Prove the tighter test-count bound for unconstrained pooling

Prove that, for sufficiently large population size n in the unconstrained pooling design with pool-inclusion probability p, the Top-k LoSc decoder identifies all k positive samples with probability at least 1−ε whenever the number of pooled tests satisfies m = O(log((n−k)/ε) / [−log(1−p(1−p)^k)]).

Background

The paper derives a rigorous but conservative sufficient bound on the number of pooled tests required by the Top-k LoSc decoder under unconstrained Bernoulli pooling. The authors then propose a sharper asymptotic scaling law involving the probability that a negative sample remains in the candidate set, namely 1−p(1−p)k.

Numerical experiments are reported to support this tighter scaling relationship for the minimum number of tests needed to attain a target failure probability ε. However, the proposed asymptotic expression is presented as a conjecture, and the paper explicitly leaves its rigorous proof unresolved.

References

There might be tighter bounds for the unconstrained and dilution-constrained pooling designs.

— Sparse Identification for Automatic Large-Scale Screening: A Constraint-Aware Framework with Ultra Fast Decoding Algorithm  (2609.21321 - Li et al., 18 Sep 2026) in Section 3, subsection “Conjectures on Tighter Bounds,” Conjecture 1 (labeled unconstrained_conjecture)

The above conjectures are supported by our experimental results, whereas rigorous proofs are left for future research.

— Sparse Identification for Automatic Large-Scale Screening: A Constraint-Aware Framework with Ultra Fast Decoding Algorithm  (2609.21321 - Li et al., 18 Sep 2026) in Section 3, subsection “Conjectures on Tighter Bounds,” Conjecture 2 (labeled dilution_constrained_conjecture)