4.5 Formal typed state-space and identity criterion

To make the negative occupancies mathematically identifiable rather than merely described as non-collapsing, define a distinct recursive type-symbol for every Hyperfunction and Infrafunction order of a given function phi.

B_phi = {tau_H,1, tau_I,1, tau_H,2, tau_I,2, tau_H,3, tau_I,3, ...}

These are basis/type labels, not scalar values. tau_I,2 is a different coordinate from tau_I,1; tau_H,2 is different from tau_H,1. Distinct type-symbols remain distinct unless a later explicit canon rule equates them.

A finite recursive occupancy state is represented by a map q_X : B_phi -> {-1, 0, +1}. The sign is the occupancy direction and the type-symbol identifies WHICH state-axis is occupied.

Equality criterion: X = Y iff q_X(b) = q_Y(b) for every recursive basis-symbol b, and their primitive-state components are also equal.

For n >= 2: q_Hn(tau_H,k)=+1 and q_Hn(tau_I,k)=-1 for every k<n; every other recursive coordinate is 0. The mirror definition applies to I_n. H1 and I1 have their four primitive components and no lower recursive coordinates because no lower H/I order exists beneath order 1.

The expression -I_k now has an identity criterion: it means negative occupancy (-1) specifically on coordinate tau_I,k. It is not a free-floating verbal negation.

Signed typed identity: (-, tau_I,k) = (-, tau_I,j) iff k=j. Also (-, tau_I,k) != (+, tau_H,j) for every k,j unless an explicit equivalence rule is added.