Intervention asymmetry

What the plate operates. Theorem 9 supplies the unified sensitivity formula $\partial {\theta }^{\ast }/\partial X=(I-D_{\theta }\Phi {)}^{-1}\cdot {\partial }_X\Phi$. The plate visualizes the per-cell magnitude of this partial across the four cells of the taxonomy from §5.0.

The closure amplification factor. $(I-D_{\theta }\Phi {)}^{-1}$ has operator norm at most $(1-{\kappa }_{\Phi }{)}^{-1}$ under contractivity, where ${\kappa }_{\Phi }=(\beta /\alpha )L_{\mathcal{D}}$. The same factor amplifies every intervention; what differs across cells is the mechanism partial. As ${\kappa }_{\Phi }\to 1$, the amplification diverges — the regime Theorem 12 classifies into three generic bifurcations, plus the pitchfork under cohort symmetry.

The N-fold asymmetry. Mode A's $\hat{R}$is not user-indexed; it enters the optimization for every user simultaneously. The fourth-cell's observation regulation is similarly unindexed. Both scale $O(1)$. Individual-scale Modes B (user-side practice on $T_i$) and C (user-side practice on $O_i$) incur the $1/N$population-averaging factor: individual practice contributes one user's share to the population average.

What the bar comparison shows.Set each cell's intervention intensity equal. Vary $N$. At small $N$, the bars look similar; the asymmetry is small. At $N=10^6$(population-scale), Mode A's bar is six orders of magnitude larger than the individual-scale Modes B/C bars. This is the framework's hardest political claim made operable.

What the proposition authorizes the prose to claim. Substituting individual-scale Modes B or C for Mode A is the analytical error that converts political work into its symptom. The framework's central political claim sits in this 2×2.