Exact Syndrome Budgets for Quantum Erasure Recovery

We study the number of independent stabilizer syndrome outcomes needed to resolve errors at known erased locations. For every matrix with paired columns, rank r, and maximum independent-pair cardinality ν, we prove the exact profile max |I|=δ rankGI = min{2δ,δ +ν,r}. One ordering of the pairs attains every entry of this profile. For an isotropic stabilizer space encoding at least one logical qudit, every inclusionwise maximal independent-pair set A satisfies d ≤|A|+1. Consequently, in the declared syndrome model, the worst-case measurement budget for δ < d erasures is exactly 2δ, including for degenerate codes. A CSS specialization gives matching and rank-partition certificates. For binary surface-code check matrices, we replace dense elimination with connected-component selection of an optimal native check set, in linear time in the sparse affected incidence input. Complete proofs, exact finite-field audits, and executable code accompany the manuscript.