The Solution Stack of General Relativity: Diffeomorphism Equivalence, Isotropy, and the Classical Geometry Regime of a Reconstruction Program

1 Introduction

1.1 The reconstruction thesis, and this paper’s regime

The organizing perspective of the wider project this paper belongs to can be stated in one sentence. Quantum gravity is not primarily the quantization of material objects in spacetime; it is the reconstruction of spacetime, matter, causality, and measurement from invariant representation structures. Under that reading, a classical spacetime is not a substrate one starts from and then quantizes. It is an output: the semiclassical shadow that any deeper reconstruction — a spin network, a tensor-network holographic code, a celestial correlator, an asymptotically safe renormalization-group trajectory, a spectral triple, a causal set — must reproduce in an appropriate limit. If that is the role of classical general relativity, then the classical theory has one job in the program: to package its own content in a form precise enough that “reproduce the classical limit” becomes a checkable statement rather than a slogan.

This paper carries out that packaging for the classical and semiclassical geometry regime. The invariant here is diffeomorphism equivalence: two Lorentzian metrics related by a diffeomorphism of the underlying manifold describe the same physical gravitational configuration, because the diffeomorphism is a relabelling of coordinates and general covariance forbids coordinates from carrying physical content. The naive way to encode this is to pass to the set of orbits, the coarse quotient $\mathrm{Sol}(\mathrm{Ein})/\mathrm{Diff}(M)$ of Einstein solutions modulo diffeomorphism. Our central claim is that this is the wrong object, and that the right one is the action groupoid, written $\mathrm{Sol}(\mathrm{Ein})/\!\!/\mathrm{Diff}(M)$ and read as a stack: its objects are solutions, its morphisms are diffeomorphisms carrying one solution to another, and — this is the point — its automorphism group at a solution $g$ is the isometry group $\mathrm{Isom}(M,g)$.

1.2 Why the coarse quotient is not enough

The reason is a theorem, not a preference. Two classical results, both from the 1970s and 1980s and both entirely rigorous, converge on it.

The first is the slice theorem of Ebin (with the infinite-dimensional manifold structure developed by Fischer and Marsden ). Near a metric $g$ the action of $\mathrm{Diff}(M)$ on the space of metrics admits a slice, and the local model for the orbit space is $(\text{slice})/\mathrm{Isom}(M,g)$. The isotropy group of the action at $g$ is exactly the isometry group. When $\mathrm{Isom}(M,g)$ is trivial the orbit space is locally a manifold; when it is not, the orbit space is locally a quotient of a linear slice by a nontrivial compact-or-noncompact group, and it fails to be a manifold there.

The second is linearization stability. Moncrief and then Fischer, Marsden, Arms and Moncrief proved that for a vacuum spacetime with a compact constant-mean-curvature Cauchy surface, the space of solutions is a smooth manifold except at solutions admitting Killing fields, where it has a quadratic singularity governed by the Killing algebra. Equivalently, such a spacetime is linearization stable if and only if it admits no global Killing vector field. The singular points of $\mathrm{Sol}(\mathrm{Ein})/\mathrm{Diff}(M)$ are precisely the symmetric solutions.

Put those together and the situation is unambiguous. The coarse orbit space is singular exactly where isotropy is nontrivial, and it is singular there because it has thrown away the isotropy. The stack keeps the isotropy as automorphism data. It is smooth — in the appropriate stacky sense — at the points where the coarse quotient tears, and the information it retains at those points, the Killing algebra of the solution, is exactly the data that has direct physical meaning: conserved Noether charges, the reducibility of the gravitational gauge algebra, and, once the classical regime is reconstructed as a limit of a deeper theory, the residual symmetry that the deeper theory must match.

1.3 Contributions and epistemic honesty

We do three things. First ([sec:framework,sec:stack]) we give precise definitions of the Lorentzian, causal, and Einstein data, and of the solution stack as an action groupoid, and we prove the isotropy identification $\mathrm{Aut}_{[\mathrm{Sol}/\mathrm{Diff}]}([g])\cong\mathrm{Isom}(M,g)$ together with the statement that the coarse quotient’s singular locus is the symmetric locus. Second ([sec:examples,sec:deformation]) we compute isotropy for the standard solution families and exhibit the isotropy jump along the Kerr-to-Schwarzschild limit as an explicit stratification, and we write down the deformation complex whose cohomology detects it. Third ([sec:connection,sec:modular]) we record the connection reformulation that bridges this regime to the spin-network regime, and we place the whole regime inside a modular weakest-link status calculus.

Throughout, every substantive claim carries an epistemic-status label. We use three ordered warrants: $\mathsf{S}$ for standard mathematics or established mathematical physics, $\mathsf{H}$ for a strong physical heuristic, and $\mathsf{P}$ for a speculative ontological extension, with $\mathsf{S}<\mathsf{H}< \mathsf{P}$. A label such as [S/H] means the claim’s warrant sits between the two. The rule that governs composition of labelled claims is that the composite inherits the worst (largest) warrant in the chain; we make this precise in 7 and use it to bound, honestly, what can be asserted when this regime is chained to the others. We also state plainly, in 9, what is not settled: the global four-dimensional derived stack of solutions, with non-proper diffeomorphism action and matter coupling, is not constructed in the literature at the generality the program would want, and we do not pretend otherwise.

1.4 Relation to prior work

The idea that the configuration space of gravity is a quotient stack $\mathrm{Met}(M)/\!\!/\mathrm{Diff}(M)$ rather than a coarse space is folklore in the mathematical-physics community and is made explicit in, for example, the nLab treatment of the moduli space of metrics and in Berktav’s construction of the moduli stack of vacuum Einstein solutions in $n$ dimensions (with a sharp stacky equivalence to a gauge theory in three dimensions). The deformation-theoretic side — a graded Lie algebra whose Maurer–Cartan equation is the vacuum Einstein equation, with linearized gravity as its first cohomology — is developed by Reiterer and Trubowitz and, in $L_\infty$ language, by Jurčo, Raspollini, Sämann and Wolf . The Hamiltonian/constraint side, where the diffeomorphism symmetry of the evolution equations is genuinely a groupoid (or Lie algebroid) and not the momentum map of any group action, is the subject of Blohmann, Fernandes and Weinstein and Blohmann and Weinstein . Our contribution is not a new theorem in any of these directions; it is a consolidation, with explicit epistemic labels and explicit isotropy computations, of why the stack — and specifically its automorphism data — is the object the reconstruction program needs, and a statement of the precise sense in which the classical regime composes with the others.

2 Mathematical framework

We fix a smooth connected oriented manifold $M$ of dimension $n=4$ unless stated otherwise; the constructions are dimension-agnostic except where a specific example fixes $n$.

2.1 Lorentzian metrics and causal structure

At a point $p$ the metric partitions $T_pM\setminus\{0\}$ into vectors that are timelike ($g(v,v)<0$), null ($g(v,v)=0$), and spacelike ($g(v,v)>0$). A time orientation is a continuous choice of one of the two components of the timelike cone at each point; we always assume $(M,g)$ is time oriented. A piecewise-smooth curve is causal if its tangent is everywhere timelike or null, and future directed if it lies in the chosen cone. The causal relation $p\preceq q$ (there is a future-directed causal curve from $p$ to $q$, or $p=q$) is the primitive causal datum; the light-cone (conformal) structure of $g$ determines $\preceq$ and conversely $\preceq$ determines $g$ up to a pointwise positive conformal factor.

The following two facts anchor the well-posedness of the classical theory and are used repeatedly; both are standard.

2.2 The Einstein equation

The initial-value formulation splits [eq:einstein] into constraints on a slice and evolution off it. On a spacelike hypersurface $\Sigma$ with induced metric $h$ and second fundamental form $K$, the Gauss–Codazzi relations turn the $00$ and $0a$ components of [eq:einstein] into the Hamiltonian and momentum constraints $$\begin{equation} R_h + (\mathrm{tr}_h K)^2 - |K|_h^2 = 16\pi G\,\rho, \qquad D^b(K_{ab}-h_{ab}\,\mathrm{tr}_h K) = 8\pi G\, j_a, \end{equation}$$ where $D$ denotes the Levi-Civita connection of the spatial metric $h$ (distinct from the spacetime connection $\nabla$ used above), $R_h$ is the intrinsic scalar curvature of $h$, and $\rho,j$ are the matter energy and momentum densities. Constraint-satisfying data $(\Sigma,h,K)$ is the input to the Cauchy problem.

7 already tells us what the correct output type of the Cauchy problem is. It is not a point of a set of metrics. It is an object of a groupoid, defined only up to the isomorphisms of that groupoid, which are exactly diffeomorphisms. The rest of the paper takes that observation seriously.

2.3 The configuration groupoid

The physical reading of 9 is the one the reconstruction program cares about. A Killing field is not a gauge redundancy. It is a physical residual symmetry: it generates a one-parameter family of isometries under which the solution is genuinely invariant, it produces a conserved Noether current for the matter it acts on, and, at the level of the gravitational gauge algebra, it makes that algebra reducible (there is a gauge transformation, generated by $\xi$, that acts trivially on $g$). Collapsing $g$ to its orbit forgets $\mathrm{Isom}(M,g)$; keeping the groupoid remembers it as the automorphism group of the object.

3 The solution stack $\mathrm{Sol}(\mathrm{Ein})/\!\!/\mathrm{Diff}(M)$

3.1 Action groupoids and quotient stacks

We recall the finite-dimensional template so that the gravitational statement is a specialization rather than a definition made up for the occasion.

The contrast with the coarse quotient is exactly [eq:autstab]. The coarse quotient set $X/G$ remembers only the orbit; it has forgotten $\mathrm{Stab}_G(x)$. Two actions with the same orbits but different stabilizers give the same coarse quotient and different groupoids. In the presence of nontrivial, jumping stabilizers, the coarse quotient is the wrong invariant precisely because [eq:autstab] is nontrivial.

12 is elementary once the objects are set up correctly, and that is the point: the content is entirely in choosing the groupoid over the set. We now show that this choice is forced by the local geometry of the solution space, not merely convenient.

3.2 The slice theorem and why the coarse quotient tears

The two theorems fit together into a single picture, which we state as the paper’s organizing proposition.

3.3 A diagrammatic summary

The structure is compactly summarized by the two maps out of the morphism space of the groupoid and the failure of the coarse quotient to be their coequalizer in a way that remembers isotropy.

Here $s(g,\varphi)=\varphi^{*}g$ and $t(g,\varphi)=g$ are source and target, the top row is the groupoid, the stack $[\mathrm{Sol}(\mathrm{Ein})/\mathrm{Diff}(M)]$ is its stackification, and the vertical map to the coarse quotient is the functor that forgets the automorphism groups. 15 says the vertical map is an isomorphism away from the symmetric locus and genuinely loses information (the isotropy groups, and with them the smoothness of the total object) on it.

4 Worked examples: isotropy and stratification

We compute $\mathrm{Isom}(M,g)$ for the standard solution families. All statements in this section are [S]; they are classical, and their point here is to make the automorphism data of 12 concrete and to exhibit an explicit isotropy jump.

21 is the concrete form of the reconstruction program’s requirement. Any deeper theory — a spin-foam amplitude, a holographic code, a renormalization-group endpoint — that claims Schwarzschild as a semiclassical limit must reproduce not just the isometry class of Schwarzschild but its $\mathbb{R}\times SO(3)$ automorphism group, and must reproduce the fact that this group is strictly larger than the automorphism group of nearby rotating solutions. A theory that produced the right coarse geometry but the wrong isotropy would be reproducing the coarse quotient, not the stack, and would be missing exactly the data — degeneracies, conserved charges, gauge-algebra reducibility — that has physical consequences.

Isotropy (automorphism) data of representative points of the solution stack. The stratification is by $\dim\mathop{\mathrm{Kill}}$; higher-symmetry solutions sit in the closures of lower-symmetry strata (21). All entries are [S].

Solution	Connected $\mathrm{Isom}_0$	$\dim\mathop{\mathrm{Kill}}$	Type
Minkowski	$\mathbb{R}^{4}\rtimes SO(3,1)^{\uparrow}$	$10$	maximally symmetric (flat)
de Sitter	$SO(4,1)$	$10$	maximally symmetric ($\Lambda>0$)
FLRW (generic $a$)	spatial $6$-dim group	$6$	homogeneous isotropic
Schwarzschild	$\mathbb{R}\times SO(3)$	$4$	static spherical
Kerr ($a\neq0$)	$\mathbb{R}\times U(1)$	$2$	stationary axisymmetric
generic vacuum	$\{e\}$	$0$	no symmetry

5 The deformation complex and the tangent stack

The linearization-stability theorem (14) is best understood as a statement about a cochain complex — the tangent complex of the solution stack at $g$ — whose cohomology organizes Killing fields, on-shell perturbations, and obstructions into a single object. This is the deformation-theoretic content of the stack, and it is the level at which the classical regime connects to the Batalin–Vilkovisky and $L_\infty$ machinery of the neighbouring regime.

5.1 The linearized theory as a two-step complex

Fix $g\in\mathrm{Sol}(\mathrm{Ein})$ vacuum. Write $DE_g$ for the linearized Einstein operator and $D\!E_g^{*}$ for its formal adjoint, and let $\delta_g^{*}:\Gamma(TM)\to\Gamma(\mathop{\mathrm{Sym}}^2 T^{*}M)$, $\xi\mapsto\mathcal L_\xi g$, be the map sending a vector field to the metric variation it generates (the linearized gauge action), and let $\delta_g:\Gamma(\mathop{\mathrm{Sym}}^2 T^{*}M)\to\Gamma(TM)$ be the (metric) divergence operator, the $L^2$ formal adjoint of $\delta_g^{*}$. Since $(\delta_g^{*}\xi)_{ab}=\nabla_a\xi_b+\nabla_b\xi_a$, its exact adjoint is $h_{ab}\mapsto -2\nabla^a h_{ab}$; we absorb the constant $-2$ into $\delta_g$, as it does not affect any cohomology below. The relevant complex is $$\begin{equation} \Gamma(TM)\;\xrightarrow{\ \delta_g^{*}\ }\;\Gamma(\mathop{\mathrm{Sym}}^2 T^{*}M) \;\xrightarrow{\ DE_g\ }\;\Gamma(\mathop{\mathrm{Sym}}^2 T^{*}M), \qquad DE_g\circ\delta_g^{*}=0, \end{equation}$$ where $DE_g\circ\delta_g^{*}=0$ is the linearized gauge invariance of the Einstein operator: it expresses that an infinitesimal diffeomorphism is annihilated by the linearized equation. Its Noether-dual companion is the linearized contracted Bianchi identity $\delta_g\circ DE_g=0$, a separate statement about the divergence of the Einstein operator; the two identities are adjoint and should not be conflated. Together they extend [eq:complex] to the four-term elliptic (Batalin–Vilkovisky field–antifield) resolution $$\begin{equation} \Gamma(TM)\;\xrightarrow{\ \delta_g^{*}\ }\;\Gamma(\mathop{\mathrm{Sym}}^2 T^{*}M) \;\xrightarrow{\ DE_g\ }\;\Gamma(\mathop{\mathrm{Sym}}^2 T^{*}M) \;\xrightarrow{\ \delta_g\ }\;\Gamma(TM), \end{equation}$$ in degrees $0,1,2,3$, whose degree-$0$ chain space carries the ghosts (gauge parameters), degree-$1$ the metric perturbations, degree-$2$ the antifields (equations of motion), and degree-$3$ the antighosts $\Gamma(TM)$, whose cohomology is $\mathop{\mathrm{coker}}\delta_g$; this is the linearized shadow of the BV complex of 24. Here “elliptic” refers to the formal principal symbol of the complex (equivalently, to its Riemannian-signature counterpart), under which [eq:complex4] is an elliptic complex; in Lorentzian signature the vacuum Einstein operator is hyperbolic — weakly hyperbolic before gauge fixing, strictly hyperbolic in harmonic gauge — and the physical evolution is governed by that hyperbolicity (7), not by ellipticity. The symbolic/elliptic reading is what makes the cohomology count below well posed; the analytic evolution is a separate, hyperbolic, matter (9). For the moduli count we read the truncation [eq:complex] as a cochain complex in degrees $0,1,2$, which gives the following cohomology.

5.2 Maurer–Cartan and the derived tangent

22 is the shadow of a richer statement: the vacuum Einstein equation is the Maurer–Cartan equation of a differential graded (or $L_\infty$) algebra, and $\mathrm{Sol}(\mathrm{Ein})/\!\!/\mathrm{Diff}(M)$ is its (formal) derived moduli.

5.3 Hamiltonian form: the constraint groupoid

There is a second, Hamiltonian, route to the same conclusion that the symmetry object of gravity is a groupoid rather than a group, and it is worth recording because it is where the diffeomorphism symmetry differs sharply from ordinary gauge symmetry.

6 The connection reformulation and the bridge to quantum geometry

The classical regime does not connect to the quantum-geometry regimes through the metric directly; it connects through a first-order, gauge-theoretic reformulation. We record it because it is the technical door from this paper’s stack into the spin-network phase space, and because it re-expresses the same isotropy invariant in gauge-theoretic language.

7 Modular composition and the weakest-link status calculus

This regime is one of twelve in a modular program. The program is deliberately not a unification: it does not assert a single master theory of which each regime is a special case. Instead it tracks how epistemic warrant propagates when regimes are chained, using a small, explicit calculus that we now state and apply.

7.1 The status calculus

This is a finite, checkable structure; it is implemented and property-tested in the accompanying Haskell development (8), where monotonicity, the unit law, and associativity are the checkable properties.

7.2 This regime’s status, and its compositions

We record three representative compositions with neighbouring regimes; each is a statement about warrant, not about existence of the bridge, and each uses $\mathrm{compose}$ (worst-component-wins), not the reported-range operation.

• Classical geometry $\circ$ gauge/BRST regime. $\mathrm{compose}(\mathsf{S/H},\mathsf{S/H})=\mathsf{H}$. This is the best-warranted bridge available to the classical regime — the perturbative BRST/BV observable algebra built on the classical solution stack (24) — yet it already lands at $\mathsf H$, because the worst component of each S/H input is $\mathsf H$. Only the pure-$\mathsf S$ structural sub-claims of the two regimes (the groupoid/isotropy identification of 12; the derived-critical-locus construction where rigorous) compose back to $\mathsf S$. This is the calculus’s core discipline: one composes the specific claims used, and two $\mathsf{S/H}$ regimes do not yield an $\mathsf{S/H}$ bridge.

• Classical geometry $\circ$ spin-network regime. $\mathrm{compose}(\mathsf{S/H},\mathsf{S/H})=\mathsf H$, and the specific bridging claim that a large-spin/coherent-state limit of a spin network reproduces a given classical $3$-geometry is independently only heuristic, consistent with $\mathsf H$.

• Classical geometry $\circ$ causal-order (indefinite) regime. $\mathrm{compose}(\mathsf{S/H},\mathsf{H/P})=\mathsf{P}$. Chaining the classical regime to a claim that presupposes genuinely indefinite (superposed) causal order pulls the composite down to speculative, no matter how careful the classical part is. This is monotonicity doing its intended work.

The twelve regimes and their standalone statuses. The weakest-link composite of all twelve is $\mathsf P$, driven by regime 12; this is the precise sense in which the program is modular rather than a single warranted theory (29).

Regime	Role	Status
1. classical geometry (this paper)	Lorentzian solution stack	$\mathsf{S/H}$
2. gauge / BV–BRST / derived	observables as cohomology	$\mathsf{S/H}$
3. spin networks	quantum spatial geometry	$\mathsf{S/H}$
4. spin foams	covariant histories	$\mathsf{H}$
5. tensor networks / QEC	holographic encoding	$\mathsf{H}$
6. positive geometries	amplitude boundaries	$\mathsf{S/H}$
7. double copy	gravity as gauge$^2$	$\mathsf{S/H}$
8. celestial holography	flat-space $S$-matrix	$\mathsf{H}$
9. asymptotic safety	UV fixed point	$\mathsf{H}$
10. factorization algebras	local-to-global observables	$\mathsf{S/H}$
11. noncommutative geometry	spectral reconstruction	$\mathsf{H}$
12. quantum causal structure	indefinite causal order	$\mathsf{H/P}$

7.3 What “reconstruction” means for this regime, made precise

We can now state the reconstruction thesis for the classical regime as a checkable requirement rather than a slogan. A candidate deeper theory $\mathcal T$ (regimes 3–12) reconstructs the classical geometry regime in a limit $\lambda\to\lambda_0$ if there is a map $$\mathrm{Real}_\lambda:\;\{\text{states/histories of }\mathcal T\}\longrightarrow \mathrm{Sol}(\mathrm{Ein})/\!\!/\mathrm{Diff}(M)$$ into the solution stack — not the coarse quotient — such that, for symmetric target solutions, $\mathrm{Real}_\lambda$ reproduces the automorphism group $\mathrm{Isom}(M,g)$ of 12 and the isotropy stratification of 1. The requirement to land in the stack, with its automorphism data, is what makes “reproduces the classical limit” falsifiable: a theory that matches the coarse geometry but not the isotropy has not reconstructed the classical regime, it has reconstructed a lossy image of it. The status of any such reconstruction claim is bounded below by $\mathrm{compose}(\mathsf{S/H}, \mathrm{status}(\mathcal T))$, which for the nonperturbative regimes 4,5,8,9,11 is $\mathsf H$ and for regime 12 is $\mathsf P$.

8 A machine-checkable model

The accompanying Haskell development models three checkable pieces of this paper: the status calculus of 28 (with monotonicity, unit, and associativity as tested properties), the isotropy table of 1 (as a lookup with dimension counts matching the Killing-field computations of 4), and the automorphism identification of 12 (as an action-groupoid whose automorphism group at an object is its stabilizer, with the isotropy-jump example of 21 as a concrete instance). The core checkable property is that $\mathrm{compose}$ is monotone and has $\mathsf S$ as unit, matching 29; a second checkable property is that the stabilizer computed from the modelled $\mathrm{Diff}$-action on the discrete example agrees with the tabulated $\dim\mathop{\mathrm{Kill}}$. The development compiles with ghc and runs a demonstration that prints the isotropy of each tabulated solution, verifies the isotropy jump along the Kerr family, and reports the weakest-link status of a few sample regime chains. We stress that this is a model of the paper’s finite combinatorial content (the status calculus and the isotropy bookkeeping), not a formalization of the analytic theorems of 3, which remain at the level of cited results.

9 Limitations and open problems

We are explicit about what is not settled, because the epistemic labels are only meaningful if the caveats behind them are stated.

9.0.0.1 Non-properness of the Lorentzian $\mathrm{Diff}$-action.

The clean slice theorem (13) uses properness of the $\mathrm{Diff}(M)$-action on metrics, which holds for Riemannian metrics on a closed manifold. For Lorentzian metrics, on non-compact $M$, or with matter, the action is generally not proper, orbits need not be closed, and a global slice theorem in the same generality is not available. The action groupoid $\mathrm{Sol}(\mathrm{Ein})/\!\!/\mathrm{Diff}(M)$ and the isotropy identification 12 survive as groupoid statements regardless — this is one of the reasons the stack, which does not require the quotient to be a nice space, is the right object — but the clean local-model statements of [thm:slice,thm:linstab] carry their compact-CMC hypotheses and should not be overclaimed for the asymptotically flat black-hole examples of 4.

9.0.0.2 No global $4$D derived stack of solutions.

Berktav’s construction of the moduli stack of vacuum Einstein solutions is explicit, but its sharpest results (the equivalence with a gauge theory) are special to three dimensions, where gravity is a Chern–Simons/BF-type theory. A fully general global, four-dimensional derived stack of vacuum solutions — handling non-generic isotropy jumps, non-compact $\mathrm{Diff}$, and matter coupling in one construction — is not established in the literature as of this writing. The graded-Lie/$L_\infty$ picture (23, ) gives the correct formal/perturbative tangent complex but not a global geometric object; completing it is the “homological PDE” program, which is open. This is precisely roadmap open-problem item 4 (“model the diffeomorphism quotient and isotropy for concrete solution families”), and we have discharged its local/isotropy half (4) while leaving the global-derived-stack half open.

9.0.0.3 Signature.

Most sharp moduli and deformation results (slice theorem, Einstein-metric moduli in the sense of Besse ) are cleanest in Riemannian signature; the Lorentzian causal theory ([thm:bs,thm:cbg]) and the moduli/deformation theory live in partly different technical worlds. We have kept the causal layer (2) and the moduli layer ([sec:stack,sec:deformation]) labelled separately rather than silently identifying them across a Wick rotation.

9.0.0.4 Physical versus gauge symmetry.

The single conceptual point on which the whole construction turns: Killing symmetries are physical residual symmetries (they source conserved charges and degeneracies), not gauge redundancies. In the stack, gauge redundancy is the groupoid’s morphisms and isotropy is an object’s automorphism group; these are different roles for what look superficially like “diffeomorphisms.” Conflating them — treating a Killing field as “pure gauge” — is exactly the error the stack exists to prevent, and it is the error a coarse quotient silently commits.

10 Conclusion

The classical regime of general relativity has, in the reconstruction program, a narrow and precise job: to package its content so that “reproduce the classical limit” is a falsifiable requirement on any deeper theory. We have argued that the package is the solution stack $\mathrm{Sol}(\mathrm{Ein})/\!\!/\mathrm{Diff}(M)$, an action groupoid whose objects are Einstein solutions, whose morphisms are diffeomorphisms, and whose automorphism group at a solution $g$ is its isometry group $\mathrm{Isom}(M,g)$ (12). The choice of the groupoid over the coarse quotient is forced, not stylistic: by the slice theorem and the linearization-stability theorems of Moncrief, Fischer, Marsden and Arms, the coarse quotient $\mathrm{Sol}(\mathrm{Ein})/\mathrm{Diff}(M)$ is genuinely singular at exactly the symmetric solutions, and it is singular there because it has discarded the isotropy that the stack keeps ([thm:linstab,prop:resolve]). We made the isotropy concrete for the standard families and exhibited the Kerr-to-Schwarzschild isotropy jump $\mathbb{R}\times U(1)\rightsquigarrow\mathbb{R}\times SO(3)$ as a stratification of the stack ([ex:jump,tab:isotropy]), and we identified the deformation complex whose $H^0$ is the Killing algebra and whose quadratic obstruction is the same data seen deformation-theoretically ([prop:cohomology,prop:mc]). We placed the regime in a modular, weakest-link status calculus ([def:status,prop:monotone]) that bounds honestly how it composes with the eleven other regimes and never inflates a chain above its weakest link. And we stated what remains open: a global four-dimensional derived stack of solutions with non-proper $\mathrm{Diff}$ and matter is not available, and the reconstruction of this regime from any nonperturbative deeper theory is, by the calculus, at best a heuristic ($\mathsf H$) claim until such a construction and such a limit are made precise.

The one sentence to carry forward is the reconstruction requirement made checkable in 7: a deeper theory reconstructs classical geometry only if its semiclassical limit lands in the solution stack, reproducing the isotropy groups and their stratification — not merely in the coarse quotient, which has already thrown that data away.