Conthereum is a concurrent Ethereum solution for intra-block parallel transaction execution, enabling validators to utilize multi-core infrastructure and transform the sequential execution model of Ethereum into a parallel one. This shift significantly increases throughput and transactions per second (TPS), while ensuring conflict-free execution in both proposer and attestor modes and preserving execution order consistency in the attestor. At the heart of Conthereum is a novel, lightweight, high-performance scheduler inspired by the Flexible Job Shop Scheduling Problem (FJSS). We propose a custom greedy heuristic algorithm, along with its efficient implementation, that solves this formulation effectively and decisively outperforms existing scheduling methods in finding suboptimal solutions that satisfy the constraints, achieve minimal makespan, and maximize speedup in parallel execution. Additionally, Conthereum includes an offline phase that equips its real-time scheduler with a conflict analysis repository obtained through static analysis of smart contracts, identifying potentially conflicting functions using a pessimistic approach. Building on this novel scheduler and extensive conflict data, Conthereum outperforms existing concurrent intra-block solutions. Empirical evaluations show near-linear throughput gains with increasing computational power on standard 8-core machines. Although scalability deviates from linear with higher core counts and increased transaction conflicts, Conthereum still significantly improves upon the current sequential execution model and outperforms existing concurrent solutions under a wide range of conditions.