In the core lesson, noise lived at scale root n and structure at scale n. With that much room, the rule finds the big lanes, and Theorem one delivers nearly a dollar of net surplus per dollar spent, with consumers unharmed.

This lesson works at the edge of that result. First, slide the large eigenvalue toward the noise, and find where recovery switches on.

Second, ask what no rule can do. The paper has two impossibility results, both built by hand.

Third, ask whether data can tell you which side of the edge you are on. The paper proposes a test, and our survey found a problem with it.

Section seven builds a test market with two equal blocks. Inside a block goods are complements, and across blocks they are substitutes. That reverses the rackets and shoes example, so the whole pattern fits in one rank one matrix. Which block a good belongs to is the unknown state.

The spectrum has three parts: a spike at minus the quantity one plus b, an all ones mode at zero where a subsidy adds no net surplus, and a bulk near minus one.

She observes this matrix plus independent Gaussian noise, with operator norm about two root n.

Her rule uses only the top eigenvector. The knob is b over root n.

Measure eigenvalues in units of root n. The noise bulk has its edge at two. The paper reports a sharp change in recovery at one, half the noise norm.

Random matrix theory has a law for this. Shifted and rescaled, the noisy matrix is a rank one spike of strength theta plus noise. Below one the spike is absorbed. Above one it appears outside the bulk, at theta plus one over theta.

Above one, the top eigenvector's squared overlap with the truth is one minus one over theta squared. Below, it tends to zero.

This is the spiked matrix transition. The paper never names it. That is our survey's finding.

Here is the check, on a log axis. The gold curve is the law. The purple dots are our survey agent's re-run of the experiment. Above the threshold they sit within about two hundredths of the curve.

The blue squares are medians read off the published figure. Same location and shape, a little flatter. We do not know why.

By our arithmetic, a Davis Kahan bound of the usual form is about four over theta, so it says nothing until theta reaches four. By then the true squared sine is already one sixteenth. Our reading: Davis Kahan proves the asymptotic theorem, and the spiked matrix law describes the figure.

This is the paper's welfare panel. The dark line is median net surplus per dollar: zero below the threshold, because spending lands on the all ones mode, and one well above.

In the published figure the fifth percentile stays a loss until about three point seven, after recovery starts. Our re-runs cross near two and a half, for reasons unknown.

The text skips the upper band. Above one on the left axis, the reversed right axis shows consumers losing.

Our survey's algebra explains both tails. The rule divides by the recovered vector's overlap with quantities. Only five percent of quantity rides on the spike, so noise can make that denominator tiny, with either sign.

Part one of Proposition three builds a market where no rule can guarantee a return. About n to the one quarter states send the same signal.

Each row is a state. In state j, direction j has welfare weight near one, the other hidden directions near zero, and all ones a small weight tau.

She picks one spending pattern for all states. Every column averages to tau, so her average gain is tau.

Tau tends to zero, so in some state her gain is at most tau.

A caution, from our reading. The error is deterministic and cancels the structure exactly. Nothing is proved about independent noise below the threshold.

Part two keeps significant structure and blocks any guarantee for consumers. The matrix is known exactly. Along all ones the eigenvalue is about minus n, so consumers get almost nothing. They gain only along a lane with a tiny eigenvalue.

Quantities have a component of size one over n on one such lane, with a hidden sign.

Her quantity data carries noise of one over root n per lane, far larger. The two signs give nearly identical data.

Whatever she spends on the hidden lane, one sign turns it against consumers. She cannot tell the signs apart, so she fails in one of the two states at least about half the time.

Section eight point two turns the theorem into a test. Start with at least four independent estimates of the market.

Run the intervention rule on one estimate. That gives a subsidy vector.

Treat a second estimate as if it were the true market, and score the subsidy vector against it: each lane's spending share times its pass through weight.

Swap roles and average. Then average over pairs, and bootstrap the pairs for a lower confidence bound.

Certify if that bound is within epsilon of one. The appendix proves the bound valid, and that under significant structure the mean tends to one. It is silent on markets without structure.

This criticism is ours, not the paper's. Its figure has a shelf at zero point eight seven where nothing is recovered.

The second estimate's eigenvalues are mostly noise, up to thirty two, so scoring gives nearly every lane a weight near one.

We simulated a market with no interactions, true return one half. The median read zero point eight, zero point eight seven, then zero point nine two as n grew.

In our runs, that floor needs the top eigenvector rule or a cutoff below the noise edge. Then large pure noise clears any fixed tolerance at the median. Heavy tailed scores make the mean based bound erratic. A cutoff above the edge certifies nothing.

Section eight point three gives the block model a consumer with quadratic utility and curvature matrix H. Then D is minus the inverse of H, so a huge eigenvalue of D is a tiny eigenvalue of H.

The plot shows utility as she buys more of everything together. Complements across categories subtract curvature in that direction. The paper requires that curvature to tend to zero from above as categories are added.

The equilibrium stays well behaved. Price rises toward the choke price, where demand hits zero, and pass through stays bounded.

Our gloss: each firm ignores the demand its price cut would create for its many complements. The paper names it double marginalization across markets.

The paper contrasts this with Pellegrino's hedonic model, where no two products are liked together, by assumption. There D is minus the inverse of B, and the smallest eigenvalue of B is at least one minus alpha. With alpha at zero point one two, no eigenvalue exceeds one point one four before normalization.

Our survey adds a general argument. With only substitutes, Perron Frobenius confines the spectrum of D between minus two and zero.

At two hundred fifty six goods and unit noise, the noise radius is thirty two. By our reading, both caps are buried.

Complements across many categories make the top eigenvalue grow like n, and it leaves the noise.

Section eight point four splits the theory into two stages. A subsidy moves prices, then prices move welfare.

The second stage survives without linearity. Welfare changes by a sum over lanes: eigenvalue, times quantity loading, times price loading.

The first stage breaks. With curved demand, pass through depends on second derivatives, and the eigenbasis of D no longer keeps the lanes separate.

The paper offers two routes and no theorem: learn the map from costs to prices as a black box, or use instruments that set prices directly.

For other network games, the stated obstacle is interaction matrices that are neither symmetric nor negative semidefinite.

Appendix D says where independent entry noise could come from. For each product pair, the authority runs one demand experiment on a distinct household. All households share one utility.

The diagonal is cheap. Every experiment involving good i measures its own price effect again, so averaging shrinks that variance by n.

Lemma nine shows the normalized error has operator norm of order root n. The main term has independent entries. The rest yield to the Frobenius norm and Markov's inequality.

By our arithmetic, signal grows like n and noise like root n, so they cross at a finite market size. The weak points are identical preferences and n squared experiments.

Last, what is new. Judging only from this paper's text, the predecessor by Galeotti, Golub and Goyal assumes the spillovers are known.

On that base sit five layers: noisy data with a guarantee in every state, recoverability without eigenvalue gaps, surplus incidence by lane, impossibility results, and a diagnostic.

By our simulation, the diagnostic run with the top eigenvector rule lacks power against a market with no structure. By our reading, Proposition three is silent on independent noise below the threshold. No theorem covers curved demand.

Read next: Appendix B point two for the two point argument, then Appendix A for the subspace proof. The reference behind the transition law is linked below.