Neutral Returns at High-Order Grazing: Sharp Cyclicity, Physical Codimension, and Weighted Crossover

Abstract

We study a neutral periodic orbit of a continuous two-field flow that is tangent to a switching seam while nearby orbits enter the second field. We separate the roles of two governing integers: the neutral-return order $n\geq2$ controls orbit count, whereas the even contact order $\nu$ controls grazing codimension, passage scale, and transverse crossover. Continuity factors the branch mismatch as $X^+-X^-=hW$; an active excursion of duration $O(q^{1/\nu})$ therefore produces a correction of order $q_+^{1+1/\nu}$. A parameter-uniform passage theorem and a cross-seam Hermite zero theorem then give the sharp fixed-stratum cyclicity: $n$ or $n+1$ periodic orbits, according to the sign coupling of smooth and active terms. A rank-$n$ physical unfolding attains the applicable bound. Contact-jet incidence gives ambient codimension $\nu-2$ for order-$\nu$ grazing and $n+\nu-2$ for simultaneous neutral grazing. Transverse contact parameters replace the central power by a weighted positive-part cap whose onset exponent crosses from $1+1/\nu$ to the generic quadratic-contact value $3/2$. The same sharp bound holds for both the exact cap and a prepared physical multiwell return, including births, mergers, and multiple entry--exit pairs. A value-only $C^1$ counterexample shows that finite cyclicity requires event-derivative information. For every even $\nu\geq4$, a closed polynomial family realizes the prescribed contact and rank conditions and the sharp periodic-orbit configurations through actual finite-time first-return maps.