Holography of Information

• Holography of Information is a principle where bulk data in gravitational systems is redundantly encoded near boundaries via emergent quantum information structures.
• It employs boundary Hamiltonians, bilocal reconstructions, and entanglement extremal surfaces to probe and recover bulk information.
• Extensions of HoI impact black-hole entropy, finite-radius reconstructions, and interdisciplinary domains such as thermodynamics and many-body theory.

Holography of Information (HoI) denotes a family of claims about redundancy, reconstruction, and encoding in which information ordinarily attributed to a bulk system is available on, or arbitrarily near, an appropriate boundary. In its strict quantum-gravity usage, HoI states that in a theory of quantum gravity a copy of all the information on a Cauchy slice is also available near the boundary of that slice, and that this redundancy is already present at low energy. In AdS/CFT and related formulations, the same idea is sharpened into the claim that geometry is an emergent organization of quantum information, with entanglement, extremal surfaces, and boundary algebras supplying the microscopic language. The term has also been extended to finite-radius reconstruction, evaporating black-hole models, sectorwise formulations in topology-summed gravity, and several broader programs in thermodynamics, many-body theory, cosmology, and information processing (Koch et al., 2022, Takayanagi, 7 Jun 2025, Gaddam et al., 2024).

1. Canonical statement and boundary observables

A canonical formulation of HoI is given by the question whether a suitable boundary Hamiltonian has a nontrivial Dirac bracket with bulk operators. In constrained Hamiltonian language, with Dirac matrix

$C_{ij}(x,y)\equiv \{\Phi_i(x),\Phi_j(y)\},$

the Dirac bracket is

$$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$

On this formulation, HoI holds when a boundary observable probes bulk data through a nonzero Dirac bracket, and fails when the corresponding bracket vanishes (Chakravarty et al., 2023).

For linearized massless gravity, the relevant boundary observable is obtained from the Gauss/Hamiltonian constraint. Integrating the constraint over a region $V$ gives

$$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$

The inverse Dirac matrix is nonlocal, with momentum-space factors like $1/\mathbf p^2$, and this nonlocality is the structural reason the boundary Hamiltonian acts on bulk operators. The resulting bracket takes the form

$$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$

so the boundary Hamiltonian brackets with bulk operators exactly as the integrated matter energy would. In this sense, bulk information is recoverable from asymptotic observables, and split states are absent (Chakravarty et al., 2023).

The same analysis yields a sharp contrast in linearized Fierz–Pauli massive gravity. The boundary operator can still be defined from the integrated constraint, but the full constraint matrix is local in momentum space rather than carrying the Gauss-law nonlocal inverse structure of the massless case. The resulting Dirac bracket with a strictly bulk insertion collapses to

$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$

The stated consequence is that asymptotic boundary observables cannot extract bulk information, local bulk operators remain hidden, and split states exist. The difference is attributed to diffeomorphism invariance: massless gravity ties physical observables to boundary dressings, whereas massive gravity explicitly breaks that redundancy (Chakravarty et al., 2023).

2. AdS/CFT formulation and bilocal holography

In AdS/CFT, HoI is translated into boundary CFT language by combining boundary vacuum cyclicity with a bilocal reconstruction of bulk fields. The gravity-side motivation uses the Reeh–Schlieder theorem together with the fact that the energy can be measured from near the boundary, so the vacuum projector belongs to the boundary algebra. The resulting claim is that every observable in the vacuum sector is a boundary observable. The AdS/CFT translation of this claim is carried out explicitly in bilocal holography for the free $O(N)$ vector model dual to higher-spin gravity in $AdS_4$ (Koch et al., 2022).

The basic collective variable is the bilocal field

$$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$

with the normal-ordered fluctuation

$$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$0

The bilocal-to-bulk map is

$$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$1

together with

$$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$2

Since $$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$3 grows with $$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$4, small-separation bilocals correspond to near-boundary bulk information, while large-separation bilocals probe deep interior bulk data (Koch et al., 2022).

The CFT statement of HoI is then an operator-product statement. Bilocals admit an OPE of the schematic form

$$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$5

where $$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$6 runs over local primaries and descendants. Because the relevant Lorentzian OPEs are convergent in the appropriate channels, a bilocal operator representing a deep-bulk excitation can be rewritten in terms of boundary-local operators. The paper emphasizes that bulk reconstruction is “fluid”: the same bulk point can be reached by infinitely many semicircles, so one can choose bilocals to avoid OPE obstructions. In that precise sense, the redundancy of information near the AdS boundary becomes an explicit CFT statement (Koch et al., 2022).

3. Entanglement, extremal surfaces, and emergent spacetime

A stronger informational interpretation identifies holographic spacetime itself as emergent from quantum information. The starting point is the black-hole entropy relation

$$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$7

which suggests that gravity organizes entropy by area rather than volume. In this view, a $$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$8-dimensional gravitational theory can be equivalent to a $$\{F(x_1),G(x_2)\}_{\text{D.B.}}=\{F(x_1),G(x_2)\}-\int dy_1\,dy_2\; \{F(x_1),\Phi_i(y_1)\}\, C^{-1}_{ij}(y_1,y_2)\, \{\Phi_j(y_2),G(x_2)\}.$$9-dimensional non-gravitational quantum theory on the boundary, and the deep reason holography works is that geometry encodes entanglement (Takayanagi, 7 Jun 2025).

The basic quantity is the entanglement entropy of a boundary subregion $V$0,

$V$1

with holographic prescription

$V$2

This is the Ryu–Takayanagi / Hubeny–Rangamani–Takayanagi bridge between quantum information and geometry. The prescription makes the area law natural, gives strong subadditivity and related entropy inequalities a geometric interpretation, and identifies the bulk dual of $V$3 as the entanglement wedge, the region enclosed by $V$4 and $V$5 (Takayanagi, 7 Jun 2025).

Quantum corrections are incorporated by the quantum extremal surface formula

$V$6

where $V$7 is the bulk entanglement entropy in the entanglement wedge $V$8. A further generalization is the island formula,

$V$9

This formula is described as crucial for black hole evaporation and the Page curve, with replica wormholes contributing to its derivation, although the Lorentzian real-time mechanism remains incompletely understood (Takayanagi, 7 Jun 2025).

The same essay gives a microscopic heuristic: each Planck-scale patch of area may be associated with entangled qubits, often heuristically like Bell pairs living on extremal surfaces. Spacetime is then a collective manifestation of a vast entanglement network, and the bulk gravitational field encodes the dynamics of entanglement among qubits. Tensor networks, especially MERA-like networks in which entanglement entropy is estimated by counting bonds crossing a minimal cut, serve as toy models of this geometry-from-circuit structure. A major open problem is the identification of the quantum circuit architecture corresponding to a given holographic spacetime, together with the relation of bulk volume or Wheeler–DeWitt action to quantum complexity (Takayanagi, 7 Jun 2025).

The informational program is extended beyond AdS by the proposal of pseudo-entropy and time-like entanglement. For initial and final states $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$0 and $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$1, one defines

$$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$2

Because $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$3 is generally non-Hermitian, pseudo-entropy may be complex. The proposal is that the imaginary part of pseudo-entropy, together with time-like entanglement and partly time-like extremal surfaces, may provide a clue to how the time direction emerges in Lorentzian or cosmological holography, particularly in dS/CFT where central charges and geodesic lengths can become complex (Takayanagi, 7 Jun 2025).

4. Finite-radius and low-dimensional realizations

HoI is not restricted to asymptotic infinity. One finite-distance formulation asks whether information inside a ball of finite radius $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$4 in flat spacetime can already be reconstructed from observables on the boundary of that ball. The setting is $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$5-dimensional Minkowski space with linearised quantum gravity minimally coupled to a free Klein–Gordon scalar, on the $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$6 Cauchy slice. States are unitary excitations of the vacuum built from normal-ordered products of $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$7 and $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$8, with a small perturbative parameter $$H_{\partial}=\frac{1}{16\pi G_N}\int_{\partial V} d^{D-2}x\; n^i\!\left(\partial_j h_{ij}-\partial_i h^k{}_k\right).$$9, and a crucial assumption is that the state has no support in a small neighborhood of the boundary where the measurements are made (Gaddam et al., 2024).

The boundary observable is the Hamiltonian inside the ball,

$1/\mathbf p^2$0

Because the theory is coupled to linearised gravity, this finite-volume Hamiltonian can be rewritten as a boundary term on the sphere $1/\mathbf p^2$1. The central result is that boundary correlators with an insertion of the finite-radius Hamiltonian determine the same support functions $1/\mathbf p^2$2 as the bulk correlators do. For the most general unitaries containing both $1/\mathbf p^2$3 and $1/\mathbf p^2$4 insertions, the result is established to first order in $1/\mathbf p^2$5; for the infinite subclass generated only by $1/\mathbf p^2$6-insertions or only by $1/\mathbf p^2$7-insertions, the result is perturbatively exact to all orders in $1/\mathbf p^2$8. The same framework treats massive and massless scalar fields simultaneously (Gaddam et al., 2024).

A second explicit realization appears in the CGHS model coupled to a massless scalar. Here the scalar decomposes into chiral components,

$1/\mathbf p^2$9

and right future null infinity $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$0 is naturally associated with the right-moving sector $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$1. The reduced radiative phase space on $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$2 has symplectic form

$$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$3

and the right Hamiltonian is

$$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$4

Adding the Bondi mass at all cuts to the radiative algebra yields a gravitationally completed asymptotic algebra, and under the usual HoI assumptions the paper proves

$$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$5

for every $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$6. The quantum state of the right-moving modes can therefore be recovered from an arbitrarily small neighborhood of the past end of $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$7, and the corresponding Page curve is flat (Dar et al., 8 Jun 2026).

Setting	Boundary data	Result
Finite ball in linearised gravity (Gaddam et al., 2024)	Correlators with $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$8 on $$\{H_{\partial},O(z)\}_{\text{D.B.}}=\int_V d^d x\,\{T_{00}(x),O(z)\},$$9	Boundary data determine the same support functions $\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$0 as bulk correlators
CGHS model (Dar et al., 8 Jun 2026)	$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$1 near $\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$2	$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$3 and the right-moving state is recoverable

These finite-radius and low-dimensional results shift HoI from an asymptotic slogan to controlled reconstruction statements. They also delimit scope: the finite-ball theorem is perturbative and low-energy, while the CGHS result reconstructs the right-moving sector rather than the full two-boundary state (Gaddam et al., 2024, Dar et al., 8 Jun 2026).

5. Sectorwise holography, islands, and factorization problems

Recent work has emphasized that HoI need not be a global statement on the full topology-summed Hilbert space. If topology-changing saddles generate genuine baby-universe or $\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$4-sector data, then the physical Hilbert space decomposes sectorwise,

$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$5

and asymptotic completeness is naturally reformulated as

$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$6

On this account, HoI survives only as a fixed-$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$7 statement unless additional input makes the sector label asymptotically accessible, aligns reconstructions across sectors, or removes the sector structure entirely (Kumar, 19 May 2026).

This refinement has consequences for black-hole interior reconstruction. In a fixed $\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$8-sector, HoI may still obstruct AMPS factorization and permit a smooth horizon. In an unconditioned topology-summed state, however, the obstruction is not automatic. The Bell-pair diagnostic uses

$\{H_{\partial},O(z)\}_{\text{D.B.}}=0.$9

and evaluates the fidelity of a sector-independent reconstruction by

$O(N)$0

The conclusion is that sectorwise smoothness does not imply sector-independent smoothness: a single global smooth interior exists only if the $O(N)$1 are aligned up to phases, if the observer can resolve $O(N)$2, or if there is no genuine $O(N)$3-structure. The resulting tension is with the expectation that horizon normalcy should be determined by local semiclassical geometry. If the Baby Universe Hypothesis holds and $O(N)$4, the obstruction disappears (Kumar, 19 May 2026).

A different factorization issue appears in the CGHS/RST comparison between HoI and the island formula. In the CGHS analysis, HoI gives a flat Page curve because the gravitationally completed asymptotic algebra already equals $O(N)$5. The authors then argue that if Hilbert-space factorization between left- and right-moving sectors,

$O(N)$6

holds in the quantum theory, then the island entropy cannot be identified with the von Neumann entropy of any one-sided asymptotic algebra. On this view, the discrepancy between HoI and islands is not exhausted by a choice between radiative and gravitationally completed algebras; it becomes a version of the factorization problem for CGHS/RST models (Dar et al., 8 Jun 2026).

These analyses jointly imply that HoI is conditional on representation, gauge structure, and sector decomposition. In that restricted sense, the principle is neither a universal substitute for nonperturbative quantum gravity nor a uniform statement about all boundary algebras.

6. Broader extensions and analogical usages

Beyond the strict gravitational principle, “holography of information” and closely related holographic language have been extended to several other domains. Some of these are presented as microscopic realizations of area-scaling information storage, others as analogical or operational encodings on lower-dimensional data.

Domain	Representative formulation	Core claim
Thermodynamics (Wei et al., 2014)	Partition function on a contour in complex parameter space	Boundary values of $O(N)$7 determine thermodynamics inside an analytic region
Cosmology (Gough, 2011)	Landauer + holographic scaling	Information energy can behave like dark energy
Gauge boundaries (Dvali et al., 2015)	Boundary Stückelberg qubits	Information capacity is controlled by the Stückelberg energy gap
Many-body criticality (Dvali, 2018)	Gapless modes on a critical level	Area-scaling memory space with “burden of memory” stabilization
Screens and computation (Putten, 2013)	Information from transition probabilities across a screen	Spherical screens encode $O(N)$8 and imply a finite computation bound
Magnonic devices (Gertz et al., 2014)	Spin-wave interference in magnetic matrices	Information is recovered from distributed interference patterns
Topological coding (Stokowski et al., 2020)	Pure braid group generators	Information is represented in a two-dimensional topological formalism
Holographic algorithms (Al-Bashabsheh et al., 2010)	Holant theorem and c-tensor	Global sums are invariant under suitable holographic basis transformations

In thermodynamic holography, the partition function

$O(N)$9

is treated as an analytic function of a complexified physical parameter, so that Cauchy’s formula reconstructs interior values from boundary data,

$AdS_4$0

The experimental link is probe-spin coherence,

$AdS_4$1

which directly measures the partition function on a vertical line in the complex plane and therefore permits holographic reconstruction of free-energy differences (Wei et al., 2014).

In the Stückelberg formulation, an information boundary forces the introduction of boundary Stückelberg degrees of freedom that act as qubits. Their information capacity is measured by

$AdS_4$2

and for a massless gauge system the gap scales as $AdS_4$3, giving

$AdS_4$4

For gravity, $AdS_4$5, so the resulting capacity scales as $AdS_4$6 and saturates the Bekenstein bound. The paper identifies this with maximal holography and proposes a correspondence between boundary Stückelberg modes and Bogoliubov modes of critical many-body systems (Dvali et al., 2015).

The many-body model of bosons on $AdS_4$7 gives an explicit microscopic realization of area-law storage. At a critical occupation number of a master mode, a level of gapless modes emerges whose degeneracy scales as

$AdS_4$8

and the binary memory space has entropy

$AdS_4$9

The distinctive dynamical claim is the “burden of memory”: when the system evolves away from criticality, patterns loaded into the gapless sector increase the energy cost of departure from the holographic state, so heavily loaded memories survive longer and information is off-loaded only by transitions between successive holographic states, with scrambling and entanglement during the transfer (Dvali, 2018).

A cosmological extension combines Landauer’s principle with holographic scaling. The information energy is defined as

$$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$0

with $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$1 from holographic scaling of information content and, at late times, $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$2. The resulting information energy density becomes approximately constant for $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$3, with effective $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$4, and is proposed as a model of dark energy with a finite acceleration epoch tied to star formation history (Gough, 2011).

Other works use holographic language more operationally. A computational-screen analysis defines information as the logarithm of the exponentially small transition probability across a screen, yielding $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$5 for a plane and $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$6 for a spherical screen, and from this derives a bound of about $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$7 future computations in the visible Universe (Putten, 2013). Magnonic holographic memory implements distributed encoding in spin-wave interference patterns within a YIG magnetic matrix, where local junction magnets modulate phases and the stored configuration is reconstructed from the output hologram (Gertz et al., 2014). A pure braid group proposal treats classical information as effectively two-dimensional and encodes symbols as pure braid generators $$\sigma(x^+,x_1^-,x_1,x_2^-,x_2)=\phi^a(x^+,x_1^-,x_1)\phi^a(x^+,x_2^-,x_2),$$8, while a proof of the Holant theorem supplies the c-tensor formalism behind holographic algorithms used in information-theoretic counting problems (Stokowski et al., 2020, Al-Bashabsheh et al., 2010).

Taken together, these broader programs suggest that “holography of information” now names both a specific gravitational redundancy principle and a wider research style in which lower-dimensional, boundary, or collective data suffice to determine a higher-dimensional or more distributed informational structure. In the narrow sense, HoI concerns bulk reconstruction in quantum gravity; in the broad sense, it names a recurrent strategy for encoding global structure in constrained boundary data.