The Emergent Geometry Sub-Programme

The structural chain

$\Pi_q \;\Longrightarrow\; V_\rho \simeq L^2(\mathbb{Z}/q\mathbb{Z}) \;\Longrightarrow\; \mathrm{Heis}_3(\mathbb{R}) \;\Longrightarrow\; L_{\mathrm{eff}} \;\Longrightarrow\; g^{\mu\nu} \;\Longrightarrow\; g^{\mu\nu} = 2\eta^{\mu\nu}.$

Five conceptually distinct stages, each resolved by a distinct group of papers: the discrete-to-continuum Mosco limit to the one-dimensional shadow $L_\Pi = -A\partial_x^2$ (Q5a, Q5a-O2, H2); the dimensional promotion via Carnot geometry and the Bass–Guivarc'h homogeneous dimension $D_{\mathrm{hom}} = 4$ (Q5b); the metric extraction and Lorentzian signature $(-,+,+,+)$ from the principal symbol of $L_{\mathrm{eff}}$ (Q5b, made unconditional by Q9); the determination of the coefficients $A_Z = A_H = 2$ by Casimir rigidity and spectral universality (Q7, Q8, Q10, U1); and the closure $A_\tau = 2 \Rightarrow g^{\mu\nu} = 2\eta^{\mu\nu}$ (Q11, W1).

Papers of the sub-programme

Discrete-to-continuum limit.

• Q5a — Mosco convergence of the admissibility Dirichlet forms to the one-dimensional shadow $L_\Pi = -A\partial_x^2$ on $L^2(\mathbb{R})$.
• Q5a-O2 — spectral atomicity of the admissible sector closes the open hypotheses [H-E1] and Conjecture [C].
• H2 — strong convergence of the rescaled Weil generators ([H2]) via a Poisson aliasing lemma and a discrete Sobolev identity.

Dimensional promotion, metric extraction and signature.

• Q5b — Carnot convergence of BFS shells, homogeneous dimension $D_{\mathrm{hom}} = 4$, metric extraction $g^{\mu\nu} \propto A_{\mu\nu}$, and Lorentzian signature $(-,+,+,+)$.
• Q9 — proves [H-lift] as a theorem, making the Q5b metric extraction unconditional.

Coefficient determination ($A = 2$).

• Q7 — identifies $H_{\mathrm{eff}} = \mathrm{Sym}^2(V_\rho)$ as the unique irreducible 3D $\mathfrak{su}(2)$-module; bridge rigidity and no-cross-terms.
• Q8 — $A_Z = C_{\mathfrak{su}(2)} = 2$ (vertical direction) by Casimir rigidity, with no free normalisation.
• Q10 — $A_H \to 2$ under spectral universality [U]; $\mathfrak{su}(2)$-isotropy of the effective form.
• U1 — proves spectral universality [U] with rate $\varepsilon(q) = O(q^{-1/2})$.

Metric closure and integrative output.

• Q11 — $A_\tau = 2$ by temporal Casimir rigidity, giving $g^{\mu\nu} = 2\eta^{\mu\nu}$.
• W1 — closes [H-w] (admissibility weight convergence) as a consequence of fibre-invariance.
• Q6b — integrative output: assembles $\Pi_q \to L_{\mathrm{eff}} \to g^{\mu\nu} \to G_{\mu\nu}$; Schwarzschild exterior and the Einstein equations as consistency conditions.

Inputs and outputs

Upstream inputs. The Weil representation $V_\rho \simeq L^2(\mathbb{Z}/q\mathbb{Z})$ as a theorem of the foundational branch; the Born–Infeld bounded-flux constraint $A_n \le c_\chi/\sqrt{\lambda_n}$; and, from the spectral admissibility sub-programme (Presentation Note 1), the spin-$\tfrac12$ sector $V_\rho \cong \mathbb{C}^2$ with $\Sigma_c(n_3) = 3$ and $\mathrm{Im}\,\mathbb{H} \cong \mathfrak{su}(2)$.

Outputs. The effective Lorentzian metric $g^{\mu\nu} = 2\eta^{\mu\nu}$ and signature $(-,+,+,+)$; the effective operator $L_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$; the three spatial directions as the image of $\mathrm{Im}\,\mathbb{H}$; and the Schwarzschild exterior and Einstein equations (via Q6b) — consumed by the gravity, gauge and fermionic papers of Branch III.

Status

The metric reconstruction is closed in the $q \to \infty$ limit: the signature is forced by elimination, and all four coefficients equal the $\mathfrak{su}(2)$-Casimir value 2. Two items remain open: the existence of an explicit finite-$q$ equivariant bridge $\phi_q: \mathrm{Sym}^2(V_\rho) \xrightarrow{\sim} W_{\mathrm{sp}}$ (no published result depends on it), and hypothesis [H1] on the full $L^2$ space (closed on the admissible sector by Q5a-O2). This note does not address the dynamics of $g^{\mu\nu}$ — the Einstein equations are the subject of the spectral gravity sub-programme.