A nominal 95% interval covering about 84.0% should get a trading desk's attention. The paper reaches that result with a 2x2 operator whose two eigenvalues remain genuinely distinct at every finite sample size. Their gap is 2kappaT^{-1/4}.
The interval comes from an oracle delta method. Its derivative is the true local derivative, which clears covariance estimation of blame. At kappa = 1, Rodríguez Domínguez derives the coverage in closed form as Phi(3.92) minus Phi(-1). Meanwhile, the underlying instrumental-variables estimator stays perfectly regular and root-T. A respectable first-stage F statistic offers no warning even as the reported object converges at T^{-1/4}. Appendix B places the perturbations inside the identified IV model, avoiding arbitrary matrix noise. The financial specification itself can therefore generate the pathology.
This counterexample carries the paper. It is a real result. The remaining question is what a desk would need to apply it, and where the specification gives out.
The estimated object
Positions and returns form a feedback loop. Persistent positions move prices through temporary cross-impact. Expected returns change, then positions respond. The paper represents this loop with A = cQ^{-1}K. Here Q measures conditional risk after resilience has played out, K is the temporary cross-impact matrix, and c expresses effective risk-bearing capacity in compatible units.
The stability margin is m = 1/2 + min Re spec(A). A positive m is subcritical; a negative m is spectrally supercritical. The boundary comes from a companion paper by the same author. This paper takes the boundary as given and studies estimated Q, K and c.
Estimation is intentionally plain. Long-horizon residuals supply the sample covariance Q. Linear IV estimates K from the price contrast after residualization on a state dictionary, with an excluded flow shifter W_t. The estimate of c is the sample mean of a normalized capacity series B_t. For p = 2, the three influence functions form an 8-dimensional vector: three entries for vech Q, four for vec K and one for c. Bartlett-HAC estimates the long-run covariance while retaining every cross-block.
Then comes the consequential choice. The paper avoids differentiating the spectral map to produce a standard error. It constructs a Wald confidence region for the 8-dimensional primitive using a chi-square cutoff, then projects the full region through m with interval arithmetic and branch-and-bound.
The result is an outward-certified [L, U] and one of three labels: SUB, SUPspec, or UNR, meaning unresolved. Numerical slop may enlarge the interval or force UNR. A sign cannot flip. As the author writes, "every unresolved box can widen the outer interval but cannot create a sign," exactly the property wanted when the result will size a book.
The confidence set also limits future capacity through a_plan = min{a_des, (1 - eta) / (2 * (1 + rho) * U_d)}. The term U_d is a certified upper bound on destabilizing intensity. The buffer eta may be any number in (0,1), while rho is a pre-specified allowance for implementation overshoot. In the simulation's economic layer, the paper sets eta = 0.10 and rho = 0.05.
Evidence from 60,000 replications
The structural Monte Carlo contains 60 cells. It combines two sample sizes, T = 250, 1000, with six spectral geometries and five distances from the boundary, running 1,000 replications in each cell. Projected-margin coverage reaches 99.992% (MC SE 0.004 pp). Pointwise delta reaches 86.510%, while a deterministic norm envelope reaches 100.000%.
Resolved mistakes separate the methods more sharply. Projected inference makes wrong resolved declarations in 0.003% of cases (two cases), compared with 4.523% for delta and 34.863% for the plug-in classifier.
Abstention pays for that safety. On an equal-weighted average of the 60 cell means, projected inference leaves 60.170% unresolved. Delta leaves 51.900% under the same averaging convention. The norm envelope leaves 91.948%, placing projection between the alternatives.
The 60.170% figure needs its grid attached. Three of the five distance cells are built for abstention: the boundary and two local sequences at plus and minus 0.5/sqrt(T). An equal-weighted overall average therefore says little about the frequency a market would produce.
Fixed-distance non-resolution at T = 250 is more informative. It is 0.00% for symmetric, complex and semisimple geometries, rising to 0.60% for nonnormal diagonalizable, 0.85% near-Jordan and 4.30% at an exact Jordan block. Every fixed-distance geometry resolves at T = 1000. Beyond closeness to the boundary, operator nonnormality drives the unresolved labels.
A separate stress test on the observed-risk cases earns more attention than most of the coverage table. The confidence region stays fixed while the branch-and-bound budget changes. Increasing the budget from 50 to 200 boxes reduces budget exhaustion from 1.000 to 0.078. Yet fixed-regime resolution remains 0.267 at 200, 400 and 1,200 boxes. Additional computation has stopped buying information. The paper does the expensive work needed to establish that distinction.
The disclosures are unusually specific. Two 32-bit seed collisions appear among 60,000 design keys. Their exact design coordinates are reported, and the runs are retained instead of repeated. The resulting standard-error component is bounded at 0.002 percentage points. Five coverage misses are identified by cell and replication index rather than rounded away.
Where implementation ends
We could not test this. Estimating K requires signed cross-asset flow X_t and the excluded shifter W_t. Our data consist of OHLCV bars, with no trade prints, signed flow or order book. The capacity primitive B_t presents an even harder obstacle. It is an application-specific normalization mapping balance sheet into an equilibrium multiplier. The paper states plainly that, without a credible N, the analysis identifies A only up to scale and cannot issue a capacity recommendation.
The abstract makes the limitation explicit: "The financial conclusions remain conditional on the identification of cross-impact, the normalization of capacity, and stability of the inputs over the action horizon." The same abstract advertises a certified capacity decision. Section 4.3 supplies a worked plan that lowers desired capacity from 1.000 to 0.441 while retaining margin 0.050. None of its three primitives comes from market inputs.
The observed-risk exercise shows the gap. It draws on daily closes for 150 equities (3 May 2010 to 23 February 2024) and 1,336 Spanish bonds (29 May 2014 to 20 December 2018). Sixty rolling Q windows are combined with three declared K geometries and five margins, producing 900 certified solver calls. Every call verifies. K and c remain imposed inputs, with observed prices identifying only the risk geometry.
No real market is classified in this paper.
The exercise still connects information quality to action. Fixed-distance resolution reaches 95.000% for equities and 38.890% for Spanish bonds. Raising the risk-uncertainty multiplier from 0.500 to 2.000 pushes median planned capacity at the exact boundary from 0.559 to 0.257 for equities. The corresponding bond figure falls from 0.095 to 0.003. Thin, noisy inputs mean lower capacity and more abstention, a direction familiar to anyone who has watched covariance estimation deteriorate on a bond panel.
The price of certification
Under the baseline loss convention, the projected-safe rule plans 0.541 of capacity. Its violation rate is 0.000%, with mean regret of 0.104. The oracle plans 0.857 with zero violations and zero regret. Certification therefore gives up roughly 37% of oracle capacity.
Among the other zero-violation policies, projected-safe fares better. The deterministic norm plans 0.342 and records regret of 0.196. Full withdrawal plans 0.000, with regret of 0.489.
Policies willing to accept violations achieve lower regret. Pointwise delta plans 0.666, incurs regret of 0.066 and violates at a 1.758% rate. Plug-in plans 0.758, with regret of 0.083 and 6.233% violations. Projected-safe leads on regret only under the high convention-loss calibration: 0.107, against 0.120 for delta and 0.271 for plug-in.
The author's own conclusion is exact: "the evidence is therefore a safety-performance tradeoff, not unconditional regret dominance." No explicit market execution cost enters the regret comparison. The calculation uses the stylized value function a - a^2/2 and a quadratic adjustment penalty around previous capacity of 0.75.
Two assumptions sit behind the 95% safety guarantee without appearing in the reported interval. First, future (Q, K) must equal the estimands inside the confidence set. Parameter drift over the action horizon falls outside that theorem. The appendix defines a predictive-drift version, though it is neither implemented nor validated.
Strong IV relevance is also assumed throughout. The design's population minimum singular value is 0.576, and the sample minimum is 0.303. No weak-IV sequence is simulated. Weak identification would break the primitive Wald region before projection through the spectral map.
Scale creates another limit. Certified branch-and-bound is implemented only for p = 2. For p > 2, the paper gives a global norm envelope of 1/2 plus or minus c_bar * k_bar / q_lower. In the paper's words, it "may increase the unresolved rate but must preserve" the outward bounds. Higher-dimensional certification is presented as another implementation of the theorem. That implementation does not yet exist.
Why the counterexample travels
The portable contribution is the failure mode. A confidence interval for a min or max eigenvalue of an estimated matrix faces the same problem. Within this model, semisimple ties and near-defective matrices emerge inside the estimated impact-risk system. Projecting the set instead of differentiating the map supplies a remedy without relying on financial content.
The capacity apparatus demands much more from the data: signed cross-asset flow, a defensible exogenous flow instrument, post-resilience returns at a documented horizon and a capacity normalization somebody will sign. A desk with a market-by-order feed and an actual balance-sheet series could begin the work. The paper acknowledges these requirements, and its restraint strengthens the case for reading it.
An implementation at p = 5 or higher would change my view. It would need certified intervals that still resolve fixed-distance cells above 0.267, the fixed-regime resolution recorded by the budget stress on the observed-risk cases.