Inverse Moduli Theory: Realizability of Degeneration Data by Compactified Moduli Spaces

Donaldson, Seiberg–Witten, Floer, and Gromov–Witten theories extract invariants or algebraic structures from compactified moduli spaces. We study the inverse problem: which prescribed gluing data and algebraic identities can be realized by compactified moduli spaces, and when can such a realization arise from Fredholm geometry? The formulation uses decorated Feynman categories, marked realization categories, and smoothing profiles that distinguish boundary, real-normal, and complex-normal degenerations.

The principal result is an obstruction to extending stabilizing bundles across the associahedron \(K_4\). A compatible family over \(K_2\) and \(K_3\) extends if and only if its glued boundary bundle extends over \(K_4\); reflecting one endpoint identification produces a Möbius summand on \(\partial K_4\) and hence a nonextendable datum. We also prove that the equations \(d^2=0\) and the \(A_\infty\) relations are precisely oriented filling conditions in dimension one, with higher extensions obstructed by ordinary smooth \(\theta\)-bordism classes. Realization categories form fibers of a cartesian fibration over gluing data, and rigid realizations satisfy descent over overlapping color sets.

Family-index, compactness, and determinant-line conditions yield further obstructions to Fredholm realization; we examine them in flow, planar, stable-curve, and stable-disk examples. The resulting framework is a sequence of realization and obstruction problems, not a universal existence theorem.