Overview
Q6b assembles the effective spacetime geometry that the admissibility framework of Foundation M yields conditionally, on a supplied Heisenberg carrier. Each step of the chain $\Pi_q \to \mathcal{L}_\Pi \to g^{\mu\nu}_{(3)} \to g^{\mu\nu} \to G_{\mu\nu}$ carries its own condition.
Q5a shows that the published admissibility forms supply no spatial limit operator: for uniformly bounded weights they converge to the zero form, and under [H-depth] and [H-w′] no common scalar normalisation yields a non-trivial toric differential operator. The operator $-A\partial_x^2$ therefore enters only as Q5b's hypothesis [H-L].
Results
- Rank-three co-metric. Under [H-L] and [H-lift], Q5b gives, in the left-invariant frame, $g^{\mu\nu} = \mathrm{diag}(-A_\tau, A_H, A_H, 0)$, with the temporal term stipulated. No lower-order term fills the central slot; a full-rank co-metric needs a new operator with a central term of homogeneous degree four and the length scale it carries (open problem Q5b-O2). No value of $A_\tau$, $A_H$ or $A_z$ is established.
- Signature. Under [H-hypred], the non-degenerate block $(-A_\tau, A_H, A_H)$ has signature $(-,+,+)$ with $\tau$ time-like; the signs are stipulated, not derived. The signature $(-,+,+,+)$ holds only for a supplied extension, for instance $\mathrm{diag}(-A_\tau, A_H, A_H, A_z)$ with $A_z > 0$.
- Newtonian exterior and conditional Schwarzschild form. For a supplied rank-four extension that is spherically symmetric and asymptotically flat in the exterior of a localised obstruction, flux conservation gives the Newtonian profile $\Phi = \Phi_0 - C/r$. The Schwarzschild form, with $r_s = 2C$, follows only with exterior homogeneity $A_r \equiv 1$ and, as external dynamical input, the closure $f_t f_r = 1$ and the exact temporal identification $f_t = 1 + 2\Phi$, $\Phi_0 = 0$. No uniqueness claim is made.
- The required extension. No extension diagonal in the left-invariant frame qualifies, and an asymptotically flat exterior (volume growth $r^3$) would not preserve the large-scale Carnot structure (volume growth $r^4$). Flat extensions exist, for example the flat metric of the exponential chart; whether the admissibility structure supplies one is open.
- Horizon (interpretation). In the conditional exterior, the degeneracy at $r = r_s$ is chart-relative; it is not an invariant breakdown of the effective operator.
- Einstein equations (conditional). The Gravity paper works with $A_g = -\nabla_g^2$ on Riemannian backgrounds; its Einstein coefficient, its sign and Newton's constant are matching data. The spectral-stationarity paper gives $G_{\mu\nu} + \Lambda_{\mathrm{ren}} g_{\mu\nu} = 8\pi G_N T^{(\Pi)}_{\mu\nu}$ under an assumed infrared response and matter functional. Applying either to the effective geometry needs the supplied extension and a Lorentzian formulation.
Interpretive outlook. The conditional results are compatible with reading spacetime geometry as an effective propagation structure rather than as primitive substrate data; they do not show that admissibility selects it.
Open directions
- [H-L]: establishing or replacing the spatial limit input, which is the open content of Q5.
- Q5b-O1: the lifting hypothesis [H-lift].
- Q5b-O2: a full-rank extension derived from the admissibility structure, with the exterior properties the Newtonian and Schwarzschild statements need.
- Q5b-O3: the values of the co-metric coefficients.
- Additional scales: whether further dimensional data enter the exterior problem.
References
Jérôme Beau. Toward Effective Spacetime Geometry from Admissible Non-Injective Projection: A Conditional Rank-Three Co-Metric and Newtonian Exterior, 2026. doi:10.5281/zenodo.19744005