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 $ν$ controls grazing codimension, passage scale, and transverse crossover. Continuity factors the branch mismatch as $X^+-X^-=hW$; an active excursion of duration $O(q^{1/ν})$ therefore produces a correction of order $q_+^{1+1/ν}$. 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 $ν-2$ for order-$ν$ grazing and $n+ν-2$ for simultaneous neutral grazing. Transverse contact parameters replace the central power by a weighted positive-part cap whose onset exponent crosses from $1+1/ν$ 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 $ν\geq4$, a closed polynomial family realizes the prescribed contact and rank conditions and the sharp periodic-orbit configurations through actual finite-time first-return maps.

Disclosure

“echanism. The os- cillatory C 1 family in proposition 7.3 identifies event structure, beyond weighted value control, as essential for finite cyclicity. Declarations Acknowledgements and generative-AI disclosure. OpenAI Codex (GPT-5, accessed August 2026; immutable snapshot identifier unavailable) assisted with language and structural editing, cross- reference and symbolic consistency checks, literature searches, and figure-code and layout editing. The au”

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2026-08-23T21:13:37.085171Z

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