Lean 4 Formal Verification of 8/10 #1stProof Problems: Complete Proofs with AI--Human Pipeline, Partial QED for Q4 & Q6

We present solutions to the ten research-level mathematics problems posed by Abouzaid, Blumberg, Hairer, Kileel, Kolda, Nelson, Spielman, Srivastava, Ward, Weinberger, and Williams in their "First Proof" benchmark (arXiv:2602.05192, February 2025). The problems span stochastic analysis, representation theory, algebraic combinatorics, polynomial inequalities, equivariant homotopy theory, spectral graph theory, lattices in Lie groups, symplectic geometry, multi-view geometry, and numerical linear algebra.

AI–Human Pipeline. The mathematical reasoning, proof construction, Lean 4 formalization, and verification in this work were carried out primarily by AI agents (large language models operating in agentic mode), with human authors providing problem selection, strategic guidance, review, and editorial oversight. This pipeline demonstrates that current AI systems can serve as the principal workforce for research-level mathematical proof, while human collaborators ensure correctness and coherence.

Results. Of the ten problems, we provide complete, rigorous proofs for eight (Q1, Q2, Q3, Q5, Q7, Q8, Q9, Q10), each accompanied by a Lean 4 formal proof skeleton that axiomatizes external deep theorems and machine-checks the logical deduction chain. For the remaining two (Q4, Q6), we give substantial partial results and reduce each to a precisely stated minimal open problem:

• Q4: proved for n ≤ 3 (all pairs), all semi-Gaussian pairs (all n), with the semi-Gaussian concavity bottleneck (A+B ≥ 0) fully closed. The general case n ≥ 4 for arbitrary p, q remains open.

• Q6: seven special graph families fully proved, conditional c=1/6 result established; the resistance-degree inequality (RDI) route proved false for general graphs. Two reduced routes remain: the multi-bin non-stuckness bridge (targeting c=1/2) and the Spectral Radius Conjecture (SRC).

For Q7 (lattices with 2-torsion), we prove an unconditional YES for all d ≥ 5 via an L-theory transfer vanishing lemma that bypasses the rational assembly obstruction even in dimensions d ≡ 0 (mod 4). Each solution has undergone multiple rounds of independent review by both AI and human reviewers.