IndisputableMonolith.Gravity.GravitationalEntanglementFromJCost
Module packaging a J-cost certificate for gravitational entanglement: a nonnegative domain cost, a positive canonical threshold, and an inhabited GravEntangCert record. Gravity theorists working in Recognition Science would cite it when linking entanglement witnesses to the RS cost functional. The content is mostly definitions and elementary positivity facts imported from Cost and Constants.
claimDefine a domain cost $C$ built from the RS $J$-cost, prove $C \ge 0$ and evaluate it at equality cases; fix a canonical positive threshold $\theta > 0$; package these into an inhabited gravitational-entanglement certificate $\mathrm{GravEntangCert}$.
background
Recognition Science measures mismatch by the unique cost $J(x) = (x + x^{-1})/2 - 1$ (equivalently $\cosh(\log x) - 1$), forced at T5 of the unified chain and obeying the Recognition Composition Law. The Cost import supplies that functional and its elementary inequalities; Constants supplies the RS tick $\tau_0$.
This gravity module lifts those primitives into a domain-level cost suitable for entanglement witnesses between gravitational degrees of freedom. The local objects are a domain cost (with nonnegativity and an evaluation identity), a canonical positive threshold against which the cost is compared, and a certificate record that bundles the pair for downstream gravity arguments.
proof idea
Definition-and-certificate module rather than a deep proof development. domainCost is introduced from the imported $J$-cost; domainCost_nonneg and domainCost_at_eq are short positivity and evaluation lemmas. canonicalThreshold and canonicalThreshold_pos fix a strictly positive comparison scale. GravEntangCert, cert, and cert_inhabited assemble those facts into an inhabited certificate structure with no heavy tactic work.
why it matters in Recognition Science
Places gravitational entanglement inside the same $J$-cost ledger used for the rest of RS physics, so entanglement thresholds sit on the same footing as the T5 uniqueness of $J$ and the RS-native constants. No downstream consumers are wired in the graph yet (used_by is empty); the module is a self-contained certificate surface for later gravity or decoherence theorems that need a nonnegative cost-versus-threshold witness. It does not itself derive Einstein equations or force $D=3$.
scope and limits
- Does not derive the Einstein field equations or Newtonian limits.
- Does not prove experimental detection of gravitational entanglement.
- Does not fix numerical values of $G$, $c$, or $\hbar$ beyond Cost/Constants imports.
- Does not establish uniqueness of the canonical threshold.
- Does not wire downstream gravity theorems; used_by is currently empty.