Fixed-residue node cubes realize the actual Fock strings

Independence of the $i$-slots. If two $i$-cells are toggled, the neighbours determining each corner's availability are of residues $i\pm 1$, and the toggled cell has residue $i$, so it is not a neighbour of any other $i$-cell; hence toggling one $i$-cell never changes the availability of another, every subset of the finitely many slots is realizable, and the orbit is a Boolean cube with $2^k$ distinct partition diagrams.

The cube is the rank-one tensor power. In decreasing-content order, the $j$-th slot's toggling multiplies by $q^{+1}$ or $q^{-1}$ according to occupancy and the iterated coproduct coefficient at slot $j$ is $q^{{\sum }_{\text{slots before }j}\text{weight}}$ for $f$ and $q^{-{\sum }_{\text{slots after }j}\text{weight}}$ for $e$; these are exactly $N_i^r$ and $-N_i^l$ by [F3], so the map from tensor bit vectors to partitions is a rank-one module isomorphism onto the cube.

The string rule, lattice and divided powers. The tensor crystal theorem [F2] gives stability of the $A_0$-span of the cube basis under the normalized string operators. On the residue basis modulo $q$, unrolling its strict/non-strict comparison cancels $+-$ in decreasing order, with $f$ acting on the first surviving $+$ and $e$ on the last surviving $-$; reversing the order cancels $-+$ and selects the last $+$ / first $-$. For divided powers, write ${ϵ}_s=1$ on an empty slot and $-1$ on an occupied slot. If $a$ empty slots are filled by $F_i^{(a)}$, the coefficient is the Laurent monomial $q^{{\sum }_{t\text{ selected}}{\sum }_{s<t}{ϵ}_s-a(a-1)\mathrm{/}2}$: the $a!$ orders of distinct toggles contribute the asymmetric factorial at $q^{-2}$, which divided by $[a]!$ is $q^{-a(a-1)\mathrm{/}2}$. Repeated toggles vanish. Likewise emptying $a$ occupied slots by $E_i^{(a)}$ has coefficient $q^{-{\sum }_{t\text{ selected}}{\sum }_{s>t}{ϵ}_s-a(a-1)\mathrm{/}2}$. These formulas prove Laurent integrality. Define the standard partition form by making cube words orthonormal. For an edge $\lambda \to \mu$ and $N=N_i(\lambda )$, its two coefficients satisfy $N_i^r+N_i^l=N-1$, so $q^{N_i^r}=q^{N-1}{q}^{-N_i^l}$. This verifies $(F_{i}x,y)=(x,q^{-1}{K}_i{E}_{i}y)$ and, by transposition, $(E_{i}x,y)=(x,q{F}_i{K}_i^{-1}y)$; together with diagonal $K_i$ this is the normalized rank-one polarization for the fixed coproduct.

Conclusion. Each fixed-residue orbit is a finite Boolean tensor cube of highest-weight-$1$ strings, the actual normalized operators preserve the raw $A_0$-partition span and induce the survivor rule on its residue classes in either content order, including $e=2$. Lattice stability precedes transport of these congruences; no equality on standard field vectors is asserted.