A state is not a history
A depth snapshot is the displayed buy and sell quantity at each price at a particular moment. It records the state visible then; it does not list the events that led to it.
Imagine two order books with the same best bid, best ask and displayed quantities. One state may have followed a period of little activity. The other may have followed repeated additions, cancellations and trades. The snapshots look the same, but the observed paths need not be.
One price, two cancellations
INTERACTIVE / QUEUE POSITION
Same depth. Different position.
Three unit orders share one price in a first-in, first-out (FIFO) queue. I is the order we observe. Choose whether to cancel U ahead of I, or V behind it.
BEFORE
Total depth 3 · Quantity ahead of I: 1
AFTER
Total depth2
Quantity ahead of I0
Cancelling U changes the quantity ahead of I from 1 to 0; total depth falls from 3 to 2.
Assumptions: one price, three displayed unit orders, FIFO and one full cancellation; no hidden quantity or new events. This example does not calculate fill probability.
The question determines the representation
A snapshot can answer a question about currently displayed prices and quantities. Studying how that state developed requires event information, with enough detail and ordering to support the question being asked. Aggregation is useful, but it is also a choice about which distinctions to discard.
Why aggregation hides the difference
Start with U, I, V, each of size one. Removing U leaves I, V; removing V leaves U, I. Both results have displayed quantity two, but the quantity ahead of I is zero in the first case and one in the second. A depth-only view groups these states together.
This group is a projection fibre: several detailed states share one reduced observation. A rule using only that observation must return the same answer for both, even though their queue-ahead quantities differ. Depth is sufficient for the total quantity here, but not for the exact quantity ahead.
Moving from this distinction to fill probability requires future event dynamics and a matching model. The example stops at state reconstruction. It explains the minimal witness in Section 5 of the preprint and is not a new empirical result.
Queue position needs its own assumptions
Displayed volume alone does not establish where a hypothetical order would sit in a queue, when it would reach the venue or whether it would execute. A model must account for the venue’s matching rules, the available feed and uncertainty about intervening activity.
Data quality comes before interpretation
Sequence gaps, inconsistent timestamps and recovery logic can change the reconstructed history. Before treating a pattern as a signal, ask whether it survives data checks and whether the information used would actually have been available at the time.
A useful discipline
State the question first. Document the feed and its limitations. Identify which information the representation removes. Then assess whether the remaining information supports the inference. A richer representation does not by itself establish predictive value or profitability.