This repository contains the source of the Emergent Geometry Presentation Note Cosmochrony paper
The Emergent Geometry Sub-Programme — Presentation Note 2.

This work is a structured entry point to the emergent geometry sub-programme of the Cosmochrony corpus, not a summary of results. It maps the constituent papers, identifies the internal phases, records the status of every result as proved, structural, numerical, or open, and states the remaining open deliverables.

Central Question

The admissibility filter $\Pi_q$ acts on the Weil representation $V_\rho \simeq L^2(\mathbb{Z}/q\mathbb{Z})$ of the Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, selecting admissible modes under the Born--Infeld bounded-flux constraint.

What effective geometry is selected in the large-$q$ limit of this structure, and what determines the numerical values of its metric coefficients?

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

Under [H-L], the sub-programme removes the postulate of a background spacetime: the effective metric is a forced consequence of admissibility, not an input. This note concerns the reconstruction of the metric only; its dynamics (the Einstein equations) belong to the spectral gravity sub-programme.

Logical 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:

1. Discrete-to-continuum — now the open hypothesis [H-L]: Q5a 3.0 withdraws the Mosco derivation of $L_\Pi = -A\partial_x^2$ (Q5a-O2 and H2 closed hypotheses of the withdrawn framework).
2. Dimensional promotion — the Carnot convergence of BFS shells and the Bass--Guivarc'h homogeneous dimension $D_{\mathrm{hom}} = 4$ promote $L_\Pi$ to a 4D operator $L_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$ (Q5b).
3. Metric extraction — the effective co-metric $g^{\mu\nu} \propto A_{\mu\nu}$ and Lorentzian signature $(-,+,+,+)$ from the principal symbol of $L_{\mathrm{eff}}$ (Q5b, unconditional via Q9).
4. Coefficient determination — $A_Z = A_H = 2$ via Casimir rigidity and spectral universality (Q7, Q8, Q10, U1).
5. Metric closure — $A_\tau = 2$ via temporal Casimir rigidity, giving $g^{\mu\nu} = 2\eta^{\mu\nu}$ (Q11; W1 closes [H-w]).

All four metric coefficients equal 2 — a single representation-theoretic datum: the eigenvalue of the $\mathfrak{su}(2)$-Casimir on the spin-1 module $\mathrm{Sym}^2(V_\rho)$.

Position in the Programme

The sub-programme sits at the interface between Branch I (axiomatic primitive) and Branch III (physical observables). It takes from Branch I the Weil representation of $\mathrm{Heis}3(\mathbb{Z}/q\mathbb{Z})$ and the Born--Infeld admissibility constraint, and from Presentation Note 1 (spectral admissibility) the facts that the admissible sector is the spin-$\tfrac12$ sector $V\rho \cong \mathbb{C}^2$ and that $\Sigma_c(n_3) = 3$ with $\mathrm{Im},\mathbb{H} \cong \mathfrak{su}(2)$. It produces the effective Lorentzian metric used throughout Branch III.

Constituent Papers

The note maps twelve constituent papers, organised by internal phase:

Phase	Papers	Central output	Status
Discrete-to-continuum	Q5a, Q5a-O2, H2	Fourier window; no-go; $L_\Pi \to$ [H-L]	C
Dimensional promotion	Q5b	$D_{\mathrm{hom}} = 4$; Carnot convergence	S
Metric extraction	Q5b + Q9	$g^{\mu\nu} \propto A_{\mu\nu}$; signature $(-,+,+,+)$	C
Coefficient determination	Q7, Q8, Q10, U1	$A_Z = A_H = 2$	P/S
Metric closure	Q11, W1	$A_\tau = 2$; $g^{\mu\nu} = 2\eta^{\mu\nu}$	S/C
Integrative output	Q6b	$\Pi_q \to L_{\mathrm{eff}} \to g^{\mu\nu} \to G_{\mu\nu}$	C

Status codes: P = proved, S = structural, C = conditional on the unestablished spatial limit hypothesis [H-L] (Q5a 3.0, Q5b 2.0).

Open Deliverables

1. Bridge existence at finite $q$. The asymptotic results ($A_Z = A_H = A_\tau = 2$) and the metric closure $g^{\mu\nu} = 2\eta^{\mu\nu}$ hold in the $q \to \infty$ limit. An explicit $\mathfrak{su}(2)$-equivariant bridge $\phi_q: \mathrm{Sym}^2(V_\rho) \xrightarrow{\sim} W_{\mathrm{sp}}$ at each prime remains an open structural problem (no published result depends on it).
2. Hypothesis [H1] on the full $L^2$ space. Closed on the admissible sector by Q5a-O2; the full-space version is not needed by any result and is listed for completeness.

Build

bash compile.sh

This runs pdflatex → bibtex → pdflatex → pdflatex on tex/EmergentGeometryNote.tex and produces out/EmergentGeometryNote.pdf.