Crisis-boundary bifurcations

What the plate operates. Theorem 12 classifies the generic codimension-1 bifurcations of the performative operator Phi when its dominant eigenvalue touches the unit circle at ${\kappa }_{\Phi }=1$. The plate visualizes the bifurcation diagrams of each regime; the reader selects which to view.

Three generic regimes, plus one under symmetry.

• Saddle-node. Eigenvalue at $+1$, quadratic non-degeneracy. Stable and unstable fixed points collide at $\mu =0$ and annihilate; above threshold the dynamics jump or diverge. Empirical signature: platform crises, hard resets, algorithm- rebuild events.
• Pitchfork (non-generic). Eigenvalue at $+1$, cubic non-degeneracy. This one is non-generic: it requires $\Phi$ to commute with a ${\mathbb{Z}}_2$ symmetry, which symmetric cohorts can supply. Where that symmetry holds, a single stable fixed point becomes unstable and two symmetric stable points appear. Empirical signature: polarized stable regimes.
• Period-doubling (flip). Eigenvalue at $-1$. Stable fixed point becomes unstable; stable 2-cycle appears. Cascade to chaos follows. Empirical signature: oscillating then chaotic engagement.
• Neimark-Sacker. Complex eigenvalue pair on the unit circle (Hopf-like). Stable fixed point becomes unstable; stable invariant circle appears with amplitude $\sqrt{\mu }$. One of the three generic codim-1 cases. Empirical signature: rage-bait cycles, viral rhythms, recurrent content patterns.

Reading the diagrams. Stable branches render as solid warm-ink curves; unstable branches as dashed oxblood. The dashed oxblood vertical line at $\mu =0$ marks the bifurcation point (i.e., the crisis boundary ${\kappa }_{\Phi }=1$). To the left: pre- crisis regime; to the right: post-crisis.

What the theorem authorizes the prose to claim. Repeated algorithm-rebuild events, monetization pivots, rage-bait cycles, viral cascade rhythms are bifurcation phenomena at the contractivity boundary the framework names — structurally classifiable trajectories of a platform in crisis.