the ( p − 1)-skeleton, we can bound its mass using Equation (3.15): Similar inequalities hold for S$_{j}$ , ∂T$_{i}$ , and ∂S$_{j}$ . In the preceding, we have accomplished the goal of projecting all currents involved from the p -skeleton to the ( p − 1)-skeleton and can now consider them as currents in the underlying space of the ( p − 1)-complex skel$_{p}$$_{−}$$_{1}$( K ). We can apply this procedure iteratively (use Conjecture 3.3.4 to localize the irregularities and then project) to push to the ( p − 2), etc. skeletons. When we subdivide each k -skeleton using Conjecture 3.3.4, the higher dimension simplices are not subdivided by default but this is easy to fix. After a k -simplex is subdivided, add a point to the interior of every ( k + 1)-simplex of which it was a face and connect the new point to every k -simplex on its boundary. This will likely generate highly irregular simplices but since we’ve already pushed the currents down beyond their dimension, it isn’t an issue. This argument continues in the same way as Theorems 3.2.4 and 3.3.1 and establishes our result with L = 2 β . Theorem 3.3.7. If T is an integral d -current in R d +1 and Conjecture 3.3.4 holds for d − and ( d + 1) -currents, then some flat norm minimizer for T is an integral current. That is, there is an integral d -current X$_{I}$ and integral ( d + 1) -current S$_{I}$ such that F ( T ) = M( X$_{I}$ ) + M( S$_{I}$ ) and T = X$_{I}$ + ∂S$_{I}$ . 72 M( T ′ $_{i}$) ≤ (2 m + 2 n + ϵ ) $^{[}$β M( T$_{i}$ ¬ M ′ ϵ \ skel$_{p}$$_{−}$$_{1}$( K )) + α$_{K}$ M( T$_{i}$ ¬ M$_{ϵ}$ \ ( M ′ ϵ ∪ skel$_{p}$$_{−}$$_{1}$( K ))) ] + M( T$_{i}$ ¬ skel$_{p}$$_{−}$$_{1}$( K )) ≤ (2 m + 2 n + ϵ )( β M( T$_{i}$ ) + α$_{K}$δ ) ≤ (2 m + 2 n + ϵ )( β M( T$_{i}$ ) + β M( T$_{i}$ )) ≤ (2 m + 2 n + ϵ )(2 β ) M( T$_{i}$ ) .