The Emergent Geometry Sub-Programme

Abstract

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 organises 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. Status revision (version 1.1). Q5a version 3.0 withdraws the first link of this chain: the canonical filtration of the admissible fibre is exactly a growing toric Fourier window, the published admissibility form converges to the zero form on it, and no common scalar normalisation produces a non-trivial toric differential operator. The existence of the spatial limit operator ${L_\Pi} = -A\partial_x^2$ is now the explicit, unestablished hypothesis [H-L] (Q5b version 2.0), and every downstream result that consumes it is conditional on [H-L]. The geometric convergence results (Carnot limit, ${D_{\mathrm{hom}}} = 4$) and the algebraic coefficient rigidities (Q7, Q8, Q10, U1) are independent of [H-L] and stand. Question Q5 is open.