The model enters a fresh MathTree episode, which begins with the existing MathTree corpus. It searches MathTree for exact counting tools and identifies what the tree still lacks. Missing pieces are proved once and added as reusable results. Those pieces grow into Lean-verified results for dense, disjoint, scaled, and combined families. The unsolved problem is narrowed to genuine overlap. The model uses MathTree to inspect failed Lean steps, repair them interactively, and retain each verified advance. This closes the complete two-divisor case and preserves stronger residual estimates for the general proof. After publication, a fresh episode recovers the accepted results under new names and uses them immediately. The new run starts ahead instead of repeating earlier proofs. The user can contribute new mathematical directions, allowing both to work together toward the episode's goal. Several ambitious routes are tested in parallel: false polynomial, topological, Fourier, and interval arguments are rejected, while a new least-common-multiple structure survives. The surviving structure is formalized in Lean. Exact subset counting and inclusion-exclusion close every family whose full least common multiple first crosses the threshold. The model reuses accepted MathTree results to isolate the exact remaining obstacle: useful counting margin must outweigh repeated coverage. Stress tests eliminate fixed trees and independent layers, leaving an adaptive transport problem. A fresh episode recovers the published chain, closes the short-period and minimal-obstruction branches, and proves a new prefix-slack result. Only global overlap transport remains open.