Irreducible Multi-Particle Holonomies in Quadratic Mode Dynamics: Subspace Geometry Beyond Single-Particle Holonomy

This preprint develops a state-level framework in which holonomy is treated as a property of cyclic subspaces rather than single-particle eigenstates or gauge fields. Within number-conserving quadratic Hamiltonian dynamics on bosonic or fermionic modes, a precise notion of single-particle irreducibility is introduced, meaning that no compatible single-particle cyclic subspace of the same dimension can satisfy a holonomic (purely geometric) condition. The main result is an explicit existence theorem showing that nontrivial, purely geometric holonomies can arise in two-particle cyclic subspaces even when the associated single-particle problem admits no holonomic cyclic subspace of the same dimension. A fully worked minimal example—two bosons in two modes—is provided, yielding a nontrivial SU(2) holonomy on an encoded two-dimensional subspace while the single-particle sector is provably non-holonomic. The paper is self-contained and includes a conservative, device-agnostic certification scheme based on loop composition and conjugacy-invariant readouts, enabling experimental verification without model-specific gauge-field assumptions.