A simulator can match NVDA's irreversibility rate while getting the sequence of book states wrong. On the same days, Yeo, Cha and Hwang's queue-reactive simulator produces 0.0517 nats per event of ordered irreversibility, against 0.0537 in the market. An order-sensitive network gives those rates. Their closeness hides a different source of asymmetry in each dataset.

The authors separate the rate into what an order-blind network finds and what a sequence model adds. The simulator's pooled, order-blind rate is 0.0510; the market's is 0.0359. Its excess of about 0.015 nats per event accounts for the near match in the ordered rates. The ordering increment runs the other way: 0.00065 for the simulator, 0.0179 for the market. As the authors put it, the simulator "builds the magnitude of the market's temporal asymmetry out of block composition rather than block order." They make this contrast their headline result in the abstract and conclusion. Clipping, architecture and scale qualify how precisely to read it.

Time irreversibility means a path's distribution can be distinguished from that of its own physical reversal. Earlier finance work measured it on scalar series such as returns and volatility. Here the object is a coarse-grained limit order book path, combining book state and order flow. The rate is a forward-to-reverse log-density ratio per event, measured in nats. The paper offers no trading payoff from the scores: they add no stable forecasting gain over standard book and recent-flow inputs. Its proposed use is to benchmark order book simulators.

The paths and the scores

The authors use Nasdaq TotalView-ITCH market-by-order data obtained through Databento, covering fifteen trading days from 2 to 23 July 2026. They replay adds, cancels, replaces and executions to reconstruct the book. An event clock divides the stream into blocks targeting B = 10 completed book updates. Each block has 37 features: book state at both ends, mechanical flow (net and gross depth change), best-quote depletion and appearance counts, price changes, and an activity clock (elapsed time, event count and completion overshoot). K = 6 blocks make a path, or about 61 realized events.

Reversal follows the book mechanics. Spread and depth retain their values but exchange positions. Mid change and net depth change, as signed quantities, reverse sign; best-quote depletion becomes appearance. For a network g, subtracting g(RW) from g(W) gives an antisymmetric path score. Under logistic-loss training, the optimum is the forward-to-reverse log density ratio. Its forward expectation is a KL divergence. The authors fit one temperature scalar on three calibration days and clip scores at ±log 99.

The pooled network averages block embeddings and takes their maximum, making it permutation invariant (8,609 parameters). A dilated temporal convolutional network, or TCN, retains their order (12,929 parameters). On daytime NVDA paths, the pooled rate is 0.0359 and the ordered rate is 0.0537. Their increment is 0.0179, ranging from 0.0171 to 0.0187 across test dates and seeds. Linear antisymmetric baselines reach 0.000368 and 0.000385. The authors locate the asymmetry in nonlinear combinations of path variables: spread times signed net depth change, an even-times-odd product, is an example a linear score cannot form.

Does block order matter?

The controls mostly say yes. Removing the activity clock leaves an increment of 0.0071. Shuffling the six blocks within each path reduces it to 0.00055. With K = 1, adjacency disappears and the larger network loses: the increment is −0.00135, negative on every test date and seed. Circular padding conceals path boundaries yet changes the increment only from 0.0179 to 0.0160.

A data-side check reaches a similar figure. The authors map blocks to coarse states, then re-chain them through a randomized Eulerian traversal that preserves every directed transition count exactly. This surrogate recovers 0.0376 of the 0.0537 ordered rate. The remaining 0.0161 is close to the architecture-based 0.0179. The same procedure removes 0.09% of the simulator's rate and 30% of the market's. Giving the simulator the market's own durations, in the QR-B variant, raises its increment only to 0.00075.

Across twelve symbols, the increment is positive on all three test dates. It ranges from 0.0038 for SOFI to 0.0184 for AAPL. SOFI combines one of the higher pooled rates with the smallest increment, just 7% of its ordered rate. Within NVDA, realized variation predicts the pathwise score difference with out-of-sample R² of 0.0004; 29 book and history inputs reach 0.045. The inference relies on sign consistency across three held-out dates and three seeds, alongside the twelve-symbol panel, which the authors treat as replication across market conditions. The 582,767 NVDA test paths overlap at a three-block stride. The authors regard dates as the independent repeats. We did not find interval estimates.

A rate tied to scale

Keep the horizon at 60 updates and alter block size: the increment is 0.0216 at (B, K) = (5, 12), 0.0179 at (10, 6), and 0.0020 at (30, 2). The abstract notes a change of more than tenfold with coarse-graining scale. In the introduction, the authors say an irreversibility rate belongs to a representation and an observation scale. Their proposed benchmark therefore reports the increment across block sizes as a scale profile, scoring a simulator on that full profile at specified B. The simulator comparison covers only B = 10, K = 6. No simulator scale profile is reported.

Architecture also affects the figure. The authors acknowledge that the increment's magnitude "depends on network size and architecture, which are not fully matched." They point to the block shuffle, K = 1 and circular padding controls. The shuffle carries the most weight because it uses the same ordered network while reducing the clock-free increment from 0.0071 to 0.00055. With no second ordered architecture reported, 0.0179 remains a target specific to this pooled and TCN pair at B = 10. The simulator comparison is less exposed to that choice, since both sources are scored with the same refitted pair.

Clipping matters too. At K = 6, the unclipped ordered rate is 0.0852 and the unclipped pooled rate is 0.0442. Their clipped counterparts are 0.0537 and 0.0359. By our subtraction, the unclipped gap is about 0.041, more than twice the clipped 0.0179. The fraction of ordered paths at the bound rises from 15% at K = 3 to 84% at K = 12. The simulator comparison uses clipped scores; we did not find unclipped simulator figures.

The simulator itself sets a limited challenge. This fully reinitializing queue-reactive variant has first-order event dynamics, holds spread constant within each day, and contains four degenerate features. The authors tested whether those features drive the finding. On one seed, removing any feature group that contains them leaves the real increment at 0.0178 or above. Learned generators remain untested, as the authors acknowledge. Scoring a learned generator in the same way would change my view of how general the result is.

Forecasting adds little

The authors also tested the scores against cumulative midpoint returns for NVDA, QQQ, SOFI and SAIA. They found no stable gain over standard book and recent-flow inputs. Baseline LightGBM has a median out-of-sample R² of 0.096 at the two-block horizon. Adding the scores changes MSE by less than 0.12% in 22 of 24 symbol and horizon cases. The authors call this exercise exploratory because its evaluation dates were not held aside.

We could not reproduce the study ourselves. Rebuilding queues and counting depletions requires event-by-event MBO messages; we have only one-minute OHLCV bars. July 2026 also lies beyond our data coverage.

For simulator calibration, the distinction is decisive. Matching the overall rate at 0.0517 against 0.0537 establishes agreement in the totals. The pooled and ordered split shows whether the sequence of blocks produced it.