4.13 Generativity: negation adds, not merely subtracts

The generative half follows from typed non-collapse. Z_n=-I_n is not zero, is not absence, is not +H_n, and is not reducible to the primitive Function/Adeptness/Usefulness/Reliability coordinates. Therefore adding Z_n enlarges the typed input-space of the Hyperfunctional state.

Z_n != 0, Z_n != +H_n, Z_n != A, and Z_n != Z_m for n != m

Once the state-output rule is allowed to depend on interactions among coordinates, the newly added coordinate can participate in interactions that were literally impossible before that coordinate existed. Those interactions can produce new dimensions of output.

Phi(x) = sum_i a_i x_i + sum_(i,j) b_ij x_i x_j + sum_(i,j,k) c_ijk x_i x_j x_k + ...

At H1 there is no Z_1 coordinate, so terms involving Z_1 cannot exist. At H2, Z_1 exists, so terms such as A*Z_1, R*Z_1, U*Z_1, F*Z_1, A*R*Z_1, and F*A*U*R*Z_1 become available. This is a literal increase in possible interaction structure.