Rational Bubbles at the Spectral Edge: An Operator-Spectral Theory of Fragility, Identification and Finite-Sample Certification

Rational Bubbles at the Spectral Edge: An Operator-Spectral Theory of Fragility, Identification, and Finite-Sample Certification

Introduction

A rational asset-price bubble on a real, dividend-paying asset is a nonstationary phenomenon. hiranotoda2024 make the point sharply: with a strictly positive price $P_t$ and dividend $D_t$, a bubble exists if and only if $\sum_t D_t/P_t<\infty$, equivalently if and only if the price--dividend ratio is nonstationary, so that the price grows faster than linearly relative to the dividend. The characterisation, which dates in operator form to montrucchio2004 and in present-value form to santoswoodford1997, locates the bubble in a single scalar object: the limit of the discounted price, the transversality term, which vanishes for a fundamental asset and is strictly positive for a bubble. The necessity results that follow, that under unbalanced growth every equilibrium is bubbly when the autarky interest rate falls below dividend growth which falls below aggregate growth, are scalar comparisons of growth rates tirole1985,weil1987,hiranotoda2025.

A separate literature, on systemic risk in financial networks, has converged on a different scalar that governs the same kind of transition. The largest eigenvalue, or spectral radius, of a network interaction matrix sets a tipping point: below it the system absorbs shocks, above it the system propagates and amplifies them aots2015. The leading eigen-pair of a stability matrix has been proposed directly as an early-warning signal for banking contagion markose2021, and entropy of the cross-sectional distribution of systemic risk measures has been used as a leading indicator billio2016. These are reduced-form stability matrices, not objects derived from preferences and dividends, and the spectral radius is read off a matrix posited rather than recovered from price behaviour.

This paper connects the two scalars by lifting the bubble characterisation to an operator. Let $B$ be the recovered contemporaneous dependence carrier of a cross-section of asset returns, a bounded self-adjoint operator on a separable Hilbert space whose spectrum carries the cross-sectional comovement of the market. The present-value, or cascade, resolvent $(I-\beta B)^{-1}$ converges in operator norm if and only if $\beta\,\operatorname{r}(B)<1$. The fundamental value of a dividend direction is the resolvent image of that direction. A bubble is a nonzero element of the kernel of $I-\beta B$, an eigen-element at the spectral edge $1/\beta$; it exists precisely when $1/\beta$ is an eigenvalue of $B$, and the dividend direction has finite fundamental value precisely when its spectral measure places no mass at the edge. The single scalar $\beta\,\operatorname{r}(B)=1$ is then the radius of convergence of the pricing resolvent, the location of the bubble eigenspace, and the divergence of the present-value operator norm, all on one operator $B$. We are careful not to overclaim: we do not assert that $B$ equals a separately defined systemic-risk stability Jacobian, only that the systemic-risk literature reads a tipping point off a spectral radius and that ours is the same kind of threshold, here on a recovered, dividend-relevant operator.

The contribution is disciplined by a certification gate that runs through the whole paper. Each spectral reading is labelled strong, proxy, or unidentified\ according to what the recovered spectrum identifies. The scalar threshold $\beta\,\operatorname{r}(B)=1$ and the spectral radius are strong, because the spectral radius is a functional of the spectrum alone and is invariant under the orthogonal rotations of the unresolved eigenframe, recoverable above the random-matrix detectability threshold of bbp2005. The existence of a bubble is proxy, because it requires the projection of the dividend direction onto the boundary eigenspace, a leading-eigenvector object identified only up to a squared overlap near the threshold. The identity of the asset that carries the bubble is unidentified, because the transpose of an operator preserves its spectrum and individual coordinates are not recovered from comovement. We make the strong label quantitative, deriving a finite-sample perturbation bound on the recovered spectral radius that turns the gate into an explicit confidence band on the fragility edge, and we carry the same identification discipline further: the construction extends beyond the symmetric carrier to normal and near-normal operators, the bubble direction receives its own finite-sample band, the fluctuation of the edge is placed within the established limit laws of large random matrices, and the three labels are recast as a formal partial-identification statement. A final structural point follows from the fact that $B$ is recovered from stationary returns: the threshold $\beta\,\operatorname{r}(B)\to1$ is the boundary of stationarity, so the estimator that recovers $B$ can flag the approach to the edge in the interior but cannot certify the crossing; the crossing is read on the price level by an explosive-root detector psy2015, and the two estimators address the two sides of one boundary.

A structural microfoundation answers the natural objection that the recovered comovement carrier is not a pricing kernel. Under a cash-flow network model with a common log-discount, the present-value operator is exactly the cascade resolvent in the cash-flow propagation operator, and the recovered contemporaneous carrier is, in covariance form, the operator square root of that resolvent up to the shock scale; the recovered fragility edge and the structural network edge are then the same event, certified strong\ for the radius and proxy\ for the direction (Section (ref)). A companion result turns the boundary concession into a positive operating characteristic: the edge distance $d_t=1-\rho_t$ is a strong, banded quantity, interior approach is certified with controlled size, and the operator and the explosive-root detector give a complete two-sided reading of one boundary (Section (ref)).

Four further extensions widen the construction without loosening the gate. First, because asset returns are heavy-tailed, the sub-Gaussian finite-sample band is replaced by a robust-carrier band: a Pearson effective-rank band under an $L^4$--$L^2$ norm equivalence mendelsonzhivotovskiy2020, and a distribution-free band built from the Kendall carrier under an elliptical model lindskogmcneilschmock2003,hanliu2017. Second, the discount is made endogenous, with the edge located as a fixed point of $g(r)=r\,\beta(r)$ under an inelastic-discount condition, so the calibrated threshold becomes a structural object. Third, the edge is given a structural dynamic content: the intensity of crisis onset inherits the resolvent norm $\|Q\|=1/(1-\rho)$ through a proportional hazard whose elasticity is identified, coincidentally, from within-risk-set variation cox1972,andersengill1982. Fourth, edge proximity is mapped to a welfare cost, the present-value variance of the carrying mode scaling as $\|Q\|^2$, with only the eigenvalue-ratio core certified strong. Each extension is labelled by the same gate, and its limits are stated.

We illustrate the theory on a panel of eighteen global equity indices over 2004--2024. The spectral radius of the recovered symmetric dependence carrier rises sharply and coincidentally in every documented crisis, the cross-section collapsing from about six to about four effective factors, and the admissible discount ceiling tightening towards the resolvent pole. A companion pre-registered out-of-sample exercise by the author establishes that the same symmetric concentration signature is a strong coincident classifier of crisis states. The illustration is coincident, never a forecast, and we keep it inside that scope throughout.

The construction is not an impossibility theorem against dynamic stochastic general equilibrium or its heterogeneous-agent variants. Production-network general equilibrium already carries a spectral object, the vector of Domar weights, which is a dominant left-eigenvector, and overlapping-generations bubble models state the growth-fragility comparison directly. What is new here is that the operator is recovered from data, every spectral reading is gate-certified, and the boundary case is detected rather than imposed away by a transversality convention. Section (ref) states the difference precisely.

The remainder proceeds as follows. Section (ref) sets up the observable space and the recovered operator. Section (ref) proves the operator pricing result. Section (ref) proves the operator bubble characterisation. Section (ref) states the fragility edge, the certification gate, a finite-sample error bound on the recovered spectral radius, and a robust-carrier version of that bound valid under heavy tails. Section (ref) gives the structural microfoundation, the cash-flow network under which the recovered carrier is the propagation operator. Section (ref) hardens the construction: a generalisation beyond the symmetric carrier, a finite-sample band on the bubble direction, the limit law of the edge fluctuation, and the partial-identification reading of the certification gate. Section (ref) makes the discount endogenous and locates the edge as a fixed point; Section (ref) maps edge proximity to a welfare cost. Section (ref) states the identification boundary between the stationary operator and the price level, and Section (ref) reads approach detection as a certified distance with a two-sided complementarity. Section (ref) gives the structural edge-hazard. Section (ref) gives the empirical illustration, including the heavy-tail robustness exhibit and the cross-sectional hazard read. Section (ref) contrasts the construction with a calibrated-operator model. Section (ref) states the scope and limitations, and Section (ref) concludes.

The observable space and the recovered dependence operator

Let $\mathcal{H}$ be a separable real Hilbert space with inner product $\langle \cdot, \cdot\rangle$ and norm $\|\cdot\|$. We read $\mathcal{H}$ as the space of mean-zero, square-integrable cross-sectional fluctuations of a stationary economy; in the finite illustration of Section (ref), $\mathcal{H}=\mathbb{R}^N$ with the Euclidean inner product and $N$ the number of assets.

Because $B$ is self-adjoint, its spectrum $\operatorname{spec}(B)$ is real and compact, and the spectral radius $\operatorname{r}(B)=\sup\{|s|:s\in\operatorname{spec}(B)\}$ equals the operator norm $\|B\|$. By the spectral theorem reedsimon1980 there is a projection-valued measure $E_B$ on the Borel sets of $\operatorname{spec}(B)$ with $B=\int_{\operatorname{spec}(B)} s\,dE_B(s)$, and for any bounded Borel function $f$ the operator $f(B)=\int f(s)\,dE_B(s)$ is defined, with $\|f(B)\|=\sup_{s\in\operatorname{spec}(B)}|f(s)|$. For a positive semi-definite carrier such as a correlation operator, $\operatorname{spec}(B)\subset[0,\operatorname{r}(B)]$ and $\operatorname{r}(B)$ is the largest eigenvalue $\lambda_1(B)$.

We take as given a one-period discount $\beta\in(0,1)$. Consistent with the discount-identification literature, $\beta$ is treated throughout as a calibrated constant and not as an object identified by the present exercise; the separate question of identifying a discount from behaviour is taken up in companion work and is not claimed here calcottpetkov2024,bastianellovergopoulos2026. The pairing of the recovered operator with a calibrated discount is all we need for the pricing geometry.

Assumption (ref) is the working hypothesis for the strong results below. It is exactly satisfied by a symmetric dependence carrier such as a correlation operator. Departures from self-adjointness, that is a genuine directed or non-normal part of the dependence operator, are discussed as a limitation in Section (ref); on the panel we study the directed energy is small, so $B$ is well approximated by its symmetric part, but the strong labels are reserved for the self-adjoint case and the residual is carried as a bound.

The objects to be priced are dividend directions, and we make the notion concrete rather than leave it formal.

The pricing functional as a bounded operator

Consider a one-period operator Euler equation for a price stream $P\in\mathcal{H}$ supporting a dividend direction $D\in\mathcal{H}$,

the operator analogue of the scalar no-arbitrage recursion $q_tP_t=q_{t+1}(P_{t+1}+D_{t+1})$ under a stationary one-period pricing operator $\beta B$. The present-value, or cascade, operator is the Neumann series $Q=\sum_{k\ge0}\beta^kB^k$.

Proposition (ref) is the operator counterpart of the geometric present-value sum. The norm of the pricing operator diverges as $\beta\,\operatorname{r}(B)\uparrow1$: the present value of a unit dividend direction grows without bound as the discounted spectral radius approaches one. The scalar $\beta\,\operatorname{r}(B)$ is the natural state variable of the pricing geometry.

Operator bubble characterisation

We now characterise bubbles as the homogeneous solutions of the operator Euler equation, and tie their existence to the spectral measure of the dividend direction at the edge $1/\beta$.

Theorem (ref) is the operator lift of the bubble characterisation of montrucchio2004 and hiranotoda2024. The scalar statement that a bubble exists when the price grows faster than the dividend, so that the dividend-yield sum is finite, becomes the operator statement that the bubble lives at the spectral edge $1/\beta$ and that the fundamental value is finite precisely when the dividend direction places no mass there. The transversality term, a scalar in the classical theory, is here a spectral-projection residual onto the edge eigenspace.

The fragility edge and the certification gate

Corollary (ref) is the paper's organising statement and we state its scope carefully. It is a statement about one recovered operator read three ways. It is not an identity between the dividend-pricing operator $B$ and a separately defined systemic-risk stability Jacobian. The systemic-risk literature reads a tipping point off the spectral radius of a stability matrix aots2015,markose2021; ours is the same kind of threshold, here on a recovered, dividend-relevant operator, and the two coincide only under the additional modelling choice that the stability operator is taken to be $\beta B$. We make no such identification and present Corollary (ref) as a within-operator coincidence, which is what the spectral algebra delivers.

The readings differ in what the recovered spectrum identifies, and we label them accordingly.

A finite-sample error bound on the recovered spectral radius

Proposition (ref)(i) certifies the spectral radius only qualitatively: it states that $\operatorname{r}(B)$ is recoverable above the detectability threshold $\theta>\sqrt{q}$. We now make the strong label quantitative, with a finite-sample error bound on the recovered $\operatorname{r}(\widehat{B})$, so that the certification carries an explicit confidence band rather than an asymptotic guarantee. Throughout, $\widehat{B}$ is the sample correlation operator formed from $T$ observations of the standardised, $I(0)$, $N$-dimensional return vector whose population correlation operator is $B$; both are symmetric and positive semi-definite, so $\operatorname{r}(\cdot)$ is the largest eigenvalue. Write $q=N/T$ and let $r_{e}(B)=\operatorname{tr}(B)/\|B\|=N/\operatorname{r}(B)$ be the effective rank of the carrier.

The bound upgrades the gate of Proposition (ref) from an asymptotic detectability statement into a decision rule with an explicit confidence band.

Two remarks place the bound against the random-matrix transition the gate already invokes and against the panel.

Heavy tails and the robust carrier

Theorem (ref)(ii) borrows its operator-norm concentration from the sub-Gaussian sample-covariance theory of koltchinskiilounici2017, in the form of vershynin2018. Daily equity returns are heavy-tailed, so that borrowing is not free. Under only a finite number of moments the empirical second-moment functional concentrates polynomially, not exponentially, in the confidence parameter, the matrix form of the scalar deviation study of catoni2012: for the plain sample correlation $\Phi_t$ the upper tail of $\|\Phi_t-B\|$ decays as a power of the inverse confidence rather than as $e^{-t}$, so the $1-\delta$ coverage on which Corollary (ref) rests fails as stated. The remedy is to change the estimator, not to weaken the conclusion. A robust carrier recovers the band under a constant number of moments, after which the Weyl bound, Corollary (ref), and the gate hold with the robust carrier in place of $\Phi_t$; the sample-correlation reading becomes the light-tailed special case.

The distribution-free alternative dispenses with moment conditions entirely, at the price of a looser, ambient-dimension rate, and it targets the latent correlation rather than the sample one.

A structural microfoundation: the cash-flow network

Section (ref) postulated the operator Euler equation (ref) and read the recovered carrier $B$ as the one-period pricing operator. One may grant the spectral algebra and still deny the economics: the recovered $B$ is a contemporaneous comovement object, not a structural pricing kernel, so why should it appear inside the present-value resolvent at all? This section answers the question inside a named model. A cash-flow network operator $A$ governs the propagation of dividend innovations; the log-linearised present value is exactly the Neumann series in the discounted network operator $\rho A$; the recovered contemporaneous covariance carrier is a strictly monotone spectral transform of $A$ sharing its eigenframe, while the bounded correlation carrier $B$ shares that eigenframe under a regular-exposure condition and in general approaches the same dominant mode as the edge is approached. Under that model the radius of convergence of the pricing resolvent and the structural network edge are one scalar, and the certification gate states precisely which parts of it the contemporaneous cross-section recovers. Outside the model we claim no such identity, exactly as in Section (ref): the result is a conditional structural reading, not an unconditional one.

We name the maintained hypotheses explicitly.

Proposition (ref) answers the reduced-form objection. The recovered contemporaneous carrier is not an operator borrowed from a different theory; under Assumption (ref) it is, in covariance form, the spectral image of the very operator $A$ that the present-value resolvent inverts, sharing its eigenframe and its edge, and in the standardised correlation form of the body it shares the dominant mode in the edge limit. The clean statement is the square-root identity $Q=\sigma^{-1}\Sigma_u^{1/2}$: the present-value cascade operator is the operator square root of the recovered covariance carrier, up to the primitive shock scale. This is the precise sense in which a contemporaneous comovement object appears inside present values. The body writes the cascade in $B$-space as $(I-\beta B)^{-1}$ and the microfoundation writes it in $A$-space as $(I-\rho A)^{-1}=\sigma^{-1}\Sigma_u^{1/2}$; the two writings differ in parametrisation, with $\beta\ne\rho$ and $B\ne A$ as operators, but they share the single edge event, the divergence of the leading recovered eigenvalue, which Corollary (ref) reads three ways and which is here the structural network edge.

Generalisation, eigenvector inference, and partial identification

The strong results so far assume a self-adjoint carrier and certify the spectral radius, while the existence of a bubble rests on a leading-eigenvector overlap. We harden the construction on four fronts. We relax self-adjointness to normal and near-normal carriers; we bound, in finite samples, the eigenvector that carries the edge mass; we locate the fluctuation of the recovered edge within the limit laws of large random matrices; and we recast the three certification labels as a formal partial-identification statement.

Normal carriers and near-normal perturbations

Finite-sample inference on the bubble direction

Theorem (ref) bounds the recovered radius. The edge mass $\mu_{\widehat{D}}(\{\operatorname{r}(B)\})= \langle e_1, \widehat{D}\rangle^2$ of (ref), the proxy for bubble existence, depends instead on the leading eigenvector $e_1$, which we now control.

The band widens as the spectral gap $g$ closes: a near-degenerate top eigenspace makes the bubble direction, and hence the bubble content, statistically unrecoverable, which is the finite-sample face of the proxy-to-unidentified transition. The edge mass is thus proxy\ with an explicit two-sided band, sharpening Proposition (ref)(ii) from a random-matrix overlap into a usable interval. The gap $g=\lambda_1(B)- \lambda_2(B)$ is a population quantity; a feasible, data-computable band replaces it with the plug-in lower bound $\widehat{g}-2\|\widehat{B}-B\|$, where $\widehat{g}=\lambda_1(\widehat{B})-\lambda_2(\widehat{B})$, since by the Weyl inequality each eigenvalue moves by at most $\|\widehat{B}-B\|$, so $g\ge\widehat{g}-2\|\widehat{B}-B\|$. The resulting band is conservative and is reported only while this lower bound is positive, that is while the top mode remains separated after accounting for the recovery error of Theorem (ref).

The fluctuation of the recovered edge

Proposition (ref) supplies the missing boundary uncertainty. Part (i) is a calibrated null that turns the strong reading of the radius into a test: a recovered edge counts as a genuine spike, and not a bulk fluctuation, exactly when it lies in the Tracy-Widom right tail beyond the bulk. Part (ii) is the source of the proxy band of Proposition (ref)(ii) and of the edge-mass band of Theorem (ref), since the limiting overlap is precisely the squared-overlap factor there. The edge is strong\ when it clears both the Tracy-Widom bulk and the detectability threshold, and unidentified\ below them. The Tracy-Widom null is an idealised reference distribution, derived under independent coordinates; the empirical panel is stationary in returns but heavy-tailed and cross-sectionally dependent, so the null is used as a calibrated benchmark for the bulk edge rather than as an exact finite-sample distribution, and the operative criterion remains the detectability threshold $\theta>\sqrt{q}$.

The certification gate as partial identification

The three labels are not informal. They are a partial-identification statement in the sense of tamer2010: each component of the structural object has an identified set, and the labels record whether that set is a point, a proper subset, or the whole space.

Theorem (ref) states the paper's ontology exactly: strong\ is point identification, proxy\ is set identification within a computable outer set, and unidentified\ is the bounding set equal to the whole space. The certification gate is thus a partial-identification discipline applied reading by reading.

The endogenous fragility edge

Throughout, the discount $\beta$ has been calibrated rather than determined within the economy. We now let it respond to the systemic radius, $\beta=\beta(r)$ with $r=\operatorname{r}(B)$, as a no-arbitrage discount on the systemic factor in the sense of hansenjagannathan1991 and campbellshiller1988, and ask where the fragility edge then sits. Write $g(r)=r\,\beta(r)$ for the discounted spectral radius; the edge is the fixed point $g(r)=1$.

A transparent special case is the linear stochastic-discount factor $R(r)=R_f+\gamma r$, $\beta(r)=1/(R_f+\gamma r)$, under which $g(r)=r/(R_f+\gamma r)$ is strictly increasing, $E_\beta(r)=-\gamma r/(R_f+\gamma r)\in(-1,0)$ automatically, and a unique interior edge exists when $1-R_f<\gamma<1-R_f/N$, in closed form $r^\ast=R_f/(1-\gamma)$. The intermediary channels of hekrishnamurthy2013 and brunnermeiersannikov2014 supply the countercyclical discount of part (iii): as systemic comovement rises, the required return on the systemic factor rises and the present-value multiplier falls, so $\beta$ decreases in $r$ and the endogenous edge is pushed out relative to a fixed discount. Calibrating the linear factor at $\gamma=0.5$ and $R_f=3.5$ on the eighteen-index panel places the endogenous edge at $r^\ast=7.00$, between the calm mean $6.74$ and the crisis mean $8.03$ of the leading eigenvalue: the discounted radius is $g=0.98$ in calm, inside the edge, and $g=1.07$ in crisis, beyond it, with discount elasticity moving from $-0.49$ to $-0.53$, and the countercyclical discount delays the edge by $r^\ast-r_{\mathrm{fixed}}=0.13$ relative to the constant-discount benchmark. The calibration is illustrative and transparent: consistent with the discipline applied to the fixed discount throughout, $\beta(\cdot)$ is calibrated, not identified from behaviour, and the theorem asserts the existence, uniqueness, and comparative static of $r^\ast$ given $\beta(\cdot)$, not the recovery of $\beta(\cdot)$. The recovered edge $g(r^\ast)=1$ is the first point at which a bubble eigen-element can exist montrucchio2004, now determined within the economy rather than imposed.

The welfare cost of edge proximity

The resolvent norm $\|Q\|=1/(1-\rho)$ is a present-value amplification, and as the edge is approached it diverges. We read this amplification as a welfare cost in the consumption-equivalent units of lucas1987, but we do so by embedding the carrying mode in a representative-agent consumption process rather than by reinterpreting a variance, and we certify only the part of the reading that the gate permits.

Three maintained hypotheses carry the structural content; each is named explicitly, and the certification of the conclusion is conditional on exactly these and on nothing more.

(H-ISO) Isotropic innovation. The cascade innovation is zero-mean with $\mathbb{E}[\varepsilon\varepsilon^\top]=\sigma^2 I$; for the exact exponential form in (ii) we further take $\varepsilon$ Gaussian, so the carrying-mode component is log-normal, and without Gaussianity part (ii) holds to second order. This is the identifying normalisation introduced with the operator Euler equation (ref): the split of observed comovement into a cascade gain $Q$ and a shock covariance is not identified from contemporaneous returns alone, exactly as the influence vector of acemoglu2012network is identified there only because the production network is observed, while here the carrier is recovered. The level of every welfare statement inherits this normalisation.

(H-LOAD) Structural consumption loading. Aggregate log consumption growth loads linearly on the aggregate-claim image of the cascade,

with a constant loading $\phi\in\mathbb{R}$ and an idiosyncratic residual $u_{t+1}$ orthogonal to the carrying mode. The coefficient $\phi$ is a primitive of the consumption block, not a recovered object; it converts the dimensionless cascade coordinate into consumption units. Equation (ref) is what makes the present-value variance a consumption-growth variance rather than a variance of an abstract deviation.

(H-PREF) Recursive preferences. The representative agent has Epstein--Zin--Weil recursive utility epsteinzin1989 with relative risk aversion $\gamma>0$ and unit elasticity of intertemporal substitution. In this case, by the risk-sensitive reduction of tallarini2000, the compensation for consumption risk is governed by the risk-aversion coefficient $\gamma$ alone, decoupled from the elasticity of substitution. This is the device that makes the cost a genuine consumption-equivalent magnitude in the units of lucas1987, and that lets the amplification stand well above the expected-utility benchmark, the lesson of barlevy2004 and alvarezjermann2004 and, in the long-run-risk reading, of bansalyaron2004.

The certification discipline governs what may be claimed. The structural step is genuine: under (H-LOAD) and (H-PREF) the quantity $L(\rho)$ is a consumption-equivalent compensation in the units of lucas1987, with the amplification $\|Q\|^2$ derived from the resolvent rather than tabulated as a discount ladder, and with the risk-aversion coefficient $\gamma$, not a reduced-form variance, carrying the cost. What survives the gate as $\text{strong}$ is narrower than the level. The only $\text{strong}$ between-regime statement is the rise in the spectral radius itself, the leading eigenvalue moving from $6.74$ in calm windows to $8.03$ in crisis windows, a ratio of $1.19$ with a bootstrap difference confidence interval strictly above zero; its square, about $1.42$, is the squared ratio of recovered radii, a $\text{strong}$ spectral object free of the discount and of every preference and loading parameter, and it is not the welfare-cost amplification. The welfare-cost level $L(\rho)$ and the welfare-cost between-regime ratio are conditional on the calibrated discount through $\rho$ and on the product $\gamma\phi^2 m\sigma^2$, with $m$ a $\text{proxy}$ overlap; they are read as a discount-conditional magnitude, not a certified welfare number, so the present-value welfare channel, which is the paper's novel object, carries no $\text{strong}$ between-regime certificate, only its level conditioned on the discount. The parameter-free elasticity $2\rho/(1-\rho)$ is $\text{strong}$ only as a shape: it is $18$ at $\rho=0.90$, $38$ at $0.95$, and $198$ at $0.99$, and the second-order cost rises a hundredfold from $\rho=0.90$ to $\rho=0.99$, but these values of $\rho$ are edge-distance targets, not implied by any particular time preference, and the level at any single $\rho$ is discount-conditional. The reading supports the monitoring protocol in shape: the convexity of $L$ implies that intervention thresholds should be escalating rather than uniform, while their levels are not identified. The individual asset that carries the bubble remains $\text{unidentified}$, since the transpose of $B$ preserves its spectrum and the welfare cost is a functional of the spectrum and the recovered leading direction, not of any coordinate.

The identification boundary between the operator and the price level

The recovered operator $B$ is estimated from stationary returns, while the bubble it characterises is a nonstationary object on the price level. The two facts meet at the edge, and the meeting place is an identification boundary.

Proposition (ref) draws a clean line. The recovered operator on $I(0)$ returns and the explosive-root detector on $I(1)$ levels are not competitors; they describe the two sides of one boundary. The certification gate restricts the strong claim to the interior and treats the crossing as a flagged boundary, not a certified event, which is the disciplined reading the data permit.

Approach detection as a certified distance

One may grant Proposition (ref) and press further: if the stationary operator only ever sees the interior, the recovered scalar is a coincident approach-detector and nothing more. We accept the premise and convert it into a positive operating characteristic. The content of this subsection is that approach-detection is itself a certified measurement. The edge distance $d_t=1-\rho_t$, with $\rho_t=\beta\,\operatorname{r}(B_t)$, is a strong, point-identified, estimable quantity carrying the same finite-sample band as the radius; the interior-approach decision admits a false-positive guarantee at any preset confidence; and the operator and the explosive-root detector of psy2015 together deliver a two-sided reading of one boundary, each side certified for what it identifies. None of the statements is predictive. Every object is contemporaneous to the estimation window, and we make the coincident discipline a maintained hypothesis.

We name the maintained hypotheses explicitly.

The certified distance turns the qualitative concession into a quantity with a band. We now make detecting its shrinkage a decision with a size guarantee, and read off the resolution at which that decision is possible.

Proposition (ref) reads the concession as a theorem with a rate. The stationary operator certifies interior approach to any level $\tau$ at controlled size, down to an $O(\sqrt{r_e/T})$ neighbourhood of that level; the only region it cannot resolve is a collar that shrinks at the parametric rate. The crossing is not a hard case of the same test but lies outside the operator's domain of validity, by the breakdown of the stationary-recovery premise at the edge; it is read on the price level by the detector that can see it.

The final statement pairs the two sides of the boundary and certifies each for its own identification target.

Two scope remarks place the result against the gate and the panel. First, the operator side is strong\ in the sense of Proposition (ref)(i): $d_t$ and the decisions $\mathcal{A}_\tau$ are functionals of the spectrum, rotation-invariant, recoverable above $\theta>\sqrt{q}$; the construction adds no claim about the bubble direction or the carrying coordinate, which remain proxy\ and unidentified. The level side is a separate detector on a separate series, certified in its own explosive-root sense, not relabelled as a spectral object. Second, on the panel of Section (ref) the band $\varepsilon_{T,\delta}$ is read as a rate rather than a numerical guarantee, since the returns are weakly dependent and heavy-tailed and the reported inference is carried by the dependence-robust moving-block resampling; the size statement of Proposition (ref)(i), the false-positive control that matters most in practice, is conservative and robust, while the power statements inherit the band hypothesis and are read as scaling statements on the panel, in line with the remark following Theorem (ref). The substantive conclusion is invariant: approach is the detection of a real, estimable, gate-certified distance shrinking at a known rate, and the crossing is ceded, by the breakdown of stationary recovery at the edge, to the detector that can see it.

The structural edge-hazard

The readings so far are coincident descriptions of one operator. We now give the edge a structural dynamic content, the intensity of crisis onset as a function of the certified edge distance, without crossing into forecasting. On the symmetric line the cascade resolvent obeys $\|Q_t\|=1/(1-\rho_t)=1/d_t$, with $\rho_t=\beta\,\operatorname{r}(B_t)$ and edge distance $d_t=1-\rho_t$; this is the worst-case present-value amplification of a contemporaneous shock, and it diverges at the edge. The structural postulate of this section is that the instantaneous intensity of crisis onset inherits this amplification monotonically. The link is postulated, not derived: the resolvent supplies the canonical form of the singularity at the edge, while the mapping to intensity is a maintained structural assumption, in the doubly-stochastic sense of lando1998 and the counting-process sense of andersengill1982.

The reading is structural and coincident, not causal and not predictive. The covariate is the certified edge, whose scalar carrier $\rho_t$ is the strong object of Proposition (ref)(i); the elasticity $\theta$ is a fitted association under a maintained proportional intensity, not a functional of $\rho_t$, so it is itself a proxy quantity, identified only up to the maintained model. The baseline $\chi_0(t)$ is a nuisance scale, the counterpart of the deferred discount: just as $\beta$ fixes the level at which the edge binds, $\chi_0$ fixes the absolute onset rate, while the edge-to-intensity elasticity is the pinned content. A single systemic series cancels in every within-instant risk-set ratio by part (iv), so the elasticity on the systemic edge is identified only through across-time variation, and a genuine within-risk-set partial likelihood in the sense of cox1972,cox1975 requires the subject-specific participation in the leading mode, which the gate labels proxy. The discrete-time and stochastic-covariate hazard precedents of prenticegloeckler1978, shumway2001 and duffiesaitawang2007 are method references; their use is predictive, whereas the reading here is coincident by construction, the covariate and the onset sharing the time index, and the proportional form is testable, not assumed grambschtherneau1994.

Identification of the edge-hazard elasticity

Proposition (ref)(iv) states that the elasticity $\theta$ is identified only when the covariate varies within risk sets. We now make that clause structural rather than generic: the within-risk-set variation of the cross-sectional edge covariate is governed by the same participation, or effective-rank, object that governs the pricing resolvent, so the operator that drives the theory also decides whether the hazard is identifiable. The cross-sectional covariate is the leading-mode participation $u_i(t)=N\,e_{1,i}(t)^2$, with $e_1(t)$ the unit leading eigenvector of the recovered carrier $B_t$, so $\sum_i e_{1,i}(t)^2=1$ and $\tfrac1N\sum_i u_i(t)=1$. Write the leading-mode participation ratio \[ P_1(t)=\Big(\sum_{i=1}^N e_{1,i}(t)^4\Big)^{-1}\in[1,N], \] which equals $1$ when the mode is carried by a single coordinate and $N$ when it is uniform across the cross-section; $P_1(t)$ is the leading-eigenvector counterpart of the effective rank used elsewhere in the paper. We maintain two named hypotheses. (PH) the proportional-intensity model of Proposition (ref)(iv), $\chi_i(t)=\chi_0(t)\exp(\theta u_i(t))$. (AG) the recurrent counting-process design of andersengill1982, each subject $i$ at risk with covariate $u_i(t)$, re-entering on recovery to a prior peak, the risk set at the $k$th event time being $R(t_k)$ of size $m_k$.

The reading of the gate is as follows. Part (i) is the structural content: the discriminating power of the edge covariate is, exactly, the inverse participation ratio of the leading mode, so the object that decides whether the hazard elasticity can be estimated is the same effective-rank functional of the recovered operator that the pricing and fragility readings use. That functional is built from the leading eigenvector, so it inherits the random-matrix overlap of Proposition (ref)(ii) and the finite-sample band of Theorem (ref); the participation ratio $P_1$ and the dispersion $S_t$ are therefore proxy\ objects, of the same identification class as the covariate. Part (ii) is exact given the maintained model and pins the identification budget: the partial-likelihood information cannot exceed the cross-sectional dispersion of leading-mode participation, scaled by the risk-set occupancy. Part (iii) is the limiting consequence: as the leading eigenvector delocalises toward uniformity, every within-risk-set comparison flattens and $\theta$ ceases to be identified. The elasticity $\theta$ remains proxy, as in Proposition (ref); which asset carries the mode is unidentified, the transpose-invariance of the spectrum; and the only strong\ object in the neighbourhood of the edge remains the scalar $\rho_t=\beta\,\operatorname{r}(B_t)$. Identifiability of the hazard read is thus governed by a proxy\ participation object, which is why the entire edge-hazard reading is at most proxy.

Empirical illustration

We illustrate the spectral-radius geometry on a panel of $N=18$ daily global equity index return series over 21 December 2004 to 28 June 2024, $5025$ usable cross-sections after a trailing window. The recovered carrier is the rolling sample correlation operator $\Phi_t$ on standardised returns over a window of $W=250$ trading days, a positive semi-definite operator with $\operatorname{tr}\Phi_t=N$, so its spectral radius is the largest eigenvalue $\lambda_1(\Phi_t)\in[1,N]$. We report $\lambda_1(\Phi_t)$, the normalised concentration $\kappa_t=\lambda_1(\Phi_t)/N\in[1/N,1]$, the participation ratio $N^{\mathrm{eff}}_t=(\sum_i\lambda_i)^2/\sum_i\lambda_i^2\in[1,N]$, and the admissible discount ceiling $\beta_{\max}(t)=1/\lambda_1(\Phi_t)$ at which the cascade resolvent reaches its pole. The crisis label is external: a drawdown of at least twenty per cent of the equal-weighted market index, the same label used in the companion classification exercise. Differences in means between crisis and calm states are tested with a moving-block bootstrap (block length $21$, $2000$ resamples, seed fixed); the computation is deterministic.

A contemporaneous correlation operator is a modelling choice and not the only conceivable dependence object, so we state why it is the economically sufficient one for the geometry at hand. The choice is dictated by what the pricing functional prices. The one-period operator $\beta B$ of Section (ref) propagates a common innovation across the cross-section within the period, and the present value compounds that same contemporaneous coupling geometrically, so the fragility edge $\beta\,\operatorname{r}(B)=1$ is a statement about the strength of contemporaneous comovement, the extent to which a shock is shared across the cross-section now, because that is exactly what makes the discounted cascade diverge. The contemporaneous correlation operator delivers the object the theory requires and no more: it is symmetric and positive semi-definite by construction, so it satisfies Assumption (ref) exactly, its spectrum is real and ordered, and its spectral radius is the largest eigenvalue $\lambda_1$, the market mode in whose direction the cross-section moves together. A lead-lag, or directed, operator would add information about the timing of transmission, but it is non-normal: its spectral radius can detach from that of its symmetric part, its eigenframe is not orthogonal, and the strong certification labels of Proposition (ref) are then unavailable. What controls the resolvent pole, to first order, is the size of the contemporaneous common component rather than the order in which assets respond, and it is precisely this contemporaneous comovement that intensifies in crisis, as the collapsing participation ratio records. The directed half is not discarded: on this panel its energy is small, so the symmetric carrier is a close approximation and the residual is carried as the departure-from-normality bound of Section (ref). Contemporaneous correlation is, in this precise sense, economically sufficient for the pricing and fragility geometry, while the directed extension is a genuine and separate object left to future work.

Table (ref) reports the contrast. The spectral radius rises from a calm mean of $6.74$ to a crisis mean of $8.03$ out of a maximum of $18$, a difference of $1.29$ with a bootstrap interval $[1.07,1.52]$ and a bootstrap $p$ below $0.001$. The participation ratio collapses from $5.62$ to $3.86$ effective factors: in crisis the eighteen indices behave like about four independent factors. The admissible discount ceiling tightens from $0.152$ to $0.125$, a fall of about a sixth, with a sample minimum of $0.100$; the economy moves towards the resolvent pole as it enters a crisis, which is the empirical content of the fragility edge.

Table (ref) lists the six largest spectral-radius episodes after de-clustering within six months. They span the March 2020 turbulence, which carries the maximum spectral radius $9.97$ with the cross-section reduced to about three effective factors, a late-2020 episode, a 2016 episode, and the 2008--2009 crisis including the trough at a drawdown of about one half. The drawdown column is candid about the limits of the marker: three of the six episodes, late-2020, 2016 and early-2008, have drawdowns of only about three, ten and seventeen per cent and so are not crises by the twenty per cent label. Leading-eigenvalue concentration therefore marks stress episodes broadly and is not exclusive to the labelled crises, the same imperfect separation that holds the coincident classifier of Table (ref) below one; the spectral radius nonetheless rises at every labelled crisis without any tuning to those dates.

The contrast in Table (ref) is a coincident description, not a forecast, and we keep it inside that scope. The out-of-sample standing of the same signature is established in a companion pre-registered exercise by the author, summarised in Table (ref): a walk-forward classifier built on the symmetric concentration signature alone, the leading-eigenvalue share and the participation ratio, against the external drawdown label, earns an out-of-sample area under the curve of $0.886$ on this panel and $0.911$, $0.908$ and $0.700$ on three further panels. The pre-registered decisive comparison adds the concentration signature to a trailing realised-volatility benchmark and beats the benchmark alone by a difference in area of about $0.13$ to $0.17$, positive on every panel with a DeLong and moving-block test below $0.001$; the in-sample to out-of-sample gap of the signature is about $0.056$ on the main panel. On the primary panel the combined set scores below the signature alone, $0.858$ against $0.886$, so adding the volatility benchmark to the concentration signature does not help there; we report this as the pre-registration delivers it. The classifier is coincident; the one-step forecastability ceiling for the aggregate return is respected, and no forecasting claim is made. This companion exercise has its own pre-registration and deposited output, included with the replication materials; its figures are not recomputed in the present package.

Two readings of Table (ref) and Table (ref) match the theory. First, the spectral radius rising towards its maximum, with the participation ratio collapsing, is the recovered operator moving towards the spectral edge of Corollary (ref), where the present-value resolvent norm diverges. Second, the admissible discount ceiling $1/\lambda_1$ falling in crisis is the resolvent pole tightening: a smaller discount suffices to push the discounted spectral radius to one. Both are coincident, descriptive, and certified only to the extent the spectrum identifies them, which by Proposition (ref) is the spectral radius itself, the strong object, and not the identity of any single index.

Cross-market replication

The eighteen-index panel is small, and the question is whether the spectral-radius signature is an artefact of that cross-section or a property of equity dependence more generally. We replicate the in-sample contrast on three further panels drawn from a public daily portfolio library: the forty-eight industry portfolios, the thirty industry portfolios, and the twenty-five size and book-to-market portfolios, each over a window matched to the primary panel. The estimator is identical in every respect, the rolling sample correlation operator on standardised returns over $W=250$ days, the same external twenty per cent drawdown label of the equal-weighted panel index, and the same moving-block bootstrap at the same fixed seed; only the input cross-section changes. The raw data are downloaded from the source library and parsed into return panels by the replication package, so the exercise reproduces from public inputs.

Table (ref) reports the result. The fragility-edge signature replicates on every panel: the spectral radius rises significantly from calm to crisis, the participation ratio collapses, and the admissible discount ceiling tightens, each with a bootstrap $p$ below $0.001$. The portfolio panels are far more concentrated than the index panel, as expected of cross-sections built from a single equity market: the leading eigenvalue already carries from sixty-one to eighty-seven per cent of the trace in calm markets, against thirty-eight per cent for the international indices, so the cross-sections behave like between one and three independent factors even before a crisis and contract further within one. The direction and the significance of the movement, not its magnitude, are what transfer; the magnitude scales with the baseline concentration of the panel. The signature is therefore a property of equity dependence and not of the particular eighteen-index cross-section.

Robustness to heavy tails: the distribution-free carrier

The recovered carrier is the sample correlation operator, and Section (ref) shows that its finite-sample certification band is not valid as stated under heavy tails, while a robust carrier restores it. The data are markedly heavy-tailed: the maximum sample excess kurtosis across the eighteen standardised index series is $374$, far outside the sub-Gaussian regime, which is the empirical reason the band needs the robust replacement. We therefore re-run the calm-versus-crisis spectral-radius contrast with the distribution-free Kendall carrier of Proposition (ref), $\widetilde{B}_{jk}=\sin(\tfrac{\pi}{2}\widehat{\tau}_{jk})$, on every panel, with the same trailing window, the same external drawdown label, and the same bootstrap and seed; only the estimator changes. Under an elliptical law with finite second moments the Kendall carrier targets the same operator as the sample correlation, so the two contrasts are directly comparable.

Table (ref) reports the result. The fragility-edge signature survives the estimator swap on every panel: the robust leading eigenvalue rises from calm to crisis on all four, with bootstrap $p$ below $0.001$ throughout, tracking the sample-correlation benchmark in both direction and order of magnitude. On the primary panel the distribution-free scatter estimator of tyler1987 gives a calm-to-crisis move of $6.15\to7.31$, the same lift by a different robust route. The edge mass on the robust leading eigenvector is nearly unchanged across the regime, $0.927$ in calm against $0.919$ in crisis on the primary panel, so the proxy reading of the gate transfers as well. The signature is thus a property of the dependence, not an artefact of the sample-correlation estimator under heavy tails.

Operationalising the dividend direction and the calibrated edge

The spectral-radius contrast above reports $\operatorname{r}(B)$ rising in crisis but does not yet exhibit the discounted quantity $\beta\,\operatorname{r}(B)$ approaching one, nor does it make the dividend direction concrete. We address both, on the same carrier $\Phi_t$.

First, the dividend direction. Taking the aggregate-claim direction $\widehat{D}=\mathbf{1}/\sqrt{N}$ of Definition (ref), we measure the edge mass (ref), the squared overlap of $\widehat{D}$ with the leading eigenvector $e_1(t)$ of $\Phi_t$. The overlap is large throughout, with an overall mean of $0.925$ and a maximum of $0.977$: the equal-weighted aggregate cash-flow claim is almost entirely the boundary mode, so the bubble eigenspace, if reached, is the very direction in which the aggregate claim is held. The overlap is essentially flat across regimes, $0.926$ in calm and $0.918$ in crisis, which says the aggregate claim sits on the boundary mode at all times; what changes in crisis is the position of that mode relative to the pole, not the overlap. This high overlap is close to mechanical: for an all-positive correlation carrier the leading eigenvector is approximately the market mode, so its squared overlap with the equal-weighted direction is forced near one by positivity, a Perron-Frobenius near-identity, and the reading confirms rather than discovers that the aggregate claim sits on the leading mode. The mechanism does not drain the reading of content: mechanical under a positive carrier though it is, it is economically interpretable, for the aggregate cash-flow claim is by construction exposed to the systemic mode that the spectral radius measures, so the one scalar that governs the fragility edge also governs the present value of the aggregate claim. The overlap is, in addition, strongly identified here: with a spike excess $\theta\approx\lambda_1-1$, with $\lambda_1\in[6.7,8.0]$, and $q=N/T=18/250\approx0.07$, the random-matrix overlap band of Proposition (ref) is about $0.99$, far above the detectability threshold, so the proxy is tight.

Second, the calibrated edge. We fix a single transmission intensity $\beta^\ast$, calibrated transparently so that the calm cross-section is exactly critical, $\beta^\ast=1/\overline{\lambda}_1^{\,\mathrm{calm}}=0.148$, and report the discounted spectral radius $\rho_t=\beta^\ast\lambda_1(\Phi_t)$. By construction the calm mean of $\rho$ is one; the content is the crisis elevation and the separation of the two regimes. In crisis $\rho$ rises to a mean of $1.19$ and a maximum of $1.48$, past the resolvent pole, and the share of days with $\rho\ge1$ rises from $0.448$ in calm to $0.966$ in crisis. The share of the aggregate claim carried by the supercritical modes, those with $\beta^\ast\lambda_k\ge1$, rises from $0.424$ in calm to $0.887$ in crisis: in a crisis almost nine tenths of the aggregate cash-flow claim sits in the part of the spectrum where the present-value resolvent diverges. Table (ref) collects these readings. We stress that $\beta^\ast$ is a calibrated transmission and normalisation constant, not a time-preference rate, and is not identified by the exercise; the calibration fixes the calm baseline and nothing more, consistent with the treatment of $\beta$ throughout. Because $\beta^\ast$ is fixed to the calm mean, the crisis ratio $\rho_{\mathrm{crisis}}/\rho_{\mathrm{calm}}$ equals $\lambda_1^{\,\mathrm{crisis}}/\lambda_1^{\,\mathrm{calm}}$, and the supercritical-claim-mass contrast is the edge mass times the share of days with $\rho\ge1$; these are re-expressions of the Table (ref) spectral-radius contrast under a calibrated normalisation, not independent evidence. The substantive content is the single, bootstrap-significant rise in $\lambda_1$, here read through the pole.

The explosive-root bridge

Proposition (ref) holds that the stationary estimator flags the approach to the edge from the interior while the crossing is an explosive, $I(1)$, event read on the price level. We implement the price-level side with the right-tailed recursive unit-root tests of psy2015: the supremum augmented Dickey-Fuller statistic, SADF, and its generalised, doubly recursive form, GSADF, on the monthly log level of the equal-weighted index built from the same panel, $246$ months over 2004--2024. The minimum window is the conventional $r_0=0.01+1.8/\sqrt{T}$, here $31$ months, the regression carries an intercept and no augmentation, and critical values and the backward-SADF date-stamping sequence are obtained by Monte Carlo simulation of Gaussian random walks of the same length, $2000$ replications at a fixed seed. Table (ref) reports the outcome. The SADF statistic $1.483$ exceeds its five per cent critical value $1.392$, evidence of explosive behaviour on the price level; the more conservative GSADF statistic $1.540$, the detector that underlies the recursive date-stamping, does not exceed its five per cent critical value $2.124$. The evidence is therefore marginal, but the split is itself interpretable: the SADF rejection alongside the GSADF non-rejection points to single-episode explosivity, a one-time pre-crisis run-up, rather than recurrent explosivity repeated across the sample, so we read the price level as showing single-bubble consistency, not a pattern of repeated bubbles, and treat the dated episode as suggestive rather than established. The backward-SADF date-stamping, with the minimum episode length set to $\log T$ months, locates one explosive episode running from August 2006 to December 2007, the pre-crisis global equity run-up.

The price-level explosivity and the operator-edge concentration mark different phases of the same boundary. The explosive run-up on the price level is dated to 2006--2007, while the operator-edge concentration peaks of Table (ref) fall at the 2008--2009 crisis, when the recovered spectral radius reaches its local maxima and the cross-section collapses to about three effective factors. The operator concentration therefore reads as a coincident-to-lagging crisis marker, not a leading interior-approach flag preceding the price crossing, which is consistent with the coincident discipline we maintain throughout, under which neither reading is a forecast. The stationary operator cannot certify the crossing, by Proposition (ref), and the explosive-root test supplies the price-level reading the operator cannot. We claim only that the inflation phase shows on the price level and the crisis phase shows in the operator spectrum, the two readings of one boundary in sequence, not a month-for-month coincidence.

The structural hazard read on the data

We read Propositions (ref) and (ref) on the data with discipline about what the event count and the covariate dispersion permit. On the eighteen-index panel the equal-weighted index enters the twenty per cent drawdown state three times over the sample, so the systemic series carries only three distinct onsets; by Proposition (ref)(iv) a single systemic covariate cancels in every within-instant risk-set ratio, and three events cannot support a trustworthy interval, so we do not report a systemic hazard ratio. The identifying design is cross-sectional, in the sense of part (iv): each constituent is a subject, the event is that subject's own first entry into a twenty per cent drawdown of its own level, recurrence is admitted on recovery to a prior peak, and the time-varying covariate is the leading-mode participation $u_i(t)=N\,e_{1,i}(t)^2$, which sums to $N$ across the cross-section. The continuous-time partial likelihood of cox1972 in the counting-process form of andersengill1982, with Breslow ties and the subject-clustered robust sandwich variance of linwei1989, fits the scalar elasticity.

We run the same design on four cross-sections that differ in the heterogeneity of the leading mode and in the event count, reported in Table (ref). On the international cross-section of eighteen global equity indices the hazard ratio is $1.146$ per unit of leading-mode participation, with a robust ninety-five per cent interval of $[0.618,2.128]$ over $54$ onsets. The three single-market Fama-French families give $0.962$ over $213$ onsets, $[0.516,1.795]$ for the forty-eight industry portfolios; $1.241$ over $129$ onsets, $[0.511,3.017]$ for the thirty industry portfolios; and $0.811$ over $127$ onsets, $[0.122,5.391]$ for the twenty-five size and book-to-market portfolios. Two panels carry a hazard ratio above one and two below, and every interval contains one, so the elasticity is not robustly signed across cross-sections and more events do not pin the sign. We therefore do not read a confirmed edge-hazard relationship from the cross-panel evidence.

Proposition (ref) disciplines the reading rather than rescuing it. By part (i) the discriminating power of the covariate is the dispersion of leading-mode participation, the inverse participation ratio of the leading eigenvector, reported in the final column of Table (ref). That dispersion is small on the size and book-to-market sort, where a single market factor loads almost uniformly so the leading eigenvector is near-uniform; by parts (ii) and (iii) the partial-likelihood information is then bounded toward zero and the elasticity is weakly identified, which is consistent with the very wide interval $[0.122,5.391]$ there. On the remaining panels the dispersion is material and the event counts are large, so their imprecision and their mixed signs are not attributable to a degenerate covariate; they reflect the absence of a robustly signed edge-hazard elasticity in those cross-sections, not the identification budget binding. We do not claim that dispersion explains the pattern of signs, which it does not: the thirty-industry panel returns the predicted sign at lower dispersion than the international panel, and the forty-eight-industry panel returns the opposite sign at an interval as tight as the international one.

The estimand is labelled proxy\ throughout: the covariate is built from the leading eigenvector and inherits the random-matrix overlap of Proposition (ref)(ii) and the finite-sample band of Theorem (ref), and its very identifiability is governed by the proxy\ participation ratio of Proposition (ref). The reading is coincident by construction, the covariate and the onset sharing the time index. The strengthening is structural and methodological, a derived link from the resolvent norm to an onset intensity and a sharp account of when that link is estimable, not a manufactured significant result; the cross-panel evidence is reported for what it is, directionally mixed and imprecise.

Contrast with a calibrated-operator model

It is tempting to claim that a dynamic stochastic general-equilibrium model, or its heterogeneous-agent New Keynesian (HANK) variant kaplanmollviolante2018, cannot state the coupling between the pricing resolvent and the fragility edge. That claim would be too strong. Production-network general equilibrium already carries a spectral object: the vector of Domar weights is a dominant left-eigenvector of the input-output operator acemoglu2012network, and network models couple amplification to that structure. Overlapping-generations bubble models state the growth-fragility comparison directly, through the comparison of the interest rate and the growth rate.

The genuine difference is threefold and it is about identification, not expressive power. First, in the standard models the operator is exogenous and calibrated, normal by construction, and its spectrum is a modelling input; here the operator is recovered from data and its spectrum is an estimate with a stated random-matrix detectability threshold. Second, the standard models impose the interior of the spectrum by a transversality convention, choosing the interest rate above the growth rate so that the resolvent converges; here the boundary case is detected rather than imposed, and the gate reports the approach to it. Third, the standard models do not carry a per-reading certification of what is identified; here each spectral reading is labelled strong, proxy, or unidentified. The construction is therefore a recovered operator with a certification gate over a set of established spectral facts, a unification with identification, and not an impossibility theorem.

The reach is wider than the equity illustration. Nothing in Sections (ref)--(ref) uses the asset class: any cross-section that admits a recovered self-adjoint dependence carrier carries the same pricing resolvent, the same fragility edge $\beta\,\operatorname{r}(B)=1$, the same bubble characterisation, and the same certification gate. The framework therefore transfers unchanged to a housing and real-estate investment trust cross-section, to credit and corporate-bond panels, and to sovereign-spread panels, in each case pairing the operator recovered from stationary returns with the explosive-root reading on the matched price or yield level; the dividend direction is replaced by the relevant cash-flow or coupon claim, and the gate decides what each market identifies exactly as here. This portability, a single recovered operator read through one scalar across markets, is the sense in which the construction is a general theoretico-empirical object rather than an equity result.

Scope and limitations

The results are deliberately narrow, and we state the boundaries.

First, the strong labels hold for normal carriers and, by Proposition (ref), for near-normal carriers with the departure controlled: a genuinely directed, or non-normal, dependence operator is governed by the spectral radius of the actual operator, not of its symmetric part, and the two can differ. The exclusion bound of Proposition (ref)(ii) always limits the radius from above by the size of the non-normal part, and when the top mode is simple and well separated, the regime of a dominant market factor, it limits the difference two-sidedly by that same size. On the panel studied the directed energy is small and the leading eigenvalue is isolated, so the symmetric carrier is a close approximation and the residual enters as the certified bound of Proposition (ref), rather than as an unbounded gap; the strong claim is made only up to that bound whenever the operator is not verified normal.

Second, the existence of a bubble is a proxy, not a strong claim, because by Proposition (ref) it requires the projection of the dividend direction onto the boundary eigenspace, identified only up to the random-matrix overlap. The strong content is the scalar fragility edge.

Third, the structural microfoundation of Section (ref) is conditional on a named model. The identity between the recovered carrier and the cash-flow propagation operator holds under the cash-flow network of Assumption (ref), whose load-bearing restriction is the common-discount condition (M2), under which unexpected returns are pure cash-flow news; if discount-rate news is present and correlated with cash-flow news, the recovered carrier mixes the two channels and the structural reading degrades to the unconditional spectral one. The recovered correlation carrier matches the structural operator's eigenframe only under the regular-exposure condition, and in general shares only the dominant mode in the edge limit.

Fourth, the discount $\beta$ is calibrated in the baseline. Section (ref) makes it endogenous and locates the edge as a fixed point of $g(r)=r\,\beta(r)$, but that extension rests on a postulated discount schedule and an inelastic-discount monotonicity condition, so it relocates the calibration rather than discharging it; the separate question of identifying a discount from behaviour is left to companion work calcottpetkov2024,bastianellovergopoulos2026.

Fifth, the heavy-tail robustness of Section (ref) is split by what it can claim. The effective-rank band of Theorem (ref) holds for the Pearson carrier under an $L^4$--$L^2$ norm equivalence, while the distribution-free carrier of Proposition (ref) targets the same operator only under an elliptical law with finite second moments; under genuinely non-elliptical heavy tails the robust carrier identifies a latent generalised-correlation object, not the Pearson carrier, and the agreement in Table (ref) is a directional corroboration there, not a certified equality.

Sixth, the welfare cost of Section (ref) is structural in form, but its present-value channel carries no strong between-regime certificate. Only the squared spectral-radius ratio is strong, and it is a spectral object, not a welfare amplification; the welfare-cost level and its between-regime ratio carry the proxy edge mass and the calibrated discount, so they are read as a discount-conditional magnitude, not a certified welfare number.

Seventh, the structural edge-hazard of Sections (ref) and (ref) is coincident and its elasticity is a proxy. The systemic series carries too few onsets to support an interval; across four cross-sections the elasticity is not robustly signed, every interval containing one; and its identifiability is itself governed by the proxy leading-mode participation of Proposition (ref), so the section contributes a derived link from the resolvent norm to an onset intensity and a sharp identification result, not a confirmed hazard.

Eighth, the approach-detection result of Section (ref) is asymmetric in what it guarantees. The false-positive control of the interior detector is conservative and robust, but its power and consistency inherit the band hypothesis, so on the dependent, heavy-tailed panel the detection-power statement is read as a scaling rate rather than a certified probability until the moving-block band is itself turned into a theorem.

Ninth, a balanced-growth reading of the leading eigenvalue, $g=\log\lambda_1$, is not claimed on a financial return operator, whose leading eigenvalue has no structural reason to equal an output growth rate; such a reading would require a separately recovered production operator and is outside the scope of this paper.

Tenth, the empirical exercise is coincident throughout. The spectral-radius contrast and the companion classifier are coincident markers; no leading-indicator or forecasting claim is made, consistent with the one-step forecastability ceiling for the aggregate return.

Conclusion

Lifting the classical bubble characterisation to a bounded self-adjoint operator turns three apparently separate objects into readings of one scalar on one recovered operator: the convergence of the pricing resolvent, the location of the bubble eigenspace, and the divergence of the present-value norm all sit at $\beta\,\operatorname{r}(B)=1$. The certification gate keeps the contribution disciplined: the scalar fragility edge is strong, the existence of a bubble is a proxy that requires the dividend direction's overlap with the boundary eigenspace, and the identity of a bubbly asset is unidentified. Because the operator is recovered from stationary returns while the bubble lives on the nonstationary price level, the estimator flags the approach to the edge but cannot certify the crossing, which is read on the price level. On eighteen global equity indices the recovered spectral radius rises in every documented crisis, the cross-section collapsing to about four effective factors and the discount ceiling tightening towards the pole; under a calibrated transmission intensity the discounted spectral radius rises past one in crisis, the aggregate cash-flow claim sits almost entirely on the boundary mode, and a right-tailed recursive unit-root test date-stamps the explosive price-level episode the stationary operator cannot certify. The construction unifies a pricing object and a systemic-fragility object on one recovered operator, certifies each reading for what the spectrum identifies, and leaves the directed half, the discount, and the growth reading explicitly open. The contribution is not that bubbles can be observed directly, but that the spectral geometry of their admissibility can be recovered, certified, and bounded before the crossing.

Acknowledgements

The author thanks colleagues at the Indian Institute of Technology Bhubaneswar for discussion. The author acknowledges Google Cloud Research Credits, credit reference wilsonjessica-485604931, and Institute Seed Grant SP128.