BlackHoleEntropyFromLedgerCert
plain-language theorem explainer
The BlackHoleEntropyFromLedgerCert structure packages positivity of the leading RS entropy term, its exact match to the Bekenstein-Hawking area law A/4, negativity of the log coefficient, and its distinction from LQG and string-theory values. Quantum-gravity researchers comparing ledger derivations to semiclassical results would cite it. The structure is defined directly as a record whose five fields are discharged by prior module lemmas on S_lead and the phi-based c_RS.
Claim. The certificate asserts that the leading entropy satisfies $S(A)>0$ for all $A>0$, that $S(A)=A/4$, that the RS coefficient $c=- (log φ)/2$ is negative, and that $c ≠ -1/2$ and $c ≠ -3/2$.
background
In the Recognition Science ledger, black-hole entropy is recovered as the count of admissible horizon states modulo σ-equivalence. The leading term is defined by S_lead(A) := A/4 in RS-native units (ℓ_P = 1). The module recovers the classical Bekenstein-Hawking formula exactly at leading order while predicting the φ-rational subleading coefficient c_RS = -log φ / 2 ≈ -0.241, which is negative and distinct from the LQG value -1/2 and the string-theory value -3/2.
proof idea
The structure is populated by direct reference to the sibling results S_lead_pos, S_lead_eq_BH, c_RS_neg, c_RS_neq_LQG and c_RS_neq_string. Each field is filled by the corresponding lemma or definition already proved in the module; the record constructor supplies the certificate with no further tactics.
why it matters
This certificate is the central data structure for Track F6 and feeds directly into the inhabited instance blackHoleEntropyFromLedgerCert. It encodes the algebraic structure of the leading-log coefficient required by the module statement, closing the leading-order match to the area law while leaving the empirical coefficient match as a hypothesis. The result distinguishes the RS prediction from both LQG and string-theory values using the eight-tick octave and φ-ladder of the framework.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.