BU98 Weak Persistent Morphism and AI Re-Activation of Process Residues under B_U

BU98 develops a B_U-based structural account of weak persistent morphism in AI mathematical discovery and cross-domain construction. Its central claim is that recent AI breakthroughs in discrete geometry, extremal combinatorics, graph construction, optimization search, and research-agent workflows reveal a shared structural mechanism: retainable process residues, compressed inside backward-mirror formal systems, can be reactivated by general-purpose reasoning models and carried through a co-admissive medium toward new minimal standing. The paper begins from the OpenAI unit-distance breakthrough. A general-purpose reasoning model generated a counterexample family using algebraic number theory, including unramified class field towers and Golod–Shafarevich-type machinery, to produce a polynomial improvement over the long-standing near-linear expectation in Erdős’s unit-distance problem. This case shows that the visible mathematical result is only the settlement surface of a deeper chain. The backward mirror preserves the citeable outcome: theorem, proof, formula, construction, and reference. Yet the processual reasoning chain behind that outcome is often compressed, fragmented, or hidden in standard arguments, adjacent literatures, failed pathways, and under-indexed structural residues. AI gains power precisely in this compressed band: it can scan visible outcomes, reactivate process residues, generate candidate structures, submit them to evaluators, and retain the fragments that survive verification. BU98 then places seven AI-related discovery cases into the same structural class, including FunSearch, the OpenAI unit-distance construction, Aletheia / Gemini Deep Think research agents, AlphaEvolve-style evolutionary coding agents, earlier machine-learning-guided mathematical intuition, extremal graph and theoretical-computer-science constructions, and LLM-driven search-verification systems. Across these cases, the same chain appears: candidate generation, evaluator filtering, residual retention, recombination, friction-cost reduction, feedback, and settlement. The core phenomenon is not mere automation of proof. It is the reactivation of latent process structure inside a verifiable construction space. Under B_U notation, this paper locks the chain: M_U \rightarrow p^w \rightarrow L_s \rightarrow G_{tl} \rightarrow C_{fric} \rightarrow Standing_{new} The universal co-admissive medium M_U carries cross-domain connectability. Weak persistent morphism p^w preserves minimal traceable continuity across domain shifts, failed attempts, perturbations, and verification windows. Soft links L_s open translinkage relations. Translinkage gradient flow G_{tl} advances path probing. The friction-cost functional C_{fric} screens paths according to cost, stability, and re-admissibility. Feedback and settlement determine whether a candidate reaches new standing. BU98 strengthens the claim that the backward mirror generally lacks explicit cognition of weak persistent morphism. Formal systems record stable achievements while frequently erasing the continuity that made them reachable. AI discovery becomes structurally powerful when it re-enters this erased zone, retrieves retainable residues, and pushes them through a shared settlement grammar. The final judgment is direct: AI mathematical discovery does not replace settlement; it expands the search field for candidates capable of settlement. Weak persistent morphism names the missing bridge between stored results, reactivated residues, verifiable construction, and new standing under B_U.