Efficient direct computation of the potential function

Determine how to compute the potential function $\Phi^{(\ell)}_s$ efficiently and directly when selecting primes for the deterministic sparse-certificate construction.

Background

The deterministic algorithm constructs, level by level, sets of primes intended to reduce the number of false candidate witnesses. The potential function Φs(ℓ)\Phi^{(\ell)}_s measures the number of tuples of false witnesses that survive the currently selected primes, and minimizing it would provide a natural way to choose the next prime.

The authors state that direct efficient computation of this potential is unclear. They therefore introduce an auxiliary estimator Zs+1(p′)Z_{s+1}(p'), prove that it approximates the relevant potential, and choose primes by minimizing that estimator instead. Thus, the paper resolves the prime-selection task indirectly but leaves direct efficient computation of Φs(ℓ)\Phi^{(\ell)}_s unidentified.

References

However, since it is not clear how to compute $\Phi{(\ell)}_{s}$ efficiently directly, we must take an alternative approach.

— Simple, Faster (min, +) Matrix Multiplication and APSP  (2610.20297 - Saha et al., 14 Oct 2026) in Section “Faster $(\min,+)$ Matrix Multiplication,” subsection “Derandomization,” paragraph following Lemma 3 (Lemma “Finding a Good Modulus”)