Schimmel Asymmetric Autocatalysis and Mirror-Lane Connections

This paper presents two mathematical results for comparing mirrored structures: reversal symmetry in recursive absolute-difference arrays and component-exchange symmetry in kinetic systems. The first theorem establishes an exact correspondence between a finite sequence and its reversal under repeated absolute differences, including the exchange of left and right boundary values. The second establishes a correspondence between paired trajectories in kinetic models satisfying a specified exchange-symmetry condition. Together, these results provide a mathematical framework for comparing paired reaction pathways, identifying invariant quantities, and distinguishing differences caused by orientation or labeling from differences in system behavior. The paper discusses applications to asymmetric autocatalysis and molecular handedness. Both mathematical results are proved under their stated assumptions. Their application to a particular chemical reaction depends on whether that reaction satisfies the relevant conditions.