PHY.4:4.3 - Turn the comparison into a constraint on the unknown relation

Specify how inputs and responses transform. If x changes by T and y by U, a proposed response function has the comparison condition:

F(T(x))=U(F(x)).

For a reversible symmetry of a relation R, the corresponding condition is R(x,y) iff R(T(x),U(y)). This permits constraining an account before choosing a direction in which to solve it.

Derive the restriction. One useful move is to hold the input fixed under some allowed transformations. The output must then remain fixed under its corresponding transformations. Another is to compare inputs related by a transformation: a value chosen at one constrains the value at the other. MATH.13 develops these mathematical consequences once the physical action has been supplied.

For example, in three-dimensional space, rotations about a nonzero vector w leave w fixed. A vector response determined only by w and unchanged physical conditions must therefore lie along w: any perpendicular component would turn. Rotations between equal-length velocities then make the scalar coefficient depend only on their length. The physical work is establishing that no additional direction or state must also be supplied; :5.1 makes those assumptions explicit.

If a proposed input was held fixed even though the physical transformation changes it, restore that input and repeat the derivation. The anisotropic case in :5.2 shows how the changed argument opens additional response directions.