the dynamics

Twelve interactive plates. Each plate is a small machine: a reader operates the apparatus result against a parameter and sees the mechanism behave. The plates are the framework's Machine surface — its claims are statable and operable.

Each plate carries a caption (the result in one sentence), a middle layer (what the controls do, what to look for), and a link back to the apparatus section that authorizes it. The plates are pedagogical companions to the formal results, which they illustrate without standing in for.

1. 01

   Foreclosure clock

   The survival probability at the reflective threshold under the closed loop. The reader watches S(θ_ref; θ_t) drop toward operational zero as the policy converges.

   → Lemma 1
2. 02

   Chrono-debt spectrum

   The foreclosed pause-time at every scale, measured against the pre-closure baseline. The reader sees D(t, τ) accumulate as τ grows toward developmental scales.

   → Definition 1.1
3. 03

   Dividual posterior

   The platform's filtering posterior contracting on the user state at the parametric rate. Variance vs tick on log-log, plus a contracting density panel.

   → Proposition 4'
4. 04

   Metric superego

   The KL distance from the user's behavioral distribution to the platform-induced target. The reader sees the monotone descent and the null-band band that the platform's measurement cannot resolve.

   → Proposition 5'
5. 05

   Captured resonance

   Two oscillators, one coupling constant. Phase-locking iff K_bm > |ω_b − ω_m|. Two dials and a phase-difference plot; the reader crosses the threshold and sees the lock.

   → Proposition 6
6. 06

   Libidinal routing

   Four registers, dependently distributed. Mutual information across registers under the platform's posterior. The reader raises off-diagonal coupling and watches I_{Π_t}(u^i; u^j) rise.

   → Proposition 7'
7. 07

   Somatic Hawkes

   Multi-mark Hawkes across S/C/P/K with cross-register coupling. Events on one track raise intensity on the others; cascades visibly emerge when α_kl is raised.

   → §3.5
8. 08

   Cohort gradient

   Developmental exposure to the converged regime. KL distance decays exponentially in exposure duration; the reader sees the long-exposure cohort start at the target.

   → Proposition 8
9. 09

   Intervention asymmetry

   Three intervention sites — R̂, T, O — and the N-fold scale gap between Mode A and the individual-scale modes. The reader compares dθ*/dX across modes.

   → Theorem 9
10. 10

    Reclamation trajectories

    Three reclamation modes plotted against the same baseline. Mode A bends the trajectory; B and C nudge initial conditions inside the closure's basin.

    → Theorem 1' + §5
11. 11

    Crisis-boundary bifurcations

    Three generic bifurcation regimes at the contractivity boundary κ_Φ = 1 — saddle-node, period-doubling, Neimark-Sacker — plus the pitchfork under cohort symmetry. The empirical signatures of each.

    → Theorem 12
12. 12

    Network coupling

    Engagement cascade across a social graph. Percolation threshold and cohort graph-amplification. The reader raises coupling and watches a cascade overtake the network.

    → Theorem 10