{"id":"cfa5387e-3b3c-46ea-bf86-dc1a86e0968d","arxiv_id":"2412.00298","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Complementary recovery in holographic codes is shown to be equivalent to preservation of Connes cocycle flow, and this holds for AdS Klein-Gordon fields in boundary diamonds and bulk wedges.","lead":"This paper proves that the key property of quantum error-correcting codes in holography, complementary recovery, is equivalent to a precise operator-algebraic condition: the encoding map must preserve the 'cocycle flow' of states. It then shows this condition holds for Klein-Gordon fields in anti-de Sitter space, giving a rigorous infinite-dimensional example of subregion-subregion duality.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.3's conclusion depends on the unstated [37, Thm 1.1] and on unproved Reeh–Schlieder separation for Ω; the paper needs to supply both for subregion–subregion duality to be established.","rationale":"Good faith read: Theorem 1.1 and Theorem 1.2 appear to be carefully proved and are substantive operator-algebraic results. My concern is not with the internal logic of those theorems, but with the bridge from them to the paper's advertised AdS/CFT application. Section 4.3's proof of Theorem 4.3 is openly a proof sketch: the 2-point function match supports the KMS/geometric-modular picture, but the actual recovery channels are imported from an external theorem [37] whose statement is not reproduced. The Reeh–Schlieder hypothesis is also asserted by citation rather than proved. These are exactly the kind of inputs that, if wrong, would break the application of Theorem 1.1. Therefore the reviewer's CONDITIONAL verdict is appropriate; I would ask the authors to state and prove the needed lemma from [37] and to give a direct argument for separation of the vacuum vector on wedge and diamond algebras before treating subregion–subregion duality as fully established.","tokens_in":26545,"tokens_out":14848,"duration_ms":157717,"concrete_test":"Write out [37, Theorem 1.1] as a self-contained lemma and verify, for N=πω(Abulk(W))'' and M=πω(Abd(D))'', each hypothesis: (i) Ω is separating for N and M (prove from the Fock/quasifree structure: no nonzero smeared field over W or D annihilates Ω); (ii) σ_t^M(N)=N, using the geometric action of the boost; (iii) any condition on the commutants needed to obtain R'. If all hypotheses hold, the lemma yields R,R' and Theorem 4.3 follows. If any hypothesis is absent or unverified, the proof has a gap; the check can be made numerical in a truncated-boson model of the wedge/diamond inclusion, where the recovery maps are constructed explicitly and condition (1) of Theorem 1.1 is tested on a spanning set.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central physical claim is not proven by the arguments displayed in §4.3. The proof identifies the bulk and boundary vacuum two-point functions (Eq. 4.26) and invokes geometric modular flow via [17], but the decisive step is the sentence 'We are therefore in position to apply [37, Theorem 1.1] to achieve complementary recovery.' [37] is not stated, so the reader cannot check its exact hypotheses or conclusion. Complementary recovery is a pair of statements — existence of R:B→A and R':B'→A' with V*R(b)V=b and V*R'(b')V=b' — and no such maps are constructed or referenced from a stated theorem. In addition, Theorem 1.1 requires Ω to be separating for B=πω(Abulk(W))'' and VΩ for A=πω(Abd(D))''; this Reeh–Schlieder input is only cited to [16,17] and would fail on some backgrounds. If [37]'s theorem needs an extra hypothesis (e.g., a specific KMS temperature, a conditional expectation, or a global commutant condition) or if separation fails on the universal cover, the advertised subregion–subregion duality does not follow.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops operator-algebraic characterizations of bulk reconstruction in AdS/CFT. The main exact result, Theorem 1.1, states that for an isometry V:H→K and von Neumann algebras A, B with a cyclic separating vector Ω for B such that VΩ is cyclic separating for A, complementary recovery of B and B′ is equivalent to (i) intertwining of Connes cocycle flows on B and B′ with those on A and A′, and (ii) equality of relative entropies on the two algebras and their commutants. Theorem 1.2 gives an approximate version: asymptotic recovery implies asymptotic cocycle intertwining, which implies asymptotic commutativity of the encoded algebras with the bulk commutants, and these three conditions are equivalent when B is hyperfinite. In Section 4 the paper applies Theorem 1.1 to Klein–Gordon fields on the universal cover of AdS, using the holography framework of Dybalski–Wrochna [30], and claims an operator-algebraic subregion–subregion duality between boundary causal diamonds and bulk causal wedges (Theorem 4.3). The final section also derives, under the modular-theoretic hypotheses, that bulk duals of type III_1 boundary factors with ergodic vacuum are either trivial or type III_1.","tokens_in":26760,"tokens_out":4971,"duration_ms":49014,"significance":"If the results are correct, Theorem 1.1 is a genuinely useful addition to the operator-algebraic quantum error correction literature: it characterizes exact complementary recovery through the physically motivated cocycle flow, and the approximate version adds a new asymptotic viewpoint. The paper also provides a detailed, largely rigorous proof of the exact theorem using standard modular theory, spatial derivatives, and ultraproduct techniques. The application in Section 4 aims at an infinite-dimensional, continuum subregion–subregion duality, a goal of clear interest to mathematical physics. However, the strongest advertised physical conclusion, Theorem 4.3, is presented as a proof sketch and depends on two unstated black-box theorems ([37, Theorem 1.1] and [51, Theorem 1.1]) as well as on unproved Reeh–Schlieder-type hypotheses; this is the main weakness of the paper.","major_comments":[{"comment":"The decisive step in the proof of Theorem 4.3 is the sentence \"We are therefore in position to apply [37, Theorem 1.1] to achieve complementary recovery of the inclusion.\" The cited theorem is never stated, so the reader cannot verify that its hypotheses are met or that its conclusion is exactly the existence of recovery channels R:B→A and R′:B′→A′ with V*R(b)V=b and V*R′(b′)V=b′. Since complementary recovery is a pair of existence statements, the proof should either state [37, Theorem 1.1] (in an appendix or in the body), construct the recovery maps explicitly, or cite a stated theorem with matching hypotheses and conclusion.","section":"§4.3 (Theorem 4.3)"},{"comment":"Theorem 1.1 requires Ω to be cyclic and separating for B=πω(Abulk(Wp,q))′′ and VΩ to be cyclic and separating for A=πω(Abd(Dp,q))′′. The paper does not prove either property. For the boundary net, the paper asserts that W∞2 defines a causal, conformally covariant pre-cosheaf and then invokes [17, Theorem 2.3(ii)], but it does not verify the axioms of that theorem (e.g., Reeh–Schlieder property, locality, modular covariance) for the algebra πω(Abd(Dp,q))′′. For the bulk wedge algebra, no argument is given that ωΩ is separating for πω(Abulk(Wp,q))′′ on the universal cover of AdS. Without these hypotheses, the modular theory of Theorem 1.1 cannot be applied to conclude subregion–subregion duality.","section":"§4.3 (Reeh–Schlieder hypothesis)"},{"comment":"The proof of Theorem 1.1 relies on two external results without stating their hypotheses: [37, Theorem 1.1(4)] is used to justify the intertwining of modular flows under the recovery map, and [51, Theorem 1.1] supplies the implication (3)⇒(1). These results are load-bearing for the equivalence claimed in Theorem 1.1. The paper should state these theorems with their exact assumptions and conclusions, or give self-contained proofs of the specific consequences used, so that the central equivalence can be checked independently of the cited papers.","section":"§3.1 (Theorem 1.1 proof)"},{"comment":"The proof that the boundary limit W∞2 is the 2-point function of a causal conformal net is only sketched. The paper shows that W2 coincides with the known AdS vacuum 2-point function, but the passage from the restricted net {πF(Abd(O))′′}O⊆Min(r) to a net satisfying the hypotheses of [17, Theorem 2.3(ii)] requires more than convergence of the 2-point function: it requires positive definiteness, locality, conformal covariance, and the Reeh–Schlieder property for the vacuum. The paper should spell out how these properties follow from the quasi-free construction and the cited references [13,17].","section":"§4.3 (boundary 2-point function)"}],"minor_comments":[{"comment":"In the statement of condition 3, the last equality reads \"SB′(ω′ψ, ω′φ)\"; the final vector should presumably be Ω, not φ. Please correct this typo.","section":"Theorem 1.1"},{"comment":"Near the end of the proof, the sentence \"Therefore, all three conditions are are equivalent\" contains a duplicated \"are\".","section":"§3.3 (Proof of Theorem 1.2)"},{"comment":"The phrase \"Theorem 1.2 (11)\" appears to refer to condition (1) of Theorem 1.2; please renumber or reword to avoid confusion with an equation number.","section":"§3.3 (Comment after Theorem 1.2)"},{"comment":"The notation \"pertinent eG0 action\" is unclear; it should presumably be the action of the universal cover of SO(2,d), e.g., Ĝ0 or the tilde version introduced earlier.","section":"§4.3"},{"comment":"Typographical issues in Section 4 include \"retarted/advanced propagators\" (should be \"retarded/advanced\") and the inconsistent use of \"M o\" and \"M o\" for the interior of M.","section":"§4"}],"recommendation":"major_revision","confidential_remarks":"The paper's reliance on [37] and [51] as unstated black boxes is somewhat concerning because both involve one of the present authors, but the results cited are plausibly standard in this research program; the main issue is the lack of a stated theorem in the manuscript, not the citation pattern itself. The most serious gap is in Section 4.3, where the advertised physical theorem is not established by the displayed arguments. I believe the gaps can be repaired by adding precise statements of the cited theorems and a careful verification of the Reeh–Schlieder hypotheses, but this requires nontrivial additional work from the authors."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The general theorems are the real substance here, and they hold up. Theorem 1.1 -- complementary recovery equivalent to preservation of Connes cocycle flow and to relative-entropy preservation -- is new and clean. The proof is detailed and uses standard modular theory plus sufficiency of subalgebras; I did not find a gap in the main line. Theorem 1.2, the approximate version via ultraproducts, is also substantive, and the type III_1 byproduct is a nice extra. The novelty over [51] is real: that paper had the (1) iff (3) equivalence, but the cocycle-flow condition (2) and the approximate intertwining result are new. The self-citations to [37] and [51] are to distinct general theorems, so I do not see a circularity problem in the abstract framework.\n\nThe soft spot is exactly where the stress-test puts it: Section 4.3. The advertised subregion-subregion duality for Klein-Gordon fields on the universal cover of AdS is not actually proven by the displayed text. The argument identifies the bulk and boundary two-point functions, asserts KMS conditions and geometric modular flow, and then says \"we are therefore in position to apply [37, Theorem 1.1].\" But [37, Theorem 1.1] is never stated, so the reader cannot check whether its hypotheses -- which may include temperature, a conditional expectation, or a global commutant condition -- are satisfied. Separately, Theorem 1.1 requires Omega to be cyclic and separating for the bulk wedge algebra and V Omega for the boundary diamond algebra; this Reeh-Schlieder-type input is only cited to [16,17], not proven for this construction. If separation fails on the universal cover, or if the black-box theorem from [37] carries extra assumptions, the physical conclusion does not follow as written.\n\nThat said, these are issues of presentation and completeness in the application section, not flaws in the core mathematics. The abstract theorems are independent of the AdS application and are worth having. The paper is honest about the proof sketch status, and the direction is clearly right.\n\nWho is this for? Operator algebraists and mathematical physicists working on holography and quantum error correction. It deserves a serious referee. I would send it to peer review, with the instruction that the referee require the authors to either state the needed theorem from [37] with enough precision and give a real proof of the separation property, or explicitly present Theorem 4.3 as conditional on those inputs. As is, I would not certify subregion-subregion duality as fully established, but the general recovery theorems are a solid contribution.","headline":"Solid general recovery theorems with a genuinely new cocycle-flow characterization; the AdS subregion-duality claim is a promising but under-specified application that needs a fleshed-out Section 4.3.","tokens_in":27335,"tokens_out":1897,"would_cite":true,"duration_ms":20056,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L10","46L55","81T40","81P45","81T20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that exact bulk reconstruction in AdS/CFT is the same as matching Connes cocycle flow between bulk and boundary algebras and as preserving relative entropies, and it uses this equivalence to establish subregion-subregion…","keywords":["AdS/CFT correspondence","bulk reconstruction","von Neumann algebras","Connes cocycle flow","relative entropy","complementary recovery","subregion-subregion duality","Klein-Gordon fields"],"falsifier":"A concrete test is to compute the kernel of the Fock-field map $u\\mapsto \\varphi_F(u)\\Omega$ for $u$ in the symplectic subspace $X_{\\mathrm{bulk}}(W_{p,q})$ using the global AdS vacuum two-point function (4.26); a nonzero kernel would mean the vacuum vector is not separating for $\\pi_\\omega(A_{\\mathrm{bulk}}(W_{p,q}))''$, so the hypothesis of Theorem 4.3 fails and the subregion-duality conclusion would not follow from this argument.","tokens_in":26302,"feed_emoji":"🌌","tokens_out":14073,"duration_ms":119133,"temperature":0.7,"pith_summary":"The paper establishes an operator-algebraic criterion for when a bulk region of anti-de Sitter space can be reconstructed from a boundary region: exact bulk recovery is equivalent to the bulk and boundary modular flows agreeing on every state through the Connes cocycle derivatives, and also equivalent to the equality of bulk and boundary relative entropies. The equivalence is proved for general von Neumann algebras with a cyclic-separating vacuum vector, and then applied to Klein-Gordon fields on the universal cover of AdS, where boundary causal diamonds and bulk causal wedges are shown to satisfy complementary recovery. A companion approximate statement ties approximate bulk recovery to the large-N vanishing of the cocycle-intertwining error, with full equivalence in the hyperfinite case. The result supplies an infinite-dimensional, field-theoretic foundation for holographic quantum error correction and identifies the kink transform with bulk cocycle flow in exact recovery settings, with boundary modular flow inducing bulk geometry and a notion of time.","feed_headline":"Bulk reconstruction equals matching Connes cocycle flow in AdS","feed_subtitle":"New proof links exact and approximate bulk recovery to relative entropy, with duality for Klein-Gordon wedges in AdS","key_machinery":"The load-bearing object is the Connes cocycle derivative $[D\\varphi:D\\omega]_t=\\Delta(\\varphi/\\omega_0)^{it}\\Delta(\\omega/\\omega_0)^{-it}$, a bounded operator that measures how the modular flow of one state is tilted relative to another; it is the state-dependent local extension of modular time. The proof of Theorem 1.1 is carried by recovery maps $R:\\mathcal{B}\\to\\mathcal{A}$ satisfying $R(b)V=Vb$, by the sufficiency theory that forces cocycles to lie in sufficient subalgebras, and by the uniqueness of the KMS condition, which pins the pushed-forward cocycle to the bulk cocycle. An analytic continuation argument transfers the intertwining identity from the real line to the imaginary axis, yielding equality of the spatial derivatives and hence of relative entropies, even when the bulk relative entropy is infinite. In the AdS application, the geometric modular action of the wedge-preserving isometry group $\\Lambda_{p,q}(t)$ on boundary diamonds and bulk wedges supplies the local notion of time, matching the algebraic cocycle flow.","core_discovery":"The paper's central discovery is Theorem 1.1. Given an isometry $V:\\mathcal{H}\\hookrightarrow\\mathcal{K}$, von Neumann algebras $\\mathcal{A}\\subseteq B(\\mathcal{K})$, $\\mathcal{B}\\subseteq B(\\mathcal{H})$, and a vector $\\Omega$ that is cyclic and separating for $\\mathcal{B}$ such that $V\\Omega$ is cyclic and separating for $\\mathcal{A}$, the following are equivalent: complementary recovery of $\\mathcal{B}$ and $\\mathcal{B}'$ for the complementary channels $V^*(\\cdot)V$ on $\\mathcal{A}$ and $\\mathcal{A}'$; the intertwining identity $V[D\\omega_\\Omega:D\\omega_\\psi]_t=[D\\omega_{V\\Omega}:D\\omega_{V\\psi}]_tV$ (and the same for the commutants) for all states $\\psi$ and all $t\\in\\mathbb{R}$; and preservation of bulk-boundary relative entropies for all states. Theorem 1.2 is the approximate analogue: a sequence of isometries admitting approximate recovery channels forces approximate cocycle intertwining, which in turn forces approximate privacy, and all three conditions become equivalent when the bulk algebra is hyperfinite. The paper then constructs a concrete instance from the symplectic solution space of the Klein-Gordon equation on the universal cover of AdS, with $\\mathcal{A}=\\pi_\\omega(A_{\\mathrm{bd}}(D_{p,q}))''$ and $\\mathcal{B}=\\pi_\\omega(A_{\\mathrm{bulk}}(W_{p,q}))''$, and shows, using the geometric modular flow of the vacuum, that the equivalent conditions hold. A by-product is that a boundary type III$_1$ factor with an ergodic vacuum forces the dual bulk algebra to be either $\\mathbb{C}1$ (with one-dimensional Hilbert space) or a type III$_1$ factor.","pith_inferences":["The same equivalence could be used as a detection tool: deviation from cocycle intertwining in an approximate code can be read as the amount of bulk locality that finite-$N$ effects destroy, which may be easier to compute than the recovery error itself.","The framework suggests a testable route to extend subregion duality beyond Klein-Gordon fields: if a holographic Hadamard state admits geometric modular flow on both sides, the same argument should yield complementary recovery for interacting or higher-spin fields.","One concrete corollary one could test in models: if a dual bulk algebra were of type III$_\\lambda$ with $\\lambda\\neq 1$, then the ergodic-vacuum assumption would have to fail in the corresponding holographic dual.","The approximate theorem suggests quantifying the $N$-dependence of the cocycle error as a proxy for the semi-classical expansion; a finite-$N$ code with exactly vanishing cocycle error would be a candidate for all-orders bulk reconstruction."],"forward_implications":["Exact complementary recovery, preservation of Connes cocycle flow, and equality of relative entropies are three co-equal descriptions of bulk reconstruction, so a holographic code that satisfies one satisfies all three.","Boundary causal diamonds in the universal cover of AdS are dual to bulk causal wedges of Klein-Gordon fields, realizing operator-algebraic subregion-subregion duality in an infinite-dimensional setting.","The kink transform conjecture gains a concrete realization: in exact recovery settings the kink transform is bulk cocycle flow, with boundary modular flow implementing bulk time and geometry.","In the large-$N$ limit, approximate recovery implies vanishing cocycle error, which in turn implies approximate privacy; for hyperfinite bulk algebras these conditions are equivalent, making the cocycle error a useful order parameter for approximate reconstruction.","The type III$_1$ ergodicity structure propagates from boundary subregions to bulk subregions: a dual bulk algebra is trivial or type III$_1$."],"supporting_citations":[{"why":"Establishes that equality of relative entropies implies complementary recovery, the (3)⇒(1) step in Theorem 1.1.","marker":"[51]"},{"why":"Supplies the operator-pushing result that turns modular-flow preservation into complementary recovery, used in the proof and in the AdS application.","marker":"[37]"},{"why":"Provides the sufficiency theory and cocycle machinery that force cocycles into sufficient subalgebras during the proof of Theorem 1.1.","marker":"[63]"},{"why":"Gives the correctability/privateness complementarity theorem on which the recovery-map arguments rest.","marker":"[22]"},{"why":"Provides the hyperfinite approximate-recovery theorem used for (3)⇒(1) in Theorem 1.2.","marker":"[32]"},{"why":"Supplies the bulk-to-boundary map and symplectic framework for Klein-Gordon fields on asymptotically AdS spacetimes used in Section 4.","marker":"[30]"},{"why":"Gives the geometric modular structure and vacuum two-point functions for the AdS Klein-Gordon field, used to identify bulk and boundary modular flows.","marker":"[16]"},{"why":"Proves geometric modular flow for boundary diamonds in conformal quantum field theory, which identifies the boundary modular automorphism group.","marker":"[17]"},{"why":"Introduces the kink transform conjecture that the paper's exact-recovery results support.","marker":"[14]"},{"why":"Defines the causal wedges, diamonds, and wedge-preserving isometry groups $\\Lambda_{p,q}(t)$ used in Proposition 4.2 and Theorem 4.3.","marker":"[70]"}],"fun_headline_variants":["Cocycle flow equals bulk reconstruction in AdS","Approximate recovery implies cocycle intertwining","Klein-Gordon wedges dual via modular flow","Type III bulk forced by ergodic boundary algebra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the global vacuum vector is cyclic and separating for the bulk wedge algebra, and that its image under the bulk-to-boundary isometry is cyclic and separating for the boundary diamond algebra; this no-local-annihilation property is taken from the literature and not proved here for the universal cover of AdS.","fun_headline_variants_meta":{"raw":{"variants":["Cocycle flow equals bulk reconstruction in AdS","Approximate recovery implies cocycle intertwining","Klein-Gordon wedges dual via modular flow","Type III bulk forced by ergodic boundary algebra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000353,"raw_usage":{"total_tokens":2012,"prompt_tokens":1129,"completion_tokens":883,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":822}},"tokens_in":745,"tokens_out":883,"duration_ms":8331,"temperature":1.0,"reasoning_tokens":822,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:32:28.363088+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test is to compute the kernel of the Fock-field map $u\\mapsto \\varphi_F(u)\\Omega$ for $u$ in the symplectic subspace $X_{\\mathrm{bulk}}(W_{p,q})$ using the global AdS vacuum two-point function (4.26); a nonzero kernel would mean the vacuum vector is not separating for $\\pi_\\omega(A_{\\mathrm{bulk}}(W_{p,q}))''$, so the hypothesis of Theorem 4.3 fails and the subregion-duality conclusion would not follow from this argument.","supporting_citations":[{"cited_title":"Holographic tensor network models and quantum error correc- tion: a topical review,","cited_arxiv_id":null,"evidence_quote":"Establishes that equality of relative entropies implies complementary recovery, the (3)⇒(1) step in Theorem 1.1."},{"cited_title":"Thermal states are vital: Entanglement Wedge Reconstruction from Operator-Pushing","cited_arxiv_id":"2005.07189","evidence_quote":"Supplies the operator-pushing result that turns modular-flow preservation into complementary recovery, used in the proof and in the AdS application."},{"cited_title":"The Large N limit of superconformal field theories and supergravity,","cited_arxiv_id":null,"evidence_quote":"Provides the sufficiency theory and cocycle machinery that force cocycles into sufficient subalgebras during the proof of Theorem 1.1."},{"cited_title":"Private algebras in quantum information and infinite-dimensional complementarity,","cited_arxiv_id":null,"evidence_quote":"Gives the correctability/privateness complementarity theorem on which the recovery-map arguments rest."},{"cited_title":"A mechanism for holography for non-interacting fields on anti-de Sitter spacetimes,","cited_arxiv_id":null,"evidence_quote":"Supplies the bulk-to-boundary map and symplectic framework for Klein-Gordon fields on asymptotically AdS spacetimes used in Section 4."},{"cited_title":"Towards a General Theory of Quantized Fields on the Anti-de Sitter Space-Time,","cited_arxiv_id":null,"evidence_quote":"Gives the geometric modular structure and vacuum two-point functions for the AdS Klein-Gordon field, used to identify bulk and boundary modular flows."},{"cited_title":"Modular structure and duality in conformal quantum field theory,","cited_arxiv_id":null,"evidence_quote":"Proves geometric modular flow for boundary diamonds in conformal quantum field theory, which identifies the boundary modular automorphism group."},{"cited_title":"Gravity dual of Connes cocycle flow,","cited_arxiv_id":null,"evidence_quote":"Introduces the kink transform conjecture that the paper's exact-recovery results support."},{"cited_title":"Algebraic holography,","cited_arxiv_id":null,"evidence_quote":"Defines the causal wedges, diamonds, and wedge-preserving isometry groups $\\Lambda_{p,q}(t)$ used in Proposition 4.2 and Theorem 4.3."}],"review_version":1}