The portable result here is that the no-manipulation restrictions execution modellers impose on impact operators sit strictly inside the region where crowding cannot support multiple equilibria. A self-confirming convention therefore requires breaking exactly the assumption most impact models start from. Rodríguez Domínguez makes that point with numbers: of 1,400 sampled economies, 18.3% of those with an indefinite symmetrised impact operator are still subcritical. The headline spectral statistic is a different matter. He states plainly that it cannot be estimated from the data he has, and the same is true for almost everyone reading this.
Two overlaps, one of them invisible in a book
Two portfolios can hold the same names on different signals, or different names on the same signals. The first pair executes against the same depth, and market impact prices that congestion. The second pair congests in information. Each one's claim on the premium shrinks as the other's capital grows, whether or not the books ever intersect. The paper makes the conditioning representation itself an equilibrium object: a subspace of driver space mapped into asset exposures by a response operator. Overlap is charged through a basis-invariant kernel, the normalised trace of the two driver-space projectors.
Scaling the impact operator moves the clearing premium and therefore the price-mediated crowding discount. It leaves the driver-space redundancy charge untouched. Holdings overlap, factor return correlation and borrow data all live downstream of the response operator. None of them can separate the two channels. The paper says separate identification needs variation that moves depth independently of driver overlap.
The first-order crowding discount is concentrated along the common speculative fund and scales with capacity and projected illiquidity. In simulation the two alignment arms separate by a factor of about three at the right of the capacity grid, over 30 economies per arm. One arm puts that fund on the least liquid direction of the impact operator, the other on the most liquid.
Dispersion of mandates does nothing on its own.
The statistic is cross-impact against deployed capacity
The threshold is tau, the negative of the smallest real part of the spectrum of capacity times cross-impact, with a critical value of one half. Below it the fundamental position path is the only square-integrable stationary one. Above it a regular destabilising direction supports a continuum of self-confirming conventions driven by extrinsic noise. The construction applies standard rational-expectations indeterminacy machinery to a recursion derived from impact and capacity rather than preferences. It carries the usual conditions: a diagonalisable reduced operator, a simple extremal eigenvalue, minus one excluded, and a destabilising direction loading at least two assets. The paper also states that the one-half boundary belongs to instantaneous impact only. With a decaying propagator the critical locus moves in a kernel-dependent way, and no numerical value is claimed.
Indefinite impact alone does not deliver multiplicity: 18.3% of the 1,400 sampled economies with an indefinite symmetrised operator are subcritical. A market can carry a direction where trading against yourself pays and support no convention at all. The dealer-hedging example keeps individual round trips strictly costly. Depth, inventory risk and hedge cost are all positive definite, and the symmetric part of the effective aggregate operator still has eigenvalues of minus 0.3541 and 2.1393.
Can it be estimated?
The abstract concedes that the threshold is not estimated in the available data, and immediately offers its replacement: an out-of-sample, driver-specific signature consistent with representation crowding, plus a statement of the identification conditions required for direct threshold measurement. The conclusion puts the empirical contribution as a set of falsifiable implications and measurement conditions for determining when markets are supercritical. Estimating tau needs signed cross-asset order flow, split into permanent and temporary blocks by the long-run impulse response in trades and quote revisions. Only the temporary inventory block belongs in the statistic. It also needs position-level effective risk capacity on the same multi-asset panel. Both halves on one panel, with a margin above the safe perturbation radius.
Over 300 economies with 200 perturbations each, relative Frobenius errors of 2% to 20% in the inputs push tau upward by 0.1% to 9% and inflate its standard deviation from 1.7% to 30%. Classification is clean above a margin of 0.3 from the boundary. It is at most 3.3% wrong for margins between 0.1 and 0.3. With the exact operator the statistic classifies 700 of 700 economies, near-defective families included. At 10% observation noise agreement falls to 0.901 for nonnormal operators. The requirement is order flow and capacity accurate enough to place a market 0.3 clear of a boundary at one half.
We could not backtest any of this. The missing piece is specific: signed cross-asset order flow with quotes, to isolate the temporary impact block, and portfolio-level deployed risk capacity on the same universe. We have daily and minute bars.
Plus 0.092, and no regime verdict
The empirical exercise is 152 S&P constituents from Bloomberg with at least 98% coverage over May 2010 to July 2023, T equal to 3,258. Against that panel sit 30 declared driver series whose out-of-sample panel R-squared is 0.19. Exposures are ranked by ridge on the first half. Residual comovement of the eight most exposed names per driver is measured on the second, and the null permutes the exposure map. Excess over that null is plus 0.092 in high-borrowing months and plus 0.047 in low ones. Fifteen of 30 drivers in the high regime and 8 of 30 in the low one clear two standard deviations of their regime-specific nulls.
The capacity comparative static does not come through. The difference of plus 0.046 fails a block permutation placebo that preserves serial dependence in daily residual correlations, p equal to 0.19. The paper reports it as a direction only.
Rodríguez Domínguez lists three limitations there himself: that insignificance, calendar clustering of the regimes, and a market-wide lending proxy used only to rank months for a panel that is a subset of the S&P universe. He also flags the deeper problem. An estimator selecting a separator on the same data the feedback contaminates recovers something partly manufactured by the mechanism it is meant to detect. Two mitigations are proposed: disjoint selection and evaluation segments, and exogenous capacity shocks from index reconstitutions or mandate changes. Neither design is exercised in the paper. The 98% coverage requirement also conditions the panel on names that survived the window, which matters less for a residual-comovement diagnostic than it would for a return series, but it is there.
What survives for a trader
Two simulation facts I would keep. Within roughly 0.05 of the boundary a formally unique market relaxes so slowly that over any shorter horizon it looks like a multiple-equilibrium market. Iterations to the fundamental equilibrium go from 7 at tau equal to 0.05 to 307 at 0.485, and exceed the cap of 200,000 exactly at 0.50.
Multiplicity does not imply drift into a convention. Least-squares learning leaves the dominant convention exactly neutral, with the E-stability eigenvalue numerically zero to machine precision in 200 supercritical economies satisfying the dominance condition. Coordination needs a device outside the learning rule.
Endogenous capacity caps amplitudes when it responds at all. The ceilings are 0.903 and 1.331 at baseline-capacity shares of 0.0 and 0.3, matched to three digits by an independent calculation of the self-consistent amplitude, the convention size that reproduces itself once capacity reprices it. At a share of 0.6 the ceiling is infinite and the largest self-consistent amplitude reaches 7.093.
The certification test is the one falsifiable observable here, and it is honest about its own resolution. Two supercritical economies with identical fundamentals run 2,500 periods and stay unconditionally indistinguishable: path correlation 0.037, stationary variances 0.099 and 0.104, first-order autocorrelations minus 0.349 and minus 0.355. Conditionally they separate, 0.062 against 0.488 and 0.510 against 0.068. The certifying separator is the driver set whose residuals pass the cross-sectional independence test. Over 40 supercritical economies of 8 assets across 4,000 periods the ordering is right in 80 of 80 runs. Mean residual dependence is 0.037 under the certifying separator against 0.194 under the alternative. The distributions still overlap: the largest true-separator score of 0.054 exceeds the smallest false-separator score of 0.044, because a destabilising direction loading two assets weakly is hard to detect at any sample size.
No strategy, no returns, no cost analysis; this is not that kind of paper, and pretending otherwise would be unfair to it. What it gives a practitioner is a reason to stop treating factor correlation as a crowding measure, and a clear specification of the dataset that would turn a taxonomy into a verdict. Those two mitigations remain unexercised, as the paper states. Run the disjoint-selection design against exogenous capacity variation and the plus 0.046 would acquire a sign you can trust. Until then the level is measured and the regime is not.
Our earlier note on a learnt conditioning representation, where the representation itself carried the Sharpe from 0.860 to 0.994 before any weighting trick, is the other side of this coin (the MINGLE review). This paper asks what happens to that gain when a crowd converges on the same drivers.