Amplification on a Recovered Network: Endogenous Risk, Spectral Concentration, and the Certified Distance to the Fragility Edge

Continuous-time macro-finance carries amplification in a scalar state, while systemic-risk measurement recovers dependence networks in reduced form. This paper builds one intermediary economy in which both are projections of a single object: the instantaneous response of returns to fundamental shocks is exactly the resolvent of a symmetric exposure-overlap operator scaled by a leverage multiplier, so the scalar liquidity spiral is the rank-one shadow of a spectral condition. Crisis dynamics reduce, under an alignment hypothesis, to one dimension along the dominant eigenvector; calm erodes the distance to the fragility edge through leverage and spectral concentration at once; a claim loading off the carrier earns an endogenous hedging premium; and resilience, the distance to the edge, admits a conditional finite-sample band with a refusal rule, while the direction of amplification is proved unidentified from symmetric-moment estimators.