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PRIMARY PROSPECTIVE RESEARCH DIRECTION · 2026-09-08
Governing question: What invariant is about to exist?
The City now exposes the prospective program through a public orchestration console: SPECTRA → Fold → Murmuration → Density → Horizon forecast → Mnemosyne seal → reveal → score → correction → next forecast. Open Horizon · Horizon — Prospective Forecast Console.
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$$ \mathcal M_t:(X_{t-1},X_t)\longmapsto(\widehat\tau_{\rm top},\Delta\widehat\beta_q,\widehat P_{H_q}). $$
The forecast must be written before the target state is revealed. It predicts the next topology-changing time, birth/death sign, post-transition Betti number, and post-transition harmonic projector.
Given observed positions $X_t=\{x_i(t)\}$,
$$ v_i(t)=x_i(t)-x_i(t-1),\qquad \widehat X_{t+1}=2X_t-X_{t-1}. $$
Then
$$ \widehat X_{t+1}\rightarrow\mathcal R_\epsilon[\widehat X_{t+1}]\rightarrow\widehat\Delta_1\rightarrow\ker\widehat\Delta_1\rightarrow\widehat P_{\rm harm}. $$
with $\widehat\beta_1(t+1)=\dim\ker\widehat\Delta_1$ and $\Delta\widehat\beta_1=\widehat\beta_1(t+1)-\beta_1(t)$.
For each pair of agents,
$$ r_{ij}=x_i-x_j,\qquad u_{ij}=v_i-v_j, $$
and an edge crossing occurs when
$$ \|u_{ij}\|^2\tau^2+2(r_{ij}\cdot u_{ij})\tau+(\|r_{ij}\|^2-\epsilon^2)=0. $$
The prospective topological event is the first positive crossing for which the Hodge-kernel rank changes:
$$ \widehat\tau_{\rm top}=\min_{\tau_{ij}>0}\{\tau_{ij}:\Delta\dim\ker\Delta_1\neq0\}. $$