equation is violated is small (in the continuous flat norm). That is, we have While Equation (3.2) can be viewed as a decomposition of P$_{δ}$ , the added error terms mean it may not be a chain on a simplicial complex. This means it cannot be used directly to bound the simplicial flat norm of P$_{δ}$ . If we use the simplicial deformation theorem to push the error terms to some complex K$_{δ}$ while preserving a pushed version of Equation (3.2), we can obtain a candidate sim- plicial decomposition of P$_{δ}$ . In order to use this to bound F$_{K}$$_{δ}$ , we must know that the deformation theorem didn’t make the small error terms large enough to matter. Unfor- tunately, the simplicial deformation theorem mass bounds rely on simplicial regularity so a sufficiently skinny simplex could mean the error terms become large. If the simplicial irregularity in K$_{δ}$ gets worse as δ ↓ 0, we will not be able to show Equation (3.1). Since we know exactly which currents we wish to push, the solution is to pick K$_{δ}$ with these in mind: make sure the complex is as regular as possible overall (independently of δ ) with any irregularities (which may be required to embed P$_{δ}$ , X$_{δ}$ , and S$_{δ}$ ) isolated in subcomplexes of small measure. By making the irregular portions small enough (so they contain a negligible portion of the error terms, even considering the possible magnifica- tion from pushing), we establish a deformation theorem variant (Theorem 3.3.6) with constant mass expansion bounds, assuming a triangulation result that lets us isolate the irregularities as described (Shewchuk’s Terminator algorithm [38] provides this in R $^{2}$). The pushed version of Equation (3.2) allows us to prove F$_{K}$$_{δ}$ ( P$_{δ}$ ) ≤ F ( T ) + O ( δ ) from which Equation (3.1) and Theorem 3.3.7 follow. 57 P$_{δ}$ = X$_{δ}$ + ∂S$_{δ}$ + ( P$_{δ}$ − T ) + ( ∂S − ∂S$_{δ}$ ) + ( X − X$_{δ}$ ) . (3.2)