The Emergent Geometry Sub-Programme

Overview

The emergent geometry sub-programme asks what geometry can be reconstructed after supplying a finite Heisenberg group, its Heisenberg–Schrödinger carrier and the associated Weil action. The admissibility filter $\Pi_q$ selects modes under the Born–Infeld bounded-flux constraint. The unconditional results concern the Carnot limit and representation-theoretic coefficient rigidities; the continuum metric interpretation additionally requires [H-L].

Under [H-L], the sub-programme reconstructs an effective Lorentzian metric rather than taking a background metric as an independent input. It sits at the interface between the foundational branch and the physical branch (gravity, gauge, fermions), whose geometric readings inherit this open hypothesis. This note concerns metric reconstruction only; metric dynamics belong to spectral gravity.

Conditional metric result. The coefficient values asserted by Q8, Q10, U1 and Q11 give, under [H-L], all four metric coefficients equal to 2, read off the eigenvalue of the $\mathfrak{su}(2)$-Casimir on the spin-1 module $\mathrm{Sym}^2(V_\rho)$. Since Q7 version 2.0 those readings rest on an identification of the measured admissible space with that module which no paper supplies, and Q7 withdraws its own support for them; the entries concerned are pending revision.

The structural chain

$\text{Supplied carrier} + [\mathrm{H\!\!-\!L}] + [\mathrm{ID}] \;\Longrightarrow\; \mathcal H_c = L^2(\mathbb{Z}/q\mathbb{Z}) \;\Longrightarrow\; \mathrm{Heis}_3(\mathbb{R}) \;\Longrightarrow\; L_{\mathrm{eff}} \;\Longrightarrow\; g^{\mu\nu} \;\Longrightarrow\; g^{\mu\nu} = 2\eta^{\mu\nu}.$

Five conceptually distinct stages, each addressed by a distinct group of papers: the discrete-to-continuum stage, now the open spatial limit hypothesis [H-L] after the Q5a version 3.2 withdrawal (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, conditional on [H-L], with [H-lift] still unestablished after Q9 2.0); the coefficient values $A_Z = A_H = 2$ asserted by Casimir rigidity and spectral universality (Q8, Q10, U1) — supports that Q7 version 2.0 no longer provides, its own numbers being Rayleigh quotients of the discrete Weil Laplacian, so those entries are pending revision; and the closure $A_\tau = 2 \Rightarrow g^{\mu\nu} = 2\eta^{\mu\nu}$ (Q11, W1), whose metric reading is conditional on [H-L] and on the identification Q7 version 2.0 records as supplied by no source.

Papers of the sub-programme

Discrete-to-continuum limit.

Q5a — v3.2: exact identification of the canonical filtration (toric Fourier window), zero-form theorem, and normalisation no-go; the spatial limit is now the open hypothesis [H-L], and Q5 is open.
Q5a-O2 — Fourier structure of the pre-saturation admissible pipeline: pure-mode fingerprints with $o(q)$ displacements (frequencies $o(q)$ for fixed block parameters, macroscopic per-$q$ under the implemented sampling, no $q$-independent limit), and the truncation status of the three-coordinate reduction; the form is $O(q^{-2})$ on the whole unit sphere (a normalisation-prefactor fact), ruling out the $q^{-1}$ coercivity, and the hypotheses [H-E1] and [C] remain open.
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 — unconditional Carnot convergence of BFS shells and homogeneous dimension $D_{\mathrm{hom}} = 4$; under [H-L], metric extraction $g^{\mu\nu} \propto A_{\mu\nu}$ and Lorentzian signature $(-,+,+,+)$.
Q9 — version 2.0 gives a free kinetic Mosco limit under [K] and compatible recovery [R]. It does not establish [H-F] $\Rightarrow$ [H-lift]; Q5b metric extraction still requires [H-L] and [H-lift].

The coefficient values ($A = 2$) and their status.

Q7 — version 2.0 states the requirements an equivariant bridge would meet and proves its rigidity under an explicit identification hypothesis $H_{\mathrm{eff}} = \mathrm{Sym}^2(V_\rho)$ that no source supplies; it determines no coefficient, and withdraws the transfer of the discrete no-cross-terms statement to the continuum symbol.
Q8 — $A_Z = C_{\mathfrak{su}(2)} = 2$ (vertical direction) by Casimir rigidity, with no free normalisation (pending revision: Q7 version 2.0 withdraws the identification and the numerical reading this draws on).
Q10 — $A_H \to 2$ under spectral universality [U]; $\mathfrak{su}(2)$-isotropy of the effective form (pending revision: Q7 version 2.0 no longer reads this as settling its bridge condition).
U1 — proves spectral universality [U] with rate $\varepsilon(q) = O(q^{-1/2})$.

Metric closure and integrative output.

Q11 — under [H-F], $A_\tau=2$ by temporal Casimir rigidity; under [H-L] and the identification Q7 version 2.0 records as supplied by no source, this reads $g^{\mu\nu}=2\eta^{\mu\nu}$.
W1 — closes [H-w] (admissibility weight convergence) as a consequence of fibre-invariance.
Q6b — under [H-L], assembles the conditional chain $\Pi_q\to L_{\mathrm{eff}}\to g^{\mu\nu}\to G_{\mu\nu}$ and obtains the Schwarzschild exterior and Einstein equations as consistency conditions.

Inputs and outputs

Upstream inputs. A supplied finite Heisenberg group with non-trivial central character, its Heisenberg–Schrödinger representation $\pi_c$ on $\mathcal H_c=L^2(\mathbb{Z}/q\mathbb{Z})$, the identification of that carrier with the admissible fibre, and the distinct associated Weil action; 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. Under [H-L], and with the coefficient values pending revision since Q7 version 2.0 withdrew the identification their derivation rests on, the effective Lorentzian metric $g^{\mu\nu}=2\eta^{\mu\nu}$, signature $(-,+,+,+)$, and 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, under the same hypothesis, the Schwarzschild exterior and Einstein equations (via Q6b) — consumed by the gravity, gauge and fermionic papers of Branch III.

Status

The metric reconstruction is conditional on the spatial limit hypothesis [H-L]: Q5a version 3.2 withdraws the Mosco derivation of the spatial limit operator, so establishing or replacing [H-L] is one of two leading open items, the other being the identification of the measured admissible space with the spin-1 module, which Q7 version 2.0 records as supplied by no source. Under [H-L] and that identification, the signature is forced by elimination and all four coefficients are read as the $\mathfrak{su}(2)$-Casimir value 2; the entries asserting those values are pending revision. Also open: the existence of an explicit equivariant bridge, at finite $q$ or in the limit, $\phi_q: \mathrm{Sym}^2(V_\rho) \xrightarrow{\sim} W_{\mathrm{sp}}$ (the typed bridge, its uniqueness and any coefficient value depend on it), and hypothesis [H1] on the full $L^2$ space (open; Q5a-O2 delimits what the three-coordinate pipeline measures and does not close [H1], [H-E1] or [C]). This note does not address the dynamics of $g^{\mu\nu}$ — the Einstein equations are the subject of the spectral gravity sub-programme.