curvatureCost
plain-language theorem explainer
curvatureCost assigns the J-cost penalty to any deviation of the density parameter from unity. Cosmologists working on the flatness problem cite it to quantify why only Ω = 1 carries zero recognition cost. The definition is a direct one-line wrapper that feeds the squared curvature term into the J-cost evaluator.
Claim. The curvature cost for density parameter $Ω$ is $J(1 + (Ω - 1)^2)$, where $J$ is the J-cost function that returns zero only at the critical density.
background
In the Recognition Science module COS-005 the flatness problem is stated as the requirement that |Ω − 1| < 10^{-60} at the Planck epoch, because any deviation grows as a²(t) and the critical density is an unstable fixed point. The RS resolution asserts that only Ω = 1 is consistent with ledger structure enforced by J-cost minimization. The J-cost function itself is supplied by the upstream definitions cost in MultiplicativeRecognizerL4 (derivedCost of the comparator on positive ratios) and cost in ObserverForcing (J-cost of a recognition event state).
proof idea
This is a one-line definition that applies the J-cost function to the expression 1 + (Ω − 1)².
why it matters
The definition supplies the cost function used by the downstream theorem flat_minimizes_cost, which proves that curvatureCost 1 is the global minimum. It fills the COS-005 claim that critical density follows from J-cost minimization and that φ-constraints lock the universe to flatness. No open scaffolding questions are attached to this declaration.
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