love_eliminates_all_waste
plain-language theorem explainer
The theorem states that for any nonzero real σ the total extraction cost of the balanced pair (σ, -σ) is strictly positive, yet drops exactly to zero once the Love operator averages the pair. Recognition Science researchers in ethics and thermodynamics would cite it to show that relational averaging removes all waste from balanced extraction. The proof is a term-mode split that checks positivity of the cosh expression for pairSystemCost and invokes the ground-state theorem for the post-Love cost.
Claim. For every real number σ ≠ 0, if C(σ, -σ) denotes the pair extraction cost before intervention and C_L(σ, -σ) the cost after Love averaging, then 0 < C(σ, -σ) and C_L(σ, -σ) = 0, where C_L(σ₁, σ₂) := 2 (cosh((σ₁ + σ₂)/2) - 1).
background
In the Thermodynamic Instability of Extraction module, pairSystemCost quantifies total pre-intervention cost for an extraction pair via the hyperbolic cosine that encodes the Recognition Composition Law. pairCostAfterLove applies the averaging operator that replaces the pair by its mean, producing 2(cosh(mean) - 1). The upstream theorem love_achieves_ground_state shows that this averaging drives any balanced pair to zero cost. The local setting treats extraction imbalance σ as a signed deviation whose marginal cost 2 sinh(σ) supplies a restoring force toward equilibrium.
proof idea
The term proof refines the conjunction into two subgoals. The left subgoal unfolds pairSystemCost, rewrites via cosh evenness, and applies linarith to the strict inequality cosh(σ) > 1 for σ ≠ 0. The right subgoal invokes love_achieves_ground_state directly.
why it matters
The declaration shows that Love removes all thermodynamic waste from balanced extraction, completing the local argument for a restoring force in the ethics module. It instantiates the Recognition Science claim that relational operators reach ground states, linking to J-uniqueness through the cosh representation of cost. No downstream uses appear yet, leaving open its integration with imported kinship or narrative structures.
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