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Multi-agent collaboration has emerged as a promising paradigm for enhancing reasoning capabilities of Large Language Models (LLMs). However, existing approaches remain largely heuristic, lacking principled guidance on what drives performance gains and how to systematically optimize multi-agent reasoning. Specifically, it remains unclear why multi-agent collaboration outperforms single-agent reasoning and which design choices contribute most to these gains, making it difficult to build better systems. We address this gap by introducing a unified theoretical framework that decomposes multi-agent reasoning gains into three conceptually independent dimensions: Exploration for diverse solution coverage, Information for high-fidelity feedback, and Aggregation for principled consensus. Through this lens, existing methods can be understood as special cases that optimize only subsets of these dimensions. Building upon this decomposition, a novel framework called PRISM (Propose-Review-Integrate Synthesis for Multi-agent Reasoning) is proposed, which jointly maximizes all three dimensions through role-based diversity, execution-grounded feedback with evidence-based cross-evaluation, and iterative synthesis with closed-loop validation. Extensive experiments across mathematical reasoning, code generation, and function calling benchmarks demonstrate that PRISM achieves state-of-the-art performance with superior compute-efficiency compared to methods optimizing partial dimensions. The theoretical framework provides actionable design principles for future multi-agent reasoning systems.