Guaranteed Slater Conditions for the Numerical Conformal Bootstrap: Fixing Instabilities in SDPB, Navigator Methods, and Skydiving

Abstract

The semidefinite programs (SDPs) of the numerical conformal bootstrap, and of S-matrix and quantum-mechanics bootstraps, are solved with primal-dual interior-point methods whose convergence guarantees rest on primal and dual Slater conditions: the existence of strictly feasible points on both sides of the duality. Many bootstrap SDPs violate these. The consequences range from slow convergence to inconclusive verdicts and the breakdown of the Skydive algorithm. We introduce "Slater tools", all acting within the PMP class accepted by SDPB. Diagnostics measure the Slater margins and, when a margin vanishes, extract the exposing certificate that identifies the degeneracy. Two independent cures follow. Transforms that provide rigorous bounds on the original SDP objective and that for the right parameter choices return the exact value. Facial reductions that use the exposing certificate to remove the degeneracies themselves. We reproduce, explain and resolve all known types of failures in a mixed-correlator 3d Ising setup. We show that the $X$-Slater condition (SDPB's primal) fails at every disallowed point in feasibility mode, holds in every navigator problem we tested, and can be guaranteed by one additional condition on the navigator function. We show that the folklore "instability remedy" of a twist gap works by removing degeneracies at the endpoints of the positivity conditions. These often stem from a mistake in factoring out the block prefactor but can also arise legitimately in channels that mix equal and unequal operator pairs. Finally we classify and remove all degeneracies of the published current-scalar bootstrap, among them one at $x=\infty$, thus incurable by any gap assumption, caused by unequal polynomial degrees within one matrix block.