Impossibility Results

The impossibility results explain why neural activations and polynomial-map step size play structural computational roles. No polynomial with a fixed number of iterates can uniformly round every point near every integer from tolerance δ to a smaller tolerance ε. No fixed polynomial map can exactly produce a phase indicator that switches from 1 to 0 at every integer counter threshold. Uniform rounding and exact phase selection are characterized as fundamental requirements for long integer computations.