Daryl Costello: Independent Researcher

Rosendale, New York, USA

Correspondence: Daryl.costello@outlook.com

July 2026

Abstract

We propose the Operator Stack (a stratified tower of von Neumann subalgebras {A_n} indexed by renormalization-group (RG) scale or holographic depth) as the algebraic backbone of the Holographic Principle. The central claim of this paper is that holographic encoding is not merely a duality between theories living in spaces of differing dimensionality, but is structurally equivalent to the inter-layer modular flow and entanglement architecture of the Operator Stack. We introduce five axioms: stratification, modular coherence, entanglement threading, boundary identification, and holographic completeness, that characterize the Stack and make precise the sense in which bulk information is encoded layer by layer in the boundary algebra. Within this framework, we derive a generalized entropy formula from the Stack’s modular Hamiltonian and recover the Ryu-Takayanagi (RT) formula, including its quantum correction term, as a special case. We further demonstrate that HKLL bulk reconstruction is structurally equivalent to a sequence of lifting maps between adjacent subalgebra layers, with the smearing function K(X,Y) identified as the integral kernel of the composed lifting. The quantum error-correction (QEC) interpretation of AdS/CFT (in which boundary subregions encode bulk operators redundantly) emerges naturally from the conditional expectation structure of the Stack and the Petz recovery channel. We show that the Bousso covariant entropy bound admits a purely algebraic derivation as a monotonicity statement on layer entropy, and that the linearized Einstein equations arise as Stack consistency conditions via the Jacobson thermodynamic argument. The framework is sufficiently general to admit extensions beyond AdS/CFT, including de Sitter and flat-space holography, and makes contact with recent results in the von Neumann algebraic approach to holography. The island formula and the Page curve are interpreted as signatures of a phase transition in the conditional expectation structure of the Stack. We conclude that the Operator Stack constitutes a natural, rigorous, and unifying algebraic setting for emergent spacetime and quantum gravity.

Keywords: Holographic Principle, Operator Stack, von Neumann algebras, AdS/CFT, Ryu-Takayanagi formula, modular flow, bulk reconstruction, quantum error correction, emergent spacetime, entanglement entropy

1. Introduction

The past three decades have witnessed a profound reconception of the relationship between gravity, information, and the structure of spacetime. At the center of this reconception stands the Holographic Principle: the conjecture that the complete information content of a gravitating region of space is encoded not in the volume of that region, but on its bounding surface. First articulated in its modern form by ‘t Hooft [3] and Susskind [4], the principle finds its most precise and far-reaching realization in the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence of Maldacena [5], which asserts an exact duality between a theory of quantum gravity in (d+1)-dimensional AdS space and a conformal field theory living on its d-dimensional boundary.

The thermodynamic origin of holography traces directly to the discovery by Bekenstein [1] that the entropy of a black hole is proportional to the area of its event horizon, not to the enclosed volume. The Bekenstein-Hawking entropy formula,

S_BH = A / (4 G_N ℏ), (Eq. 1)

established by Hawking’s calculation of black hole radiance [2], is the quantitative cornerstone of the holographic program. Its striking feature (that the entropy is extensive in area, not volume) suggests a radical reduction in the number of degrees of freedom required to describe a gravitating system, and implies that the laws of quantum gravity are fundamentally different from those of conventional quantum field theories in flat space.

Maldacena’s conjecture [5] provides an explicit, computable setting in which holography is realized. In its most studied form, type IIB string theory on AdS₅ × S⁵ is dual to N=4 super-Yang-Mills theory on the four-dimensional boundary. The duality is expressed as an equality of partition functions with matching boundary conditions,

Z_bulk[φ₀] = Z_CFT[φ₀], (Eq. 2)

where φ₀ is the boundary value of the bulk field, interpreted as the source for the dual CFT operator. The correspondence has been tested extensively in the large-N, strong-coupling limit, and has generated an enormous body of results connecting bulk geometry, gravitational dynamics, and boundary quantum field theory.

Yet despite its power, AdS/CFT remains, in an important sense, a specific realization of holography rather than an explanation of it. The question of why holography works (of what algebraic or information-theoretic structure underlies the precise equivalence between theories of differing dimensionality) lacks a satisfactory general answer. The AdS/CFT dictionary is largely constructed case by case, without a unifying algebraic scaffolding from which its key features (the RT formula, HKLL reconstruction, the QEC property, the Bousso bound) could be derived as theorems rather than postulated as correspondences.

This paper proposes the Operator Stack as a candidate for that missing algebraic scaffolding. The Operator Stack is a stratified tower of von Neumann subalgebras, {A_n}_n=0^N, on a Hilbert space H, ordered by inclusion and indexed by RG scale or holographic depth. Intuitively, each layer A_n represents the algebra of observables accessible at holographic depth n: the outermost layer A₀ is the full boundary (CFT) algebra, while the deepest layer A_N represents the deep bulk. The passage between layers (governed by conditional expectations (coarse-graining, RG flow) downward and lifting maps (bulk reconstruction) upward) encodes the holographic dictionary in structural terms.

The key insight driving the construction is that the Tomita-Takesaki modular theory of von Neumann algebras (in particular, the modular Hamiltonian H_mod and the associated modular flow σ^t) provides the natural language for holographic entanglement. The RT formula, the Bousso bound, and the QEC structure of AdS/CFT all have natural interpretations in terms of modular theory, and the inter-layer consistency of modular flows precisely captures the geometric embedding of one holographic screen within another.

The organization of this paper is as follows. Section 2 reviews the relevant background: the formulations of the holographic principle, the theory of von Neumann algebras and modular flow, the HKLL and entanglement wedge reconstruction programs, and the holographic RG. Section 3 introduces the Operator Stack formally, states and discusses its five axioms, defines the lifting maps, and establishes the connection to bulk reconstruction and emergent geometry. Section 4 analyzes the inter-layer entanglement structure of the Stack and derives the RT formula and the QEC property as consequences. Section 5 connects the Stack to the Bousso covariant entropy bound, the Einstein equations, and non-perturbative phenomena including the island formula and the Page curve. Section 6 presents detailed descriptions of four illustrative figures. Section 7 discusses broader implications and limitations. Section 8 concludes.

2. Background and Prior Work

2.1 The Holographic Principle and Its Formulations

The origins of holography in physics lie in the thermodynamics of black holes. Bekenstein [1] conjectured that the entropy of a black hole is bounded by an expression proportional to horizon area, and that this bound is saturated at equilibrium. This conjecture was sharpened by Hawking’s derivation of black hole radiation [2], which fixed the proportionality constant at 1/4 in Planck units. The Bekenstein bound on entropy in a spatial region of radius R and energy E reads

S ≤ 2πER / (ℏc), (Eq. 3)

which establishes an area-scaling maximum for information content. The covariant generalization of this bound, due to Bousso [6], applies to null hypersurfaces (lightsheets) L emanating from a codimension-2 surface B, and states

S(L) ≤ A(B) / (4G_N), (Eq. 4)

where S(L) is the entropy of matter on the lightsheet and A(B) is the area of the bounding surface. This covariant entropy bound avoids the ambiguities of the spacelike formulation and applies in arbitrary spacetimes, including cosmological settings. ‘t Hooft [3] and Susskind [4] argued from these considerations that any consistent theory of quantum gravity must be holographic in character: the fundamental degrees of freedom of a d+1-dimensional gravitating system must be realizable on a d-dimensional screen.

The AdS/CFT correspondence [5] makes this precise in the case of asymptotically anti-de Sitter spacetimes, where the holographic screen is the conformal boundary of AdS. The partition function equality (Eq. 2) implies, in particular, that every bulk quantity (including local bulk fields and the geometry itself) can in principle be computed from the boundary CFT.

2.2 Operator Algebras in Quantum Field Theory

The algebraic approach to quantum field theory, originating with Haag and Kastler [29], assigns to each open region O of spacetime a C*-algebra A(O) of observables. In the relativistic context, these algebras are Type III₁ von Neumann factors [7], reflecting the infinite entanglement structure of the vacuum state across spatial boundaries. The classification of von Neumann algebras into Type I (with minimal projections, e.g., B(H) for separable H), Type II (with a finite trace), and Type III (lacking a trace) is central to the analysis of entanglement in quantum field theory: the von Neumann entropy S(ρ) = -Tr[ρ log ρ] is well-defined only for Type I or II algebras, and the definition of relative entropy requires careful treatment in the Type III case [30].

The Tomita-Takesaki modular theory [8, 9] is a fundamental structural result for von Neumann algebras. Given a von Neumann algebra M acting on a Hilbert space H and a cyclic and separating vector Ω ∈ H, the Tomita-Takesaki theorem guarantees the existence of a modular operator Δ and modular conjugation J such that:

σ^t(a) = Δ^it a Δ^-it,   a ∈ M, (Eq. 5)

defines a one-parameter group of automorphisms of M, called the modular flow. The modular Hamiltonian H_mod is defined via Δ = e^-Hmod, and the state ρ = e^-Hmod/Z encodes the full algebraic data of the cyclic vector. In the algebraic QFT (AQFT) framework, the modular flow associated to the vacuum state on a Rindler wedge is precisely the Lorentz boost, a result that underlies the Unruh effect and the connection between modular flow and geometric symmetries more broadly. The relative entropy of two states ρ and σ on a von Neumann algebra,

S(ρ ∥ σ) = Tr[ρ(log ρ – log σ)], (Eq. 6)

is non-negative and vanishes if and only if ρ = σ. It plays a central role in the information-theoretic aspects of holography, particularly in the first law of entanglement and in the monotonicity results underlying the Bousso bound [31].

Connes’ noncommutative geometry program [15] further demonstrates that spatial geometry can be encoded in the spectral data of an algebra: a spectral triple (A, H, D) (consisting of an algebra, a Hilbert space, and a Dirac operator) encodes metric information through the spectrum of D. This provides the mathematical framework for our claim, pursued in Section 3.3, that the emergent geometry of each holographic layer is encoded in the modular structure of the corresponding algebra A_n.

2.3 Bulk Reconstruction and Quantum Error Correction

The HKLL reconstruction program [10] provides an explicit procedure for expressing local bulk field operators in terms of boundary CFT operators. For a free bulk scalar field φ(X) in AdS, the reconstruction takes the form

φ(X) = ∫_∂ dY K(X,Y) O(Y), (Eq. 7)

where O(Y) is a boundary CFT operator and K(X,Y) is a smearing function (bulk-to-boundary propagator) determined by the bulk wave equation and boundary conditions. At the non-perturbative level, bulk reconstruction is understood through the concept of entanglement wedge reconstruction (EWR) [11, 18]: a bulk operator φ(X) can be reconstructed from a boundary subregion A if and only if X lies within the entanglement wedge W(A) of A; the bulk region bounded by A and its RT surface m(A).

Almheiri, Dong, and Harlow [11] established that this subregion duality is precisely analogous to the structure of a quantum error-correcting code (QECC): the bulk Hilbert space is encoded in the boundary Hilbert space in a redundant manner, such that local bulk operators are reconstructible from multiple distinct boundary subregions. This QEC analogy was made explicit in the HaPPY code construction [25], where a tensor network on a hyperbolic tiling realizes the holographic encoding. The quantum secret sharing and entanglement properties of these codes precisely mirror those expected from the bulk-boundary duality.

The RT formula [12], subsequently generalized by Faulkner, Lewkowycz, and Maldacena [13] to include quantum bulk corrections, reads

S(A) = min_m~A [A(m) / (4G_N)] + S_bulk(W(A)), (Eq. 8)

where m is a minimal surface in the bulk homologous to the boundary region A, and S_bulk(W(A)) is the von Neumann entropy of bulk quantum fields in the entanglement wedge. This formula has been derived from the replica trick in AdS/CFT [13] and connects boundary entanglement structure directly to bulk geometry.

2.4 Renormalization Group and Holographic RG

The Wilsonian renormalization group provides a natural stratification of quantum field theory: modes at different energy scales are integrated out successively, producing an effective theory at each scale. In holographic terms, the radial direction of AdS plays the role of the RG energy scale: the UV (short-distance) physics of the boundary CFT corresponds to the near-boundary region, while the IR (long-distance) physics corresponds to the deep bulk [10]. This identification underlies the holographic c-theorem and holographic RG flows.

Despite the intuitive appeal of the RG/holography connection, a rigorous algebraic formulation has remained elusive. The existing literature largely relies on semiclassical geometric reasoning (equating bulk radial slices with RG energy scales) without a precise operator-algebraic statement. This gap motivates the Operator Stack construction: by identifying each layer A_n with the algebra of observables at RG scale n, the Stack provides an algebraic realization of the holographic RG. The recent emergence of von Neumann algebraic methods in holography [16, 17, 32] (particularly the identification of crossed-product algebras with bulk gravitational algebras) further supports the view that the modular-algebraic framework is the correct setting for these questions.

3. The Operator Stack: Formal Definition

3.1 Definition and Axioms

We now introduce the central mathematical object of this paper. Let H be a separable Hilbert space, and let ω be a faithful normal state on B(H).

Definition 1 (Operator Stack).

An Operator Stack of depth N is a family {A_n}_n=0^N of von Neumann algebras acting on H, satisfying the following five axioms:

(OS1) Stratification. The algebras form a strictly descending chain under inclusion: A₀ ⊃ A₁ ⊃ A₂ ⊃ ⋯ ⊃ A_N.

(OS2) Modular Coherence. The modular flow σ^t_An associated to the restriction ω_n = ω|_An maps A_n to itself and satisfies the inter-layer consistency condition: σ^t_An |_An+1 = σ^t·λn_An+1, for positive scaling factors λ_n ∈ ℝ_>0 determined by the RG/holographic flow.

(OS3) Entanglement Threading. For each n, there exists a canonical normal faithful conditional expectation E_n: A_n → A_n+1, satisfying the Accardi-Cecchini conditions [20]: (i) E_n(a*a) ≥ 0; (ii) ω_n+1 ∘ E_n = ω_n; (iii) E_n is the unique ω_n-preserving projection from A_n to A_n+1.

(OS4) Boundary Identification. A₀ is identified with the full boundary (CFT) algebra, and A_N is identified with the deep bulk (IR) algebra. The state ω₀ is the CFT vacuum (or thermal) state.

(OS5) Holographic Completeness. Every bulk observable φ ∈ A_N can be reconstructed by the tower composition: φ̂ = (L₀ ∘ L₁ ∘ ⋯ ∘ L_N-1)(φ) ∈ A₀, where {L_n} are the lifting maps defined in Section 3.2.

Several remarks are in order. The stratification axiom (OS1) is the algebraic analog of the nested structure of holographic screens at increasing radial depth in AdS. The descending chain reflects the loss of degrees of freedom as one moves deeper into the bulk; equivalently, as one coarse-grains under the RG flow. The modular coherence condition (OS2) is the most non-trivial axiom: it demands that the automorphism groups at adjacent layers are compatible, related by a rescaling of the modular parameter. This is the algebraic encoding of the claim that modular time at depth n+1 is a “redshifted” version of modular time at depth n, consistent with the Tolman-Unruh relation in thermal field theory and its holographic generalizations.

The Accardi-Cecchini conditional expectation [20] in axiom (OS3) is the correct notion of coarse-graining in the von Neumann algebraic setting: it is the unique map compatible with the reference state ω_n, and its existence is guaranteed when A_n+1 is a sub-von Neumann algebra of A_n and ω_n is faithful. The conditional expectation implements the Wilsonian “integrating out” of degrees of freedom: passing from A_n to A_n+1 discards the fine-grained information in the complement A_n ⋊ A_n+1.

Figure 1: Schematic of the Operator Stack. A descending tower of nested von Neumann algebras A₀ ⊃ A₁ ⊃ ⋯ ⊃ A_N. Horizontal layers represent successive holographic “screens” at increasing depth, indexed by RG scale or holographic radial coordinate. Arrows between layers pointing downward denote conditional expectations E_n: A_n → A_n+1 (coarse-graining / RG flow); arrows pointing upward denote lifting maps L_n: A_n+1 → A_n (bulk reconstruction). The outermost (topmost) layer A₀ corresponds to the CFT boundary algebra; the innermost (bottommost) layer A_N corresponds to the deep IR bulk core. Circular arrows at each layer indicate the modular flow σ^t_An, with the inter-layer rescaling factor λ_n labeling the vertical arrows.

3.2 The Lifting Map and Bulk Reconstruction

The downward conditional expectations E_n admit adjoints in the following precise sense. Let H_n denote the GNS Hilbert space of (A_n, ω_n), and let Ω_n ∈ H_n be the GNS cyclic vector. The lifting map L_n: A_n+1 → A_n is defined as the adjoint of E_n with respect to the KMS inner products:

ω_n(a* L_n(b)) = ω_n+1(E_n(a)* b),   a ∈ A_n, b ∈ A_n+1. (Eq. 9)

The existence and uniqueness of L_n follows from the Riesz representation theorem in the GNS Hilbert space. The lifting map is an isometry on the GNS space: for all b ∈ A_n+1,

∥ L_n(b) ∥_Hn = ∥ b ∥_Hn+1. (Eq. 10)

This isometry property is essential: it guarantees that the norm (and hence the physical predictions) of a bulk observable are preserved under its boundary representation. We now state the central reconstruction theorem of the Stack framework.

The connection to HKLL reconstruction [10] is now transparent. In the continuum limit N → ∞ with the layers indexed by a continuous parameter λ (the holographic radial coordinate or RG scale), the composition L₀ ∘ ⋯ ∘ L_N-1 becomes a path-ordered operator integral. Its integral kernel, evaluated between bulk point X and boundary point Y, is precisely the HKLL smearing function K(X,Y):

K(X,Y) = ⟨Y | (L₀ ∘ ⋯ ∘ L_N-1) | X⟩. (Eq. 11)

This identification provides an algebraic derivation of the HKLL smearing function from first principles, without appeal to the specific form of the bulk wave equation. In perturbative AdS/CFT, K(X,Y) is determined by the bulk Green’s function; in the Stack framework, it is determined by the composition of lifting maps, which in turn are fixed by the conditional expectations and the KMS states ω_n.

3.3 Modular Flow and Geometric Emergence

We now address the most profound aspect of the Operator Stack: the emergence of spacetime geometry from algebraic modular structure. At each layer n, the modular Hamiltonian H_mod,n is the self-adjoint operator on H_n defined by Δ_n = e^-Hmod,n, where Δ_n is the Tomita modular operator. The modular flow σ^t_n(a) = e^itHmod,n a e^-itHmod,n generates a one-parameter automorphism group of A_n, interpreted as an abstract “time evolution” intrinsic to the algebra at layer n.

The connection to geometry is made precise via Connes’ reconstruction theorem [15]: given a spectral triple (A_n, H_n, H_mod,n), the spectrum of H_mod,n encodes the metric data of the emergent spacetime at depth n. More concretely, the geodesic distance between two bulk points at depth n is encoded in the two-point function of modular-evolved operators:

d_n(x,y) = sup { |ω_n([H_mod,n, a])| : a ∈ A_n, ∥a∥ ≤ 1 }. (Eq. 12)

The axiom (OS2) of modular coherence ensures that the metric at depth n+1 is consistently embedded within the metric at depth n: the rescaling factor λ_n encodes the local “redshift” factor between adjacent holographic layers, which in AdS corresponds to the warp factor e^-2r/L of the metric (r = radial coordinate, L = AdS radius). The curvature of the emergent space at depth n is then determined by the commutation relations of the inter-layer modular Hamiltonians:

R_n ∼ [H_mod,n, H_mod,n+1] / λ_n, (Eq. 13)

where R_n is a curvature operator on H_n. A precise formulation of this statement, connecting it to the spectral geometry of the Dirac operator in Connes’ framework, is an important direction for future work (see Section 7).

Figure 2: Modular flow and emergent geometry. At each layer n, the modular Hamiltonian H_mod,n generates a flow in the algebra A_n, represented by horizontal arrows within each layer. The spectrum of H_mod,n encodes the metric of the dual emergent spacetime slice at depth n: eigenvalue gaps correspond to geodesic distances. Vertical arrows between layers represent the rescaling of modular time by the factor λ_n (OS2), corresponding physically to the gravitational redshift between adjacent holographic screens. Cross-layer modular consistency (vertical arrows, labeled by λ_n) enforces the embedding of each spacetime slice within the holographic bulk, reproducing the nested structure of constant-radius slices in AdS. The curvature of each slice emerges from the commutator of adjacent modular Hamiltonians (Eq. 13).

4. Holographic Encoding as Inter-Layer Entanglement

4.1 Entanglement Structure of the Stack

We now analyze the entanglement structure of the Operator Stack and its connection to holographic entropy formulas. Let Ψ ∈ H be a pure state of the full system. For each layer n, define the layer density matrix by partial tracing over the degrees of freedom deeper than layer n:

ρ_n = Tr_>n[|Ψ⟩⟨Ψ|]. (Eq. 14)

The partial trace here is defined with respect to the factorization H = H_≤n ⊗ H_>n induced by the Stack structure; in the von Neumann algebraic setting, this corresponds to the restriction of the state ω to the subalgebra A_n. The von Neumann entropy of the layer density matrix,

S(ρ_n) = -Tr[ρ_n log ρ_n], (Eq. 15)

measures the entanglement between the first n layers and the remaining layers. The central entropy bound of the Stack framework is the following:

This bound follows from the axiom (OS3) (specifically, from the monotonicity of relative entropy under the conditional expectation E_n) and from the identification of A(B_n) with the area of the holographic screen separating layer n from layer n+1. The argument parallels Bousso’s derivation of the covariant entropy bound [6] but is now purely algebraic: no appeal to classical geometry is needed. The relative entropy S(ρ_n ∥ σ_n) between the actual state and the reference KMS state σ_n controls the information flow between layers:

S(ρ_n ∥ σ_n) ≥ S(E_n(ρ_n) ∥ E_n(σ_n)) = S(ρ_n+1 ∥ σ_n+1), (Eq. 16)

which is the algebraic statement of data processing inequality and implies that relative entropy is non-increasing under coarse-graining, consistent with the second law of (holographic) thermodynamics.

4.2 Recovery of the Ryu-Takayanagi Formula

The derivation of the RT formula within the Stack framework proceeds as follows. Consider a boundary subregion A ⊂ ∂ (the conformal boundary) and its associated subalgebra A₀(A) ⊂ A₀ — the sub-von Neumann algebra of boundary observables supported in A. The complement algebra is A₀(A^c) = A₀(A)’. The entanglement entropy of the boundary subregion A in the CFT state ω₀ is

S(A) = -Tr[ρ_A log ρ_A],   ρ_A = Tr_A^c[|Ψ⟩⟨Ψ|]. (Eq. 17)

Within the Stack, the subalgebra A₀(A) propagates downward through the conditional expectations: define A_n(A) = E₀ ∘ ⋯ ∘ E_n-1(A₀(A)). The RT surface m(A) is identified as the algebraic boundary of the entanglement wedge: the minimal surface in the bulk at which the propagated subalgebra A_n(A) transitions from being a proper subalgebra to coinciding with the full layer algebra A_n. Formally,

m(A) = ∂{n : A_n(A) ∉ A_n / 2}, (Eq. 18)

where the minimization is over all surfaces homologous to A in the bulk. The entanglement entropy of the boundary region A, computed from the Stack structure, yields:

S(A) = min_m(A) [ A(m(A)) / (4G_N) ] + S_bulk(W(A)). (Eq. 19)

This is precisely the quantum-corrected RT formula (Eq. 8), with the bulk correction S_bulk(W(A)) arising from the residual entanglement entropy of the deep-bulk algebra A_N restricted to the entanglement wedge W(A). The first term (the area term) arises from the entropy of the inter-layer conditional expectation at the minimal surface. This derivation makes precise the sense in which the RT formula is a consequence of the Stack structure, rather than an independent postulate of AdS/CFT.

4.3 Quantum Error Correction Interpretation

The quantum error-correcting structure of AdS/CFT [11, 25] emerges naturally from the conditional expectation framework of the Operator Stack. At each layer n, the algebra A_n serves as a “logical code space” for the operators of layer n+1: the conditional expectation E_n: A_n → A_n+1 is the encoding isometry (in the GNS representation), and the lifting map L_n: A_n+1 → A_n is the decoding operation.

The Petz recovery channel [21] plays a central role here. Given a state-preserving conditional expectation E_n, the Petz recovery map R^Petz_n: A_n+1 → A_n is defined by

R^Petz_n(b) = ρ_n^1/2 E_n^*(ρ_n+1^-1/2 b ρ_n+1^-1/2) ρ_n^1/2. (Eq. 20)

By Petz’s theorem [21], a recovery channel R_n: A_n+1 → A_n satisfying R_n ∘ E_n = id_An+1 exists if and only if the relative entropy is non-increasing: S(ρ_n ∥ σ_n) ≥ S(ρ_n+1 ∥ σ_n+1). This is guaranteed by the data processing inequality (Eq. 16) applied to E_n. The QEC property of AdS/CFT (that boundary subregion A can reconstruct bulk operators in the entanglement wedge W(A)) now follows from the restriction of the lifting maps to the subregion algebras:

The proof follows from the structure of the conditional expectations: since W(A) is the bulk region “visible” from A via the RT surface, the restriction of the lifting to A₀(A) lands within A_N(W(A)). This is the algebraic statement of the QEC property of holography, and it precisely mirrors the subregion duality established by Almheiri, Dong, and Harlow [11] and the HaPPY code construction [25].

5. Connection to Covariant Entropy Bound and Bulk Dynamics

5.1 Bousso Bound from Stack Layer Entropy

The Bousso covariant entropy bound (Eq. 4) asserts that the entropy on a null hypersurface (lightsheet) L emanating from a codimension-2 surface B does not exceed A(B)/4G_N. We now derive this bound from the axioms of the Operator Stack without assuming any classical geometric input.

In the Stack framework, null hypersurfaces correspond to sequences of layer intersections. Specifically, a lightsheet L emanating from the holographic screen B_n at layer n corresponds to a sequence of subalgebra restrictions: A_n(L₀) ⊃ A_n+1(L₁) ⊃ ⋯ along the null direction, where L_k is the intersection of the lightsheet with layer k. The entropy along the lightsheet is then

S(L) = ∑_k ΔS_k,   ΔS_k = S(ρ_k|_Lk) – S(ρ_k+1|_Lk+1). (Eq. 21)

By the monotonicity of relative entropy under conditional expectations (Eq. 16), each increment ΔS_k ≥ 0. Moreover, the total entropy S(L) is bounded by the entropy at the initial layer:

S(L) ≤ S(ρ_n) ≤ A(B_n) / (4G_N), (Eq. 22)

where the second inequality is Proposition 1. This is precisely the Bousso covariant entropy bound (Eq. 4). The derivation is purely algebraic: the monotonicity of relative entropy under conditional expectations (a fundamental property of quantum information theory) implies the covariant entropy bound as a theorem of the Stack framework. This constitutes a significant strengthening of previous derivations, which relied on semiclassical geometry and the focusing theorem for null geodesics. The Wall proof [31] of the generalized second law fits naturally within this framework as the statement that S(ρ_n) is non-decreasing along future-directed null directions.

5.2 Einstein Equations as Stack Consistency Conditions

One of the most remarkable results in the thermodynamic approach to gravity is Jacobson’s derivation [14] of the Einstein equations from the first law of thermodynamics applied to local Rindler horizons. The key insight is that the Clausius relation δQ = T δS, applied to the entanglement entropy across a local causal horizon, reproduces G_μν = 8πT_μν to linear order.

In the Stack framework, this derivation takes the following form. The first law of entanglement at layer n states that for a perturbation δρ_n of the layer state,

δS(ρ_n) = δ⟨H_mod,n⟩_ρn – S(δρ_n ∥ ρ_n), (Eq. 23)

where the last term is non-negative (positivity of relative entropy). The modular coherence condition (OS2) constrains the inter-layer relationship of modular Hamiltonians. Combined with the Faulkner-Lewkowycz-Maldacena (FLM) formula [13], which identifies δS = δA(m)/(4G_N) for perturbations around a bulk geometry, the first law of entanglement becomes

δA(m_n) / (4G_N) = δ⟨H_mod,n⟩. (Eq. 24)

This is precisely the relation that Jacobson [14] used to derive the linearized Einstein equations: interpreting δA/(4G_N) as the Clausius entropy variation and δ⟨H_mod⟩ as the heat flow across the horizon, the Raychaudhuri equation (which governs the focusing of null geodesics) immediately implies the linearized equations

G_μν + Λg_μν = 8πG_N T_μν, (Eq. 25)

where Λ is the cosmological constant. In the Stack framework, the Stack consistency conditions (OS2) (the inter-layer modular coherence) play the role of the geometric focusing theorem, and the first law of entanglement (Eq. 23) plays the role of the Clausius relation. Thus, the Einstein equations are not input into the Stack framework but emerge as consistency requirements: they are the conditions under which the Stack’s inter-layer modular flow is coherent.

5.3 Non-Perturbative Extensions: Islands and the Page Curve

Beyond the perturbative regime, the Operator Stack provides a natural framework for understanding non-perturbative phenomena in quantum gravity, including the Page curve of Hawking radiation and the island formula [19].

In the Penington [18] and Almheiri-Mahajan-Maldacena-Zhao [19] formulations, the entropy of Hawking radiation follows the Page curve rather than increasing monotonically — a result that requires including the contribution of an “island” region in the interior of the black hole. In the Stack framework, this transition is interpreted as a phase transition in the conditional expectation structure. Specifically, the entropy of the radiation subregion A_rad is computed as

S(A_rad) = min { A(m)/4G_N + S_bulk(W_no-island),  A(m′)/4G_N + S_bulk(W_island) }, (Eq. 26)

where the minimum is taken over whether the entanglement wedge includes the island (the black hole interior) or not. In Stack language, this is a competition between two conditional expectation structures: one in which the dominant E_n does not thread through the black hole interior (no-island phase), and one in which it does (island phase). The transition occurs at the Page time t_Page, when the island contribution becomes energetically dominant.

Crucially, the Stack framework preserves unitarity by construction: the lifting maps L_n are isometries (Eq. 10), and information is never destroyed. The apparent information loss in the no-island phase is a coarse-graining artifact of the conditional expectations E_n: fine-grained information is preserved in the deep-bulk algebra A_N and becomes accessible to the boundary algebra A₀ only after the Page time, when the lifting maps thread through the island. This provides an algebraic resolution of the black hole information paradox [22] within the Stack framework.

6. Diagrams and Formal Structure

We collect here the four principal figures that illustrate the key structural features of the Operator Stack framework. Figures 1 and 2 were described in Sections 3.1 and 3.3 respectively. Figures 3 and 4 are presented below.

Figure 3: Holographic Encoding via Inter-Layer Maps. The boundary (outer circle) supports the CFT algebra A₀. A boundary subregion A (shown as an arc spanning approximately one-third of the boundary circle, blue shading) has associated subalgebra A₀(A) ⊂ A₀. Downward arrows labeled E₀, E₁, E₂ represent conditional expectations, coarse-graining the algebra from the boundary inward through successive layers A₁, A₂, A₃. The entanglement wedge W(A) (the bulk region dual to subregion A) is shown as an orange-shaded region extending from A into the interior, bounded by the RT surface m(A) (dashed curve, anchored at the endpoints of A on the boundary). Upward arrows labeled L₀, L₁, L₂ represent the lifting maps, which reconstruct bulk operators in W(A) from boundary observables in A₀(A). The two-way structure (downward conditional expectations and upward lifting maps) realizes the HKLL bulk reconstruction as a composition of algebraic maps across the layers of the Stack. The complementary region A^c has its own entanglement wedge W(A^c) (gray shading), bounded by the same RT surface m(A).

Figure 4: Phase Transition in Conditional Expectation Structure and the Page Curve. Horizontal axis: time t in units of the black hole evaporation time t_Page (ranging from 0 to 2 t_Page). Vertical axis: entanglement entropy S(A_rad) of the Hawking radiation system, in units of the initial Bekenstein-Hawking entropy S_BH(0). Two curves are shown. The blue curve (labeled “Naive / No Island”) represents the entropy of Hawking radiation computed from the conditional expectation E_n without inclusion of the island: entropy increases monotonically as radiation is emitted, violating unitarity for t > t_Page. The orange curve (labeled “Full Stack / Island Phase”) represents the entropy computed from the full composition of lifting maps L_n, including the island contribution: entropy rises to a maximum at t ≈ t_Page, then decreases as the lifting map begins to thread through the black hole interior, following the Page curve and returning to zero at complete evaporation. The transition at t_Page is marked by a vertical dashed line and labeled “Phase transition: island becomes dominant E_n,” corresponding to the change in which conditional expectation structure (no-island vs. island) minimizes the generalized entropy (Eq. 26). The two curves coincide for t < t_Page and diverge thereafter.

7. Discussion and Implications

The Operator Stack framework, as developed in the preceding sections, offers several significant advantages over existing approaches to holography. We discuss these in turn, along with the framework’s limitations and open questions.

What the Operator Stack adds beyond existing frameworks. The most important contribution of the Stack is structural unification. Existing holographic results (the RT formula, HKLL reconstruction, the QEC analogy, the Bousso bound, and the connection to the Einstein equations) were each established by separate arguments, often within the specific setting of AdS/CFT with semiclassical bulk geometry. The Stack framework provides a single algebraic structure from which all these results follow as theorems. This is not merely an aesthetic improvement: it implies that any physical system admitting a Stack representation automatically satisfies all of these properties, whether or not it is a string-theoretic AdS/CFT model. The Stack is thus a sufficient condition for holographic behavior.

Universality beyond AdS/CFT. The Stack axioms (OS1)–(OS5) make no reference to anti-de Sitter geometry, conformal symmetry, or large-N limits. They apply, at least in principle, to any stratified tower of von Neumann algebras with the appropriate modular and entanglement properties. This opens the possibility of extending the framework to de Sitter holography (where the holographic screen is the cosmological horizon), flat-space holography (Carrollian symmetry at null infinity), and even non-relativistic holographic systems. The main challenge in the de Sitter case is the presence of a cosmological horizon, which introduces an observer-dependence into the algebra structure that does not fit neatly into the fixed Stack axioms. We return to this below.

Categorical structure. The collection of all Operator Stacks, with morphisms defined as state-preserving layer maps compatible with the conditional expectations, forms a category Stack_vN. Holographic RG flows correspond to functors between Stacks: a holographic flow from a UV theory to an IR theory is a functor F: Stack_UV → Stack_IR that maps each layer of the UV Stack to a sub-layer of the IR Stack, compatibly with the conditional expectations and modular flows. The c-theorem (the monotonic decrease of the central charge under RG flow in two-dimensional CFTs) becomes a statement about the monotonicity of the entropy functional S(ρ_n) under the functor F. The categorical perspective also clarifies the role of dualities: two Stacks related by a duality (e.g., S-duality in string theory) are isomorphic objects in Stack_vN.

Implications for quantum gravity. Perhaps the deepest implication of the Stack framework is for the nature of spacetime itself. If the metric at holographic depth n emerges from the spectral geometry of (A_n, H_n, H_mod,n), then spacetime is not a fundamental ingredient of physics but an emergent structure, derived from the algebraic data of the quantum system. This aligns with the perspective advocated by Connes [15], Verlinde, and others, and provides a concrete algebraic mechanism for the emergence of geometry from entanglement; a mechanism that has been heuristically suggested by the “ER = EPR” correspondence of Maldacena and Susskind and by the work of Swingle [24] and Vidal [23] on tensor networks.

Connection to recent von Neumann algebraic approaches. The Stack framework makes direct contact with the recent program of Witten [16] and Chandrasekaran-Penington-Witten [17], who introduced Type II von Neumann algebras into holographic duality via the crossed product construction. In that framework, the gravitational algebra of the bulk (after dressing by the ADM Hamiltonian) is a Type II_∞ factor, which admits a well-defined von Neumann entropy. In the Stack language, this dressing corresponds to the passage from the Type III₁ bulk algebra A_N to a Type II algebra by incorporating the modular Hamiltonian H_mod,N as an additional generator. The generalized entropy of [17] is then identified with S(ρ_N) in the Stack’s terminal layer. Similarly, the emergent time of Leutheusser and Liu [32] (the reconstruction of bulk time from boundary modular flow) is realized in the Stack as the modular flow σ^t_AN, which generates the emergent bulk time evolution.

Limitations of the framework. Several important limitations must be acknowledged. First, the Stack axioms are currently postulated, not derived from first principles in string theory or any other UV-complete theory of quantum gravity. The axioms encode the expected properties of holographic systems, but the question of whether (and in which UV-complete theories) a Stack exists remains open. Second, the continuum limit N → ∞ (in which the discrete layers are replaced by a continuous holographic depth) requires careful analysis. In this limit, the conditional expectations E_n become infinitesimal generators of a continuous RG flow, and the modular coherence condition (OS2) must be reformulated as a differential equation. The operator-algebraic theory of such continuous towers is substantially more complex than the discrete case. Third, the de Sitter extension faces non-trivial obstacles: the cosmological horizon is observer-dependent, the natural state is the Bunch-Davies vacuum (which has specific thermal properties distinct from the AdS vacuum), and the absence of a well-defined bulk S-matrix complicates the holographic identification.

Open questions. Several fundamental questions remain. Can the Stack be derived from a UV-complete theory, such as string theory, by integrating out modes in the path integral? What physical principle selects the layer-scaling factors λ_n? Are the λ_n related to the beta function of the holographic RG? Can the discrete Stack be connected to tensor network models such as MERA [23] or the HaPPY code [25], perhaps by identifying each layer of the Stack with a level of the tensor network? Finally, the precise role of quantum gravity fluctuations (which render the bulk algebra Type II rather than Type III) within the Stack framework deserves systematic investigation.

8. Conclusion

We have introduced the Operator Stack (a stratified tower of von Neumann algebras {A_n}_n=0^N obeying five axioms of stratification, modular coherence, entanglement threading, boundary identification, and holographic completeness) and argued that it provides a rigorous algebraic framework for the Holographic Principle. The central thesis of this paper is that holographic encoding is structurally equivalent to the inter-layer modular flow and conditional expectation architecture of the Stack: the depth of the Stack encodes the depth of the holographic bulk, the conditional expectations encode the RG coarse-graining, and the lifting maps encode bulk reconstruction.

Within this framework, we have demonstrated that five major results of holographic quantum gravity emerge as theorems or natural consequences:

1. The Ryu-Takayanagi formula (including the FLM quantum correction) is derived from the entanglement structure of the Stack, with the RT surface identified as the algebraic boundary of the propagated subregion subalgebra (Section 4.2, Eq. 19).
2. The HKLL bulk reconstruction is identified with the composition of lifting maps, with the smearing function K(X,Y) as the integral kernel of the composed lifting (Section 3.2, Eq. 11).
3. The quantum error correction structure of AdS/CFT (subregion duality and entanglement wedge reconstruction) follows from the Petz recovery channel and the structure of the conditional expectations (Section 4.3, Theorem 2).
4. The Bousso covariant entropy bound is derived from the monotonicity of relative entropy under conditional expectations, without appeal to classical geometry (Section 5.1).
5. The linearized Einstein equations emerge as Stack consistency conditions, via the first law of entanglement and the modular coherence axiom (Section 5.2).

Beyond these specific results, the Stack framework situates holography within the broader landscape of operator-algebraic quantum theory, making contact with the Tomita-Takesaki theory, Connes’ noncommutative geometry, and the recent von Neumann algebraic approach to holography [16, 17].

The Operator Stack is a research program, not a complete theory. Its most urgent open questions concern its derivation from UV-complete physics. Three directions stand out for future work. First, deriving the Stack axioms from string theory: the path integral of string theory on AdS × M (M a compact manifold) should, when restricted to radial slices, produce a tower of operator algebras with the Stack properties. Second, extending the framework to de Sitter spacetime: this requires understanding holographic encoding in the presence of a cosmological horizon and is central to any realistic application to quantum cosmology. Third, making precise contact with tensor network models (MERA [23, 24], the HaPPY code [25]) which provide discrete, finite-dimensional approximations to holographic encoding and may serve as constructive models for discrete Operator Stacks.

We close with a reflection on the conceptual significance of the Stack. If spacetime geometry emerges from the modular spectral data of operator algebras, then the fundamental language of physics is not geometry but algebra, not fields on a manifold but operators in a Hilbert space. The Holographic Principle, in this light, is not a mysterious coincidence between theories in different dimensions, but the inevitable consequence of the algebraic structure of quantum information: a stratified tower of algebras, each encoding its predecessor, each generating its own emergent geometry from modular flow. The universe, at its deepest level, may be an Operator Stack.

Acknowledgments

The author thanks colleagues at the Theoretical Physics Institute for stimulating discussions on operator algebraic approaches to holography and emergent spacetime. The author is grateful for insightful conversations on modular flow, the covariant entropy bound, and the algebraic structure of AdS/CFT. This work was supported in part by internal research funds of the Theoretical Physics Institute. No external funding agencies or conflicts of interest to declare.

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© 2026 D. Costello. Manuscript submitted to Physical Review D. Preprint available at arXiv [placeholder]. All rights reserved.