On the Mathematical Consistency of the Collective Vacuum Model

This work examines whether the principal structural assumptions of the Collective Vacuum Model (CVM) can be realized within a mathematically consistent single-field framework. A single complex matrix-valued collective field[ M(X)∈ GL(3, C)]is used to accommodate vacuum geometry, internal structure, collective evolution, and topological sectors within one field-state space. Two distinct bilinear forms are employed: an indefinite real trace form for the induced pseudo-Riemannian geometry and a positive Hermitian form for physical kinetic terms. The central phase of the matrix field is treated as a local (U(1)) gauge redundancy. The physical vacuum is represented as a five-dimensional submanifold of the matrix state space rather than as an isolated field configuration. A local realization with Lorentzian signature ((-++++ )) and a homogeneous and isotropic three-dimensional spatial sector is constructed. For a composite-metric action in which the metric is induced by the collective field, normal variations lead to an embedding-type field equation. Within a homogeneous and isotropic warped background, a sufficient rank-three condition on the extrinsic-curvature projections is shown to force the Einstein branch. An explicit local weak-deformation example demonstrates that the required rank condition can hold on an open region of parameter space. Sufficient local conditions for positive quadratic normal-sector kinetics are also identified, while the limitations associated with higher-derivative structure, constraint propagation, and the full Hamiltonian analysis are stated explicitly. The purpose of the work is not to establish uniqueness or necessity of the construction, but to demonstrate the existence of a non-empty class of single-field realizations satisfying the principal local consistency requirements of the CVM.