« Financial Interactions and Collective States: Description of Dynamic Collective States », working papers (Avril 2026)

In previous works (Gosselin and Lotz 2025a,b), we introduced the notion of collective states to describe capital allocation mechanisms among investors, firms, and banks across a sectoral space. A collective state is defined as a decomposition of the economy into sub-collective states—groups of sectors characterized by cross-sectoral stakes, disposable capital, and sectoral returns—which jointly determine the phase of each group. These states are intrinsically unstable: small perturbations in capital or returns may propagate nonlinearly across sectors, inducing feedback effects, recompositions, and defaults. Collective states must therefore be understood as elements of a fully interconnected dynamical system. In Gosselin and Lotz (2026), we outlined the properties required of a dynamic formalism capable of capturing the multiplicity of sub-collective states, their interactions, and their transitions. We proposed a field-based framework in which a collective state is decomposed into a basis state—describing the partition of stakes among agents—and associated fiber states—capturing the multiple configurations of capital and returns compatible with that partition. Deviation states are modeled as realizations of two classes of fields: basis deviation fields and fiber deviation fields.

The present work develops this formalism into a full field theory of interacting collective states. For each possible sectoral decomposition, two fields encode both phase configurations and fluctuations, leading to an infinite-dimensional structure encompassing all potential group formations and transitions. We examine three complementary approaches—transition, stable-state, and operator formulations—to characterize mergers, dissociations, and phase shifts of sub-collective states. We then construct an autonomous field theory for fiber deviations, allowing multiple equilibria and internal transitions within groups. Finally, we develop the complete fibered field theory, showing that basis and fiber transitions mutually trigger one another, generating an endogenous and persistent exploration of the infinite space of collective states. The resulting system is inherently unstable, with discontinuous transitions constituting the core mechanism of its long-term evolution.