Does a consistency loss beat the many-body expansion?

The fragment-2-rdm-sheaf construction imposes overlap consistency as an in-training penalty . The incumbent way to stitch fragment quantities is the additive many-body expansion (MBE) applied post hoc. Is the in-training constraint actually better, and if so, where does the gain come from?

The question

Constraint versus correction

MBE corrects a sum of fragment values by adding pair, triple, … residuals after each fragment is computed independently. The sheaf loss instead couples fragments during training, so overlaps are reconciled by a minimum-variance projection onto the section space before any assembly. Does coupling-in-training beat correction-after in accuracy, in transferability, or only in sample efficiency? Or does it merely reproduce a low-order MBE with extra machinery?

What would settle it

• A regime where the consistency-trained model extrapolates to new sizes and new conformers but a matched MBE model does not, isolating the constraint rather than the capacity. This demands controlling for parameter count and .
• A decomposition showing the projection captures cross-fragment correlation that a truncated MBE misses, i.e. the harmonic/curl content the additive sum drops.
• A null result is that at fixed context radius the two agree to within statistical error, so the loss buys only convenience.

This is the make-or-break gate for the whole fragment-sheaf programme. If the consistency loss is not measurably better than MBE, the construction is elegant but thin.