A wide mix on headphones that turns thin on a phone speaker is one of the most common ways a mix falls apart. It is also one of the most avoidable, once you know where width lives and which tools put it there.
Where width lives
Any stereo signal can be split in two: the part that is identical in both channels, the Mid, and the part that differs, the Side. Everything you hear as width is in the Side. A mono speaker, a club system summed to mono, a phone, a Bluetooth speaker: they all play the Mid and throw the Side away.
That gives a simple rule. Anything you add to the Side channel costs you nothing in mono, because mono never hears it. Anything you change in the Mid changes the mono mix, whether you meant to or not.
Why most wideners break in mono
The classic ways of making a signal wider all work by making the two channels different from each other: a short delay on one side, an inverted copy mixed back in, an all-pass phase shift, or a chorus with opposite modulation on each channel. They do sound wider. They also rewrite the Mid, because a delayed or phase-shifted copy no longer adds up cleanly with the original. Sum to mono and you get comb filtering: some frequencies cancel, others double, and the sound turns hollow. The width was borrowed from the mono mix rather than added to it.
Checking in mono after every widening move is the usual advice, and it is good advice. But it treats the symptom. The cause is that the tool wrote to the Mid.
Width that folds back to nothing
Airy takes the rule literally: it never writes to the Mid. It measures how much width the signal already has, band by band, as the coherence between the two channels, and it synthesises Side content to fill the gap up to the width you ask for. Because the added material lives only in the Side, the mono sum of the output is the mono sum of the input, sample for sample. Not corrected afterwards, not approximately: by construction.
You can check that yourself below. The interface is the plugin, rendered offline in sync with the sound. Widen the group as far as you like, then press MONO: the output folds to mono in your browser, the way a phone speaker would. Then press BYPASS with MONO still on. If the plugin never touched the Mid, the two states should be the same signal. They are.
The mono arpeggio is the harder case. Its Side channel is digital silence to begin with, so there is nothing to widen. GENERATE builds a Side channel from the signal's own spectrum. At 100 % the arpeggio fills the room; fold it to mono and the original comes back exactly.