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The question is what electrons and photons actually do in that regime and what experiments would see.","posed_since":"","precise":"Assuming the loop expansion in a constant crossed field fails for $\\alpha \\chi^{2/3} >> 1$, construct a nonperturbative description, for example self-consistent Schwinger-Dyson equations for the dressed electron and photon propagators with a controlled truncation, an effective running coupling, or a semiclassical strong-coupling expansion. Give the resulting dressed dispersion relations and the rates of photon emission and pair creation, and show that the scheme reproduces the perturbative results for $\\alpha \\chi^{2/3} << 1$.","problem_ref":null,"references":"","settled_by":"A nonperturbative framework whose truncation error is controlled, verified against known one- and two-loop results, and yielding testable rates.","status_note":"","title":"What theory describes QED at $\\alpha \\chi^{2/3}$ much larger than 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