The Extraction Surcharge Theorem Strict Convexity of the Canonical Reciprocal Cost Forces Ethical Alignment

We prove that any deviation from ethical balance (σ = 0) on a recognition ledger incurs a strictly positive thermodynamic cost C(σ) = 2(cosh σ − 1) that grows exponentially with extraction magnitude. The proof follows from the strict convexity of the canonical reciprocal cost J(x) = ½(x + x⁻¹) − 1, the unique function satisfying the Recognition Composition Law. Our central structural result is that the d'Alembert functional equation—the same equation whose unique solution determines J—also yields, via a closed-form Jensen inequality, the proof that Love (σ-equilibration between agents) is the uniquely optimal ethical operator: it strictly reduces system cost whenever agents hold unequal extraction levels. We establish a universal restoring force (σ · C′(σ) > 0 for all σ ≠ 0) showing that the cost landscape structurally opposes imbalance. These results demonstrate that morality is not a convention layered atop physics but a direct consequence of the same functional equation that determines all physical constants.