A stop-loss order is an instruction to sell automatically once the price falls through a level chosen in advance, and a margin liquidation is the same thing imposed rather than chosen. Traders and margin engines tend to pick the same levels, so a market quietly carries an uneven reservoir of committed selling spread across price. This paper models that reservoir as a density and asks the obvious question about it: when a decline starts setting it off, what decides whether the selling peters out or feeds itself?
Combining the density with the market's price impact gives an exact answer. The fall comes to rest at the first level where the resistance accumulated on the way down matches the shock that began it. Two quite different transitions fall out of that rule. Raise the height of the density and the behaviour changes smoothly through a critical point, with the cascade behaving like a branching process and its size distribution carrying the exponent three halves that such processes are known for. Raise the shock instead and the change is abrupt. In a book built from stated margin rules and round-number levels, a shock a little over a capacity of 1.130 per cent of price turns a 2.87 per cent decline into a 14.48 per cent one.
Where the selling sits matters as much as how much of it there is. Two books holding identical volume differ by a factor of 2.73 in expected shortfall depending on how that volume is clustered, and the clustered one is the quieter of the two on average. Concentration moves risk rather than creating it, which is an uncomfortable property for anyone who measures risk by averages.
The quantities the theory needs are then measured rather than assumed. Using public order-book and open-interest archives around the deleveraging of 10 October 2025, price impact is recovered to within 11.42 per cent from two independent constructions, the bid side is shown to have thinned by a factor of 25.49 against 4.75 on the ask, and the multiplier is placed between 1.13 and 1.44, above the critical point. Across seven contracts, the impact measured before the event ranks the falls that followed perfectly, while the quantity liquidated ranks them backwards. That is the pattern cross-margin implies and isolated margin does not.
What survives from all of this is something usable. A fragility index built from the density alone ranks the levels that cascade with a rank correlation of 0.949, and loses nothing when three quarters of the order book is hidden from it. Across 1,883,520 minutes of price history the model's round-number prediction holds in both directions and gets stronger the rounder the level.