Most mathematical research agents target problems with fixed statements. In applied‑mathematics conjecture research, however, investigators often need to revise candidate questions or conclusions as the inquiry proceeds, coupling problem formulation with proof development. To address this, we introduce PACE, a research‑proof agent that supports such adaptive workflows.\
PACE operates through a structured research loop that selects mathematical objectives and either generates or challenges candidate mechanisms. The full prover builds proofs from these mechanisms, while an independent verifier checks them against the original target.\
A central component is the Evidence‑Tracking Memory, which records assumptions and dependencies that arise during research and flags evidence for revalidation when a candidate or condition changes. This memory enables task‑specific context composition that keeps verified premises, exploratory material, and unresolved obligations distinct.\
To evaluate proof construction and conjecture research, we present ResearchProofBench, an applied‑mathematics benchmark focused on optimization. The benchmark comprises 51 known‑proof tasks (with 331 hidden checkpoints) and six open‑problem cases. Known tasks assess reconstruction of published results, while open cases assess mathematical investigation and new proof development.\
On the 51 known tasks, PACE achieves an aggregate Proof Score of 75.16 and completes 20 full proofs. In conjecture research, PACE produces conditional results whose submission review scores exceed those of baseline systems.\
Review: PACE demonstrates the potential of tightly coupling problem formulation with proof generation, especially in adaptive applied‑mathematics research. Its memory mechanism and independent verification provide strong support for interpretability and reliability, and the approach can be extended to more complex mathematical domains.