What question did this study set out to answer?

This research aims to explore the limitations of projection-based approximations in many-body systems and their effects on correlation dynamics.

March 8, 2026Open Access

Single Obstruction Theory: Projection Nonlinearity, ℓ¹ Rigidity, and Density Scaling

Key Points

This research aims to explore the limitations of projection-based approximations in many-body systems and their effects on correlation dynamics.
Proved projection non-dilability of nonlinear projections with linear semigroups.
Analyzed ℓ¹ coboundary norm as a unique diagnostic of obstructions.
Utilized finite-range non-commuting lattice interactions to study density scaling.
Established that nonlinear projections cannot be conjugate to any linear semigroup.
Identified ℓ¹ coboundary norm as the sole additive faithful diagnostic for defects.
Found that obstruction density remains constant over a significant time interval regardless of system size.

Abstract

Projection-based approximations such as mean-field theory and product-state ansätze simplify many-body systems by discarding correlations. This paper studies the structural obstruction created by such projections. We prove three results: Projection non-dilability: a nonlinear projection composed with a linear semigroup cannot be conjugate to any linear semigroup. Categorical rigidity: the ℓ¹ coboundary norm is the unique additive faithful diagnostic of the resulting defect. Finite-time density scaling: for finite-range non-commuting lattice interactions the obstruction density remains nonzero over an O(1) time interval independent of system size. The framework uses Banach-valued presheaves over finite posets and connects geometric projection curvature, sheaf-theoretic obstruction, and correlation growth in many-body dynamics.

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