flat_minimizes_cost
plain-language theorem explainer
Flat geometry minimizes the curvature cost in the Recognition Science ledger. A cosmologist using the J-cost functional would invoke this result to show why deviations from Ω=1 incur positive cost. The argument reduces directly to the non-negativity of Jcost and its vanishing at argument 1.
Claim. Let $J$ denote the J-cost function. Then $J(1) ≤ J(1 + (0.01)^2)$, where the curvature cost is defined by curvatureCost(Ω) = $J(1 + (Ω - 1)^2)$.
background
The module addresses the flatness problem: observed Ω = ρ/ρ_c lies within 0.0002 of unity, yet standard cosmology treats Ω=1 as an unstable fixed point whose deviations grow as a²(t). In Recognition Science the ledger structure forces Ω=1 as the sole consistent value because any curvature raises J-cost. The auxiliary definition curvatureCost(Ω) := Jcost(1 + (Ω-1)²) encodes this penalty directly from the J-cost functional. Upstream lemmas establish Jcost(1)=0 and Jcost(x)≥0 for all x>0.
proof idea
The term proof unfolds curvatureCost, simplifies the squared deviation at Ω=1 to zero, rewrites Jcost(1)=0 via Jcost_unit0, applies Jcost_nonneg to the positive argument 1.0001, and closes the inequality by norm_num.
why it matters
The result supplies the cost-minimization step required by the module's RS solution to the flatness problem, showing that ledger consistency alone selects Ω=1. It sits downstream of J-uniqueness (T5) and the Recognition Composition Law, and upstream of any claim that φ-constraints lock the universe to flatness without invoking inflation. No open scaffolding remains for this specific inequality.
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