Recursive Reasoning Exploration: Mathematical Research through Evolving Reasoning Graphs : case study: An Exact Certificate-Degree Theorem for Two-Term Tensor Decompositions : Phase III of the Math Instruments Program

Abstract

Large collections of manuscripts and research notes offer ways to reuse arguments, while making it harder to retain hypotheses, check claims, and understand how results were obtained. We propose Recursive Reasoning Exploration (RRE), a framework for cumulative mathematical research through evolving reasoning graphs. RRE links claims and jointly required premises to research targets, attempted arguments, counterexamples, checks, and revisions. Each investigation can generate further targets, and its supported outcomes become available to subsequent investigations. This is Phase III of the Math Instruments program: identifying mathematical instruments, reconstructing reusable reasoning, and organizing recursive research from that reasoning. The framework supports human-directed, collaborative, and goal-directed autonomous exploration, with open-ended exploration as a further mode. Its records also provide material for learning how mathematical investigations develop. A complexity research case study establishes an exact theorem: for every complex tensor of order n >= 2 and rank greater than two, the minimum Nullstellensatz certificate degree for its original two-term decomposition equations is 3n. This includes an infinite family whose every flattening has rank at most two. The exploration develops an explicit boundary certificate, a balanced transfer lemma, and an exhaustive argument using classical rank-two geometry. A parallel branch gives bounds 3m + 3 <= delta_m(I_m, N) <= 3m + 6 for every nonzero nilpotent m by m matrix. Six graph revisions record the progression from inherited results to these theorems, with exact arithmetic checks and a replayable exploration history. We present the RRE representation and protocol, the mathematical proofs, visualizations of graph evolution, and a research agenda for shared human-AI exploration, cumulative reuse, and education.