Odd-Sum Minkowski Inclusion in Reversal-Limited Peak Gain

For a fixed discrete-time linear time-invariant system with a scalar output and fixed disturbance amplitude limits, we establish a law for peak-gain increments across reversal bounds. On a finite horizon, the gain increment from one additional reversal may increase between consecutive bounds, but cannot increase when the starting bound rises by two. We prove this law by recombining binary sequences while preserving their sum at each index and balancing their transition counts within two. At each interior bound, three consecutive constrained gains determine the exact gap to the tightest upper bound from a linear penalty on the reversal count with an optimized nonnegative weight. Explicit truncation bounds extend both results to the infinite horizon for stable systems with real, absolutely summable impulse responses. For multiple inputs, both results hold when the bound limits the sum of channelwise reversal counts. The worst-case output peak equals the maximum sum of channel peak gains scaled by their disturbance amplitude limits over admissible allocations.