A certified lower bound on the quantum-capacity threshold of the depolarizing channel
The noise threshold below which the qubit depolarizing channel retains positive quantum capacity has been studied since 1996. The classic constructions come with exact finite formulae, but in every reported threshold to date -- most recently the record of Agarwal et al. -- the final step, the sign of the coherent information at the reported point, is a floating-point evaluation. We give the first exact-arithmetic, independently machine-checkable positivity proofs in this regime: an explicit 45-copy rank-two input state $A$, published as a 1,472-byte witness, with certified $I_c>0$ at the exact rational $p = 16239/250000 = 0.064956$ (per-Pauli convention; total error rate $3p = 0.194868$) -- beyond both the best value printed in the Agarwal et al. paper ($0.064657$) and the strongest state in the same authors'public repository ($\approx 0.064911$), both numerical. Because the depolarizing family is a semigroup under composition, fixed-input coherent information is nonincreasing in $p$ on $[0,1/4]$, so a single certified point extends to the entire interval below it. The same machinery confines the positivity boundaries of $A$ and of the strongest public state of Agarwal et al. to disjoint rational intervals separated by more than $1/25000$ -- to our knowledge the first proven ordering of the positivity boundaries of two explicit competing code states. Every computational claim in the paper reduces to a finite list of big-integer comparisons, checkable by a dependency-free few-hundred-line verifier whose soundness rests on three self-contained half-page lemmas. Payloads, certificates, and verifier accompany the paper as a supplementary artifact.