Against the backdrop of advancing the "dual carbon" goals and accelerating global climate governance, preventing cross-market risk contagion among carbon, energy, and financial markets induced by climate policy uncertainty is of great significance. A four-dimensional risk linkage analysis framework of "CPU-carbon-energy-finance" was constructed. The quantile vector autoregressive model and frequency domain decomposition method were employed to examine the risk spillover effects of climate policy uncertainty under normal and extreme conditions from both the time and frequency dimensions. The results show that climate policy uncertainty plays a pivotal role in risk transmission and exhibits a pronounced state-dependent characteristic. It acts as a net risk receiver under normal conditions but turns into a net risk transmitter under extreme scenarios, with its impact significantly amplified during the COVID-19 pandemic and geopolitical conflicts. This role reversal is closely associated with declining risk appetite, heightened liquidity constraints, and the reshaping of policy expectations. Short-term spillovers consistently dominate the transmission process, reflecting the rapid market response to information. However, medium- and long-term spillovers are markedly intensified under extreme market states, indicating that extreme events can awaken the market's "long-term memory" and lead to cross-period risk contagion. The energy market serves as a critical transmission node. During downturns, coal and natural gas reinforce risk concentration, while during upturns, oil becomes a primary spillover source and strengthens its linkages with the carbon market and commodity futures. The findings provide empirical evidence for constructing a state-dependent risk defense system, optimizing the design of energy transition pathways, and facilitating a smooth energy transition.