Effective Spacetime Geometry from Admissible Non-Injective Projection

Overview

Q6b is the geometric companion to Q6a. While Q6a derives the gauge group $G_\Pi$ from the fibre invariants of the non-injective projection, Q6b analyses the base geometric structure — the effective Lorentzian manifold on which those gauge fields propagate — and closes the geometric chain from the admissibility filter to the Einstein tensor.

Starting from the Q5a–Q5b chain (Mosco convergence → effective operator $\mathcal{L}_{\mathrm{eff}}$ → Lorentzian metric from principal symbol), Q6b identifies the resulting metric explicitly as $g^{\mu\nu} = \mathrm{diag}(-2,2,2,2)$, establishes the Schwarzschild metric from flux conservation, and shows that the Einstein equations emerge as consistency conditions of the spectral entropy functional.

[H-lift] closed. The [H-lift] hypothesis, previously carried from Q5b, has been proved unconditionally in Q9 via generator suppression at rate $O(q^{-1/2})$. All geometric results of Q6b are therefore unconditional with respect to [H-lift].

Core contributions

Effective metric with Lorentzian signature: the admissibility filter $\Pi_q$ selects degrees of freedom whose continuum limit carries a metric of Lorentzian signature $(-,+,+,+)$, derived from the principal symbol of $\mathcal{L}_{\mathrm{eff}}$ (Q5b Theorems 5.2 and 6.1).
Fully determined co-metric: combining Q6b with Q8 (Casimir rigidity: $A_z = 2$), Q10 (spectral universality: $A_H = 2$), and Q11 (temporal Casimir rigidity: $A_\tau = 2$), the effective co-metric is fully explicit: $g^{\mu\nu} = \mathrm{diag}(-2,\,2,\,2,\,2)$.
Schwarzschild metric from flux conservation: in the presence of a localised stationary obstruction with spherical symmetry, flux conservation through admissible spheres forces the unique stationary exterior solution to be the Schwarzschild metric. Uniqueness follows from the Born–Infeld admissibility bound, not symmetry alone.
Horizon as projection degeneracy: the Schwarzschild horizon $r = r_s$ is the locus where the principal symbol of $\mathcal{L}_{\mathrm{eff}}$ becomes degenerate. The underlying admissible structure remains regular.
Hamilton–Jacobi propagation structure: once $g^{\mu\nu}$ is reconstructed, the eikonal equation $g^{\mu\nu}\partial_\mu S\,\partial_\nu S = 0$ is the Hamilton–Jacobi equation for massless propagation in the emergent geometry (effective reformulation, not a primitive layer).
Einstein equations as consistency conditions: via the spectral entropy functional $S_\Pi[g] = \frac{1}{2}\log\det'\mathcal{L}_\Pi$, the renormalised metric variation produces $G_{\mu\nu}$ at the infrared two-derivative order (Gravity paper). The Einstein equations emerge as consistency conditions, not microscopic laws.
Chain closure: Q6b supplies the middle link of the geometric chain $\Pi_q \xrightarrow{\text{Q5a}} \mathcal{L}_\Pi \xrightarrow{\text{Q5b}} g^{\mu\nu} \xrightarrow{\text{Gravity}} G_{\mu\nu}$, identifying the symbol-extracted metric of Q5b with the metric entering the variational argument of the Gravity paper.

Geometry from the admissibility filter

The admissibility filter $\Pi_q$ acts as a projective selection mechanism on the relational substrate, retaining only those configurations compatible with the spectral admissibility constraints. Q6b analyses the geometric structure of the image of this filter.

The effective operator $\mathcal{L}_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$ has a principal symbol $\sigma_2(\mathcal{L}_{\mathrm{eff}}) = A^{\mu\nu}(x)\xi_\mu\xi_\nu$ whose non-degenerate part defines the effective metric tensor $g^{\mu\nu}(x) \propto A^{\mu\nu}(x)$. Q5b Theorems 5.2 and 6.1 establish the Lorentzian signature $(-,+,+,+)$ from the asymmetry between the central and horizontal directions in the Heisenberg group.

The Mosco convergence of Q5a ensures that $\mathcal{L}_{\mathrm{eff}}$ converges in the appropriate functional-analytic sense as $q \to \infty$, giving a well-defined limiting geometry. Subsequent papers (Q8, Q10, Q11) pin down the coefficients to $g^{\mu\nu} = \mathrm{diag}(-2,2,2,2)$, yielding a structural determination of $G_N$ when combined with the Gravity paper.

Complementarity with Q5b. Q5b derives the Lorentzian metric combinatorially from BFS stratification; Q6b derives it analytically from the admissibility filter. Both give the same Lorentzian signature and are mutually consistent.

Schwarzschild geometry and the horizon

The effective geometry of Q6b is defined in homogeneous, quasi-isotropic regimes. In the presence of a localised stationary obstruction with spherical symmetry, the admissibility structure selects a specific exterior metric through a uniqueness argument based on flux conservation.

The flux conservation equation $\frac{1}{r^2}\frac{d}{dr}\!\left(r^2 A^r(r)\frac{d\Phi}{dr}\right) = 0$ (conservation of admissible flux through spheres) forces $\Phi(r) = \Phi_0 - C/r$, which translates via the operator–metric correspondence into the Schwarzschild metric coefficients $g_{tt} = -f(r)$, $g_{rr} = f(r)^{-1}$ with $f(r) = 1 - r_s/r$.

Uniqueness follows from the combined constraints of flux conservation and the bounded Born–Infeld admissibility bound, which excludes additional dimensional scales (no cosmological constant, no charge) at the level of the present construction.

The Schwarzschild horizon at $r = r_s$ is interpreted as the locus where the principal symbol $\sigma_2(\mathcal{L}_{\mathrm{eff}})$ becomes degenerate. The underlying admissible substrate remains regular; only the projected effective geometry encounters a degeneracy.

Relation to the Cosmochrony programme

Q6b occupies the interface between the geometric programme (Q5a, Q5b) and the dynamical programme (Q7–Q13). Its role is to provide the effective geometric framework assumed by later papers:

Q7: uses the effective geometry of Q6b to study the spatial bridge problem between $H_{\mathrm{eff}}$ and the three horizontal directions of $\mathrm{Heis}_3$.
Q8–Q11: pin down the co-metric coefficients ($A_z$, $A_H$, $A_\tau$), completing $g^{\mu\nu} = \mathrm{diag}(-2,2,2,2)$ and yielding a structural determination of Newton's constant $G_N$ via the Gravity paper.
Q12: the admissible principal bundle $P_{G_\Pi}(M, G_\Pi)$ established by Q6b is the geometric setting for the vertical variation that yields Yang–Mills equations.
Q13: addresses the coupled Einstein–Yang–Mills back-reaction between the gauge fields of Q6a and the effective geometry of Q6b.

Together Q5b and Q6b constitute the complete geometric foundation of the Q-series: Q5b provides the BFS-stratification derivation, Q6b provides the admissibility-filter derivation and connects to the Einstein tensor.

Open directions

Closing Q5a hypotheses: rigorous proofs of H1, H-w, H-E1, and Conjecture C in Q5a would remove all remaining conditionality from the geometric results of Q6b.
No-scale rigidity: the exclusion of additional dimensional scales in the Schwarzschild uniqueness argument (Remark 3.3) rests on a structural claim about the Born–Infeld bound $c_\chi$. Making this rigorous requires showing that no independent dimensional parameter violating A3–A4 closure can appear in the admissibility structure.
Coupling to matter: the extension of the effective geometry framework to include projected matter fields is deferred to future work.

References

Jérôme Beau. Effective Spacetime Geometry from Admissible Non-Injective Projection, 2026. doi:10.5281/zenodo.20257944