Common torus for iterated integer coverings

Let $J$ be a nonsingular integer matrix and suppose finitely many nonnegative integer levels $i$ occur. Put $i_0=\min i$ and $H=p_{J^{i_0}}(Y)$ . Then every level variable has the representation

p_{J^i}(Y)=p_{J^{i-i_0}}(H).

Thus the torus with coordinate $H$ is a common torus on which all these level functions can be compared, added, and multiplied by pullback.

Bounded levels and compatibility

If $0\le i-i_0\le D$ , only finitely many matrices $J^{i-i_0}$ occur. The chain rule therefore gives uniform constants at each fixed derivative order for these pullbacks. A function defined on the common torus need not descend to an individual higher level: that would require additional invariance under the relevant deck translations.

When the active levels vary between slow-coordinate neighborhoods, local formulas must have equal pullbacks to the original $Y$ -torus on overlaps. This is the compatibility condition defining one global function; merely giving a smooth formula on each local common torus does not ensure it.