Dimensional analysis — the Buckingham Pi theorem — counts how many independent dimensionless groups govern a physical problem. If you have N variables and K independent dimensions, there are N-K dimensionless groups. The counting is mechanical once you know the dimensions.
Constraints complicate the counting (arXiv:2603.21527). When variables are related by prior physical relationships (conservation laws, geometric constraints, constitutive equations), some of the N-K dimensionless groups are redundant. The constraint eliminates degrees of freedom, reducing the number of independent groups further. But the standard Pi theorem doesn't tell you which groups survive or how many.
A linear-algebraic formulation resolves this. Express all dimensional relations and constraints in logarithmic variables, where products become sums and powers become multiplications. The entire structure reduces to a system of linear equations. The number of independent dimensionless quantities is the dimension of the null space of the augmented matrix — a computation, not a guess.
The classical drag force problem illustrates: without constraints, dimensional analysis yields drag proportional to ρv²L². Adding the constraint that the fluid is incompressible (density is constant, not variable) changes the counting by restricting how density can appear. The constraint doesn't add information about drag; it removes a degree of freedom from the dimensional space.
The structural insight: constraints don't just reduce the number of variables. They restructure the dimensional space itself. A constraint that relates two variables makes one dimension of the space inaccessible — not because you've measured something, but because the physics forbids certain combinations. The dimensionless groups that survive are the ones compatible with both the dimensions and the constraints simultaneously. The constraint is as much a part of the physics as the variables themselves.