capacity limit as witness, applies reflexivity, and reduces the two sides by rewriting with the light-cone identity and ring normalization.
Claim. For $R_s > 0$ let $A := 4$π$R_s^2$ be the horizon area. Define $S_{BH} := A/(4$τ₀²$c²)$. Then there exists $N$ such that $N$ equals the ledger capacity limit evaluated at $A$ and ℓ₀ and $S_{BH} = N/4$.
background
The module derives the Bekenstein-Hawking entropy from the ledger capacity limit that encodes maximum recognition flux through the horizon. HorizonArea Rs is defined as 4πRs². The constants satisfy ℓ₀ = c τ₀ by the lemma c_ell0_tau0, with ell0 and tau0 the fundamental length and tick duration in RS-native units. LedgerCapacityLimit A ell0 supplies the integer count of recognition units that saturates the flux at area A.
proof idea
The tactic proof introduces A and S_BH, then uses LedgerCapacityLimit A ell0 as the witness N. Reflexivity dispatches the first conjunct. The second conjunct unfolds both sides, rewrites with c_ell0_tau0, and finishes by ring_nf.
why it matters
The result embeds the standard Bekenstein-Hawking formula inside the Recognition Science ledger framework, confirming that black-hole entropy is one quarter the maximum ledger capacity. It realizes the module objective that S_BH = A/(4ℓ_p²) arises from maximum recognition flux. No downstream theorems are recorded yet; the declaration closes a definitional step linking the event horizon to the eight-tick recognition cycle.
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