Examine the logical coherence between the following political ideologies and their systemic mechanisms:
1. Evidence 1: Libertarians assert that individuals should be rewarded strictly according to their merit and effort in a free market. Inference 1: Therefore, Libertarian theory upholds open and fair competition as the necessary engine to achieve societal equality.
2. Evidence 2: Socialist doctrine dictates that massive economic disparities inherently corrupt democratic political systems. Inference 2: Consequently, Socialists believe that political, social, and economic inequalities are inextricably linked.
3. Evidence 3: Liberal philosophy advocates for a minimal welfare baseline, leaving the remainder of wealth distribution to the state apparatus. Inference 3: Thus, Liberals agree with Socialists that public control over essential resources is the only path to true justice.