Holography and Quantum Gravity

Holography relates a gravitational theory in a spacetime with boundary to a nongravitational quantum theory associated with that boundary. Quantum gravity is broader: it asks for a mathematically and physically complete framework whose observables, states, dynamics, semiclassical limit, and empirical consequences are all defined. This volume teaches both subjects by attaching every conclusion to a specific dictionary, observable, state or ensemble, approximation hierarchy, and test. A successful calculation is never promoted silently into an exact duality, a unique nonperturbative definition, or evidence about nature.

Helpful background. Large-N Limits, Normalizations, and Orders of Limits supplies the expansion language used throughout. Conformal OPE data and large-N sparse spectra supply boundary inputs. Duality claims and evidence distinguish a dictionary from a proof. Thermal Density Operators and the KMS Condition and Entropy of a Regulated Subregion supply imported statistical and information-theoretic notions. Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity and the gravity-regime taxonomy mark the low-energy boundary of this volume.

Choose the claim before choosing the method

A determinate holographic or quantum-gravity question starts with the tuple

\mathfrak C= (\mathcal T_1,\mathcal T_2;\mathcal H,\mathcal A,\rho; \mathcal B,\boldsymbol\lambda,\boldsymbol\epsilon;\mathcal O),

where the two descriptions $\mathcal T_{1,2}$ , Hilbert-space or algebraic sector $(\mathcal H,\mathcal A)$ , state or ensemble $\rho$ , boundary and global data $\mathcal B$ , parameters $\boldsymbol\lambda$ , control parameters $\boldsymbol\epsilon$ , and observable $\mathcal O$ are fixed. The claim must then state whether it is a definition, theorem in a restricted model, protected equality, perturbative or saddle expansion, numerical result, conjectural dictionary, or empirical constraint.

This discipline matters because several famous statements have the same informal name but different logical content. The GKPW relation is a generating-functional dictionary with normalization and boundary-condition data. Large- $N$ factorization is an asymptotic statement about correlators. The Ryu–Takayanagi formula is a leading semiclassical entropy prescription in a specified state and geometry. An island saddle is a contribution to a generalized-entropy calculation. None alone proves a complete microscopic equivalence or an exact unitary evaporation process.

Use the following question to choose an entrance:

Present task	Begin with	Stop when you can state
evaluate a proposed duality	claims, observables, and evidence	both theories, every global datum, observable map, limit, evidence class, and falsifier
infer a semiclassical bulk from CFT data	large-N bulk criteria	which criteria are necessary, which are only suggestive, and the finite-gap or finite- $N$ error
compute an AdS correlator	AdS dictionary → holographic renormalization → Witten diagrams	boundary condition, source normalization, counterterm scheme, contour, and perturbative order
connect a holographic model to string theory	top-down constructions	decoupling limit, compact space, flux, KK scale, string scale, loop parameter, and consistent truncation
study horizons, response, or transport	thermal phases → real-time holography → holographic matter	ensemble, state preparation, causal prescription, observable, and evidence ceiling
reconstruct a bulk observable	bulk reconstruction → entropy geometry → entanglement wedges and QEC	dressing, region, algebra, code subspace, recovery norm, and approximation error
assess black-hole information claims	microstates → path integrals → information and islands	microscopic quantity, saddle prescription, factorization assumption, entropy result, and what remains uncomputed
compare quantum-gravity programs	consistency and swampland → canonical and loop programs → safety and discrete programs	defined objects, physical observables, continuum limit, semiclassical recovery, and falsifier
interpret a proposed signal	quantum cosmology → phenomenology	mechanism, nuisance model, likelihood, alternatives, and the strongest attainable inference

The twenty-nine chapters

Each chapter owns one layer of the argument. The order below is the canonical navigation order, but the routes are goal-dependent rather than a single compulsory sequence.

Claims, dictionaries, and controlled bulk calculations

Chapter	Central question
Quantum-Gravity Claims, Observables, and Evidence	What makes a duality, completion, or empirical claim logically and operationally definite?
Large N and Semiclassical Bulk Criteria	Which independent spectral and correlator conditions support a weakly coupled, approximately local bulk?
AdS Geometry, Boundary Problems, and the Dictionary	Which patch, boundary condition, field/operator map, global sector, and normalization define an AdS observable?
String, Brane, and Top-Down Constructions	How do decoupling, flux, compactification, KK modes, and corrections establish controlled embeddings?
Nonperturbative String- and M-Theory Definition Proposals	Which object each matrix or string-field construction defines, and what completeness test it meets?
Holographic Renormalization and Radial Dynamics	How are finite variational data, one-point functions, Ward identities, and anomalies extracted at an AdS boundary?
Witten Diagrams and AdS Perturbation Theory	How do propagators, contact and exchange diagrams, loops, and counterterms produce normalized boundary correlators?
AdS Scattering, Mellin Methods, and Bulk Locality	When do Mellin poles, Regge behavior, gap hierarchies, and flat limits support a bulk scattering interpretation?

Thermal physics, reconstruction, and information

Chapter	Central question
Thermal Phases and AdS Black Holes	Which ensemble and saddle determine a phase, entropy, instability, or finite- $N$ smoothing effect?
Lorentzian, Nonequilibrium, and Chaotic Holography	Which contour, state, horizon condition, and time window define real-time response and scrambling?
Holographic Matter, Transport, and Model Building	What mechanism a gravitational model demonstrates, and why that need not identify a microscopic material?
Bulk Reconstruction and Gravitational Dressing	Which boundary algebra reconstructs which dressed bulk observable, in which perturbative region?
Holographic Entropy and Quantum Extremal Geometry	How do RT, HRT, FLM, QES, replicas, and higher-derivative functionals differ and connect?
Modular Response, Relative Entropy, and Emergent-Gravity Claims	Under which semiclassical assumptions do information-theoretic identities imply canonical energy or field equations?
Entanglement Wedges and Holographic Quantum Error Correction	What is exact in toy codes, leading order in gravity, or approximate at finite $N$ ?
Holographic Complexity Proposals and Diagnostics	What CV, CA, path-integral, and tensor-network proposals compute, and where their nonuniqueness enters?

Controlled laboratories and black holes

Chapter	Central question
AdS3/CFT2 and Three-Dimensional Gravity	Which exact two-dimensional CFT data control BTZ and three-dimensional gravity, and what remains unsettled about pure gravity?
AdS2, JT Gravity, SYK, and Random Matrices	Which near-AdS2 observables are captured by the Schwarzian, SYK, topology expansion, or an ensemble completion?
Black-Hole Microstates and Stringy Entropy	When does an index equal a degeneracy, and how do brane counts and corrections compare with gravitational entropy?
Wormholes, Gravitational Path Integrals, and Ensembles	Which contour and theory define a topology sum, and whether its connected term belongs to a fixed theory or an ensemble?
Black-Hole Information, Islands, and Interiors	What Page curves and islands establish, and what microscopic dynamics, decoding, interiors, and endpoints they leave open?

Beyond AdS and comparative quantum gravity

Chapter	Central question
Higher-Spin and Vector-Model Holography	How do higher-spin symmetries and vector-model data define a non-Einstein holographic regime?
de Sitter and Cosmological Holography	Which wavefunction, in-in, or static-patch object a proposed cosmological dictionary relates?
Flat-Space, BMS, and Celestial Holography	How are null-infinity data, soft sectors, celestial transforms, and completeness claims distinguished?
Quantum-Gravity Consistency and Swampland	Which statements are conditional results, conjectures, examples, counterexamples, or phenomenological constraints?
Canonical, Loop, Spin-Foam, and Group-Field Quantum Gravity	How are constraints quantized, physical observables defined, and continuum and semiclassical recovery tested?
Asymptotic Safety, Causal, and Discrete Quantum-Gravity Programs	What evidence a truncation, triangulation, causal set, or random geometry gives for a continuum quantum theory?
Quantum Cosmology and Singularity-Resolution Programs	Which state, clock, probability rule, approximation, and spacetime criterion make a cosmological prediction or resolution claim meaningful?
Quantum-Gravity Phenomenology and Comparative Status	Which observations can constrain a mechanism, and why a bound or anomaly rarely selects a unique UV completion?

Claim and handoff matrix

The same evidence word can license different conclusions in different domains. This matrix provides the volume-wide comparison basis; each chapter supplies the detailed assumptions and counterexamples.

Domain	Object and observable	Controlled hierarchy	Strongest licensed conclusion	Characteristic falsifier or failure	Canonical continuation
AdS/CFT dictionary	renormalized boundary correlators and bulk boundary data	large $N$ , large coupling or protected sector, fixed boundary conditions	equality or asymptotic match for stated observables	normalization, global form, or boundary-condition mismatch	Chapters 2–8
top-down construction	brane decoupling limit and compactification spectrum	$g_s$ , $\alpha'/L^2$ , KK scale, backreaction	controlled gravitational regime of a specified construction	no scale separation or inconsistent truncation	Chapters 4–5
thermal holography	partition function, free energy, response functions	saddle expansion, derivative expansion, finite- $N$ corrections	phase or response in the specified ensemble and time window	wrong saddle, unstable mode, or nonuniform late-time limit	Chapters 9–11
reconstruction	dressed bulk operator and boundary algebra	perturbation theory, code subspace, region, recovery error	reconstruction within the stated algebra and accuracy	complement detects the operator or error fails to shrink	Chapters 12 and 15
entropy geometry	regulated entropy or relative entropy	semiclassical $G_N$ , loop order, replica continuation	extremal-surface or modular statement under its hypotheses	competing saddle, replica ambiguity, or missing counterterm	Chapters 13–14
low-dimensional laboratory	Schwarzian/JT/BTZ/SYK or protected brane sector	low energy, large $N$ , genus or index expansion	exact or controlled result in that model	extrapolation beyond the model or ensemble	Chapters 17–19
gravitational path integral	boundary amplitude and chosen integration cycle	saddle and topology expansion	asymptotic amplitude for the declared contour	negative mode, contour ambiguity, or factorization conflict	Chapter 20
black-hole information	fine-grained radiation entropy and reconstructable algebra	semiclassical gravity plus bath and replica assumptions	Page-type entropy transition or wedge statement	microscopic spectrum/dynamics fails to realize assumptions	Chapter 21
non-AdS holography	wavefunction, S-matrix, null-boundary data, or celestial correlator	asymptotic, soft, loop, or analytic-continuation control	proposed dictionary component or consistency test	incomplete observable algebra or failed unitarity/completeness	Chapters 22–24
consistency program	charge spectrum, moduli limit, or black-hole process	EFT control and stated quantum-gravity assumptions	theorem conditional on a duality, or a named conjectural constraint	controlled counterexample or failed hypothesis	Chapter 25
alternative QG program	constraints, histories, discrete structures, or RG flow	regulator, truncation, refinement, and continuum scaling	property of a defined model or convergent sequence	anomaly, no continuum limit, or no semiclassical sector	Chapters 26–28
phenomenology	detector-level observable and likelihood	EFT mechanism, calibration, nuisance and statistical errors	bound, sensitivity, anomaly, or detection of the stated effect	competing explanation or loss of robustness	Chapter 29 and Research

Shared conventions and recurrent limit checks

The site uses the Lorentzian signature $(+,-,\ldots,-)$ , natural units, and the curvature convention

[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma.

Euclidean gravitational pages state their positive-definite metric, action, integration cycle, boundary terms, and continuation back to a specified Lorentzian observable. The normal orientation and extrinsic-curvature sign are always local data. AdS pages usually write the bulk dimension as $d+1$ and the boundary dimension as $d$ ; departures are stated explicitly.

Four limit-order questions recur:

Is $N\to\infty$ taken before the late-time, high-energy, or large-entropy limit?
Is the string scale removed before or after the low-energy and flat-space limits?
Is a replica continuation performed before choosing the dominant saddle, and is that continuation unique?
Is regulator removal uniform in the region, state family, operator norm, or observable being claimed?

If changing the order changes the result, the claim includes the order. “Semiclassical,” “nonperturbative,” “emergent,” and “resolved” are not substitutes for an error estimate or operational test.

Live questions and evidence ceilings

Several topics are active disputes rather than settled conclusions. Finite- $N$ entanglement-wedge reconstruction, for example, must distinguish exponentially small reconstruction corrections in one approximate-code framework Witten 2026 from the argument that ordinary finite- $N$ holographic CFTs do not realize the stronger shared-logical-operator form of holographic QEC Terashima 2026. These are competing research claims, not a consensus verdict inserted into a textbook formula.

Likewise, the claim that baby-universe Hilbert spaces are one-dimensional under swampland assumptions McNamara and Vafa 2020 must be read alongside constructions arguing for encoded closed-universe sectors under additional entanglement and final-state assumptions Antonini et al. 2025. A recent landscape exclusion argument explicitly assumes a brane-worldvolume origin for stable AdS duals Bedroya and Steinhardt 2025; its conclusion must retain that premise.

The examples in this section were checked through 10 August 2026. The durable lesson is the comparison method, not a frozen verdict. Current bounds, newly proposed counterexamples, priority disputes, and confidence assessments continue in the holography and quantum-gravity Research map. No page in this volume should be read as reporting a confirmed empirical detection of quantum gravity.

Check your route

A correlator claim. You are given a Mellin amplitude with exchange poles. Before calling it a local bulk amplitude, identify the Gamma-function convention, contact ambiguity, Regge growth, spectral gap, flat-limit scaling, and finite-gap remainder. The shortest route is Chapters 2, 3, 7, and 8.

An information claim. You are given a semiclassical Page curve. A complete analysis still asks which radiation algebra factorizes, which saddle and analytic continuation were used, whether a fixed theory or ensemble was computed, what decoding task is defined, and which microscopic endpoint dynamics is absent. The shortest route is Chapters 13, 15, 20, and 21.

A phenomenology claim. You are given an energy-dependent time delay. A licensed inference requires an EFT mechanism, source and detector systematics, intrinsic-emission nuisance model, population likelihood, independent channel, and comparison with Lorentz-invariant alternatives. The shortest route is Chapters 25 and 29, with the experimental record kept in Research.

Chapter structure diagrams

Each chapter diagram is embedded once on its chapter overview, where the surrounding prose and semantic comparison table supply its full context. The compact list below preserves direct access without repeating twenty-nine wide diagrams on this page.

Open the chapter-diagram index

Volume claim-domain and handoff table

The table below is the linear, searchable counterpart to the volume route map. Each row names the object, required declarations, approximation and evidence ceiling, a falsifier, the failure condition, and the chapter that owns the detailed treatment.

Claim domains, failure conditions, and destinations across Volume XV
Domain and destination	Object and observable	State, ensemble, and conventions	Approximation, status, and evidence timing	Uncertainty, counterevidence, and falsifier	Failure condition	Licensed conclusion
Quantum-Gravity Claims, Observables, and Evidence	dictionary; saddle result; phenomenological proposal	Declare both theories and parameter map; state, contour, and expansion order; mechanism, nuisance model, and data; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: theory pair and global data → state, algebra, and observable → claim class and regime → evidence and falsifier → bounded conclusion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “round-trip one protected observable; vary saddle and limit order; compare independent alternatives” check is counterevidence to the promoted claim.	exact equivalence; nonperturbative completion; evidence for a unique UV theory	A quantum-gravity claim becomes testable only after its theories, observable, regime, evidence class, and falsifier are fixed.
Large N and Semiclassical Bulk Criteria	factorization; large gap; finite-N sector	Declare operator normalization and N scaling; single-trace sector and gap definition; order of N, time, and energy limits; use the volume conventions unless the page states a local replacement.	Conditional theorem or structural result. Control chain: normalized large-N data → factorization and spectrum → gap and coupling hierarchy → locality and correction tests → semiclassical-bulk criterion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “connected-correlator hierarchy; Regge and finite-gap tests; exponential and recurrence checks” check is counterevidence to the promoted claim.	geometric locality; a sufficient local bulk; exact late-time behavior	Factorization, sparsity, a higher-spin gap, and controlled corrections are independent criteria; large N alone does not imply Einstein gravity.
AdS Geometry, Boundary Problems, and the Dictionary	scalar mode; current or stress tensor; state dictionary	Declare mass, patch, and boundary condition; gauge group, charges, and counterterms; Euclidean cap or Lorentzian initial data; use the volume conventions unless the page states a local replacement.	Dictionary entry or correspondence claim. Control chain: AdS patch and conformal boundary → boundary and global conditions → asymptotic field modes → renormalized source and response → operator dictionary. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “normalizability and flux test; Ward identity and two-point normalization; causal and inner-product check” check is counterevidence to the promoted claim.	a branch outside its stability window; complete global equivalence; all heavy states have smooth geometries	An AdS/CFT entry is defined by geometry, boundary conditions, global sector, quantization branch, and normalization—not by a mass-dimension formula alone.
String, Brane, and Top-Down Constructions	brane decoupling; lower-dimensional truncation; supergravity regime	Declare charges, energy scaling, and asymptotics; retained fields and compact geometry; curvature, string scale, and string coupling; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: worldsheet and brane data → decoupling or near-horizon limit → flux and compact factors → truncation and corrections → top-down dictionary. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “open/closed and near-horizon limit; uplift every lower-dimensional solution; alpha-prime and loop estimates” check is counterevidence to the promoted claim.	a complete dictionary at finite cutoff; the complete compactified theory; a nonperturbative definition	Top-down control requires separate checks of decoupling, flux quantization, Kaluza-Klein scales, truncation, string corrections, and loops.
Nonperturbative String- and M-Theory Definition Proposals	matrix quantum mechanics; matrix or string field model; AdS/CFT as definition	Declare large-N limit and compactification sector; gauge fixing, background, and observables; exact boundary theory and global data; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: defining variables and action → Hamiltonian or path integral → observable and sector map → spacetime recovery tests → typed definition claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “spectrum and scattering benchmarks; unitarity and spacetime-recovery tests; bulk dictionary completeness” check is counterevidence to the promoted claim.	manifest covariant completeness; all nonperturbative backgrounds; a definition of arbitrary quantum gravity	BFSS, BMN, IKKT, matrix strings, string field theory, and AdS/CFT define different objects and satisfy different completeness tests.
Holographic Renormalization and Radial Dynamics	divergent on-shell action; one-point function; radial flow	Declare radial coordinate and induced fields; source normalization and counterterm action; cutoff surface and boundary conditions; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: Fefferman-Graham cutoff data → radial Hamilton-Jacobi recursion → local and finite counterterms → one-point functions and Ward identities → renormalized observable. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Hamilton-Jacobi cancellation; Ward and anomaly identities; canonical transformation check” check is counterevidence to the promoted claim.	scheme-independent finite contact terms; a bare bulk coefficient; ordinary Wilsonian RG without qualifications	Holographic renormalization fixes divergences but leaves declared finite schemes, boundary conditions, and contact terms in the physical answer.
Witten Diagrams and AdS Perturbation Theory	contact diagram; exchange diagram; loop diagram	Declare vertex, measure, and source normalization; field spectrum and propagator branch; regulator, counterterms, and contour; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary sources and bulk action → propagators and vertices → AdS diagram integral → OPE and analytic checks → perturbative boundary correlator. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “crossing and short-distance check; pole and conformal-block decomposition; unitarity cut and renormalization check” check is counterevidence to the promoted claim.	a unique bulk Lagrangian; a flat-space amplitude; an exact finite-N correlator	A Witten diagram is an AdS perturbative contribution whose normalization, boundary conditions, contact terms, and loop counterterms remain explicit.
AdS Scattering, Mellin Methods, and Bulk Locality	Mellin pole; flat-space limit; bulk locality	Declare Gamma measure, contour, and normalization; wavepackets and AdS-radius scaling; gap, Regge bound, and EFT window; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: CFT and Mellin data → exchange poles and contacts → Regge and gap controls → flat-space scaling limit → bounded locality claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “recover OPE data and residues; unitarity and normalization check; finite-gap remainder estimate” check is counterevidence to the promoted claim.	a unique local particle exchange; finite-radius S-matrix equality; exact locality at finite gap	Mellin poles, bounded growth, a large gap, and a controlled flat limit support different parts of a bulk-scattering interpretation.
Thermal Phases and AdS Black Holes	Hawking-Page transition; charged or rotating saddle; higher-derivative entropy	Declare boundary topology and canonical ensemble; chemical potentials and regularity; covariant action and boundary terms; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: ensemble and boundary data → Euclidean gravitational saddles → action, entropy, and charges → stability and finite-N checks → thermal phase claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “renormalized free-energy crossing; first law and Hessian stability; Wald and Euclidean checks” check is counterevidence to the promoted claim.	a sharp finite-N transition; global stability in every ensemble; microscopic state counting	A holographic phase is an ensemble-specific saddle comparison; classical dominance does not determine exact finite-N spectra or recurrences.
Lorentzian, Nonequilibrium, and Chaotic Holography	retarded response; quasinormal mode; scrambling or plateau	Declare source convention and infalling condition; background, channel, and boundary condition; operator ordering, N, and time limits; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: real-time contour and state → Lorentzian bulk geometry → initial and horizon conditions → poles, response, and chaos tests → time-window conclusion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “flux and Kramers-Kronig checks; pole and convergence test; OTOC or spectral benchmark” check is counterevidence to the promoted claim.	all Schwinger-Keldysh correlators; nonlinear thermalization time; exact finite-N irreversibility	Real-time holography depends on contour, state preparation, causal boundary conditions, and a declared time window; horizons do not fix exact late time.
Holographic Matter, Transport, and Model Building	transport coefficient; infrared scaling phase; bottom-up model	Declare current normalization and Kubo limit; matter content and boundary conditions; operator map and parameter calibration; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: bulk matter action and state → charged or scaling background → horizon and boundary response → transport and robustness tests → bounded mechanism claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Ward, positivity, and horizon checks; irrelevant-deformation stability; competing models and data residuals” check is counterevidence to the promoted claim.	a universal material value; microscopic phase identification; top-down UV completion	A holographic matter model can demonstrate a mechanism or universality class without identifying the microscopic theory of a material or plasma.
Bulk Reconstruction and Gravitational Dressing	HKLL field; dressed observable; subregion reconstruction	Declare free equation, patch, and normalizable modes; anchoring and gauge convention; region, code sector, and norm; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary region and algebra → bulk equation and smearing → interaction and gravitational dressing → causal and error checks → reconstructable observable. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “boundary limit and commutator check; Gauss-law and boundary-charge check; complement and overlap consistency” check is counterevidence to the promoted claim.	exact finite-N local operator; strict compact support; state-independent global locality	Bulk reconstruction is always relative to a region, algebra, dressing, state sector, perturbative order, and error notion.
Holographic Entropy and Quantum Extremal Geometry	RT or HRT surface; FLM or QES; replica derivation	Declare state, region, homology, and causal setting; bulk entropy scheme and Newton coupling; integer replicas and continuation rule; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: regulated state and entropy → extremal or generalized functional → replica and surface selection → maximin and correction checks → geometric entropy claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “maximin and competing-surface check; generalized-entropy extremization; cone regularity and saddle competition” check is counterevidence to the promoted claim.	exact finite-N entropy; microscopic factorization; unique nonperturbative continuation	RT, HRT, FLM, and QES are distinct prescriptions with different orders in gravity, saddle choices, and renormalization requirements.
Modular Response, Relative Entropy, and Emergent-Gravity Claims	entanglement first law; relative-entropy equality; field-equation inference	Declare reference state and modular operator; boundary and bulk algebras plus area term; all balls or shapes and local dynamics assumptions; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: boundary relative entropy → modular flow and first law → bulk canonical energy → positivity and shape checks → conditional gravity inference. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “linear-response identity; second-order positivity check; constraint and boundary-term check” check is counterevidence to the promoted claim.	nonlinear field equations; exact equality for all states; a unique quantum-gravity action	Information identities constrain gravity only after the dictionary, code sector, semiclassical expansion, and boundary terms are fixed.
Entanglement Wedges and Holographic Quantum Error Correction	toy tensor code; semiclassical wedge; finite-N recovery	Declare finite Hilbert spaces and exact isometry; code subspace, center, and area term; operator class and error norm; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: code subspace and algebras → encoding and recovery maps → wedge and area-center data → error norm and finite-N test → qualified QEC claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “erasure-recovery identity; relative entropy and complementary recovery; optimal recovery and complement test” check is counterevidence to the promoted claim.	a theorem about gravity; exact finite-N isometry; consensus on a shared logical algebra	Exact toy-code recovery, leading semiclassical wedge reconstruction, and finite-N holographic QEC are different statements with different algebras and errors.
Holographic Complexity Proposals and Diagnostics	complexity equals volume; complexity equals action; complexity bound	Declare slice, length scale, and subtraction; WdW patch and null-boundary terms; gate set or bulk prescription; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: boundary complexity task → CV, CA, or path proposal → regulator and counterterms → growth and switchback tests → proposal comparison. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “late-growth and formation checks; joint, counterterm, and switchback check; counterexample and normalization test” check is counterevidence to the promoted claim.	the same observable as action; scheme-independent complexity; a universal operational theorem	CV, CA, path-integral, and tensor-network prescriptions are inequivalent conjectures whose divergences and operational meaning must be compared.
AdS3/CFT2 and Three-Dimensional Gravity	Brown-Henneaux symmetry; BTZ saddle; pure gravity proposal	Declare boundary falloffs and charge normalization; ensemble, identifications, and spin; spectrum, topology sum, and global form; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: CFT2 and global data → AdS3 boundary conditions → Virasoro or Chern-Simons structure → BTZ and modular checks → three-dimensional gravity claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “central extension and integrability; thermodynamic and modular check; unitarity and modular consistency” check is counterevidence to the promoted claim.	a complete Hilbert space; all heavy CFT states are BTZ geometries; established nonperturbative theory	AdS3/CFT2 combines unusually strong symmetry with global and spectral subtleties; BTZ control does not settle pure quantum gravity.
AdS2, JT Gravity, SYK, and Random Matrices	Schwarzian mode; SYK interface; matrix completion	Declare near-AdS2 boundary condition and coupling; disorder, large N, and conformal window; ensemble and spectral density; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: near-AdS2 boundary data → JT and Schwarzian mode → SYK low-energy matching → topology and matrix ensemble → model-specific conclusion. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “thermodynamic and correlator check; bilocal saddle and soft-mode matching; genus and late-time comparison” check is counterevidence to the promoted claim.	complete bulk spectrum; unique duality to JT gravity; the unique fixed Hamiltonian	JT, Schwarzian dynamics, SYK, and random matrices overlap in controlled limits but do not provide one unique fixed-theory completion.
Black-Hole Microstates and Stringy Entropy	BPS index; brane microstate count; microstate geometry	Declare charges, chamber, and fermion signs; decoupling limit and charge map; smooth solution family and quantization; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: charges and protected sector → index versus degeneracy → brane count and attractor data → correction and emission checks → microstate claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “wall-crossing and growth check; Cardy or localization comparison; charge, moduli, and measure test” check is counterevidence to the promoted claim.	actual degeneracy in every regime; typical non-BPS geometry; enumeration of the full Hilbert space	Protected indices, actual degeneracies, typical states, and non-BPS microstates must remain distinct when compared with black-hole entropy.
Wormholes, Gravitational Path Integrals, and Ensembles	Euclidean wormhole; replica wormhole; factorization statement	Declare boundary data, measure, and cycle; replica boundary conditions and continuation; fixed theory, ensemble, or alpha sector; use the volume conventions unless the page states a local replacement.	Conditional theorem or structural result. Control chain: boundary conditions and contour → saddles and negative modes → topology or replica sum → factorization and alpha-sector tests → completion claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “negative-mode and steepest-descent check; saddle competition and entropy check; multi-boundary connectedness test” check is counterevidence to the promoted claim.	a convergent gravitational path integral; microscopic evaporation dynamics; automatic ensemble averaging	A wormhole contribution is defined only with a contour and theory; connected boundaries do not by themselves establish an ensemble or a fixed-theory completion.
Black-Hole Information, Islands, and Interiors	Page curve; island saddle; interior reconstruction	Declare unitary benchmark and radiation factorization; bath coupling, QES functional, and replicas; code sector, state dependence, and algebra; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: evaporation model and radiation algebra → replicas and generalized entropy → island and wedge selection → Page, decoding, and interior tests → microscopic ceiling. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “early- and late-time entropy check; competing-saddle and renormalization test; overlap, commutator, and decoding tests” check is counterevidence to the promoted claim.	a microscopic S-matrix; unique nonperturbative completion; observer-independent exact local operators	Islands can produce semiclassical Page curves and radiation wedges without supplying microscopic evaporation dynamics, decoding, interiors, or endpoints.
Higher-Spin and Vector-Model Holography	higher-spin dictionary; alternate boundary condition; symmetry breaking	Declare boundary model, parity phase, and global data; scalar branch and double-trace flow; coupling and anomalous dimensions; use the volume conventions unless the page states a local replacement.	Dictionary entry or correspondence claim. Control chain: vector-model large-N data → higher-spin algebra and fields → boundary conditions and correlators → breaking and tensionless tests → non-Einstein bulk regime. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “three-point and Ward-identity tests; critical/free endpoint comparison; gap and locality diagnostics” check is counterevidence to the promoted claim.	local Einstein gravity; unchanged bulk theory at all orders; automatic string completion	Higher-spin/vector-model dualities define controlled non-Einstein regimes; restoring a large higher-spin gap is a separate dynamical problem.
de Sitter and Cosmological Holography	late-time wavefunction; in-in correlator; static-patch proposal	Declare state, boundary data, and phase convention; closed-time contour and operator ordering; observer algebra and horizon state; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: de Sitter state and observer patch → wavefunction or in-in observable → analytic continuation and dictionary → entropy and static-patch tests → proposal status. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “Ward and analytic-continuation check; reality and cutting checks; entropy and finite-system consistency” check is counterevidence to the promoted claim.	ordinary Euclidean CFT probabilities; the same object as a wavefunction coefficient; complete global de Sitter holography	Late-time wavefunctions, in-in correlators, and static-patch observables are different objects; no single continuation supplies a complete de Sitter dual.
Flat-Space, BMS, and Celestial Holography	asymptotic symmetry; celestial transform; celestial OPE	Declare falloffs, charges, and phase space; basis, contour, and infrared dressing; operator basis, loop order, and regulator; use the volume conventions unless the page states a local replacement.	Proposal or conditional construction. Control chain: null-infinity states and data → BMS, soft, and dressing sectors → celestial transform and OPE → loop, unitarity, and completeness tests → flat-holography status. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “soft theorem and memory triangle; inverse transform and covariance; factorization and unitarity checks” check is counterevidence to the promoted claim.	a complete holographic dual; a standalone local CFT; nonperturbative completeness	A flat-space S-matrix, its celestial transform, BMS charges, and a complete celestial dual are distinct layers of a proposed dictionary.
Quantum-Gravity Consistency and Swampland	no global symmetry; weak gravity or distance conjecture; de Sitter or landscape claim	Declare exact AdS/CFT or other stated premise; charge normalization or moduli metric; compactification and EFT assumptions; use the volume conventions unless the page states a local replacement.	Named conjecture under stated hypotheses. Control chain: quantum-gravity premise and black holes → charge, tower, and moduli constraints → examples and conditional derivations → counterexamples and dated tests → typed consistency claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “boundary current and bulk gauge check; extremality, towers, and examples; controlled examples and counterexamples” check is counterevidence to the promoted claim.	an assumption-free theorem about all gravity; a proven universal inequality; settled absence of all vacua	No-global-symmetry results and swampland conjectures have different hypotheses and evidence; examples do not erase counterexamples or dates.
Canonical, Loop, Spin-Foam, and Group-Field Quantum Gravity	area or volume spectrum; spin-foam amplitude; semiclassical recovery	Declare kinematical Hilbert space and operator choice; boundary state, model, and refinement; state family and coarse graining; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: constrained phase space → connections and spin networks → Hamiltonian, foam, or group-field dynamics → continuum and semiclassical tests → physical-observable claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “gauge and diffeomorphism treatment; simplicity, asymptotic, and anomaly checks; Einstein limit and fluctuation scaling” check is counterevidence to the promoted claim.	physical measurable discreteness; established continuum dynamics; full low-energy phenomenology	Kinematical discreteness, constraint solutions, spin-foam amplitudes, continuum limits, and classical recovery are separate achievements.
Asymptotic Safety, Causal, and Discrete Quantum-Gravity Programs	asymptotic-safety fixed point; triangulation or causal-set limit; spectral dimension flow	Declare effective action, gauge, regulator, and truncation; ensemble, measure, and refinement rule; diffusion operator and averaging; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: RG or discrete defining data → regulator, truncation, or refinement → continuum scaling observables → unitarity and causality tests → UV-completion claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “critical-exponent and truncation stability; finite-size scaling and observable recovery; scale and discretization stability” check is counterevidence to the promoted claim.	a proven UV completion; Einstein gravity in all observables; physical spacetime dimension by itself	A fixed point in a truncation or a continuum-looking phase in a discrete model is evidence that requires regulator, convergence, and unitarity control.
Quantum Cosmology and Singularity-Resolution Programs	Wheeler-DeWitt solution; loop-cosmology bounce; singularity resolution	Declare inner product, boundary condition, and clock; difference equation and effective regime; operational criterion and observable; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: cosmological state and constraint → clock and boundary condition → minisuperspace or effective dynamics → decoherence and full-theory tests → qualified resolution claim. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “constraint and probability-current check; density bound and state-spread check; BKL, perturbation, and extension tests” check is counterevidence to the promoted claim.	a unique cosmological prediction; resolution in the full inhomogeneous theory; all notions of singularity are removed	A wavefunction, effective bounce, bounded variable, geodesic extension, and full singularity resolution are different claims.
Quantum-Gravity Phenomenology and Comparative Status	modified propagation; gravity-mediated entanglement; compact-object or cosmological signal	Declare EFT operator, source model, and distance; locality, mediation, and noise assumptions; waveform or primordial model and nuisance priors; use the volume conventions unless the page states a local replacement.	Model-specific calculation or conditional result. Control chain: mechanism and detector observable → data and nuisance model → likelihood and systematics → competing explanations → bound, anomaly, or detection. Sources are cited on the destination page; literature checked through 10 August 2026.	Track omitted corrections, alternate branches, and competing definitions. A failed “multi-energy and multi-source likelihood; classical-channel and decoherence controls; cross-channel and population tests” check is counterevidence to the promoted claim.	evidence for one UV completion; selection of a quantum-gravity program; confirmed quantum gravity without alternatives	Quantum-gravity phenomenology begins with a mechanism and likelihood; sensitivity, bounds, anomalies, and detections have sharply different meanings.

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References

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