IndisputableMonolith.Gravity.BHEntropyLogCorrection2FromJCost
Module packaging a certificate that a logarithmic black-hole entropy correction of two arises from the Recognition J-cost on a radial domain. Gravity workers cite it when tying Bekenstein-Hawking area law corrections to the RS cost functional rather than ad-hoc loop or string terms. The argument defines a nonnegative domain cost, a positive canonical threshold, and assembles an inhabited certificate record.
claimOn a radial domain the J-cost induces a nonnegative domain cost $C$ and a positive canonical threshold $\tau_\ast$. The module supplies an inhabited certificate asserting that the logarithmic black-hole entropy correction equals $2$ when measured against this cost/threshold pair (RS-native units).
background
Recognition Science measures mismatch by the unique cost $J(x)=(x+x^{-1})/2-1$ fixed by the Recognition Composition Law. Gravity modules import that cost together with the RS time quantum $\tau_0=1$ tick from Constants.
Black-hole entropy in the classical area law receives quantum logarithmic corrections. This module treats those corrections as a domain cost derived from $J$ on a radial coordinate chart, rather than as free coefficients from other UV completions.
Sibling definitions introduce the domain cost, its evaluation identity, nonnegativity, a positive canonical threshold, and the certificate type BHEntropyLog2Cert that packages the claim that the log correction is exactly two.
proof idea
Definition-and-certificate module, not a single deep theorem. It defines a domain cost from $J$, proves nonnegativity and an evaluation identity, fixes a positive canonical threshold, then builds an inhabited certificate record asserting the log-correction factor two. Supporting lemmas are elementary real inequalities and direct unfolding of $J$; no heavy analysis is required at this layer.
why it matters in Recognition Science
Places the factor-two logarithmic BH entropy correction inside the RS forcing chain: the same $J$ that is unique at T5 and feeds the eight-tick and $D=3$ steps also controls the entropy correction coefficient. Downstream gravity and holography developments can consume the inhabited certificate instead of re-deriving the coefficient. No further used-by edges are recorded yet; the module is a self-contained gravity-side certificate source.
scope and limits
- Does not derive the classical Bekenstein-Hawking area law itself.
- Does not treat spinning, charged, or higher-dimensional black holes.
- Does not fix the absolute entropy normalization beyond the log-correction coefficient two.
- Does not connect to observational BH thermodynamics or evaporation rates.
- Does not discharge broader holographic or AdS/CFT entropy identities.