Spectral Pricing: Bayesian Learning and the Explore-Exploit Frontier via the Latent Framework

A seller facing unknown demand must balance exploration (learning the demand curve) against exploitation (maximizing immediate revenue). We organize the explore-exploit tradeoff through the Latent Number \(\rho\) of the demand function. Under a one-step multiplicative template on variance proxies (successive variances satisfying \(\mathrm{Var}_t = \rho\,\mathrm{Var}_{t+1}\)), iterating yields geometric contraction \(\propto \rho^{-t}\); the companion script proves matching one-step geometric templates for per-period regret (Theorem 5) and exploration horizons (Theorem 8). The narrative uses the stylized scalings \(R(T) \propto \log(T)/\log(\rho)\) and \(\tau^ \propto \log(T)/\log(\rho)\) as standard bandit-style targets motivated after those templates—not as new uniform theorems proved here. Spectral language motivates truncating to \(N^\) components with error of order \(\rho^{-N}\) at the level of the encoded algebraic templates. Airline-style versus grocery-style scenarios (\(\rho \approx 4\) vs.\ \(\rho \approx 10\)) appear as illustrative calibrations with \(5\times\) exploration and \(4\times\) relative regret ratios fixed as hypotheses in the script. Fifteen lemmas in the companion proof file, machine-verified in the Lean 4 real-arithmetic kernel, zero user axioms.