The Emergent Geometry Sub-Programme

The emergent geometry sub-programme of the Cosmochrony corpus addresses a single central question: how does the admissible Weil–Heisenberg fibre give rise to an effective four-dimensional Lorentzian geometry? Starting from the admissibility filter $\Pi_q$ acting on the Weil representation $V_\rho$ of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, the sub-programme closes the following chain: \[ \Pi_q \;\Longrightarrow\; V_\rho \simeq L^2(\mathbb{Z}/q\mathbb{Z}) \;\Longrightarrow\; \mathrm{Heis}_3(\mathbb{R}) \;\Longrightarrow\; L_{\mathrm{eff}} \;\Longrightarrow\; g^{\mu\nu} \;\Longrightarrow\; g^{\mu\nu} = 2\eta^{\mu\nu}. \] This note maps eleven constituent papers (Q5a, Q5a-O2, Q5b, Q6b, Q7–Q11, U1, W1, H2) across five internal phases and records the status of every result. The remaining open items are not obstructions to the admissible-sector metric closure. They concern, respectively, the finite-$q$ bridge formulation and the extension of the Q5a Mosco tightness control from the admissible sector to the full function space.