Observables from Gluing: Factorization Algebras, Cosheaves, and the Local-to-Global Reconstruction of Quantum Observables

1 Introduction

1.1 The reconstruction thesis for observables

This paper is one entry in a modular program whose governing perspective is that quantum gravity is less the quantization of matter placed inside a fixed spacetime than the reconstruction of spacetime, matter, causality, and measurement from invariant representation structures. Each regime in the program isolates one such invariant. The regime treated here isolates the invariant that governs measurement: not a single global algebra of observables posited at the outset, but a rule that assigns to every region of spacetime the observables one can measure inside it, together with the compositional law by which measurements in small regions combine into measurements in large regions.

The mathematical content of that law is captured by two closely related structures. The first is the factorization algebra of Costello and Gwilliam , which assigns to each open set $U$ of a manifold a cochain complex $\mathscr{F}(U)$ of observables and, to each finite family of pairwise disjoint opens $U_{1},\dots,U_{n}$ inside a larger open $V$, a structure map $$\begin{equation} \mathscr{F}(U_{1})\otimes\cdots\otimes\mathscr{F}(U_{n})\;\longrightarrow\;\mathscr{F}(V), \end{equation}$$ subject to associativity, symmetry, and a local-to-global gluing axiom formulated over a distinguished class of covers. The second is the net of algebras of algebraic quantum field theory (AQFT) in the Haag–Kastler tradition  and its locally covariant refinement due to Brunetti, Fredenhagen, and Verch , which assigns to each spacetime region a $C^{*}$- or $*$-algebra $\mathcal{A}(U)$ subject to isotony, Einstein causality, and (in the covariant version) functoriality across all globally hyperbolic spacetimes at once.

The structural point that organizes the paper is the contrast in [eq:strmap-intro] with the way a sheaf behaves. A sheaf reconstructs a global section from local sections by a limit: sections that agree on overlaps glue to a unique global section, so the global datum is a constraint satisfied by local restrictions. A factorization algebra reconstructs its global value by a colimit: the value on a large region is built up out of, and generated by, the values on small regions, glued by the composition maps [eq:strmap-intro]. Observables compose; they do not merely restrict. This is the precise sense in which the observable content of a field theory is reconstructed local-to-global.

1.2 What is proved here, and what is cited

The literature on factorization algebras is substantial and technical, resting on homotopical algebra, functional analysis, and the theory of $\infty$-categories. Rather than reproduce that machinery, this paper does three things.

First, we assemble a careful, notation-complete account of the definitions (2): the colored operad of disjoint opens, prefactorization algebras, Weiss covers, the factorization/cosheaf condition, the descent contrast between colimits and limits, the relation to $\mathrm{E}_n$-algebras, and the AQFT-net axioms. This section states the deep theorems (the $\mathrm{E}_n$ recognition principle, the free-field construction, the AQFT comparison theorems) with proof sketches and explicit citations, and it is honest about which parts are being cited rather than proved.

Second, we prove in full a finite, self-contained reconstruction theorem (20 in 3). A locally constant, unital prefactorization algebra on the real line determines an associative product, and the associativity of that product is a consequence of the associativity of the structure maps for three disjoint intervals. We formulate this for a finite combinatorial model of intervals in which the statement becomes an exhaustive finite computation, so that the proof is complete and machine-checkable. This is the paper’s flagship illustration of the reconstruction thesis: an associative algebra — the prototype of a one-dimensional observable algebra — is reconstructed from purely local gluing data.

Third, we exhibit accompanying Haskell code (4.4) implementing the finite model. The code builds the poset of intervals, the structure maps, and the recovered product, and it verifies associativity, unitality, and the weakest-link status calculus by exhaustive evaluation over a finite basis. A passing check is therefore a proof for the finite model, not a random sample.

1.3 Epistemic labels

Throughout, claims are tagged with one of three warrants drawn from the project’s status calculus (3.1): [ S ] for a standard mathematical or mathematical-physics statement (a proved theorem or an established correspondence), [ H ] for a strong heuristic physical reading, and [ P ] for a speculative ontological extension. Composite labels such as [ S/H ] record that a claim mixes warrants. The labelling is not decorative. It is governed by a monotone composition rule (worst warrant wins), so that chaining regimes together can never raise the epistemic status of a conclusion above that of its weakest input. The reconstruction theorem, the $\mathrm{E}_n$ recognition principle, the free-field construction, and the factorization-algebra/AQFT comparison theorems are all [ S ]. The reading of renormalizability and anomaly-freedom as the vanishing of a cohomological quantization obstruction is [ H ]. The existence of a genuinely background-independent, nonperturbative quantum-gravity factorization algebra is [ P ] and open.

1.4 Organization

2 develops the framework. 3 states and proves the main results. 4 works the one-dimensional example in detail, including the free particle and a locality/gluing consistency check, and describes the Haskell model. 5 places the regime among its neighbours in the program and runs the weakest-link calculus explicitly. 6 records limitations and open problems. 7 concludes.

2 Mathematical framework

Fix a ground field $\Bbbk$ of characteristic zero; in the physical applications $\Bbbk=\mathbb{C}$. We write $\mathrm{Vect}$ for the category of $\Bbbk$-vector spaces and $\mathrm{Ch}$ for the category of cochain complexes of $\Bbbk$-vector spaces, each a symmetric monoidal category under the tensor product over $\Bbbk$. Let $M$ be a smooth manifold and $\mathrm{Op}(M)$ its poset of open subsets, ordered by inclusion.

2.1 Disjoint opens and the factorization operad

The combinatorics of [eq:strmap-intro] is that of a colored operad. Recall that a (symmetric, colored) operad $\mathcal{O}$ consists of a set of colors, sets of operations $\mathcal{O}(c_{1},\dots,c_{n};c)$ with inputs colored $c_{1},\dots,c_{n}$ and output colored $c$, an action of the symmetric group $\Sigma_{n}$ permuting inputs, identity operations, and an associative, equivariant composition.

An operation exists exactly when the inputs sit disjointly inside the output. This encodes the geometric fact that observables supported in disjoint regions can be multiplied together and the product lands in any region containing all of them.

2.2 Prefactorization algebras

For $\mathcal{C}=\mathrm{Vect}$ or $\mathrm{Ch}$ this is the setting of Costello and Gwilliam . The single-open maps $m^{V}_{U}$ make $\mathscr{F}$ a covariant functor $\mathrm{Op}(M)\to\mathcal{C}$; the multi-input maps make it much more.

2.3 Weiss covers and the factorization condition

An ordinary open cover is too coarse to detect the compositional structure of observables, because it does not see how finitely many points are jointly distributed. The right notion is due to Weiss.

A Weiss cover is in particular an open cover (take $|S|=1$), but it is far finer: it must contain, for every $k$, opens large enough to hold any $k$ chosen points. The collection of all opens diffeomorphic to a disjoint union of finitely many balls is the prototypical Weiss cover of a manifold. Weiss covers are the covering families of a Grothendieck topology on $\mathrm{Op}(M)$, the Weiss topology.

Given a prefactorization algebra $\mathscr{F}$ and a Weiss cover $\mathfrak{U}$, the structure maps assemble into a simplicial (Čech-type) object whose $p$-th term records observables on $p$-fold “intersections” in the factorization sense, $$\begin{equation} \check{C}(\mathfrak{U},\mathscr{F})_{p} =\bigoplus_{i_{0},\dots,i_{p}}\mathscr{F}\!\left(U_{i_{0}}\cap\cdots\cap U_{i_{p}}\right), \end{equation}$$ with the structure and restriction maps as faces. Its (normalized, totalized) realization is the factorization Čech complex, and there is a canonical map $\check{C}(\mathfrak{U},\mathscr{F})\to\mathscr{F}(U)$ induced by the structure maps.

The equivalence [eq:fa-glue] is the local-to-global axiom. It says the global observables on $U$ are generated, up to coherent homotopy, by observables supported in the members of any factorizing cover, with the generation carried out by the composition maps [eq:strmap-intro]. In the physical reading, a measurement on $U$ is a homotopy colimit — a coherent gluing — of measurements localized in smaller regions.

2.4 Descent: colimits against limits

It is worth isolating the descent statement in elementary terms, because it is the formal shadow of the physics. Let $U=U_{1}\cup U_{2}$ with $U_{1},U_{2}$ open and suppose $\{U_{1},U_{2}\}$ together with $U_{1}\cap U_{2}$ generate a factorizing cover (in dimension one this holds for two overlapping intervals). For a factorization algebra valued in $\mathrm{Ch}$, [eq:fa-glue] specializes to an exact triangle $$\begin{equation} \mathscr{F}(U_{1}\cap U_{2})\longrightarrow \mathscr{F}(U_{1})\oplus\mathscr{F}(U_{2}) \longrightarrow \mathscr{F}(U)\xrightarrow{\ [1]\ }, \end{equation}$$ the factorization analogue of a Mayer–Vietoris sequence, with the arrows built from the structure maps rather than from restrictions. In cohomology this expresses $H^{\bullet}\mathscr{F}(U)$ as a colimit of the local pieces. For a sheaf one would instead obtain the dual triangle with a product in the middle and the map reversed; the direction of the arrows is exactly what distinguishes the two regimes.

2.5 Relation to $\mathrm{E}_n$-algebras

On Euclidean space the factorization condition has a purely algebraic shadow. Call a factorization algebra $\mathscr{F}$ on $\mathbb{R}^{n}$ locally constant if for every inclusion of opens $D\subseteq D'$ each diffeomorphic to a ball, the structure map $\mathscr{F}(D)\to\mathscr{F}(D')$ is an equivalence.

7 is the conceptual template for the finite reconstruction theorem of 3: it says that in one dimension the entire factorization structure is equivalent to an associative algebra, so recovering an associative product from one-dimensional gluing data is not an accident but the shadow of an equivalence of categories.

2.6 AQFT nets and locally covariant field theory

The AQFT tradition packages the same physics with a different emphasis: unital $C^{*}$- or $*$-algebras, Lorentzian signature, and causality built in as a primitive axiom.

The move from 8 to 9 is the AQFT counterpart of the reconstruction thesis: the theory is defined not on subregions of one fixed spacetime but functorially on the category of all admissible spacetimes, so that which spacetime one works on becomes part of the data glued over rather than a fixed background. This is the sense in which locally covariant AQFT builds in a form of background independence at the level of the site.

2.7 Observables of a free theory

The running example that makes both formalisms computable is a free (Gaussian) theory. We recall the construction in the factorization language  in the form we need.

Let $E$ be the sheaf of fields (sections of a graded vector bundle on $M$), with a free BV action whose linearized equations of motion are governed by a formally self-adjoint Green-hyperbolic operator $P$. Write $E_{c}(U)$ for compactly supported fields on $U$. The classical observables are $$\begin{equation} \mathrm{Obs}^{\mathrm{cl}}(U)=\left(\mathrm{Sym}\big(E_{c}(U)^{\vee}[-1]\big),\, d\right), \end{equation}$$ the (completed) symmetric algebra on the shifted dual of compactly supported fields, with $d$ the Chevalley–Eilenberg/BV differential encoding the equations of motion. The quantum observables $\mathrm{Obs}^{\mathrm{q}}(U)$ are a deformation of [eq:classobs] over $\Bbbk[[\hbar]]$ whose differential is $d+\hbar\,\Delta$ for the BV Laplacian $\Delta$ built from the propagator of $P$; the structure maps implement the (normal-ordered) operator product. The following is the free-theory output we cite.

2.8 Factorization homology and global invariants

There is a global counterpart to the local assignment $\mathscr{F}$: an operation that integrates a factorization algebra over a manifold to produce a single invariant, in the way ordinary homology integrates a local coefficient system. For a locally constant factorization algebra on $\mathbb{R}^{n}$, equivalently an $\mathrm{E}_n$-algebra $A$ by 7, and for a framed $n$-manifold $M$, the factorization homology $$\begin{equation} \int_{M} A \;=\; \operatorname*{colim}_{\,(\mathrm{Disk}_{n}\downarrow M)} A \end{equation}$$ is the homotopy colimit of $A$ over the category of framed embeddings of finite disjoint unions of disks into $M$ . It is the value one would assign to “all of $M$” by gluing the local $\mathrm{E}_n$-structure, and it satisfies an Eilenberg–Steenrod-type characterization: it takes disjoint unions to tensor products and satisfies an excision (collar-gluing) axiom $$\begin{equation} \int_{M} A \;\simeq\; \int_{M_{0}} A \;\otimes_{\int_{\mathbb{R}\times P} A}\; \int_{M_{1}} A \end{equation}$$ whenever $M=M_{0}\cup_{\mathbb{R}\times P}M_{1}$ is a decomposition along a collar $\mathbb{R}\times P$. Excision [eq:excision] is the global face of the same local-to-global principle carried locally by [eq:fa-glue]: a global invariant is assembled from the invariants of the pieces, glued over the collar, by a (relative) tensor product rather than a fibre product.

Two special cases anchor the picture. When $A$ is the (suitably interpreted) free commutative algebra, $\int_{M}A$ reduces to a form of the ordinary homology of $M$; this is the sense in which factorization homology generalizes singular homology. When $M=S^{1}$ and $A$ is an associative ($\mathrm{E}_1$) algebra, factorization homology computes the Hochschild homology, $$\begin{equation} \int_{S^{1}} A \;\simeq\; \mathrm{HH}_{\bullet}(A), \end{equation}$$ the derived trace of $A$ . The circle is the worldline of a particle wrapped in time, and [eq:HH] says the corresponding global observable is the trace of the algebra recovered on the line — the partition function of the one-dimensional theory. This is the point of contact with the cobordism/TQFT reading of the program: factorization homology over a closed manifold plays the role of a state-sum partition function, computed by gluing local algebraic data.

We record the one-dimensional consequence for later use, since it connects directly to the reconstruction theorem of the next section.

3 Main results

We now isolate the reconstruction phenomenon in a form that is fully proved rather than cited. The strategy is to pass to a finite combinatorial model of the line, in which local constancy becomes an exact identification and the homotopy colimit of 5 becomes an ordinary colimit computable by hand.

3.1 The S/H/P status calculus

We first record the composition rule that governs the epistemic labels, since one of our results is about it. Let $(\{S,H,P\},\le)$ be the totally ordered set with $S<H<P$, read as increasing speculativeness.

14 is elementary but load-bearing: it is the reason the program is modular rather than unified. Chaining the present regime to another can never launder a speculative input into a standard conclusion. We use it explicitly in 5, and it is one of the properties checked by the accompanying code.

3.2 A finite model of the line

Fix an integer $N\ge 1$ and let $L=\{1,2,\dots,N\}$ be a finite ordered set of sites, a discretization of an interval of the real line. By an interval we mean a nonempty set of the form $[a,b]=\{a,a+1,\dots,b\}$ with $1\le a\le b\le N$; the empty set is also admitted as the empty interval $\varnothing$. Two intervals are disjoint if they share no site, and we say $[a,b]$ lies to the left of $[c,d]$ if $b<c$. By convention the empty interval $\varnothing$ is disjoint from every interval and is regarded as lying both to the left of and to the right of every interval; consistently with $\mathscr{F}(\varnothing)=\mathbf{1}$, it contributes the unit under the structure maps and its placement carries no ordering content. The ambient interval is $L=[1,N]$.

Local constancy lets us transport every $\mathscr{F}(I)$ to a single object. Fix a nonempty interval and call the transported object $A:=\mathscr{F}([1,1])$; for any nonempty interval $I$ we obtain a canonical isomorphism $\varphi_{I}\colon\mathscr{F}(I)\xrightarrow{\ \sim\ }A$ by composing the (invertible) single-open maps along a chain of inclusions $[1,1]\subseteq\cdots\subseteq I$ or their inverses. The associativity axiom makes $\varphi_{I}$ independent of the chosen chain: any two chains of inclusions between the same two intervals give the same composite, because each elementary square of inclusions commutes by the compatibility axiom of 2.

3.3 Reconstruction of an associative algebra

We now read off a binary product from the ordering of the line.

3.4 Comparison with AQFT nets

We record the bridge theorems relating the two formalisms, which we cite rather than prove. They are the rigorous justification for treating factorization algebras and AQFT nets as two presentations of the same local-to-global observable content.

3.5 Quantization as a cohomological obstruction

The passage from classical to quantum observables is not automatic. In the factorization/BV formalism it is controlled by a sequence of cohomology classes.

4 Worked example: the line

We make 20 concrete, exhibit a locality/gluing check, connect to the free particle, and describe the accompanying code.

4.1 A group-algebra model

Take $\mathcal{C}=\mathrm{Vect}$ over $\Bbbk$ and fix a finite group $G$. Define a finite prefactorization algebra on $L=[1,N]$ by $$\mathscr{F}(I)=\Bbbk[G]\ \text{for every nonempty interval }I,\qquad \mathscr{F}(\varnothing)=\Bbbk,$$ with single-open maps $m^{J}_{I}=\mathrm{id}_{\Bbbk[G]}$ (so $\mathscr{F}$ is locally constant), unit $u\colon\Bbbk\to\Bbbk[G]$ the inclusion $1\mapsto e$ of the identity element, and binary structure map, for $I_{1}$ to the left of $I_{2}$, $$m^{J}_{I_{1},I_{2}}(x\otimes y)=x\cdot y \quad (x,y\in\Bbbk[G]),$$ the convolution product of $\Bbbk[G]$, extended to $n$-ary maps by the ordered product $x_{1}\cdots x_{n}$ following the left-to-right order of the intervals.

25 is a concrete witness of 20: the group algebra is reconstructed from the line, and its associativity is the associativity of the ternary structure map. For a nonabelian $G$ the recovered algebra is noncommutative, which is why the left-to-right ordering of intervals is essential.

4.2 A locality/gluing consistency check

Local constancy and the structure maps must be mutually consistent: transporting a product computed in a small ambient interval up to a large one must agree with the product computed directly in the large one. This is the finite avatar of the factorization gluing axiom [eq:fa-glue], and it is a nontrivial constraint that a careless assignment can fail.

The point of stating 26 separately is diagnostic: it is exactly the identity that fails if one tries to define the structure maps inconsistently (for example by letting the product depend on the length of the ambient interval), and it is the identity the code checks by exhaustive enumeration over $G$ and over admissible interval configurations.

4.3 The free particle and the Weyl algebra

10 identifies the cohomology of the quantum observables of the free particle on the line with the Weyl algebra. We indicate the finite shadow of this statement to connect the abstract construction with 20.

For the free particle the classical observables on an interval are generated by the value and derivative of the field at a point, i.e. by a position mode $q$ and a momentum mode $p$, and the classical factorization product is commutative. The BV Laplacian $\Delta$ deforms the product; the deformation is measured by the propagator, and at first order in $\hbar$ it introduces the commutator $$\begin{equation} [q,p]=\mu(q\otimes p)-\mu(p\otimes q)=\hbar, \end{equation}$$ the canonical commutation relation. The associative algebra recovered by 7 on the line is thus the Weyl algebra $\mathrm{W}=\Bbbk\langle q,p\rangle/([q,p]-\hbar)$, and its associativity is the associativity manufactured in 20. The left-to-right ordering of intervals encodes the operator ordering, which is exactly why the recovered product is noncommutative even though the classical one was commutative: noncommutativity of $\mathrm{W}$ is the $\hbar$-linear shadow of the ordering built into $\mathrm{E}_1$. The Weyl algebra itself admits no finite-dimensional model: over a field of characteristic zero it is simple, and taking the trace of $[q,p]=\hbar\,\mathbf{1}$ would force $0=\dim(V)\,\hbar$, so it has no nonzero finite-dimensional representation. A finite model instead captures the exponentiated (Weyl-form) relations $UV=e^{2\pi i/k}VU$, whose finite solutions are the algebras of finite Heisenberg groups, that is finite-dimensional group algebras of the kind used in 4.1. It is to these finite group algebras — not to a quotient of the algebraic Weyl algebra, which does not exist — that 20 and the accompanying code apply verbatim.

4.4 The Haskell model

The accompanying code, in src/factorization-algebras/, implements the finite theory of [sec:finite-model,sec:group-model] and checks its claims by exhaustive computation. It is organized as follows.

• Core.hs defines intervals on $L=[1,N]$, disjointness and left-of ordering, the group $\mathbb{Z}/k$ and its group algebra over the rationals as association lists (avoiding the hidden containers package), the single-open and $n$-ary structure maps, local-constancy transport, and the recovered product $\mu$ of 18.

• Properties.hs states the checkable claims as deterministic, exhaustive tests: associativity of $\mu$ over all triples of basis elements (20); left and right unitality; the gluing-consistency identity of 26 over all admissible interval configurations up to a cutoff; equivariance of the binary map under swap combined with the abelian case; and the weakest-link status calculus of 14 (associativity, commutativity, the unit law for $S$, and monotonicity) over all triples in $\{S,H,P\}$.

• Main.hs runs the demonstrations — printing the recovered product table for a small group, exhibiting the reconstruction of associativity, and running the full verification suite — and exits nonzero if any check fails.

Because every check ranges exhaustively over a finite basis and uses exact rational arithmetic, a passing run is a proof of the corresponding finite claim, not a sampled estimate. The code compiles with ghc using only the base package.

5 Relation to other regimes and the weakest-link calculus

The factorization/AQFT regime sits at a definite place in the modular program, and its relations to neighbouring regimes are governed by 14. We make three of them explicit.

5.1 Gauge and constraint structure (BV–BRST)

The immediate predecessor is the BV–BRST regime of gauge and constraint structure, whose invariant is that physical observables are the degree-zero cohomology $H^{0}(Q_{\mathrm{BV}})$ of a gauge-resolving differential on a single background. The factorization regime globalizes this: the quantum observables $\mathrm{Obs}^{\mathrm{q}}(U)$ form a factorization algebra whose cohomology $H^{0}(\mathrm{Obs}^{\mathrm{q}}(U))$ recovers the BV–BRST gauge-invariant observables on the region $U$, now assembled local-to-global across $M$ rather than computed once around a fixed background . Both the BV–BRST identification and the factorization packaging are [ S/H ]: standard mathematics for the perturbative/local statement, heuristic as claims about the observables of interacting quantum gravity. By 14 the composite is [ S/H ], unchanged; the two regimes reinforce without either laundering the other’s status.

5.2 Holographic encoding (tensor networks and QEC)

A second neighbour reconstructs measurement by redundant encoding: bulk observables represented on boundary subregions via quantum-error-correcting codes, with entanglement-wedge reconstruction as the dictionary. The local subalgebra structure of a factorization algebra or AQFT net is the algebraic substrate on which such reconstruction statements are phrased, and the modular/Type III structure of the local algebras  is the natural meeting point. Entanglement-wedge reconstruction itself carries status [ H ] (a semiclassical holographic statement), so by 14 any composite claim built from the factorization regime and entanglement-wedge reconstruction is at best [ H ]. This is an honest ceiling: the algebraic side is [ S/H ], but the composite inherits the weaker holographic warrant.

5.3 Asymptotic boundary data (celestial holography)

A third neighbour reconstructs the $S$-matrix from conformal data on the celestial sphere and asymptotic (BMS) symmetry representation theory. This is a different local-to-global scheme — observables organized by null infinity rather than by a Weiss cover of a bulk region — and its overall warrant is [ H ]. Composing it with the factorization regime again yields [ H ] by 14. The three neighbours realize the same reconstruction thesis by structurally different means (compositional gluing here, redundant encoding there, asymptotic-symmetry representation theory in the third), and the weakest-link calculus keeps the composite honest in each case.

5.4 Background independence and the ceiling

The factorization-algebra assignment lives on a fixed manifold $M$; the locally covariant AQFT refinement (9) improves this to a functor over the category $\mathsf{Loc}$ of all globally hyperbolic spacetimes, which is background independence at the level of the site but not at the level of a fluctuating metric. A genuinely background-independent, nonperturbative quantum-gravity observable algebra — of the kind one would want to extract from a spin-foam state sum or an asymptotically safe fixed point — is not known to be a factorization algebra or a cosheaf at all. Any claim that packages such an object as a factorization algebra therefore composes a [ P ] (speculative, currently unconstructed) input with the [ S/H ] factorization machinery, and by 14 lands at [ P ]. This is the precise sense in which the reconstruction thesis is, in this regime, proved where the theory is free or perturbative and open where it is gravitational.

6 Limitations and open problems

We collect the limitations honestly, separating what is established from what is programmatic.

6.0.0.1 Lorentzian signature.

The Costello–Gwilliam construction is developed most cleanly in Euclidean signature and in a formal ($\hbar$-adic) perturbative sense; the propagator that defines the BV Laplacian is a Euclidean Green’s function. The Lorentzian theory requires the time-ordered/Green-hyperbolic technology of perturbative AQFT and of the time-orderable prefactorization algebras of Benini–Perin–Schenkel and Benini–Musante–Schenkel . The comparison theorems (22) are the current bridge, and they are restricted to free (or linear BV) theories. Extending the equivalence to interacting Lorentzian theories is open.

6.0.0.2 Interacting and renormalized observables.

For interacting theories the quantum factorization algebra exists only after renormalization, and its very existence is the content of the obstruction theory of 24. The structure maps then carry the operator product expansion, and there is no closed-form finite model of the kind we used in 4. The finite reconstruction theorem 20 applies to the associative (one-dimensional, or free) shadow, not to the full interacting higher-dimensional theory.

6.0.0.3 Background dependence.

Even the locally covariant refinement fixes the class of spacetimes and treats the metric as background data of the site, not as a quantized field. Packaging a fluctuating-metric, background-independent observable algebra — the actual object of quantum gravity — as a factorization algebra is not achieved and is flagged [ P ].

6.0.0.4 Analytic subtleties.

The homotopy colimits of 5 and the completed symmetric algebras of [eq:classobs] require care with functional-analytic completions (nuclear or convenient/bornological vector spaces) and with the Weiss topology’s non-standard covers. Our finite model deliberately sidesteps these by working with finite-dimensional algebras, where colimits are ordinary and completions are trivial; the price is that it captures the algebraic skeleton, not the analysis.

6.0.0.5 Open problems.

Three problems mark the boundary of the current state of the art.

1. Extend the free-theory comparison (22) to interacting, gauge-fixed (BV-quantized) perturbative gravity, matching factorization-algebra and AQFT observables order by order in $\hbar$.

2. Decide whether any background-independent observable algebra — from a spin-foam state sum, a loop-quantum-gravity kinematical setup, or an asymptotically safe fixed point — can be presented as a factorization algebra or a cosheaf of observables at all. This is not done, and should be treated as open rather than assumed.

3. Develop the modular-theoretic (Type III, relative-entropy) bridge between AQFT local algebras and the holographic-QEC reconstruction of 5.2, which is currently underexplored.

7 Conclusion

The observable content of a quantum field theory is not a single global algebra handed down at the outset. It is a rule assigning to each region the measurements possible there, together with a compositional law by which local measurements generate global ones. Factorization algebras make that law precise as a cosheaf condition over the Weiss topology, gluing by colimits (observables compose) rather than by limits (sections restrict); AQFT nets make it precise as a functor with built-in causality, and the two are provably the same for free theories. In one dimension the entire structure collapses to an associative algebra, and we proved, for a finite combinatorial model in which every step is an exhaustive computation, that the associative product and its associativity are reconstructed from the factorization gluing data alone. The accompanying Haskell code checks that reconstruction, together with unitality, gluing consistency, and the weakest-link status calculus, exactly on the finite model.

Read against the program’s governing thesis, the regime supplies the measurement layer of the reconstruction of physics from invariant representation structures: it is the mechanism by which what can be measured is assembled from the compositional structure of local data. The honest boundary of the achievement is sharp. The reconstruction is a theorem for free and perturbative theories, a strong heuristic for the identification of renormalizability with a cohomological obstruction, and an open problem — flagged speculative and governed by the weakest-link calculus — for the genuinely background-independent, gravitational case.