Bilateral Trade Under Heavy-Tailed Valuations: Minimax Regret without a Variance Bound

In contextual bilateral trade under full feedback, the posted price does not affect which valuations are observed. We show that in this model such action-independent feedback removes the polynomial adaptation penalty familiar from heavy-tailed bandits: fully parameter-free algorithms attain the oracle minimax T-exponents up to logarithmic factors, with no knowledge of the moment order p in (1,2) or its scale σ p, and -- in the nonparametric case -- none of the effective Hölder smoothness β in (0,1]. The statistic that makes model selection possible is a paired squared-loss difference, whose noise-square term cancels exactly, leaving noise damped by the candidate gap. The resulting bilateral-trade regret rates are new. Trader valuations have bounded conditional densities and heavy tails -- finite p-th moments for some p in (1,2), with possibly infinite variance. An epoch-based algorithm with truncated means achieves regret (T(2-p)/p) in the parametric model and (T 1-2β(p-1)/(βp + d(p-1))) when the market value function is β-Hölder, with matching Ω(times) lower bounds -- under a mild nondegeneracy condition -- via Assouad's method and a fixed-support mixture construction -- characterizing the minimax rate in T up to logarithmic factors over the effective smoothness range β in (0,1], interpolating between the classical nonparametric rate at p = 2 and the trivial linear rate as p to 1+. The enabling structural step extends the self-bounding property of Bachoc et al. (ICML 2025) from bounded to real-valued valuations: within our conditionally independent, conditionally centered noise model, bounded conditional densities and finite first moments suffice for the expected regret of any price π to satisfy E[g(m,V,W) - g(π,V,W)] Lm-π squared -- no second moment is needed.