From Local Rewards to Recursive Gains: Hybrid AI Search for C₄-Free Hypercubes

Erdős Problem #86 asks how many edges of the $n$-dimensional hypercube can be retained without forming a $4$-cycle. AI-assisted search for recursive constructions must decide both where to search in this exponentially large space of coupled constraints and what to value, since immediate edge count can misrank seeds by their recursive potential. We present a hybrid system with two modes over compact symmetry-quotient states. A frozen language model proposes typed search actions that constraint programming executes and an independent verifier certifies; a low-dimensional mode saturates within one dimension and hands its verified seed pool to a high-dimensional mode, which a jump cycle reaches on already-computed recursive values. Both modes share the proposer, the verifier and a fitted head; the high-dimensional mode adds an analytic, exactly computed recursive advantage under the classical Brass–Harborth–Nienborg (BHN) lift to the score, and these values feed back into repair objectives and retention. The archive shows that a recursive state does not fix graph structure and that seeds with fewer edges can produce stronger descendants: a searched $Q_{15}$ seed starts 1,024 edges below the classical seed yet exceeds its same-template rollout by 12,288 edges after one lift and stays ahead at the next two depths. Combining complementary seed lineages yields a construction family that provably matches or exceeds the refined BHN benchmark for every $n\ge 7$. Across $9\le n\le 16,384$, six independently verified seeds realize the archive envelope, with strict improvements in 13,805 dimensions (84.3%). These results connect structural exploration to high-dimensional construction guarantees through template-specific recursive value.