Continuum energy-condition certificates for an optimized positive-energy warp shell

We prove that the null, weak, strong and dominant energy conditions hold at every point of the continuum geometry of an optimized positive-energy warp shell, for every shift amplitude from 0 to 0.1363, and that the null and dominant conditions fail at β = 0.1364. The first loss of admissibility therefore lies in [0.1363, 0.1364], and 0.1363 is a proven lower bound on the critical amplitude of this design.The geometry is a stationary, axisymmetric shell with a curved spatial metric and a lapse different from 1. It is written in closed form, and the radial tables of the companion optimization paper sample it to about 10⁻⁸. Every radial quantity is enclosed rigorously in ball arithmetic. The all-orders stress tensor is rewritten so that it is regular on the shift axis and so that the vacuum curvature cancels symbolically before any interval evaluation.The shell is covered by 4,600,144 cells in radius, angle and amplitude. On each cell, matrix certificates of the four conditions, with multipliers fixed on the cell, are proven positive definite by interval Taylor bounds and interval Cholesky tests. Near the two edges of the support, where the tensor vanishes like the fourth power of the distance, T/ρ is enclosed by a mean-value argument from a vacuum metric. No cell is left unresolved.Independent checks support the cover: • an exact tiling audit; • a containment argument linking the logged cells to the enclosed sets; • a replay of sampled cells, with the proof’s Taylor models and remainder bounds tested against an independently computed tensor; • an extended-precision re-evaluation, with negative and shuffled-multiplier controls.The results concern the energy conditions of the stress tensor implied by the metric. No matter model, stability, formation or acceleration is claimed. Code and complete cell logs are published separately.