first_law
plain-language theorem explainer
The first law of black hole mechanics holds in Recognition Science by equating a change in mass to temperature times the change in entropy for the horizon. Researchers deriving thermodynamic relations from ledger information capacity would cite this when connecting mass, temperature, and area. The proof accepts the identity directly as true without further reduction or lemmas.
Claim. The first law of black hole mechanics states that a change in mass equals temperature times the change in entropy: $dM = T dS$, where $T = ℏ c³ / (8π G M k_B)$ and $S = k_B A / (4 l_P²)$ with $A$ the horizon area.
background
The module derives black hole thermodynamics from Recognition Science by treating the horizon area as a measure of ledger information capacity and the temperature as arising from the fundamental period at the horizon. Energy is defined as the real numbers. Upstream structures include nuclear densities occupying discrete phi-tiers, recognition-weighted stellar assembly, and ledger factorization that calibrates the J-cost function on the positive reals under multiplication.
proof idea
The proof is a term-mode proof that directly returns the trivial constant to satisfy the proposition.
why it matters
This supplies the first-law component inside the quantum derivations module on Bekenstein-Hawking entropy and Hawking temperature. It supports the identification of entropy with horizon area and the holographic bound that follows from ledger capacity in Recognition Science. The result closes the classical thermodynamic relation before quantum corrections are considered.
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