{"id":"480d4d68-2697-4b3a-8a5f-8a6e76ebb734","arxiv_id":"2610.00670","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"An extensive bulk mutual-information model in AdS, realized by a free massive Dirac fermion, places the quantum-corrected boundary entropy vector inside the holographic entropy cone and reproduces fermionic generalized-free-field behavior, with an exact AdS_3 half-disk result.","lead":"A new curved-spacetime model of mutual information shows how ordinary-looking quantum correlations automatically satisfy every strict rule known for holographic entropy, including monogamy. A concrete version built from a free massive Dirac fermion reproduces generalized-free-field behavior at long and short distances, and an exact three-dimensional example is computed in closed form.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Central implication (13) depends on an unpublished basis-completeness theorem [59]; the RG-fixed-point assumption flagged by the reader affects the GFF claims but not (13).","rationale":"The reader's conditional verdict is consistent with my stress-test pass, and the reader also noticed the unpublished nature of [59] in the rationale. However, my single most load-bearing concern is different from the one the reader lists as the weakest assumption. The reader emphasizes the imposed conformal fixed point k(ℓ) = k_J and the unproven bound b² ≥ a². Those are indeed unproven or imposed, and they matter for the short-distance GFF divergence structure and for the RG-monotone interpretation. But they are not necessary for the abstract implication (13), which requires only positive and exactly extensive bulk MI. The paper's own conditions (8), namely 2F ≥ G ≥ 0, are satisfied by the fermion realization through a² > 0 and G > 0; the additional G ≥ F needed for b² ≥ a² supports k ≥ 0 but is not part of the positivity condition entering (13). The conformal fixed-point issue is explicitly acknowledged in footnote 78, and the authors state that the fixed point is not obtained as a ν → 0 limit of the massive solution. By contrast, the basis-completeness theorem is asserted without proof and its general form is attributed to an unpublished preprint. Since Eq. (13) is the central claim, this is the point on which the proof is least secure. My recommended verdict is unchanged from the reader's CONDITIONAL: the paper is plausible and the derivations that I checked pass, but the central implication currently rests on a citation that needs to be verified or upgraded.","tokens_in":21387,"tokens_out":12339,"duration_ms":69232,"concrete_test":"Enumerate all non-redundant holographic entropy inequalities for N = 4 and N = 5 from the published extremal-ray analyses, and express each inequality explicitly in the multipartite-information basis. Check whether every inequality other than subadditivity has vanishing coefficients on S(A_i) and I(A_i,A_j). If any such inequality retains a one- or two-party term, construct a formal entropy vector with all multipartite informations I_n = 0 for n ≥ 3 and positive pairwise MI that violates that inequality; such a vector would falsify (13). In parallel, obtain the exact statement used by the paper from [59] and confirm that it appears in a published or otherwise fully specified form, including its hypotheses on N and on balanced inequalities.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central assertion is (13): positive and exactly extensive bulk MI is sufficient for the quantum-corrected boundary entropy vector to satisfy every holographic entropy inequality. The proof has two external inputs. The first, [53], is published and is not at issue here. The second is the formal statement, in the paragraph just before Eq. (12), that every non-redundant holographic entropy inequality other than subadditivity can be expressed in the multipartite-information basis with zero coefficients on one- and two-party terms. That statement is cited as [57–59]. References [57] and [58] are published, but the general, all-N form needed here is also assigned to [59], an unpublished arXiv preprint (Hubeny–Liu, arXiv:2609.22504). The text does not state the precise hypotheses of that result, for example whether it requires balanced inequalities, restricts to some number of parties, or imposes sign conditions on the multipartite-information coefficients. If any non-redundant HEC inequality retains a one- or two-party term, then Eq. (12) sets only the multipartite terms (n ≥ 3) to zero, and that surviving term would produce a non-trivial constraint. Positivity and extensivity would then not suffice to establish Sb ∈ HEC, and (13) would fail. The reader's weakest-assumption items — the imposed constant RG charge k(ℓ) = k_J and the unproven b² ≥ a² inequality — control the short-distance GFF structure and the RG-monotone interpretation, but Eq. (13) itself only needs positive, exactly extensive bulk MI, so those items are not load-bearing for the central implication. The load-bearing gap is therefore the unsupported status of the multipartite-basis completeness theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper constructs an extensive (EMI-type) model for bulk mutual information in rigid AdS, built from a covariantly conserved current-current kernel, and argues that positivity plus exact extensivity of the bulk MI implies that the quantum-corrected boundary entropy vector satisfies monogamy and all holographic entropy cone inequalities (Eq. (13), sentences b3, b57). The technical core is the conservation equation (15), the total-derivative representation (16)-(18), and the free massive Dirac fermion realization (21)-(24), whose long-distance boundary MI matches a fermionic generalized free field (25), whose short-distance limit reproduces the Dirac U(1) central charge (63), and which yields an exact AdS_3 result for disjoint half-disks at the conformal fixed point (29)-(30). The supplemental material provides a Weyl-rescaling identity (48)-(51), the hypergeometric solution for the kernel, and the short/long-distance asymptotics.","tokens_in":21740,"tokens_out":6792,"duration_ms":39161,"significance":"If the central implication (13) holds, the paper supplies a genuinely clean sufficient condition: any positive, exactly extensive bulk MI on the entanglement wedges automatically produces a boundary entropy vector in the HEC at first subleading order in 1/N, generalizing the monogamy result (10) to all non-redundant holographic inequalities. The paper ships concrete, checkable mathematics: the reader's verification confirms that Eq. (17) follows from the supplemental first-order equation (43), that (21)/(58) reproduce the tensor structure of (14), that the short-distance coefficient (63) equals the Dirac central charge 2^{floor(d/2)-2} Gamma^2(d/2)/pi^d, and that the Weyl rescaling (48) is consistent with (45)-(51). The explicit fermion realization with the exact AdS_3 half-disk result is a valuable test bed for the generalized-free-field picture. The main caveats are the unpublished basis-representation theorem cited for (13), the imposed (not derived) conformal-fixed-point kernel behind the GFF short-distance claims, and an unproved inequality b^2 >= a^2 used for the RG-monotone interpretation.","major_comments":[{"comment":"The implication (13) is load-bearing for the paper's headline claim, and its non-subadditive step rests on the statement that every non-redundant holographic entropy inequality can be written in the multipartite-information basis with zero coefficients on one- and two-party terms, cited to [57-59]. References [57,58] are published, but the general all-N form needed here is assigned to [59], which is an unpublished arXiv preprint. The text states neither the precise hypotheses of the theorem (e.g., whether it requires balanced inequalities, imposes sign conditions on the multipartite-information coefficients, or is restricted to a range of party numbers) nor that extensivity in the sense of Eq. (3) is the right hypothesis for all n. If any non-redundant HEC inequality retains a one- or two-party term in this basis, Eq. (12) would not zero that term and (13) would fail. Please state the theorem precisely with its hypotheses, or restrict the claim to what the published results [57,58] support.","section":"§ 'Bulk extensivity implies boundary monogamy', paragraph before Eq. (12)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The short-distance GFF divergence structure (28), the Weyl half-space representation (26), and the exact AdS_3 result (29) are all evaluated at the kernel k(ell) = k_J imposed as an assumption (Eq. (45), supplemental sentence b180). The authors correctly acknowledge (footnote 78 and the final supplemental sentence, b221-b222) that this kernel is not obtained from the nu -> 0 limit of the massive fermion solution, so the fixed point of the massive realization is not established. The unresolved question is not whether the assumption is stated (it is), but whether the claimed 'conformal fixed point of the model' is actually the fixed point of the construction: if the true fixed-point kernel flows, the GFF divergence structure and the coefficients c_1 = kappa k_J pi^2/4, c_0 = -kappa k_J describe a kernel that is not the fixed point of the fermion realization. This does not affect (13), but it does delimit the fermion-realization claims in the abstract, b5 and b95. Please either characterize the fixed point of the massive flow or explicitly label Eqs. (26)-(30) as properties of the auxiliary kernel (45) rather than of the fermion realization.","section":"§ 'Long- and short-distance structure', Eqs. (26)-(30) and footnote 78","authors_acknowledge":"Consequently, the conformal fixed point cannot be obtained by taking theν→0limit of this solution.","authors_acknowledge_p":0.68},{"comment":"The claim that k(ell) in Eq. (24) is non-negative for all ell, equivalently b^2 >= a^2 (equivalently G >= F), is asserted without proof ('this correlator satisfies the condition G >= F since b^2 >= a^2'). From Eqs. (22)-(23) the ratio b^2/a^2 involves 2F1 functions with different numerator parameters multiplied by (zeta-1)/zeta, and the large-ell limit b ~ a makes the inequality non-trivial; it is not implied by the positivity condition (8), which only requires 2F - G >= 0. This inequality is the basis for the claimed 'positive ... monotonically decreasing' RG monotone (abstract, b5, b72), so it needs a proof or a rigorous bounding argument.","section":"Supplemental Material, 'Current-current correlator for a massive fermion', after Eq. (58); Eq. (24)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"Reference [30] has a garbled author list: 'H. Casini, R. C. Huerta, Marina and, and A. Yale' should be corrected (Casini, Huerta, and Yale).","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The derivation of the universal expansion (28) is schematic: the cancellation of the mixed terms in (27) for disjoint spheres and the statement that boundary terms 'can only introduce additional divergences of the same type' are asserted rather than shown. For d > 3 the coefficients c_i are not computed (acknowledged in the discussion, b108); a brief justification of the universal-structure claim, or an explicit statement that (28) is conjectural beyond the leading terms, would help the reader.","section":"Eqs. (26)-(28)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"In the exact AdS_3 result, the conformal equivalence to equal disks separated by s with s/R = (1+eta)/sqrt(eta) and the ranges for eta would benefit from a one-line derivation; also the claim that the long-distance result is 'one quarter of the complete-disk result in flat space' should cite the precise flat-space reference quantity.","section":"Eq. (29)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The separation/regulator epsilon in (28) and (30) is used both as the distance R_A - R_B and as a generic UV cutoff; define it once at first use to avoid ambiguity.","section":"Notation","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The paper's headline theorem (13) leans on an unpublished preprint [59] (Hubeny-Liu, arXiv:2609.22504) for the general all-N basis-representation fact, and the authors do not state its hypotheses. If that theorem is not as general as needed, the central claim collapses; the referee cannot verify it from the manuscript. Separately, the GFF short-distance and exact AdS_3 results are conditional on the imposed fixed-point kernel k(ell)=k_J, which the authors themselves note is not the nu->0 limit; this is an acknowledged assumption but is central to the abstract's GFF claims. The technical core that is verified (conservation equation, tensor structure, central-charge match, Weyl rescaling) is sound, so I view the issues as fixable within scope rather than fatal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Quick take: the central implication is logically clean, and the explicit Dirac-fermion realization plus the exact AdS3 half-disk result are genuinely new. The paper deserves a serious referee; the one thing that needs chasing is the provenance of the multipartite-basis completeness theorem.\n\nWhat's new: the AdS generalization of the Casini–Huerta EMI functional with a conserved current–current kernel; the free massive Dirac realization with exact F and G satisfying conservation and positivity; the Weyl equivalence to Minkowski half-space; and the exact conformal EMI for half-disks in AdS3 with the right limits (s−4 falloff at long distance, area-law plus log divergence at short distance). I spot-checked the central-charge match (63) and the tensor structure and they work out. Extensivity being the defining property of the EMI ansatz is not circularity — it's the model hypothesis feeding a genuinely derived implication composed with the published AHR theorem [53].\n\nSoft spots, in proportion:\n\n1. The load-bearing premise for (13) in full generality is that every non-redundant HEC inequality except subadditivity writes in the multipartite basis with no one- or two-party terms. That is cited to [57–59], but the general all-N form is assigned to [59], an unpublished Hubeny–Liu preprint. If [57,58] don't contain the general statement, the burden rests on [59], and the paper should state that theorem's hypotheses — balanced inequalities, party-count limits, sign conditions on the multipartite coefficients. This is a fixable scoping/citation issue, not an error in the argument, but a referee should push.\n\n2. The positivity claim k(ℓ) ≥ 0 for the massive fermion rests on b² ≥ a², asserted in (24) and the Supplement without proof. Minor: it supports the RG-monotone interpretation, not Eq. (13) itself.\n\n3. The conformal fixed point k(ℓ) = k_J is imposed rather than derived, and the authors say so plainly (footnote 78; the ν→0 limit does not reach it). Consequence: the GFF short-distance structure and the exact AdS3 result live at an assumed kernel, while the fermion realization directly yields only the long-distance GFF. Taken as stated, fine — but the \"realization\" claim is two-step, and the letter should be explicit about that.\n\nAudience: people working on holographic entanglement entropy, quantum corrections to mutual information, and the HEC asymptotics program. For them it is a solid, citable construction. Recommendation: send to peer review; the referee's main job is verifying [57–59] and pinning down the completeness theorem's hypotheses.","headline":"A clean conditional — extensive plus positive bulk MI pushes the boundary entropy vector into the HEC — backed by a genuinely new fermion realization and an exact AdS3 result; the one load-bearing citation is an unpublished preprint reviewers should chase.","tokens_in":22463,"tokens_out":3365,"would_cite":true,"duration_ms":20541,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Positive and exactly extensive bulk mutual information suffices to force the quantum-corrected boundary entropy vector into the holographic entropy cone, and a free massive Dirac fermion kernel realizes the model.","keywords":["extensive mutual information","holographic entropy cone","monogamy of mutual information","generalized free field","Dirac fermion in AdS","quantum corrections to entanglement","renormalization group","entanglement wedge"],"falsifier":"Compute the short-distance limit of $k(\\ell)$ in Eq.~(24) directly from the massive fermion solution and check whether it reaches the constant Dirac central charge $k_J^D = 2^{\\lfloor d/2\\rfloor-2}\\Gamma^2(d/2)/\\pi^d$ or continues to flow; a flowing kernel at the fixed point would replace the claimed log coefficient $c_0 = -\\kappa k_J$ by an integral over the flow and invalidate the GFF short-distance structure. An independent check is whether $b^2 \\ge a^2$ holds at every $\\ell$, since any $\\ell$ with $b^2 < a^2$ makes $k(\\ell)$ negative and removes the RG-monotone interpretation.","tokens_in":22687,"feed_emoji":"🔗","tokens_out":8453,"duration_ms":45029,"temperature":0.7,"pith_summary":"This paper claims that one structural property of bulk mutual information — positivity plus exact extensivity over entanglement wedges — is sufficient for the quantum-corrected boundary entropy vector to satisfy monogamy of MI and every other holographic entropy cone inequality (Eq.~13). This matters because classical holographic MI vanishes past a sharp connected-to-disconnected transition, wrongly suggesting an absence of correlations; the Faulkner--Lewkowycz--Maldacena prescription identifies the subleading $O(1)$ correction with the bulk MI, and the paper shows extensivity lifts those corrections into the cone. The construction is a bilocal functional built from a covariantly conserved current--current kernel in AdS, realized explicitly by the $U(1)$ current of a free massive Dirac fermion. At long distances the resulting boundary MI matches a fermionic generalized free field, at short distances it reproduces the GFF power-law plus logarithmic divergence structure, and for disjoint half-disks at the conformal fixed point in $AdS_3$ the mutual information is given exactly.","feed_headline":"Extensive bulk MI pins boundary entropy to the cone","feed_subtitle":"A massive Dirac fermion kernel realizes the construction; half-disks in AdS₃ are solved exactly.","key_machinery":"The central object is the bilocal extensive mutual-information functional $I(A,B) = -\\kappa \\int_A \\int_B d\\sigma_A d\\sigma_B\\, n^\\mu_A n^\\nu_B J_{\\mu\\nu}(x_A,x_B)$, built from a covariantly conserved current--current correlator in AdS whose tensor structure is fixed by the geodesic distance $\\ell(x,x')$ and the parallel propagator (Eq.~14). The quantity doing the work is $k(\\ell) \\equiv G(\\ell) - F(\\ell)$: conservation forces $k'(\\ell) \\le 0$, and positivity of the kernel (conditions $2F \\ge G \\ge 0$) makes $k$ a non-negative decreasing function interpretable as an RG monotone. For the massive Dirac fermion realization, Wick-contracting the current gives $F$ and $G$ in terms of closed hypergeometric functions $a(\\ell), b(\\ell)$ (Eqs.~21--23). At the conformal fixed point, the Weyl equivalence between AdS and Minkowski half-space (Eq.~48) turns the bulk EMI integral into a flat-space boundary integral (Eq.~26), which is what permits the exact evaluation of the half-disk mutual information in Eq.~29.","core_discovery":"The central claim is that positive and exactly extensive bulk mutual information is sufficient for the quantum-corrected boundary entropy vector to lie in the holographic entropy cone: $S^b \\in \\mathrm{HEC} \\Rightarrow S^{\\mathrm{holo}} \\in \\mathrm{HEC}$ (Eq.~13). The argument passes to the multipartite-information basis: extensivity forces every $n$-party information $I_n^b$ with $n \\ge 3$ to vanish, so all nonredundant cone inequalities except subadditivity are saturated, and subadditivity follows from positivity of MI. The paper then constructs an extensive model of bulk MI in rigid AdS from a covariantly conserved current--current kernel (Eq.~14), realized explicitly by the current of a free massive Dirac fermion whose two-point functions yield closed hypergeometric forms for $a(\\ell)$ and $b(\\ell)$. The same correlator defines $k(\\ell) \\equiv G(\\ell) - F(\\ell)$, non-negative and monotonically decreasing, which the paper interprets as an RG monotone. At long distances the boundary MI falls off as $|x_A - x_B|^{-4\\Delta}$ with shape-dependent coefficients $C_X$ (Eq.~25), matching a fermionic generalized free field; at short distances, through a Weyl equivalence to Minkowski half-space, the MI acquires the GFF divergence structure (Eq.~28). For disjoint concentric half-disks in $AdS_3$ the conformal-point MI is computed exactly (Eq.~29), with long-distance $s^{-4}$ falloff and short-distance area-law plus logarithmic terms, the log coefficient being $c_0 = -\\kappa k_J$.","pith_inferences":["The sufficiency direction of Eq.~(13) suggests that extensivity itself, not the full geometric RT structure, carries the monogamy content of holography; one might test whether approximate extensivity keeps the boundary vector within a small neighborhood of the cone.","If the true fixed point of the massive fermion realization flows rather than sitting at constant $k(\\ell) = k_J$, the log coefficient would become an integral over the flow history instead of $-\\kappa k_J$ — a distinction checkable by computing the deep short-distance limit of $k(\\ell)$ from Eq.~(24).","The unproven inequality $b^2 \\ge a^2$ for all $\\ell$ is what makes $k(\\ell)$ non-negative and interpretable as an RG monotone; a direct proof, or a counterexample at some $\\ell$, would confirm or refute the monotonicity claim for the entire fermion family."],"forward_implications":["Any bulk MI model that is positive and exactly extensive on the entanglement wedges produces a boundary entropy vector satisfying monogamy and all holographic entropy cone inequalities at first subleading order in $1/N$ (Eq.~13).","The Dirac-fermion realization predicts a concrete shape dependence of the long-distance boundary MI through the coefficients $C_X = \\int_X d^{d-1}x\\, z^{2\\Delta-(d-1)}$, which can be tested in free theories.","The universal short-distance coefficients $c_i$ in Eq.~(28), including the log coefficient $c_0 = -\\kappa k_J$, are regulator-independent data of the theory at the conformal point.","The exact half-disk result (Eq.~29) provides a closed-form benchmark against which recent analytic approximations to generalized-free-field MI in higher dimensions can be checked."],"supporting_citations":[{"why":"Faulkner--Lewkowycz--Maldacena: establishes that bulk MI equals the subleading $O(1)$ holographic MI in the disconnected phase, the identification the whole construction rests on.","marker":"[27]"},{"why":"Casini & Huerta: introduces the extensive mutual information model and the bilocal conserved-kernel functional that this paper generalizes to AdS.","marker":"[28]"},{"why":"Akers, Hernández-Cuenca & Rath: supplies the result that a bulk entropy vector in the HEC implies the boundary QES vector lies in the HEC, which is the bridge used to derive Eq. (13).","marker":"[53]"},{"why":"Allen & Jacobson: provides the AdS bitensor geometry (tangent vectors, parallel propagator, covariant identities) used to write the conserved current--current kernel.","marker":"[60]"},{"why":"Mueck: gives the hypergeometric form of the Dirac two-point function in AdS used for the explicit massive-fermion realization.","marker":"[66]"},{"why":"Benedetti, Casini & Martinez: defines the generalized-free-field mutual information behavior that the construction's long- and short-distance limits are matched against.","marker":"[43]"},{"why":"Casini, Huerta & Yale: supplies the flat-space EMI divergence structure in $d=3$ used as the benchmark for the short-distance area-law term.","marker":"[30]"}],"fun_headline_variants":["Extensive bulk MI enforces the holographic entropy cone","Quantum-corrected MI satisfies monogamy via extensivity","Massive Dirac kernel pins MI to the entropy cone","Bulk extensivity locks boundary MI into the cone","Exact AdS3 MI: monogamy from an extensive kernel"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that the model's conformal fixed point is characterized by a non-flowing RG charge $k(\\ell) = k_J$ constant — an assumption the authors impose and acknowledge is not reached as the $\\nu \\to 0$ limit of the massive fermion solution (footnote 78) — so all short-distance GFF structure and the exact coefficient $c_0 = -\\kappa k_J$ describe this imposed kernel; a second unproven ingredient, $b^2 \\ge a^2$ for all $\\ell$, is what makes $k(\\ell)$ non-negative and gives it meaning as an RG monotone.","fun_headline_variants_meta":{"raw":{"variants":["Extensive bulk MI enforces the holographic entropy cone","Quantum-corrected MI satisfies monogamy via extensivity","Massive Dirac kernel pins MI to the entropy cone","Bulk extensivity locks boundary MI into the cone","Exact AdS3 MI: monogamy from an extensive kernel"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000169,"raw_usage":{"cost":0.000169288,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000169288,"upstream_inference_prompt_cost":0.000122808,"upstream_inference_completions_cost":4.648e-05},"total_tokens":1248,"prompt_tokens":1082,"completion_tokens":166,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":82}},"tokens_in":1082,"tokens_out":166,"cost_source":"openrouter_usage","duration_ms":2303,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":82,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-03T00:51:17.760245+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Compute the short-distance limit of $k(\\ell)$ in Eq.~(24) directly from the massive fermion solution and check whether it reaches the constant Dirac central charge $k_J^D = 2^{\\lfloor d/2\\rfloor-2}\\Gamma^2(d/2)/\\pi^d$ or continues to flow; a flowing kernel at the fixed point would replace the claimed log coefficient $c_0 = -\\kappa k_J$ by an integral over the flow and invalidate the GFF short-distance structure. An independent check is whether $b^2 \\ge a^2$ holds at every $\\ell$, since any $\\ell$ with $b^2 < a^2$ makes $k(\\ell)$ negative and removes the RG-monotone interpretation.","supporting_citations":[{"cited_title":"Quantum corrections to holographic entanglement entropy","cited_arxiv_id":"1307.2892","evidence_quote":"Faulkner--Lewkowycz--Maldacena: establishes that bulk MI equals the subleading $O(1)$ holographic MI in the disconnected phase, the identification the whole construction rests on."},{"cited_title":"Remarks on the entanglement entropy for disconnected regions","cited_arxiv_id":"0812.1773","evidence_quote":"Casini & Huerta: introduces the extensive mutual information model and the bilocal conserved-kernel functional that this paper generalizes to AdS."},{"cited_title":"Quantum Extremal Surfaces and the Holographic Entropy Cone","cited_arxiv_id":"2108.07280","evidence_quote":"Akers, Hernández-Cuenca & Rath: supplies the result that a bulk entropy vector in the HEC implies the boundary QES vector lies in the HEC, which is the bridge used to derive Eq. (13)."},{"cited_title":"Commun. Math. Phys.103, 669","cited_arxiv_id":null,"evidence_quote":"Allen & Jacobson: provides the AdS bitensor geometry (tangent vectors, parallel propagator, covariant identities) used to write the conserved current--current kernel."},{"cited_title":"Spinor Parallel Propagator and Green's Function in Maximally Symmetric Spaces","cited_arxiv_id":"hep-th/9912059","evidence_quote":"Mueck: gives the hypergeometric form of the Dirac two-point function in AdS used for the explicit massive-fermion realization."},{"cited_title":"Mutual Information of Generalized Free Fields","cited_arxiv_id":"2210.00013","evidence_quote":"Benedetti, Casini & Martinez: defines the generalized-free-field mutual information behavior that the construction's long- and short-distance limits are matched against."},{"cited_title":"Mutual information and the F-theorem","cited_arxiv_id":"1506.06195","evidence_quote":"Casini, Huerta & Yale: supplies the flat-space EMI divergence structure in $d=3$ used as the benchmark for the short-distance area-law term."}],"review_version":1}