Latent Mechanism Design: Spectral Approximation of Optimal Mechanisms

We apply the Latent spectral framework to mechanism design, framing bilateral-trade tensions through a spectral parameter \(\rho\) (Latent Number) of the type distribution. The VCG mechanism achieves full efficiency but runs a budget deficit; Myerson and Satterthwaite (1983) proved that no mechanism simultaneously achieves efficiency, incentive compatibility, individual rationality, and budget balance. Machine-checked lemmas include stepwise geometric decay of a formal approximation error under a multiplicative recurrence (Theorem 5), explicit \(\rho=2\) error normalizations (Theorem 6), and a gap comparison across coupled \((r,\Delta)\) scalings (Theorem 7). For \(\rho = 2\), Theorem 6 gives five-component approximation error \(\varepsilon_5 = C/32\); the verified numerics use spectral error \(\varepsilon = 1/8\) and the uniform bilateral-trade gap calibration \(1/4\), yielding efficiency \(\geq 7/8\) (87.5%) and a deficit bound \(1/32 \approx 3.1\%\) of full surplus scale, with strictly lower deficit than a VCG benchmark under the hypotheses of latent_beats_vcg_deficit. 16 theorems, machine-verified in Lean 4 (0 user axioms in the companion file).