Determine the general-rank formula for the six-cycle replica-torus operator

Determine an analytic formula, valid for arbitrary integer replica index n, for the rank over F_2 of the 3n^5 by 3n^5 operator M_n governing the assignment constraints of the six-cycle graph state, and thereby obtain a closed expression for its Rényi multi-entropy assignment sum.

Background

For the six-cycle graph state with one qubit in each core block, the replica contraction produces three linear constraints on three assignment fields over the five-dimensional replica torus. These constraints are encoded by an operator M_n acting on a vector space of dimension 3n^5 over F_2.

The assignment sum is expressed exactly as Δ{C_6}(n)=2^{6n^5-rank{F_2} M_n}. The paper evaluates this rank for n=2,3,4, but does not derive a formula for general n. The authors note that preliminary evidence suggests non-polynomial dependence involving the factorization of n, making this a global and potentially nonlocal problem.