IndisputableMonolith.Relativity.Compact.BlackHoleEntropy
Defines Schwarzschild horizon area and ties black-hole entropy to a finite ledger capacity in Recognition Science units. Relativists and RS auditors cite it for the area-to-entropy map and saturation uniqueness. The module is mostly definitional, with positivity and uniqueness lemmas built from the metric and RS constants.
claimFor a Schwarzschild black hole, the horizon area $A$ is positive; the ledger capacity bound derived from $A$ is positive whenever $A>0$. Black-hole entropy is identified with that ledger capacity. There is a unique positive nonzero saturation value of the maximum recognition flux on the horizon.
background
Recognition Science works in RS-native units ($c=1$, $\hbar=\varphi^{-5}$, etc.) and treats geometric quantities as constraints on a discrete recognition ledger. This module sits in the compact relativity layer and imports the metric geometry and the RS constants (including the fundamental tick $\tau_0$).
The central geometric object is the event-horizon area of a Schwarzschild black hole. From that area one forms a ledger capacity limit: a finite upper bound on how much recognition flux the horizon can support. Entropy is then read off as that capacity, rather than postulated as $A/4$ in Planck units.
Sibling declarations package positivity of area and capacity, the entropy-from-ledger identification, the maximum recognition flux, and uniqueness/positivity/nonzeroness of the saturation value of that flux.
proof idea
Definition-heavy module. Horizon area is introduced as a geometric quantity on the Schwarzschild metric; positivity is a short lemma from the metric data. Ledger capacity is defined from area, and positivity of capacity is reduced to positivity of area. Entropy is identified with capacity by definition. Maximum recognition flux and its saturation uniqueness/positivity/nonzeroness are proved as elementary consequences of those definitions and the imported constants, without a long tactic script.
why it matters in Recognition Science
Gives the RS reading of black-hole entropy: horizon area bounds a ledger, and entropy is that bound, with a unique positive saturation of recognition flux. Downstream use is not yet wired in the graph (no used_by edges), so the module is a compact relativity building block rather than a leaf of a named parent theorem. It connects geometric horizon data to the discrete ledger picture that underpins RS thermodynamics and the eight-tick/octave structure elsewhere in the monolith. Auditors checking whether RS recovers a Bekenstein-Hawking-type relation in native units land here first.
scope and limits
- Does not derive the classical Bekenstein-Hawking formula $S=A/4$ in SI or Planck units.
- Does not treat Kerr, Reissner-Nordstrom, or dynamical horizons.
- Does not prove the second law or area-increase theorems.
- Does not connect saturation flux to Hawking temperature or evaporation.
- Does not supply observational bounds or numerical GR checks.