Exact transport theory of dendritic competition and arrest

We solve exactly the transient two-dimensional advection diffusion problem generated by one and two semi-infinite internal Dirichlet lines, representing slender dendrites advancing into a supersaturated liquid while fixing the solute concentration on their surfaces. For a single dendrite, the mixed boundary value problem is reduced to a Wiener-Hopf equation, factorized explicitly, and inverted to the time domain by the Cagniard-de Hoop method, yielding the single-layer density in closed form in terms of the complex error function. For two parallel dendrites separated by a lateral distance h, the interaction enters through the transcendental kernel (1+e^{-κh})^{-1}, whose lower Wiener-Hopf factor is represented by a logarithmic Cauchy integral, and the resulting finite distance feeding field is reduced to a regularized oscillatory integral. The exact two-dendrite solution reveals a structural obstruction to purely local growth laws: the leading inverse square-root tip singularity is universal, with identical amplitudes for both dendrites independently of spacing, even though the neighbor modifies the feeding field at every finite distance behind the tip. Competition therefore cannot be encoded in any local tip amplitude alone. This obstruction forces a nonlocal coupling variable, which we identify as a finite tip zone functional that retains the subleading near tip structure responsible for competitive depletion. The interaction further resolves into a discrete hierarchy of exponentially screened modes with dominant screening length h/π, which sets the natural scale of competitive suppression. These exact transport results provide the analytical backbone for a reduced theory of dendrite arrest and winner selection, and are validated against independent finite-difference computations.