Effective Spacetime Geometry from Admissible Non-Injective Projection

Core contributions

• Effective metric under [H-L]: conditional on [H-L], 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).
• Co-metric coefficients under [H-L] and the missing identification: conditional on [H-L], 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 reads $g^{\mu\nu} = \mathrm{diag}(-2,\,2,\,2,\,2)$. Since Q7 version 2.0, those coefficient values additionally rest on an identification of the measured admissible space with the spin-1 module that no paper supplies, and the entries asserting them are pending revision.
• Schwarzschild metric under [H-L]: conditional on [H-L], 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 under [H-L]: conditional on [H-L], 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 under [H-L]: 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 under [H-L]: conditional on [H-L], 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.
• Conditional chain: under [H-L], 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, under [H-L], the effective metric tensor $g^{\mu\nu}(x) \propto A^{\mu\nu}(x)$. Under that same hypothesis, Q5b identifies the Lorentzian signature $(-,+,+,+)$ from the asymmetry between the central and horizontal directions in the Heisenberg group.

Under [H-L], $\mathcal{L}_{\mathrm{eff}}$ converges in the appropriate functional-analytic sense as $q \to \infty$, giving a well-defined limiting geometry by hypothesis. Subsequent papers (Q8, Q10, Q11) assert the coefficients $g^{\mu\nu} = \mathrm{diag}(-2,2,2,2)$; those values depend on an identification Q7 version 2.0 records as supplied by no source, and are pending revision. Newton's constant $G_N$ remains a renormalised matching datum in the Gravity paper; it is not structurally determined here.

Schwarzschild geometry and the horizon

Conditional on [H-L], 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). Under [H-L], 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-dimensional spatial sector; the horizontal distribution of $\mathrm{Heis}_3$ is two-dimensional and is kept distinct from that sector.
• Q8–Q11: assert the co-metric coefficients ($A_z$, $A_H$, $A_\tau$), read as $g^{\mu\nu} = \mathrm{diag}(-2,2,2,2)$ and pending revision in the cascade Q7 version 2.0 opens; the Gravity paper retains Newton's constant $G_N$ as a renormalised matching datum.
• Q12: on the conditional base geometry and for supplied compact gauge data, the admissible principal bundle is the setting for the vertical variation yielding Yang–Mills equations.
• Q13: addresses the coupled Einstein–Yang–Mills back-reaction between the gauge fields of Q6a and the effective geometry of Q6b.

Under [H-L], Q5b and Q6b provide the geometric foundation consumed downstream: Q5b supplies the BFS-stratification extraction, while Q6b connects that conditional geometry to the Einstein tensor.

Open directions

• Establishing [H-L]: the spatial limit hypothesis is not established; proving or replacing [H-L] would remove the remaining conditionality from the geometric results of Q6b — this is the open content of Q5.
• 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