A Chain- and Diagram-Level Semantics for Morphological Calculus Refinement, monodromy, and bivector orbit decompositions

Abstract

Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line. Sommen's examples reveal two basic difficulties: a single space may admit several such expressions, and scalar addition and multiplication suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics in terms of script chain complexes. The cell-count polynomial satisfies \[ \mathcal M_S(t)=\mathcal P_{S,\Bbbk}(t)+(1+t)\mathcal B_{S,\Bbbk}(t), \] where \(\mathcal P\) is the Poincar\'e polynomial and \(\mathcal B\) records boundary ranks. Over \(\mathbb Z\), Smith labels distinguish unit pairs---which model refinement overhead in the explicit subdivisions treated here---from non-unit pairs carrying torsion. We separate Cartesian products from bundles, derive the exact monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction for diagrams of script complexes. Mapping cones, joins, and double mapping cylinders appear as reduced models of this diagrammatic semantics. We apply this construction to the bivector discrepancies left open in Sommen's calculations, using finite CW models after unit-sphere normalization. In dimension four, Hodge decomposition identifies the unit bivector sphere with \(S^2*S^2\); the angular excess in Sommen's calculation is the contractible relative complex produced by inserting the rank-two midpoint. In dimension five, the \(SO(5)\)-action on \(S(\Lambda^2\mathbb R^5)=S^9\) has principal orbit \(SO(5)/T^2\) and singular orbits \(\widetilde G_2(\mathbb R^5)\) and \(\mathbb{CP}^3\). For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from the sphere homology by seven unit-labelled pairs.