Discrete Symmetries, Symmetry Breaking, and Exact Zero-Pole Dynamics in Colliding Massless Rational Pulses

Discrete symmetries constrain the motion and real axis crossings of the interference zeros in the meromorphic continuation of a wave field, thereby constraining the singular structure of its logarithmic derivatives. We consider two counter-propagating second-order rational pulses that satisfy the one-dimensional massless wave equation exactly. With q=rexp(iφ) denoting the relative complex amplitude, the field admits two antilinear reflection symmetries: real q preserves collision centered spacetime inversion followed by complex conjugation, whereas |q|=1 preserves fixed time spatial reflection followed by conjugation up to an overall phase. The balanced in phase state q=1 is the nondegenerate intersection of these symmetry manifolds; q=-1 produces global cancellation on the collision slice. For q=1, the two interference zero branches lie on the imaginary axis and cross the real axis at ct=x₀±l, on opposite sides of the pulse center collision. Away from the symmetry manifolds, the zero trajectories deform continuously and the associated pairing constraints are lost, while the crossing conditions remain available in closed form. A logarithmic complex action representation yields local momentum, energy, transport ratio, and a second-order complex quantum potential without altering the underlying wave dynamics. Near an isolated non-characteristic moving zero, the leading simple pole factors cancel in the transport ratio, whereas the second-order term develops a double pole. The leading real axis response therefore is scaled as d_min^(-2). The reference finite window fit gives an exponent of -1.885 (ρ=-0.995), and the fitted exponent approaches -1.998 as the fitting interval is restricted toward the isolated zero regime. These results establish an exact benchmark relating antilinear symmetry, complex zero-pole geometry, and real axis differential amplification.