docs/CLOSURE_LEMMA.md

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 $m=3, k=3$: $$\mathrm{\mid }{M}_3(G_3)\mathrm{\mid }=\varphi (3)\times \mathrm{\mid }\{b:{\mathbb{Z}}_3\to {\mathbb{Z}}_3\mid \gcd (\sum b,3)=1\}{\mathrm{\mid }}^2=2\times {18}^2=648|M\_3(G\_3)|\; =\; \backslash phi(3)\; \backslash times\; |\backslash \{b:\; \backslash mathbb\{Z\}\_3\; \backslash to\; \backslash mathbb\{Z\}\_3\; \backslash mid\; \backslash gcd(\backslash sum\; b,\; 3)=1\backslash \}|^2\; =\; 2\; \backslash times\; 18^2\; =\; 648$$

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 $m=4$.