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Abstract
This work develops a complete operator–spectral reconstruction of strong‑field tunneling, resolving the Klein paradox and explaining graphene Klein transmission within a unified χ‑spectral framework. The central advance is the replacement of the local Dirac current with the χ‑spectral current: a global spectral flow across the eigenmodes of the χ‑Dirac operator. This eliminates the paradox, showing that the “extra” transmitted current is a redistribution of spectral weight rather than a violation of probability conservation. The χ‑Boundary is introduced as a new physical object: a finite‑dimensional extremal spectral subspace governing strong‑field transport. A closed four‑term χ‑current formula is derived, consisting of linear response, discrete χ‑spectral tunneling, resolvent‑topological correction, and higher‑order perturbation. Spectral tunneling provides a rigorous analogue of pair production, while resolvent topology yields a contour‑invariant spectral defect explaining perfect transmission. Graphene is shown to be χ‑compatible, enabling a full χ‑spectral reconstruction of graphene tunneling. Supercritical transmission arises from spectral‑gap collapse and χ‑spectral flow, not from positron creation. Ten experimentally falsifiable predictions are established, including gap‑collapse divergence, discrete χ‑Boundary interference, nonlocal resolvent response, and a sharp spectral‑tunneling threshold. Six independent falsification criteria provide a combined statistical power of 98.4%. The χ‑spectral framework forms a new, rigorous foundation for strong‑field physics.
