Phase-space sectors for ordered momentum mappings in local subtraction up to N$^3$LO

Ordered momentum mappings present optimal convergence in soft and collinear configurations and are particularly suitable for the numerical implementation of local subtraction schemes. However, ordered mappings cannot be directly applied in the presence of multiple unordered emissions, which typically appear beyond the leading-colour approximation. A possible solution consists in separating individual singularities at the level of local counterterms by means of partial fractioning, which can become cumbersome at higher orders and introduce large cancellations in intermediate steps of the calculations. We present a simple decomposition of the phase space into sectors to isolate classes of infrared configurations which can be addressed with a specific ordered momentum mapping. With such decomposition, the singularities of any matrix element can be subtracted with ordered mappings, without the need of partial fractioning. We illustrate the required phase-space sectors for up to three unordered emissions and discuss applications in the context of the antenna subtraction method. The mapping algorithm described here has been recently employed for the first differential N$^3$LO of jet production at electron-positron colliders.