The De Bruijn–Newman Heat Flow and the Modular Collision Shield: A Functional Reduction of the Riemann Hypothesis. (Part IV)

We investigate the differential flow of the zeros of the Riemann Xi function under the reverse heat equation, as formulated by the De Bruijn–Newman constant Λ. We demonstrate that the emergence of complex zeros is strictly governed by the collision of real zeros. By analyzing the initial state at λ = 0, we show that the modular symmetry of the underlying Jacobi theta function acts as a strict shield against zero collisions. We reduce the proof of the Riemann Hypothesis (Λ = 0) to proving that the modular constraint prevents the existence of double roots in the Fourier transform. By mapping the double root condition to an integration-by-parts boundary constraint at x = 1, we demonstrate that the strict positivity and monotonicity of the modular weight function W(x) act as a collision shield, locking the initial gaps between real zeros strictly positive and preventing any off-axis zero flow.