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 admissibility axioms do not select the carrier (an explicit $\mathfrak{S}_3$ countermodel). The unconditional results concern the Carnot limit and the homogeneous dimension $D_{\mathrm{hom}} = 4$; the representation-theoretic statements fix the shapes of invariant forms, not coefficient values; the continuum metric interpretation additionally requires [H-L] and [H-lift].

Under [H-L] and [H-lift], the sub-programme extracts an effective co-metric from an operator rather than taking a background metric as an independent input; that co-metric has rank three, and its completion is open. 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.

Coefficient status. No metric coefficient has an established value. Q8 version 2.0 shows that $\mathfrak{su}(2)$-invariance fixes an invariant form only up to a free positive factor, so no coefficient is read off the Casimir eigenvalue 2, and that the commutator supplies no central coefficient; Q10 version 2.0.1 controls the ratio $A_H/A_Z$ only under named inputs; Q5b version 2.3 leaves the central slot empty; Q11 version 2.0 proves that spatial $\mathrm{SU}(2)$ invariance leaves $A_\tau$ independent of the spatial coefficients. Version 2.2 of the presentation note accordingly states its endpoint as $g^{\mu\nu} \propto \eta^{\mu\nu}$, valid only for a supplied rank-four extension satisfying the hypotheses of Q11 Corollary 6.1 (including a supplied negative ordering term), after rescaling $\tau$, with its overall scale free.

The structural chain

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

The presentation note (version 2.3) states its final step as $g^{\mu\nu} \propto \eta^{\mu\nu}$, a target that holds only for a supplied rank-four extension satisfying the hypotheses of Q11 Corollary 6.1, including a supplied negative ordering term (Q5b-O2 open), after rescaling $\tau$, with an overall scale that no source fixes: the central slot is empty (Q5b, Q8) and no coefficient value is fixed (Q8, Q10, Q11).

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, structural); the metric extraction from the principal symbol of $L_{\mathrm{eff}}$, an operator Q5b poses under [H-L] and [H-lift] with a supplied temporal term, giving a rank-three co-metric whose non-degenerate block has signature $(-,+,+)$ (Q5b, conditional on [H-L], [H-lift] and [H-hyp]); the coefficient values, for which Q8 and Q10 establish that the Casimir fixes no coefficient, that the commutator supplies no central one, and that the ratio $A_H/A_Z$ is controlled only under named inputs, while U1 version 2.1 leaves [U] open; and the temporal coefficient, which Q11 version 2.0 shows spatial $\mathrm{SU}(2)$ invariance leaves free, so that one datum coupling the temporal and spatial normalisations is missing.

Papers of the sub-programme

Discrete-to-continuum limit.

Q5a — v3.3: exact identification of the canonical filtration (toric Fourier window); a zero-form theorem for uniformly bounded weights; and, under [H-depth] and [H-w′], no common scalar normalisation yields a non-trivial toric differential limit. A conditional dual-window limit remains, the spatial limit is 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; for bounded weights 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 via a Poisson aliasing lemma and a discrete Sobolev identity: the statement of [H2] of Q5a version 2.0, which discharges no hypothesis of the current Q5a.

Dimensional promotion, metric extraction and signature.

Q5b — unconditional Carnot convergence of BFS shells and homogeneous dimension $D_{\mathrm{hom}} = 4$; under [H-L] and [H-lift], a rank-three co-metric $\mathrm{diag}(-A_\tau, A_H, A_H, 0)$ whose non-degenerate block has signature $(-,+,+)$ under [H-hyp]; the full-rank extension Q5b-O2 and the coefficient values are open (version 2.3).
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 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 — version 2.0: the commutator supplies no central coefficient; a constant one needs a term of homogeneous degree four and a length scale; for a supplied fully invariant target, $A_H = A_Z = 2\lambda$ with $\lambda$ free.
Q10 — version 2.0.1: a sharp ratio lemma and a conditional bound $|A_H/A_Z - 1| \le 2\varepsilon/(1-\varepsilon)$ under [U] and three named geometric inputs; an absolute value needs an independent scale; no value derived.
U1 — proves an exact finite-Heisenberg-generator obstruction; version 2.0 withdraws its proof of [U] and the rate $O(q^{-1/2})$.

Temporal coefficient and integrative output.

Q11 — version 2.0: under a supplied action trivial on the ordering line and spin-one on space, every invariant form is $\alpha\,k_\tau^2 + \beta\,Q_{\mathrm{sp}}$ with independent scalars, so spatial $\mathrm{SU}(2)$ invariance leaves $A_\tau$ free and no invariance condition fixes a Lorentzian co-metric up to scale; under the inputs of Q5b and Q8 the co-metric is $\mathrm{diag}(-A_\tau, 2\lambda, 2\lambda, 2\lambda)$ with two independent scales, and one datum coupling them is open.
W1 version 2.0.1 — withdraws its former proof of [H-w]; retains a conditional bound for a distinct partial-sum proxy. The bridge to Q5a response weights and [H-w] remain open.
Q6b version 2.1 — under [H-L] and [H-lift], assembles the chain $\Pi_q\to L_{\mathrm{eff}}\to g^{\mu\nu}\to G_{\mu\nu}$, conditional at each arrow: a rank-three co-metric, a leading Newtonian exterior for a supplied spherically symmetric and asymptotically flat extension with an asymptotically homogeneous exterior, a Schwarzschild form only with external dynamical input (no uniqueness claim), an interpretive horizon reading, and a conditional relation to the Einstein equations; no coefficient value.

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 [H-lift], a rank-three effective co-metric whose non-degenerate block has signature $(-,+,+)$ under [H-hyp], with no established coefficient value, and the effective operator $L_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$; the reading of the three directions of $\mathrm{Im}\,\mathbb{H}$ as spatial remains an open bridge (Q7 Conjecture 4.7); and, conditionally on these hypotheses together with a supplied rank-four Lorentzian extension of the co-metric (Q5b-O2, open), the leading Newtonian exterior of Q6b, with the Schwarzschild form only under external dynamical input, and a conditional relation to the Einstein equations — 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 2.1 records as supplied by no source. Under [H-L], [H-lift] and [H-hyp], the non-degenerate block of the co-metric has signature $(-,+,+)$; the central slot needs a new operator (Q8), and no coefficient value is established. 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.