# The Closure Lemma in Moduli Space $M_k(G_m)$

## Theorem Statement
The space of valid $k$-Hamiltonian decompositions for the group $G_m = \mathbb{Z}_m^3$ (fiber-structured case) is a torsor under the group of 1-cocycles $H^1(\mathbb{Z}_m, \mathbb{Z}_m^2)$.

Specifically, for $m=3, k=3$:
$$|M_3(G_3)| = \phi(3) \times |\{b: \mathbb{Z}_3 \to \mathbb{Z}_3 \mid \gcd(\sum b, 3)=1\}|^2 = 2 \times 18^2 = 648$$

## The Lemma
Given any $k-1$ level assignments $b_0, b_1, \dots, b_{k-2}$ such that each individual $b_c$ satisfies the single-cycle condition $\gcd(\sum b_c, m)=1$, the final level $b_{k-1}$ is uniquely determined (up to gauge) by the fiber bijection constraint.

## Verification for $m=3$
This lemma has been exhaustively verified for $m=3$. The total solution count of 648 exactly matches the formula $\phi(m) \times N_b^{k-1}$, where $N_b$ is the number of valid level-transfers.

## Structural Generalization ($m > 3$)
For larger $m$, the "torsor" structure suggests that solutions are not scattered randomly but form a structured lattice in the cohomology space. The "Closure" implies that once $k-1$ colors are fixed to be Hamiltonian, the $k$-th color is forced to "close the loop" of the short exact sequence $0 \to H \to G \to G/H \to 0$.

This observation reduces the search space for $P1$ and $P2$ from $|S_k|^{m^3}$ to $|H^1(Z_m, Z_m^2)|^{k-1}$, a compression of over $10^5 \times$ for $m=4$.