IndisputableMonolith.Foundation.Entanglement_Cost_RS
Foundation module that defines Recognition Science entanglement cost via a domain-cost functional and a canonical positive threshold, plus nonnegativity lemmas and an inhabited certificate type. Cost and constants theorists cite it when binding entanglement to the J-cost layer. Argument is definitional: short algebraic sign proofs and structure packing, not a deep derivation.
claimThe module defines a domain cost $C_{\mathrm{dom}}$, proves $C_{\mathrm{dom}}\ge 0$, introduces a canonical entanglement threshold $\theta_*>0$, and packages an inhabited certificate asserting the RS entanglement-cost bound.
background
Recognition Science measures discrepancy with the unique J-cost $J(x)=(x+x^{-1})/2-1$ (forcing step T5), obeying the Recognition Composition Law. The imported Cost layer supplies that functional infrastructure; Constants supplies the RS-native tick $\tau_0=1$.
This module specializes those primitives to entanglement. It introduces a domain cost that evaluates RS cost on domain data, records its nonnegativity, and fixes a canonical positive threshold against which entanglement cost is compared. A small certificate type then packages the numerical claim for later ledger or verification use.
No new forcing-chain step is proved here; the setting is purely the cost-and-threshold interface needed before entanglement can sit on the same footing as other RS observables.
proof idea
Definition-first module. Domain cost and the canonical threshold are introduced as definitions; nonnegativity of domain cost and positivity of the threshold are short algebraic lemmas inherited from the Cost layer and constant positivity. The certificate structure and its inhabited instance are pure packaging: they wrap the bound as a Prop-carrying record so downstream code can demand a witness without re-proving the sign facts.
why it matters in Recognition Science
Gives entanglement a first-class RS cost bound in native units, aligned with the J-cost and tick conventions used across the foundation. Downstream graph edges are currently empty, so the module is a leaf awaiting consumers in measurement, ledger, or entanglement layers. It does not itself close a T0–T8 step, but it is the natural place those layers will look when they need a certified lower bound on entanglement cost rather than an ad-hoc real parameter.
scope and limits
- Does not derive entanglement cost from the Recognition Composition Law.
- Does not equate the RS threshold to laboratory entanglement monotones.
- Does not advance any T0–T8 forcing obligation.
- Does not tie the bound to the mass ladder, alpha band, or eight-tick dynamics.
- Does not claim uniqueness of the threshold beyond positivity.