black_hole_entropy_one_statement
plain-language theorem explainer
The black-hole entropy one-statement theorem asserts that the leading RS ledger entropy equals A/4, that the log-correction coefficient equals minus half the logarithm of the golden ratio, and that this coefficient differs from both the loop-quantum-gravity and string-theory canonical values. Quantum-gravity phenomenologists would cite the result when testing discrete-geometry predictions against semiclassical entropy formulas. The proof is a single term that assembles the equality lemma, reflexivity on the coefficient definition, and two named
Claim. For every positive real $A$, the leading-order entropy extracted from the Recognition Science ledger satisfies $S_ {lead}(A) = A/4$. The coefficient of the leading logarithmic correction is $c_{RS} = - (log phi)/2$, where phi is the golden ratio. This coefficient is distinct from the loop-quantum-gravity value $-1/2$ and the string-theory value $-3/2$.
background
The module recovers the Bekenstein-Hawking entropy from the discrete RS ledger as the count of admissible horizon states modulo sigma-equivalence. In RS-native units the leading term is defined by the sibling S_lead(A) := A/4. The coefficient c_RS is fixed to minus half the logarithm of the golden ratio, and the module distinguishes it from the LQG and string-theory predictions. Upstream results supply the correction-factor definition of c_RS, the canonical arithmetic object, the active edge count per tick, and the seven-axiom reduction to four structural conditions.
proof idea
The proof is a term-mode construction that packages the lemma S_lead_eq_BH for the area-law equality, reflexivity to confirm the explicit form of c_RS, and the pair of lemmas c_RS_neq_LQG together with c_RS_neq_string to separate the coefficient from the two literature values.
why it matters
This declaration supplies the algebraic core of the BlackHoleEntropyFromLedger module and directly supports the combined formula S_RS(A) = A/4 + c_RS log A. It realizes the phi-rational structure forced by T5 J-uniqueness and T6 self-similar fixed point inside the entropy sector. The result provides a concrete falsifier for the leading-log coefficient and closes the algebraic part of Track F6, leaving only the empirical match to semiclassical gravity as an open hypothesis.
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