Conduction Intensity and Structural Fragility of Heston Volatility Surfaces

We develop an operator-theoretic measure of structural fragility for implied volatility surfaces under the Heston stochastic volatility model. The conduction intensity, defined as the operator norm of the Fr\'echet derivative of the implied volatility mapping, captures worst-case parameter sensitivity of the entire surface. We prove Fr\'echet differentiability via a three-link chain (characteristic function analyticity, Fourier integral regularity, Black--Scholes inversion) with explicit bounds on the Riccati function derivatives, establish a first-order duality with local model uncertainty, and provide five main results. First, is functionally independent from the industry-standard triple (ATM vol, skew, butterfly): linear regression yields R²=0. 129, meaning 87\% of -information is invisible to standard monitors. Second, the Heston conduction matrix has effective rank two: the (v₀, v) subspace explains >99. 7\% of total conduction variance, a consequence of ATM dominance quantified by the wings-to-ATM Vega ratio bound w/₀ₓ₌<0. 03. Third, a direction-crossing theorem: the most dangerous perturbation direction rotates continuously from eₕ䃐 to eₕ as increases, with critical ^* satisfying _ (2w (^*, T) -1) \, dT=0, numerically ^* 3. 25. Fourth, a high- fragility paradox: strong mean-reversion renders the model extremely sensitive to v, with ^ (v) ^0. 52. Fifth, the Feller boundary 2=ᵥ² produces Lipschitz-continuous analytic robustness, contrasting with the hard transitions found in CEV and CDO models. Stylized analyses of the 2018 Volmageddon and 2020 March crisis demonstrate that raw misleadingly decreases during crises (Vega-inflation bias). The v₀-normalized =₀ corrects this, while a directional anomaly indicator amplifies 3. 69-fold (2018) and 4. 2-fold (2020) before the crash.