An Exclusion Floor in Incompressible Media: A Conditional Variational Closure in D+1 Sectors

We consider a constrained continuum medium in Euclidean space R^D with velocity field ν subject to the incompressibility condition div(ν)=0. When this constraint is imposed variationally by a scalar Lagrange multiplier P, the resulting extended constrained formulation contains D kinematic components together with one scalar constraint field. The corresponding off-shell field count is therefore D+1.

We then introduce an explicit mesoscopic closure intended for the saturation regime of the constrained medium. The closure is formulated through a quadratic load-imbalance functional on a coarse-grained control cell. Its minimizer represents a stationary neutral state of the extended D+1-sector system, in which no sector sustains a privileged share of the total saturation load. This yields E_P = E_tot/(D+1), where E_P is the load carried by the incompressibility sector.

To connect this stationary load partition with a geometric exclusion fraction, we further identify the incompressibility-sector load with the pressure-volume work required to sustain an excluded core at saturation, and the total cell load with the saturation pressure acting over the full control-cell volume. Under these constitutive identifications, the excluded-volume fraction is fixed algebraically as Π = V_core / V_cell = 1/(D+1). For D = 3, this yields Π = 1/4.