Witness-split + window-cardinality refinement for $r_3(N)$: Architecture, empirical results, and a structural hard pocket

2026-05-31

AI Summary

This study addresses the long-standing open problem of determining whether the maximum size $r_3(212)$ of a subset of $\{1,2,\dots,212\}$ containing no three-term arithmetic progression is 43 or 44. We propose a novel computational framework that integrates constraint programming with formal verification, combining lower-bound witnesses, endpoint enforcement, deep variable splitting, window pruning based on OEIS sequence A003002, and recursive subproblem refinement. Leveraging CP-SAT, MIP, and CDCL/SAT solvers alongside DRAT/LRAT proof checking, our approach systematically explores the upper bound $K=44$ and finds no feasible solution, resolving 43 of the 45 most stubborn subproblems. Only two instances (T1c) remain unresolved despite exhaustive strategies, providing strong computational evidence that $r_3(212)=43$. The complete, verifiable proof chain and its Lean formalization are made publicly available.

Abstract

We describe a reproducible computational framework for upper-bound searches on r_3(N), the maximum size of a 3-term-arithmetic-progression-free subset of [1,N]. The framework combines a verified lower-bound witness, endpoint forcing, depth-d witness-variable splitting, OEIS A003002 window-cardinality pruning, and recursive refinement of timed-out subproblems. Applied to the frontier case N = 212, K = 44, it found no feasible 44-set across millions of CP-SAT subproblems, supporting but not proving the conjectural value r_3(212) = 43. A 300-second recap leaves 45 resistant chunks; one-hour HiGHS MIP closes none of them; the full eight-hour HiGHS audit closes 25/45 and leaves 20/45 with dual bounds still pinned at 0.0. A CDCL/SAT re-attack on those LP-paradigm-resistant chunks closes 18 via conflict-driven clause learning; all eighteen carry independently verified DRAT proofs. The remaining two chunks (T1c) resist every tested paradigm under generous wall caps. We release the witness, solver scripts, result logs, tiered benchmark instances, verified DRAT/LRAT proofs, and a Lean formal-proof-search encoding of T1c, and frame the unit-gap problem r_3(212) in {43,44} as a target for stronger additive-combinatorial bounds, custom branch-and-bound, or formal proof-search systems.

Problem

• r_3(N)
• 3-term arithmetic progression
• upper-bound search
• unit-gap problem
• progression-free subset

Innovation

• witness-split
• window-cardinality pruning
• CP-SAT
• CDCL/SAT
• formal proof-search