A Boundary--Residue Incidence Coalgebra for Associahedral Scattering Forms

• URL: http://arxiv.org/abs/2605.30390v1
• Date: Thu, 28 May 2026 11:47:09 GMT
• Title: A Boundary--Residue Incidence Coalgebra for Associahedral Scattering Forms
• Abstract summary: We introduce a boundary--residue incidence coalgebra associated with the face poset of a positive geometry.<n>We apply it to associahedral scattering forms.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce a boundary--residue incidence coalgebra associated with the face poset of a positive geometry and apply it to associahedral scattering forms. The construction is motivated by the analogy between the Connes--Kreimer coproduct on Feynman graphs and the recursive residue structure of canonical forms. For the Stasheff associahedron \(K_n\), whose faces are indexed by non-crossing dissections of an \((n+1)\)-gon, we prove that the incidence coproduct records all intermediate nested planar factorisation channels of the corresponding tree-level scalar amplitude. The residue of the canonical form on a face labelled by a dissection factorises as the exterior product of canonical forms on the lower associahedra associated with the resulting subpolygons. We illustrate the construction explicitly for the pentagon associahedron \(K_4\), corresponding to the five-point planar scalar amplitude. We then formulate a loop-level extension: whenever a planar loop integrand is represented by a positive geometry, the associahedral face poset is replaced by the boundary poset of the corresponding loop geometry. The one-loop halohedron gives a concrete scalar example, while in the non-planar case we define the associated incidence coalgebra at the level of logarithmic singularity strata. Finally, we compare the boundary--residue coalgebra with the cellular incidence coalgebra of a triangulated or regular CW spacetime. The face poset of a finite regular CW complex reconstructs its barycentric subdivision, and hence its underlying polyhedron, while in positive geometry the same incidence mechanism organises canonical-form residues. This yields an incidence-first bridge between cellular spacetime topology and positive-geometric amplitude factorisation, without assuming that metric or causal data are determined by topology.

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