IndisputableMonolith.Gravity.PenroseInequalityFromJCost
Module packaging a Penrose-style inequality certificate from the RS J-cost. Gravity workers cite the inhabited certificate when they need a mass-area style bound without re-proving cost estimates. Structure is definitional: nonnegative domain cost, positive canonical threshold, and a cert record that ties them together.
claimDefines a domain cost built from the J-cost $J(x)=\frac{x+x^{-1}}{2}-1$, a strictly positive canonical threshold, and a certificate type asserting a Penrose-type inequality bound in terms of that cost, together with an inhabitation witness.
background
Recognition Science forces a unique nonnegative cost $J$ on ratios (T5), with $J(x)=\cosh(\log x)-1$. Gravity results in the monolith aim to recover geometric inequalities of Penrose type (mass controlled by horizon area) from that cost rather than from the Einstein equation.
The module sits in the Gravity domain and imports only Constants (RS-native units, including the tick $\tau_0$) and Cost. Sibling definitions introduce a domain-restricted cost, its evaluation identity, and nonnegativity; a canonical threshold with a positivity lemma; and a certificate structure that packages those facts.
No curvature or ADM machinery is assumed here. The local setting is purely the cost calculus needed to state a Penrose-shaped bound in RS units.
proof idea
Definition-and-certificate module, not a deep derivation. It introduces domainCost with an evaluation lemma and a nonnegativity proof, fixes canonicalThreshold with a positivity proof, then bundles both into the PenroseIneqCert record and supplies an inhabited instance. Argument structure is interface packaging: the mathematical content is the cost/threshold pair, not a multi-step geometric estimate.
why it matters in Recognition Science
Inserts a J-cost route to a Penrose-type bound into the RS gravity stack. Classically the Penrose inequality links ADM mass to horizon area under dominant energy; here the same shape of claim is witnessed from the forced cost $J$ (T5) and the Cost library alone. Downstream use is not yet wired in the graph (no used_by edges), so the module functions as an interface layer for later black-hole or horizon theorems. It keeps gravity claims dependent only on Constants and Cost, isolating them from the full forcing chain (T6--T8) until those links are added.
scope and limits
- Does not prove the classical GR Penrose inequality or any spacetime curvature estimate.
- Does not derive Einstein equations, ADM mass, or horizon area formulas.
- Does not fix numerical mass or area values in SI or RS units.
- Does not invoke eight-tick, phi-ladder, or D=3 forcing steps.
- Certificate inhabitation is packaging, not a dynamical existence theorem.