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ENGAGE THE LAB
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Murmuration is the City’s distributed-topology laboratory. Agent geometry generates a relational complex; the complex generates a Hodge operator; the operator’s harmonic kernel carries topological information.
$$ X(t)\rightarrow\mathcal R_\epsilon[X(t)]\rightarrow\Delta_q(t)\rightarrow\ker\Delta_q(t)\rightarrow P_{H_q(t)} $$
At each frozen frame, the direct SPECTRA bridge is
$$ \Gamma_{q,t}=I-\epsilon_t\Delta_q(t)^2,\qquad \Gamma_{q,t}^{\,n}\rightarrow P_{\ker\Delta_q(t)}. $$
Across a moving swarm the exact object is non-autonomous:
$$ \Phi_T=\Gamma_{T-1}\Gamma_{T-2}\cdots\Gamma_0. $$
The recovered canonical source uses $N=50$, Vietoris–Rips scale $\epsilon=0.45$, seed 42, and $\Delta_1=B_1^TB_1+B_2B_2^T$.
| State | Edges | Triangles | $\beta_0$ | $\beta_1$ |
|---|---|---|---|---|
| Annulus | 150 | 150 | 1 | 1 |
| Packed disk sample | 224 | 411 | 2 | 1 |
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Errata retained. At $\epsilon=0.45$ the documented packed-disk realization has $\beta_1=1$, not zero. Its rejection in the original demo is therefore a persistence/coherence-gate rejection, not simply disappearance of the H1 kernel.
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A deterministic continuous reconstruction between the two canonical source states produced zero-mode rank events at frames 63, 73, 86, and 94. Every birth/death changed $\operatorname{rank}P_{harm}$ and produced nonzero harmonic-projector motion.