Convex Projections and Dual Certificates for Energy-Condition Violations in Warp-Metric Stress Tensors

For a prescribed spacetime metric, we test whether its Einstein stress tensor satisfies the classical energy conditions using convex optimization with independently checkable certificates. Via the S-lemma, the null, weak and strong energy conditions become one-parameter linear matrix inequalities at each event, valid for every Hawking–Ellis type; violations carry separating certificates whose dual cone consists of causal mixtures. Adding covariant conservation on the fixed background, a weak-duality construction gives a verified lower bound on the distance to the nearest conserved, admissible tensor field.Applied to the Alcubierre and Natário drives, every admissible tensor field lies at a relative distance of at least 0.72, and we prove this tends to exactly 1/√2 for slow bubbles because momentum density, not negative energy, carries the obstruction. We also derive exact bounds on the positive matter needed to restore the energy conditions. For the positive-energy warp shell of Fuchs et al., rebuilt from its public source, all four energy conditions hold for every observer at the published shift amplitude of 0.02 and fail together at 0.0304, where a null constraint binds in the shell’s equatorial plane.The results concern admissibility on a fixed background, not realizable matter or geometries. Code, data and certificates are linked as supplementary material.