Blockchain technology has been introduced to tackle the data issues arising from centralization in carbon management. However, current blockchain systems remain inefficient for carbon management in construction projects, falling short in scalability, security, and high throughput. Moreover, blockchain mainly serves as a database, so it is not easy to shape stakeholders' low-carbon behaviors actively and directly. Thus, this paper designs a novel incentive mechanism (i.e., consensus mechanism with incentive design) in blockchain, Proof of Carbon Reduction (PoCR), to enhance blockchain system performance and promote low carbon practices among stakeholders. To this end, carbon reduction credit (CR credit) is first proposed to evaluate stakeholders' carbon performance via the reporting and crowd-wisdom subsystems. Second, the optimized Byzantine Fault Tolerance (BFT) with a novel block leader election algorithm and a fair reward allocation method is proposed for the transaction consensus process in the blockchain to reduce communication complexity while enhancing consistency using cryptographic techniques. Third, controlled experiments are conducted and research findings show that PoCR can achieve (1) effective low-carbon incentive, providing financial returns for low-carbon performance and enhancing income fairness, (2) excellent efficiency, reducing the quadratic message complexity of typical BFT-based protocols to 0(mn), and increasing the throughput over 10 % (Delegated BFT) and 25 % (Algorand) while reducing latency over 80 % (Practical BFT) and 20 % (Algorand), (3) enhanced security, outperforming existing protocols in anti-attack and anti-corruption capabilities. This paper contributes to both blockchain design and low-carbon incentive design, paving the way for decentralized low-carbon management in construction.