A First Proof Sprint

2026-02-14

AI Summary

This work addresses the challenges of low efficiency and insufficient reliability in verifying and repairing research-level mathematical proofs during compressed proof sprints. We propose a multi-agent collaborative framework that integrates rapid draft generation, adversarial verification, targeted repair, and explicit provenance tracking. By decomposing propositions into dependency roadmaps to localize proof gaps, and incorporating structure-aware validation with inter-layer switching strategies, our approach distinguishes between mathematical reasoning and quality control states, enabling heterogeneous yet traceable proof artifacts. The framework has yielded progress on several open problems: an existence pathway for Problem 3, a constrained solution for Problem 5, conditional validity for Problem 10, and an unconditional Kₙ result for Problem 6 with c₀ = 1/3. Node-level verification artifacts are produced throughout, significantly enhancing the calibratability and reliability of the proof process.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Replanning and Plan Repair

Application Category

• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Agentic search

Abstract

This monograph reports a multi-agent proof sprint on ten research-level problems, combining rapid draft generation with adversarial verification, targeted repair, and explicit provenance. The workflow uses wiring-diagram decompositions of claim dependencies to localize gaps and coordinate reviewer-driven revisions. Final outcomes are heterogeneous but explicit: the manuscript distinguishes mathematical status from QC-validation status. Mathematically, Problem~3 has a validation-complete existence path under the scoped criterion used here (uniqueness/irreducibility treated as optional), Problem 5 is solved in a scope-limited form for $F_O$-local connective spectra, Problem 10 is conditional under clearly stated assumptions (with explicit necessity counterexamples when assumptions are dropped), and Problems 4 and 6 are partial with named remaining obligations in the general case (including an unconditional $K_n$ result for Problem 6 with $c_0 = 1/3$). Problem 7 is treated as provisionally closed via the rotation-route theorem chain, pending independent ledger re-check. At the QC layer, Problems~7 and~9 have node-level validation artifacts but still contain unresolved verifier gaps. The main methodological result is that structure-aware verification and layer-switching strategies improve reliability and calibration in compressed proof sprints.

Problem

• mathematical proof
• verification gaps
• existence and uniqueness
• conditional assumptions
• local connective spectra

Innovation

• multi-agent proof sprint
• wiring-diagram decomposition
• adversarial verification
• structure-aware verification
• layer-switching strategies