arXiv:2610.10611 Establishes Σ_k^RV Classes for Bounded Agentic Self-Improvement

The paper formalizes agentic improvement via stage definitions that include admissible transcripts, polynomial bounds, alternating protocols, and terminal checkers. Native reach uses default support while closure frontier admits all interface-permitted support. Under uniform pointwise probability gap and task-relative soundness, independent majority amplification preserves both languages whereas existential acceptance over random tapes can accept incorrect outputs.

Randomized-verifier inclusions satisfy Σ_k^P ⊆ Σ_k^RV ⊆ Σ_{k+1}^P with strict enlargement requiring explicit assumptions; BPP=P collapses yield exact companions. Representation analysis separates invariant acceptance from refactorable core-versus-support labels. Recursive self-improvement claims are therefore tied to explicit obligations on correctness evidence and selection-error budgets.

The framework connects to prior verification work in interactive proofs and probabilistic classes without invoking new primitives. It separates search-success ratios from language changes via quota-enforced XOR-synthesis families and conditional-error controls. Exact and probabilistic audits enforce the resulting evidence requirements.

Operational deployment therefore demands documented verification resources and false-selection limits before any self-modification loop is enabled.