The Emergent Geometry Sub-Programme

Overview

The emergent geometry sub-programme answers a single central question. The admissibility filter $\Pi_q$ acts on the Weil representation $V_\rho \simeq L^2(\mathbb{Z}/q\mathbb{Z})$ of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, selecting admissible modes under the Born–Infeld bounded-flux constraint: what effective geometry is selected in the large-$q$ limit, and what determines its metric coefficients?

The sub-programme removes the postulate of a background spacetime: the effective Lorentzian metric is a forced consequence of admissibility, not an input. It sits at the interface between the foundational branch (the Weil representation and the Born–Infeld constraint) and the physical branch (gravity, gauge, fermions), which all depend on a derived effective spacetime. This note concerns the reconstruction of the metric only — its dynamics, the Einstein equations, belong to the spectral gravity sub-programme.

All four metric coefficients equal 2. The spatial, vertical, horizontal, and temporal entries are governed by a single representation-theoretic datum: the eigenvalue of the $\mathfrak{su}(2)$-Casimir on the spin-1 module $\mathrm{Sym}^2(V_\rho)$. The conformal factor is not a choice of units; after absorbing it, the effective metric is exactly Minkowski.

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.