radiationEntropyNaive
radiationEntropyNaive computes the entropy carried away by radiation as the product of a black hole's initial Bekenstein-Hawking entropy and its evaporated mass fraction. It supplies the linear growth segment of the Page curve in Recognition Science models of evaporation before the turnover. The definition performs direct multiplication on the two fields of the EvolvingBlackHole record.
claimFor an evolving black hole with initial entropy $S_0$ and evaporated fraction $f$, the naive radiated entropy is $S_0 f$.
background
Recognition Science treats black hole evaporation through ledger conservation, where entanglement entropy counts shared defect entries between the black hole and its radiation. The EvolvingBlackHole structure records the initial Bekenstein-Hawking entropy together with the evaporated fraction subject to the constraints $S_0 > 0$ and $0 ≤ f ≤ 1$. Upstream, entropy is defined as total defect count, yielding zero entropy for the minimum-defect initial state.
proof idea
One-line definition that multiplies the initial entropy field by the evaporation fraction field of the supplied EvolvingBlackHole record.
why it matters in Recognition Science
This definition supplies the pre-Page-time branch to pageEntropy, which assembles the full Page curve. It realizes the early-time linear growth segment in the QG-004 derivation of the Page curve from ledger dynamics. The construction supports the information-preservation claim that late radiation restores the initial entropy through entanglement with early quanta.
scope and limits
- Does not model the decrease in radiation entropy after Page time.
- Does not incorporate higher-order corrections from the phi-ladder or defect distributions.
- Does not verify the Page time location at exactly half evaporation.
- Does not address firewall or singularity issues.
Lean usage
def exampleUse (bh : EvolvingBlackHole) : ℝ := radiationEntropyNaive bhformal statement (Lean)
61noncomputable def radiationEntropyNaive (bh : EvolvingBlackHole) : ℝ :=
proof body
Definition body.
62 bh.S_initial * bh.evaporation_fraction
63
64/-! ## The Page Curve -/
65
66/-- The Page time: when half the initial entropy has been radiated.
67 This occurs at evaporation_fraction = 1/2. -/