Observables as Cohomology: BV–BRST Complexes and Derived Critical Loci as the Gauge Layer of Reconstructed Spacetime

1 Introduction

1.1 The reconstruction thesis, restricted to the gauge layer

This paper is one module of a deliberately modular program whose governing sentence is:

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.

Twelve regimes carry that program, each responsible for one layer. The present module is the gauge, descent, and observable layer. Its single organizing claim is short:

Observables are gauge-invariant cohomology classes.

The word “reconstruction” is doing real work here. In the textbook picture one starts with a gauge field, notices that many field configurations describe the same physics, and then quotients or gauge-fixes to remove the redundancy so that quantization can proceed. The redundancy is treated as an obstacle. In the reconstruction picture the redundancy is data. What is physical is not a point in a quotient set but a class in a cohomology theory, and the cohomology retains exactly the information a naive quotient throws away: the stabilizer of a configuration, the relations among gauge generators, the relations among those relations. For gravity this retained information is the isometry (Killing) content of a background, and it is not decoration. It controls degeneracies, conserved charges, and the reducibility structure of the perturbative expansion.

The mathematical statement of “physical content is a class, not a point” is that the correct object is a derived quotient, or equivalently a groupoid or stack, rather than a coarse quotient. At the level of a single action functional $S$ on a space of fields, the correct on-shell gauge-invariant object is the derived critical locus $\mathrm{dCrit}(S)$: the derived intersection of the graph of $dS$ with the zero section of the cotangent bundle. Its ring of functions, with a canonical differential, is precisely the classical BV complex. The degree-zero cohomology of that complex is the algebra of on-shell gauge-invariant functionals. This is the technical core of the module, and the rest of the paper unpacks it.

1.2 What is standard, what is heuristic, what is speculative

The program attaches to every dictionary entry a status drawn from an ordered set of three warrants, $\mathsf{S} < \mathsf{H} < \mathsf{P}$: S for a standard mathematical or mathematical-physics correspondence, H for a strong heuristic physical representation, P for a speculative ontological extension. Statuses compose by the worst component, so no chain of reasoning can be advertised at a status better than its weakest link. We take this discipline seriously and tag each result. To preview the honest accounting:

• The identification of the BV antibracket with the $(-1)$-shifted Poisson bracket on $\mathrm{dCrit}(S)$, the nilpotency $Q_{\mathrm{BV}}^2=0$ as the classical master equation, and the computation of $H^0$ as invariant functions on the critical locus, are all [ S ]: established mathematics.

• The identification of that $H^0$ with the physical observable algebra of the interacting quantum theory is [ S/H ]: it holds when a tower of cohomological obstructions vanishes order by order in $\hbar$ , and that vanishing is the rigorous content of “renormalizable and anomaly-free.”

• Any claim that this observable algebra extends to a nonperturbative, background-independent theory of quantum gravity is [ H ] at best and, when chained to the more speculative regimes of the library, [ P ]. We do not make such a claim; we bound it.

1.3 Contributions

Three things in this paper are ours rather than a rehearsal of the literature.

First (8), a fully explicit finite-dimensional BV model whose every cohomology group is computed by hand and, independently, by an executable Haskell program that checks the nilpotency of the BRST and BV differentials by exhaustive evaluation on a basis. This discharges, for a finite model, the project’s formalization item.

Second (9), a clean statement in the representation-stack language of the library of the fact that a background with a nontrivial isometry group forces a nontrivial ghost-for-ghost tower whose lowest reducibility degree is dual to the isometry Lie algebra. This ties the classical slice-theorem picture of the gravitational moduli stack to the perturbative-quantum ghost tower, and it is the BV-level shadow of “isotropy is retained, not erased.”

Third (11), a worked application of the weakest-link status calculus that places this regime among the other eleven, including the sharp computation that BV-BRST composed with an indefinite-causal-order regime lands at speculative status no matter how careful the BV construction is.

1.4 Relation to prior work

The antibracket and antifield formalism are due to Batalin and Vilkovisky , with the reducible (linearly dependent generator) case in the second paper. The homological reading, Koszul–Tate resolution and BRST differential, is developed in Henneaux and Teitelboim and, for the local cohomology $H(s|d)$, in Barnich, Brandt and Henneaux . The functional and locally covariant formulation appropriate to field theory on curved spacetime is due to Fredenhagen and Rejzner . The derived-geometric foundation, in which $\mathrm{dCrit}(S)$ carries a canonical $(-1)$-shifted symplectic structure, is Pantev, Toën, Vaquié and Vezzosi ; its symplectic reduction and the comparison with classical BV is Anel and Calaque . The factorization-algebra package, in which the quantization of a classical BV theory is controlled by an obstruction cocycle, is Costello and Costello–Gwilliam . The $L_\infty$ repackaging of the data is Jurčo, Raspollini, Sämann and Wolf . For gravity specifically, the BV-BFV treatment of the Einstein–Hilbert action is Cattaneo and Schiavina , building on the boundary axioms of Cattaneo, Mnev and Reshetikhin . Our job is to assemble these into the reconstruction narrative and to tag the seams honestly.

1.5 Outline

2 recalls the representation-stack setting and the status calculus. 3 sets up graded geometry and the antibracket. 4 builds the BRST complex from ghosts and the Chevalley–Eilenberg differential. 5 builds the Koszul–Tate resolution and states the classical master equation. 6 gives the derived-critical-locus picture and the $(-1)$-shifted symplectic structure. 7 states the main results on observables as cohomology. 8 works the finite abelian and nonabelian examples. 9 treats the reducible gravitational case. 10 discusses the BV Laplacian, the quantum master equation and the obstruction. 11 composes this regime with the other eleven. 12 lists limitations and open problems. 13 concludes.

2 The representation-stack setting and the status calculus

We record the ambient structure the module inherits, so the paper is self-contained.

2.1 Representation entries and the realization pipeline

A representation entry is a tuple $E=(M,P,\tau,\sigma)$: a mathematical object $M$, a physical target $P$, a translation datum $\tau$ relating them, and an epistemic status $\sigma$. A collection of entries forms a representation prestack, a pseudofunctor $\mathrm{Rep}_{\mathrm{phys}}\colon \mathrm{Dom}^{\mathrm{op}}\to\mathrm{Gpd}$ from a category of domains to groupoids. It is a representation stack for a Grothendieck topology when locally defined, pairwise-compatible entries glue to a global entry uniquely up to coherent isomorphism. The point of insisting on a stack rather than a mere presheaf of sets is that gluing is by equivalence, not equality, so automorphism data (gauge redundancy, stabilizers, isotropy) is retained. Every regime of the library is, in its own way, an instance of the slogan $$\begin{equation} \text{physical content} \;=\; \text{groupoid/stack},\qquad \mathrm{Aut}_{[X/G]}(x)\;\simeq\;\mathrm{Stab}_G(x). \end{equation}$$ The present module is the instance in which $X$ is a space of fields, $G$ is a gauge group (or, infinitesimally, a gauge Lie algebroid), and the homotopical quotient $[X/G]$ intersected with the shell $\{dS=0\}$ is the derived critical locus.

2.2 The status calculus

2 is the formal reason the framework is modular rather than unified. Chaining regimes never improves the epistemic warrant of the composite beyond the weakest link. We will invoke this repeatedly, and it is checked in the companion code by an exhaustive test over the (finite) status set.

3 Graded geometry and the antibracket

3.1 Graded manifolds of fields

Fix a field $\Bbbk$ of characteristic zero (we take $\Bbbk=\mathbb{R}$ or $\Bbbk=\mathbb{Z}$ depending on whether we track signs numerically). Physical BV data live on a $\mathbb{Z}$-graded manifold; for a self-contained account we present the local, finite-dimensional algebraic model and treat the field-theoretic case as its formal or local-functional completion, following .

We assign to generators several integer gradings that will be used in bookkeeping. The ghost number $\mathrm{gh}$ is the total $\mathbb{Z}$-grading. It is refined into the pure ghost number $\mathrm{pgh}\ge 0$, carried by ghosts, and the antifield number $\mathrm{afn}\ge 0$, carried by antifields, with $$\begin{equation} \mathrm{gh}\;=\; \mathrm{pgh}- \mathrm{afn}. \end{equation}$$ The Grassmann parity is $\mathrm{gh}\bmod 2$ for the standard field content; we keep parity and ghost number distinct because for reducible theories they can differ in intermediate constructions.

3.2 The odd symplectic form and the antibracket

The defining feature of BV geometry is an odd Poisson bracket of degree $+1$, the antibracket. Concretely, the field content is doubled: to every generator $\phi^A$ (fields, ghosts, ghosts-for-ghosts) one adjoins an antifield $\phi^{*}_A$ of complementary degree, $$\begin{equation} \mathrm{gh}(\phi^{*}_A) \;=\; -\mathrm{gh}(\phi^A) - 1,\qquad \varepsilon(\phi^{*}_A) \;=\; \varepsilon(\phi^A)+1 \pmod 2, \end{equation}$$ where $\varepsilon$ is Grassmann parity. The antibracket of two functionals $F,G$ on the doubled space is $$\begin{equation} (F,G) \;=\; \frac{\partial^{R} F}{\partial \phi^A}\, \frac{\partial^{L} G}{\partial \phi^{*}_A} \;-\; \frac{\partial^{R} F}{\partial \phi^{*}_A}\, \frac{\partial^{L} G}{\partial \phi^A}, \end{equation}$$ with a sum over $A$, where $\partial^{R}$ and $\partial^{L}$ denote right and left derivatives. The bracket carries ghost number $+1$ and reverses Grassmann parity.

3.3 The BV differential

7 is the linchpin: the master equation is precisely the integrability condition making $Q_{\mathrm{BV}}$ a differential, so that cohomology is defined. The companion Haskell program verifies both the master equation and $Q_{\mathrm{BV}}^2=0$ for the finite model of 8 by exhaustive evaluation, giving an independent machine check of 7 in that case.

4 The BRST complex: ghosts and gauge invariance

We now specialize to the geometry of a gauge symmetry and recover the classical BRST complex as one half of the BV complex. Throughout, the gauge symmetry is generated by a Lie algebra (or Lie algebroid) $\mathfrak{g}$ acting on a space $X$ with coordinate ring $\mathcal{O}(X)$.

4.1 The Chevalley–Eilenberg model of invariance

10 is the cohomological reconstruction of the naive quotient: the invariant functions $\mathcal{O}(X/G)$ are recovered as $H^0$, but the full cohomology retains the group-theoretic data $H^\bullet(\mathfrak{g})$ that the coarse quotient forgets. This is [eq:stackslogan] at the level of functions.

4.2 Field-antifield doubling and the classical BRST charge

To couple the BRST differential to the dynamics one adds antifields and requires the master equation. Write $\phi^i$ for the original fields with antifields $\phi^{*}_i$, and $c^\alpha$ for ghosts with antifields $c^{*}_\alpha$. The minimal BV action for an irreducible gauge theory with action $S_0(\phi)$ invariant under $\delta_\epsilon \phi^i = R^i{}_\alpha(\phi)\epsilon^\alpha$ is $$\begin{equation} S \;=\; S_0(\phi) \;+\; \phi^{*}_i R^i{}_\alpha(\phi) c^\alpha \;-\; \tfrac12 c^{*}_\gamma f^{\gamma}{}_{\alpha\beta}(\phi) c^\alpha c^\beta \;+\; \dots, \end{equation}$$ where the dots denote terms required when the gauge algebra closes only on-shell. The master equation [eq:cme] for [eq:minimalbv] unpacks, order by order in antifield number, into the statement that $S_0$ is gauge invariant, that the $R^i{}_\alpha$ generate a (possibly open) algebra with structure functions $f^\gamma{}_{\alpha\beta}$, and the higher Jacobi-type constraints on the structure functions .

5 The Koszul–Tate resolution and the classical master equation

The BRST differential imposes gauge invariance. Its partner, the Koszul–Tate differential, imposes the equations of motion homologically, resolving the critical locus.

5.1 Assembly: the full BV complex as a bicomplex

The BRST differential $s$ (pure ghost number $+1$, antifield number $0$) and the Koszul–Tate differential $\delta$ (pure ghost number $0$, antifield number $-1$) assemble, together with correction terms, into the single BV differential $Q_{\mathrm{BV}}=(S,-)$ of total ghost number $+1$. Expanding in antifield number, $$\begin{equation} Q_{\mathrm{BV}} \;=\; \delta \;+\; s \;+\; (\text{higher-order in antifields}). \end{equation}$$ The bigrading is displayed in 1. The horizontal direction is pure ghost number (increased by $s$); the vertical is antifield number (decreased by $\delta$). $Q_{\mathrm{BV}}$ moves one step up-and-right in total ghost number.

6 The derived critical locus and shifted symplectic structure

13 is the classical statement; its clean modern home is derived algebraic geometry. We record the dictionary.

6.1 Derived critical locus

Let $X$ be a smooth affine scheme (or the formal/local model of a space of fields) and $S\in\mathcal{O}(X)$ a function. The ordinary critical locus is $\mathrm{Crit}(S)=\{dS=0\}$, the intersection of the graph $\Gamma_{dS}\subset T^{*}X$ with the zero section $X\subset T^{*}X$. When this intersection is not transverse, which is exactly the gauge-theoretic situation, the honest scheme $\mathrm{Crit}(S)$ loses information. The derived intersection retains it.

The square $$\begin{equation} \text{[commutative diagram --- see PDF]} \end{equation}$$ is a homotopy pullback. Taking $\pi_0$ (classical truncation) recovers $\mathrm{Crit}(S)$, while the higher homotopy encodes the failure of transversality, which is the gauge/ghost data.

6.2 Why the derived object is the right invariant

The derived critical locus is the honest model of “the space of solutions modulo gauge, with stabilizers retained.” Its tangent complex at a solution $\phi_0$ is a three-term complex $$\begin{equation} \mathbb{T}_{\mathrm{dCrit}(S),\phi_0}\colon\quad \mathfrak{g}\xrightarrow{\ R\ } T_{\phi_0}X \xrightarrow{\ \mathrm{Hess}(S)\ } T^{*}_{\phi_0}X \xrightarrow{\ R^{\dagger}\ } \mathfrak{g}^{*}, \end{equation}$$ in degrees $-1,0,1,2$: the degree $-1$ part is the gauge directions (ghosts), the degree $0$ the field fluctuations, the degree $1$ the equations-of-motion constraints (antifields), the degree $2$ the Noether identities (antighosts). The cohomology of [eq:tangentcomplex] at a symmetric background is where the reconstruction story becomes concrete: a nonzero $H^{-1}$ is exactly a stabilizer (a Killing vector, for gravity), and the framework refuses to discard it. We make this precise in 9.

7 Main results: observables as cohomology

We collect the module’s central statements. 13 already gave the classical observable algebra; we now state the observable dictionary, the gauge-fixing independence, and the reconstruction proposition.

19 is the technical statement of “the redundancy is not physical.” The observable is the cohomology class; the gauge-fixed representative is a choice with no invariant meaning. This is the module’s version of the library-wide slogan that a physical datum is an equivalence class, not a representative.

8 Worked examples

We compute three examples explicitly. The first is small enough that every cohomology group is written out and checked by the companion Haskell code; it is the model whose nilpotency the code verifies.

8.1 A finite abelian gauge system

Let $X=\mathbb{R}^2$ with coordinates $(q^1,q^2)$, and let $\mathfrak{g}=\mathbb{R}$ act by translation in $q^2$: $\rho(T)=\partial/\partial q^2$. This is the toy of a “pure gauge direction.” Introduce one ghost $c$ (odd, $\mathrm{pgh}=1$), its antifield $c^{*}$ ($\mathrm{afn}=1$), and antifields $q^{*}_1,q^{*}_2$ ($\mathrm{afn}=1$).

8.1.0.1 BRST part.

The action of the invariant functions is $\mathcal{O}(X)^{\mathfrak{g}}=\mathbb{R}[q^1]$, functions of $q^1$ alone. The BRST differential is $s\, a = c\,\partial_{q^2}a$, and $s\,c=0$ (abelian, no structure constants). By 10, $$\begin{equation} H^0(s) = \mathbb{R}[q^1],\qquad H^k(s)=0 \ (k\ge 1), \end{equation}$$ since $\mathbb{R}[q^1,q^2]\xrightarrow{\partial_{q^2}}\mathbb{R}[q^1,q^2]$ has kernel $\mathbb{R}[q^1]$ and is surjective (every polynomial has a $q^2$-antiderivative), so the ghost number $1$ cohomology vanishes.

8.1.0.2 Dynamics.

Take $S_0(q)=\tfrac12 (q^1)^2$, gauge invariant (it does not depend on $q^2$). Equations of motion: $\partial_{q^1}S_0=q^1=0$, $\partial_{q^2}S_0=0$. The nontrivial Koszul–Tate action is $\delta q^{*}_1 = -q^1$, $\delta q^{*}_2=0$. The critical locus is $\{q^1=0\}\cong\mathbb{R}$ (the $q^2$-axis), and $$\begin{equation} \mathcal{O}(\mathrm{Crit}S_0) = \mathbb{R}[q^1,q^2]/(q^1) = \mathbb{R}[q^2]. \end{equation}$$ Since $\delta q^{*}_2=0$ identically, $q^{*}_2$ is a $\delta$-cocycle that is not a boundary: the equation of motion $\partial_{q^2}S_0\equiv 0$ is a Noether identity (it is the statement that $S_0$ is $q^2$-independent). To restore acyclicity we add a second-generation antifield $\eta$ with $\delta\eta = q^{*}_2$; this is the ghost-for-ghost of the trivial reducibility $1\cdot\partial_{q^2}S_0=0$. With $\eta$ adjoined, $H_{k>0}(\delta)=0$ and $H_0(\delta)=\mathbb{R}[q^2]$.

8.1.0.3 Full BV cohomology.

Combining, the degree-zero BV cohomology is gauge-invariant on-shell functions: $$\begin{equation} H^0(Q_{\mathrm{BV}}) = \big(\mathbb{R}[q^2]\big)^{\mathfrak{g}} = \big(\mathbb{R}[q^2]\big)^{\partial_{q^2}} = \mathbb{R}, \end{equation}$$ the constants: on the critical locus $\{q^1=0\}$ the only surviving coordinate is $q^2$, and it is pure gauge, so the physical observable algebra is one dimensional. This is the expected answer: a single free “radius” coordinate $q^1$ that is fixed to $0$ on shell, and a single gauge coordinate $q^2$ that carries no invariant. The example exhibits, in miniature, both a Koszul–Tate resolution with a Noether identity and a nontrivial gauge quotient.

8.2 A nonabelian Lie-algebra model

Let $\mathfrak{g}=\mathfrak{su}(2)$ with structure constants $f^\gamma{}_{\alpha\beta}=\varepsilon_{\alpha\beta\gamma}$, acting on $X=\mathbb{R}^3$ by the vector representation, $\rho(T_\alpha)= \varepsilon_{\alpha\beta\gamma}q^\beta\partial_{q^\gamma}$ (rotations). The invariant ring is $$\begin{equation} \mathcal{O}(X)^{\mathfrak{g}} = \mathbb{R}[r^2],\qquad r^2=(q^1)^2+(q^2)^2+(q^3)^2, \end{equation}$$ the radial functions. By 10, $H^0(s)=\mathbb{R}[r^2]$. The higher cohomology is $H^\bullet(\mathfrak{su}(2))\otimes\mathbb{R}[r^2]$ away from the origin; $H^\bullet(\mathfrak{su}(2))=\Lambda(x_3)$ is an exterior algebra on a single degree-$3$ generator (the Cartan $3$-form), reflecting $H^\bullet(\mathrm{SU}(2))=H^\bullet(S^3)$. Thus $$\begin{equation} H^0(s)=\mathbb{R}[r^2],\quad H^3(s)\supset \mathbb{R}[r^2]\cdot x_3,\quad H^1(s)=H^2(s)=0 . \end{equation}$$ The companion code builds this $s$ for $\mathfrak{su}(2)$ symbolically and checks $s^2=0$ on a basis of $\mathcal{O}(X)\otimes\Lambda^{\le 3}\mathfrak{g}^{*}$, giving a machine confirmation of 9 for this case.

8.2.0.1 Adding a potential.

With $S_0=\tfrac14(r^2-1)^2$ (invariant), one has $\partial_{q^i}S_0=(r^2-1)q^i$, so the critical locus is the unit sphere $\{r^2=1\}$ together with the origin $\{q=0\}$. On this locus the invariant coordinate $r^2$ satisfies $r^2(r^2-1)=0$, hence $$\begin{equation} H^0(Q_{\mathrm{BV}}) \;=\; \mathbb{R}[r^2]\big/\big(r^2(r^2-1)\big) \;\cong\; \mathbb{R}\oplus\mathbb{R}, \end{equation}$$ the locally constant functions on the two invariant critical values $r^2\in\{0,1\}$. The physical observable algebra is two-dimensional, exactly as the reconstruction picture predicts.

8.3 Comparison with the derived critical locus

For the nonabelian model, $\mathrm{dCrit}(S_0)$ on the quotient stack $[\mathbb{R}^3/\mathrm{SU}(2)]$ has $\pi_0 = \{r^2=1\}/\mathrm{SU}(2)\sqcup\{0\}$, a point plus the origin, and its tangent complex at a generic point of the sphere is $\mathfrak{su}(2)\to \mathbb{R}^3 \to \mathbb{R}^3 \to \mathfrak{su}(2)^{*}$ with $H^{-1}=0$ (the action is locally free away from the origin) but $H^{-1}\ne 0$ at the origin (full stabilizer $\mathfrak{su}(2)$). The jump in $H^{-1}$ at the origin is the isotropy jump that a coarse quotient would erase; the derived object records it. This is the finite model of the gravitational statement of 9.

9 The reducible gravitational case

We now state the module’s substantive physical result. Consider perturbative gravity around a fixed background metric $\bar g$ on a globally hyperbolic $M$. The gauge symmetry is infinitesimal diffeomorphism, $\delta_\xi g_{\mu\nu}= \mathcal{L}_\xi g_{\mu\nu}=\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu$, generated by vector fields $\xi\in\mathfrak{X}(M)$, which play the role of the gauge Lie algebra $\mathfrak{g}$. The ghost is a fermionic vector field $c^\mu$; the BV action is the Einstein–Hilbert action extended by the antifield couplings $g^{*}_{\mu\nu}\mathcal{L}_c g^{\mu\nu}$ and the ghost self-coupling $c^{*}_\mu c^\nu\partial_\nu c^\mu$ .

9.1 Reducibility from Killing vectors

At a background with $\mathfrak{k}\ne 0$, the gauge generator $R\colon\xi\mapsto \mathcal{L}_\xi\bar g$ has a kernel: Killing vectors generate no field variation. This is a reducibility of the gauge symmetry, and it is exactly the situation 11 and 12 handle by introducing ghosts-for-ghosts.

9.2 Interpretation

24 is the sense in which “isotropy is retained, not erased” propagates from the classical moduli stack into the perturbative observable algebra. A coarse quotient of metrics by diffeomorphisms would represent Schwarzschild by a single point and forget its four-dimensional isometry group. The BV-BRST complex refuses: the four Killing vectors appear as four ghosts-for-ghosts, and they control the degeneracy structure of the linearized theory (zero modes of the kinetic operator, would-be gauge modes that are not gauge). This is the concrete mechanism behind the library’s recurring pattern that physical content is a stack with retained automorphisms, not a bare quotient set.

10 Quantization: BV Laplacian, quantum master equation, obstruction

The classical results above are [ S ]. Quantization is where the honest status downgrades to [ S/H ], and we are careful to say why. Throughout this section we extend the scalars of 3 from $\Bbbk=\mathbb{R}$ to $\Bbbk=\mathbb{C}$ (equivalently $\mathbb{R}[i]$): the path-integral weight $e^{iS_\hbar/\hbar}$ and the $i\hbar\,\Delta$ term in the quantum master equation below require the imaginary unit, so the graded-commutative algebra $\mathcal{O}(\mathcal{F})$ is complexified before quantization.

This is the precise, citable sense in which “renormalizable and anomaly-free” becomes “a cohomology class vanishes.” It is also the precise place the module’s status downgrades: the classical observable algebra $H^0(Q_{\mathrm{BV}})$ is [ S ], but its survival as the observable algebra of the interacting quantum theory is contingent on $H^1(Q_{\mathrm{BV}})$-valued obstructions vanishing, which is a perturbative, renormalization-scheme-dependent statement, hence [ S/H ].

11 Composition with the other eleven regimes

The library is modular. Its regimes compose by the weakest-link status calculus of 2, and this section makes the compositions explicit for the gauge layer. The dictionary status of the BV-BRST/derived-critical-locus entry is [ S/H ] (assumptions: perturbative or derived local model; limitations: nonperturbative global completion is model dependent).

11.1 Within QG-II: factorization algebras

The natural continuation is the factorization-algebra regime, which globalizes the single-background BV complex into a local-to-global assignment of observable complexes $U\mapsto \mathrm{Obs}(U)$ over spacetime, with $H^0(\mathrm{Obs}(U))$ recovering the gauge-invariant observables on the region $U$ . Both entries are [ S/H ]; their composite for perturbative, local gauge theory remains [ S/H ]. Composed with any claim of a fully background-independent factorization algebra of quantum gravity, the composite degrades to [ H ], because no such construction is established.

11.2 Down to QG-I: the classical moduli stack

The classical regime represents $\mathrm{Sol}(\mathrm{Ein})/\!/\mathrm{Diff}$ as a stack with isotropy $\mathrm{Isom}(M,\bar g)$. 24 is the BV-level image of this. Both are [ S/H ]; the bridge claim “the derived critical locus is the perturbative model of the classical moduli stack” is itself [ S/H ] locally (17) but only [ H ] as a global 4D statement, since a global derived-stack construction of $\mathrm{Sol}(\mathrm{Ein})/\!/\mathrm{Diff}$ in four dimensions is not settled (12).

11.3 Across to QG-III and QG-VII: the perturbative wall

Composing the gauge layer with the spin-foam regime (status [ H ], genuinely nonperturbative) or the asymptotic-safety regime ([ H ]) yields, by 2, $$\begin{equation} \mathrm{compose}(\mathsf{S/H},\mathsf{H}) = \mathsf{H}. \end{equation}$$ The weak link is BV-BRST’s own limitation clause, “nonperturbative global completion is model dependent.” This is not pessimism about the mathematics; it is the statement that a chain reaching a nonperturbative regime cannot advertise better than heuristic status even if the BV part is rigorous.

11.4 The sharpest case: QG-X, indefinite causal order

The proposed causal-order regime carries status [ H/P ]: the process-matrix mathematics is well-defined ([ H ]) but the claim that quantum gravity dynamically realizes indefinite causal order is [ P ]. Composing, $$\begin{equation} \mathrm{compose}(\mathsf{S/H},\mathsf{H/P}) = \mathsf{P}. \end{equation}$$ A paper attempting “BV quantization of gravity on an indefinite causal background” must present its conclusion as speculative regardless of how careful the BV construction is, because the worst warrant in the chain is $\mathsf{P}$. This is the monotonicity of 2 doing exactly the work it is meant to do: bounding claims from above. The composition is verified in the companion code by exhaustive evaluation over the status set.

11.5 Summary table

12 Limitations and open problems

We are explicit about what this module does not establish.

1. Perturbative only. The entire construction is perturbative or, at best, formal/local in the derived-geometric sense. The Costello–Gwilliam quantization theorem lives in perturbative renormalization; there is no nonperturbative, background-independent BV complex for quantum gravity. Every claim that reaches beyond the perturbative regime is tagged [ H ] or worse.

2. Kerr ghost-for-ghost not carried out. 25 counts ghosts-for-ghosts by isometry dimension, which is rigorous, but a fully explicit assembly of the perturbative gravitational BV complex around Kerr (rotating, lower symmetry, where the reducibility interacts with the ergoregion and horizon structure) is, to our knowledge, not written down in one place. The closest available starting points are the BV-BFV data for the Einstein–Hilbert action of Cattaneo and Schiavina and the boundary/corner axioms of Cattaneo, Mnev and Reshetikhin , neither of which specializes the reducibility structure to a stationary axisymmetric background with a horizon. We flag the Kerr assembly as the concrete original computation this module invites. [ H ]

3. Global 4D derived moduli stack. A global (not merely local/formal) construction of $\mathrm{Sol}(\mathrm{Ein})/\!/\mathrm{Diff}$ as a derived stack in four dimensions, accounting for the non-compact diffeomorphism group, isotropy jumps, and matter coupling, is not settled. The 2019 “Stacks in Einstein Gravity” construction is $3$D and Chern–Simons-equivalent; a $4$D follow-up matching the generality this library wants is not established. [ H/P ]

4. Cut/regulator complementarity. Whether the BV-BRST cohomology differential and the coaction/cobracket decomposition of the amplitude regime are literally adjoint functors in a suitable category is open (roadmap item 2); we state it as a structural analogy, not a theorem. [ P ]

5. Formalization gap. The companion Haskell code checks nilpotency for finite models by exhaustive evaluation. A machine-checked proof of $Q_{\mathrm{BV}}^2=0$ for the general BV complex (not a fixed finite model), in a proof assistant, remains future work (roadmap item 7). The finite check is [ S ]; the general formal proof is not yet done.

13 Conclusion

The gauge layer of the reconstruction program has a clean invariant: observables are the degree-zero cohomology of the BV differential $Q_{\mathrm{BV}}=(S,-)$ on the derived critical locus $\mathrm{dCrit}(S)$. The derived critical locus is the homotopical model of the on-shell gauge quotient; it carries the PTVV $(-1)$-shifted symplectic structure whose bracket is the BV antibracket; its degree-zero cohomology is the gauge-invariant on-shell functionals; and its tangent complex retains, in negative degree, exactly the stabilizer data (Killing vectors, for gravity) that a coarse quotient would erase. The reconstruction reading is not decorative: gauge redundancy is not noise to be removed before quantization but invariant representation structure whose cohomology reconstructs what can be measured.

We have been careful to tag each step. The classical cohomological statements are standard mathematics ([ S ]). Their survival into the interacting quantum theory is contingent on a vanishing obstruction in $H^1(Q_{\mathrm{BV}})$ and is therefore heuristic and perturbative ([ S/H ]). Any extension to a nonperturbative, background-independent, or indefinite-causal-order regime is speculative and is bounded, not asserted, by the weakest-link status calculus. The two computations we offer as our own contribution, the fully explicit finite BV model with its machine-checked nilpotency, and the ghost-for-ghost tower dual to the isometry algebra on a symmetric gravitational background, are both firmly in the [ S ]/[ S/H ] range, and they are exactly the pieces that make the slogan “physical content is a stack, not a quotient” concrete at the level of perturbative quantum gravity. What remains open, the Kerr computation, the global 4D derived moduli stack, the adjunction with the amplitude regime, and the proof-assistant formalization, is stated plainly as open.

Acknowledgements

This work is part of the YonedaAI modular quantum-gravity representation library. It reuses the shared status calculus and dictionary of that project and cites external literature verified against arXiv and journal metadata.