A strengthening of the MCFL-ness of $O_2$

We study positive-data learning of bounded-fan-out linear multiple context-free grammars under a fixed explicit finite monoid homomorphism $h$. The main obstacle beyond the context-free case is that an MCFG nonterminal derives a tuple whose components may be placed in a surrounding sentence in different orders. We introduce sentence-interface types as finite external control objects for such tuple occurrences. A type records the permutation of tuple components in the final sentence together with the $h$-values of the boundary intervals between them. For reduced working binary linear nondeleting MCFG presentations whose string languages satisfy $(f,h)$-tuple substitutability, we build a typed refinement, a finite characteristic sample, and a canonical positive-data learner. Once the sample contains this characteristic sample and remains contained in the target language, the learner reconstructs the language exactly. Consequently, for fixed fan-out bound $f$ and fixed explicit $h$, the resulting class is identifiable in the limit from positive data. Moreover, the hypothesis associated with any given finite sample is constructible in polynomial time for fixed $f$ and fixed $h$, including output size. Thus sentence-interface control is the finite mechanism that lifts fixed-$h$ distributional reconstruction from context-free grammars to bounded-fan-out linear MCFGs.

Let $L\subseteqΣ^*$ and fix a morphism $h:Σ^*\to M$ into a finite monoid. We study exact factorization and canonical presentation in the relative syntactic congruence $θ_{L,h}:=\equiv_L\cap\ker h$. We separate unique factorization from finite direct presentation. An exhaustively computer-checked $36$-element quotient has a unique exact prime factorization for every live non-unit class, yet its valid prime-return rules contain an infinite family, so unique factorization does not imply the finite relative presentation property (FRP), even for a finite quotient. We lift the same defect to a nonregular context-free language with an infinite relative quotient and finite prime spectrum. To isolate the obstruction, we introduce the finite-state relative presentation property (FSRP), in which canonical valid right-hand-side languages are represented by finite residual controllers, and prove $\mathrm{FRP}\subsetneq\mathrm{FSRP}$. We then introduce prime-target left-division determinism (PTLD), which implies unique exact factorization, tail exactness, tail determinism, and a quadratic bound on valid rules. A nonregular deterministic context-free example with a finite group observer satisfies PTLD while lying outside every fixed $(k,\ell)$-substitutable class. Finally, for fixed $h$ we give a strong positive-data learner for the canonical PTLD presentation with polynomial-time hypothesis updates and a finite characteristic sample, together with a limit reconstruction of the canonical FSRP controller from weakly behaviorally correct CFG-valued learners.

We report on an experiment in which Claude Opus~4.6, equipped with a suite of Model Context Protocol (MCP) tools for the Rocq proof assistant, autonomously proved 10 of 12 problems from the 2025 Putnam Mathematical Competition. The MCP tools, designed with Claude by analyzing logs from a prior experiment on miniF2F-Rocq, encode a "compile-first, interactive-fallback" strategy. Running on an isolated VM with no internet access, the agent deployed 141 subagents over 17.7 hours of active compute (51.6h wall-clock), consuming approximately 1.9 billion tokens. All proofs are publicly available.