4.6 Strict-distinctness theorem for higher orders

Let n>m. H_n contains +tau_H,m and -tau_I,m because m<n. H_m contains neither coordinate because its direct recursion contains only orders k<m. Therefore their occupancy maps differ on at least those coordinates.

n > m => q_Hn(tau_H,m)=+1 while q_Hm(tau_H,m)=0

Therefore n>m => H_n != H_m. The same proof gives I_n != I_m. This is now a theorem from the identity criterion rather than a stipulation.

Define direct recursive richness rho(X)=|support(q_X)| for finite orders. Then rho(H_n)=rho(I_n)=2(n-1), so n>m => rho(H_n)>rho(H_m).

The recursive call-tree measure S_n=3^(n-1) supplies a second, unfolded structural-richness measure. Both direct width and unfolded recursive complexity strictly increase with n.