{"id":"3c38149f-432d-44d3-8b0c-4cccf0e687ec","arxiv_id":"2411.19207","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The upper bound of holographic conditional mutual information equals twice the entanglement of state-constrained purification and diverges in the many-interval limit, revealing abundant multipartite entanglement.","lead":"By maximizing the conditional mutual information in holography, the authors show its upper bound is set by a new quantity, the entanglement of state-constrained purification, and that the maximum grows without limit as the conditioning region is split into more pieces. The result exposes a dimensional divide: four-party entanglement among three fixed regions is sparse in AdS3/CFT2 but abundant in higher dimensions.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The load-bearing step is the weaker disconnectivity condition (Sec. 4.1), not the m<=5 MPT extrapolation; if any connected E beats all disconnected E, Eq. 4.17 fails. The MPT pattern only affects finite-m rates.","rationale":"The reader's weakest-assumption pick (MPT diagram zigzag maximality verified only up to m=5) is a legitimate gap, but it is not the most load-bearing premise for the paper's central claim. The central claim as stated is Eq. 4.17 plus the divergence/saturation of -I3. Those results are supported by Section 4.2's explicit construction using the weaker disconnectivity condition, not by the finite-m maximum pattern. Even if the zigzag pattern fails at large m, one can still take the explicit sequence of E intervals in the two gaps and approach 2min(SA,SB); the divergence and saturation conclusions survive. What would actually break the central claim is a failure of the weaker disconnectivity condition: if there exists a configuration with I(A:E)>0 whose CMI exceeds that of every configuration with I(A:E)=I(B:E)=0, then the upper bound I(A:B|E) <= 2EoSP(A:B) is false and Eq. 4.17 collapses. The proof of the weaker disconnectivity condition in Section 4.1 is a diagrammatic sketch. Its key step, that enlarging the internal gap increases CMI until the next phase transition, is asserted from sign counting in Figure 17 rather than derived from an explicit expression for the CMI as a function of the gap. The proof also does not explicitly verify that constraints I and II (ABE fully connected, E fully disconnected) are preserved throughout, although those constraints are needed to use formula (4.1) and to derive the bound (4.15). A concrete numerical check in AdS3/CFT2 for small m would settle whether the monotonicity and disconnectivity claims actually hold, and would therefore test the foundation of Eq. 4.17 without relying on the large-m pattern extrapolation. For these reasons I partially agree with the reader: the MPT extrapolation is a real gap, but the disconnectivity condition is the more load-bearing one, and the verdict should remain CONDITIONAL rather than stronger, since the identified concern is a testable but not yet conclusively demonstrated step.","tokens_in":45166,"tokens_out":29301,"duration_ms":261450,"concrete_test":"For AdS3/CFT2 with fixed intervals A,B, use the exact cross-ratio CMI expressions (2.9) for m=1 and (2.14) for m=2, and analogous formulas for m=3,4, to numerically track CMI while applying the Sec. 4.1 splitting procedure to an interval Ei with I(A:E)>0. Verify that for each intermediate gap size the CMI is nondecreasing and that the final configuration has I(A:E)=I(B:E)=0 with CMI no smaller. Repeat for about 100 random initial configurations per m<=5. Independently, globally optimize CMI over all 2m endpoints (e.g., differential evolution) without imposing I(A:E)=0 and compare with the best disconnected value; if the unconstrained optimum exceeds the disconnected optimum for any m<=5, the weaker disconnectivity condition is false and Eq. 4.17 is not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Eq. 4.17, sup_E I(A:B|E) = 2EoSP(A:B), requires that the supremum over all E is attained among configurations satisfying the disconnectivity condition I(A:E)=I(B:E)=0. This is exactly the weaker disconnectivity condition of Sec. 4.1. Its proof is a diagrammatic argument: after splitting an interval Ei and enlarging the internal gap, it asserts 'when we enlarge the gap, CMI increases' based only on the sign pattern of RT surfaces in Figure 17, not on an explicit inequality. The proof also does not track whether the entanglement wedge of ABE remains fully connected and whether E remains fully disconnected, which are needed for formula (4.1) and the bound (4.15). If the monotonicity claim fails in some phase, a connected E could have CMI larger than 2EoSP, invalidating the central claim. The reader's flagged MPT-pattern extrapolation (Sec. 3.1, m<=5) is a real gap, but it only affects the finite-m maximum and the specific divergence rates (linear/quadratic/quartic in m); the existence of divergence and saturation to 2min(SA,SB) is already established by the explicit construction in Sec. 4.2, which does not rely on the zigzag maximality. Thus the disconnectivity condition is the more load-bearing premise.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the maximum of holographic n-partite information when n-1 boundary regions are fixed and the remaining region E is varied. For the tripartite case it proposes a multi-entanglement phase transition (MPT) rule, introduces MPT diagrams, derives a generating polynomial for the maximum CMI as a function of the number m of intervals in E, and identifies a divergence of -I3 as m tends to infinity. It then proves, in a weaker form, a disconnectivity condition and claims sup_E I(A:B|E) = 2 EoSP(A:B), with EoSP the entanglement of state-constrained purification; in AdS3/CFT2 for distant intervals this is 2 min(S_A,S_B), saturating the Araki-Lieb upper bound. For I4 and I5 the paper obtains finite upper bounds in AdS3/CFT2 and divergent bounds in higher dimensions, and interprets these results as evidence that all bipartite entanglement in holography emerges from tripartite global entanglement.","tokens_in":45509,"tokens_out":5014,"duration_ms":58611,"significance":"If correct, the results are striking: any two distant small regions in a holographic state can be fully tripartite-entangled with a third region, and the conditional mutual information can saturate its information-theoretic upper bound. The paper's strengths include the absence of free parameters fitted to the target quantities, the extensive explicit evaluations reported in Figures 14-15 and Tables 1-3, and the concrete, falsifiable prediction for the m-dependence of the maximal CMI encoded in the generating polynomial (3.5). The central theorems, however, are not proven with full rigor: the MPT rule and the disconnectivity condition are argued through geometric phase-transition and monotonicity pictures, and several limits are asserted rather than derived. The conclusions are plausible and interesting, but the load-bearing inequalities need to be established more carefully before the claims can be regarded as secure.","major_comments":[{"comment":"The text states that the multi-entanglement phase transition rule is 'proved' by an iterative hill-climbing argument, but the proof paragraph concludes that the rule 'naturally holds in practice.' No theorem shows that a local maximum of CMI as a function of the 2m endpoints must occur at a configuration where 2m phase transitions occur simultaneously, nor that the coordinate-ascent iteration converges to the global maximum. Because the finite-m maximum values in Section 3 and the generating polynomials in Appendix A rely on this rule and on the zigzag diagram being the global maximizer, this gap is load-bearing. Please either provide a rigorous proof or explicitly state the rule as a conjecture whose verification is limited to the small-m cases.","section":"Sec. 2.2"},{"comment":"The proof of the weaker disconnectivity condition rests on the assertion that 'when we enlarge the gap, CMI increases,' justified only by the sign pattern of the RT surfaces in Figure 17. No explicit inequality is given, and the proof does not track whether the entanglement wedge of ABE remains fully connected and whether E remains fully disconnected during the splitting. These properties are required for formula (4.1) and the bound (4.15). If the monotonicity claim fails in some phase, a connected E could yield CMI larger than 2 EoSP(A:B), invalidating Eq. (4.17). Please supply a quantitative proof of the monotonicity, or state the disconnectivity condition as an assumption and test it numerically in each phase used.","section":"Sec. 4.1, Fig. 17"},{"comment":"The derivation of the upper bound I(A:B|E) ≤ 2 S_B relies on the limits in Eq. (4.4), which are asserted rather than derived. In particular, the claim that lim_{m→∞} S_{Gap_{m+1} B Gap_{m+2}} = S_B requires specifying how the intervals E_i are chosen and how the UV cutoff ε is taken relative to m; the paper elsewhere (Eq. (3.14)) uses m ∝ 1/ε. The limiting procedure must be made precise before the saturation statement can be accepted.","section":"Sec. 4.2, Eqs. (4.3)-(4.6)"},{"comment":"The claim that the zigzag MPT diagram gives the global maximum for every m is verified only for m ≤ 5, and Table 1 presents the m=4 comparison at a single cross ratio (CR=1/3). The generating polynomial (3.5) and the linear/quadratic/quartic divergence rates in Section 3.2 assume this pattern for all m. If a different diagram overtakes the zigzag one at large m, the finite-m maximum values and the specific rates would change (although the existence of divergence may still follow from the explicit construction in Section 4.2). Please either prove the zigzag maximality or explicitly restrict the divergence-rate claims to the zigzag family.","section":"Sec. 3.1, Table 1 and Figs. 12-13"},{"comment":"The disconnectivity condition for I4 is argued through Figure 20, but at a crucial step the text states that ruling out diagram (1.1) 'seems difficult' and instead uses the purification argument with F. This is an acknowledgment that the proof is incomplete. Because the upper bound (5.11) and the finiteness/divergence dichotomy in Section 5.3 depend on this condition, the argument needs to be completed or the condition must be stated as a conjecture.","section":"Sec. 5.2, Fig. 20"}],"minor_comments":[{"comment":"The notation is confusing: in the text I denotes exp(CMI/2), while the CMI itself appears in Eq. (1.1); please define the plotted quantity explicitly in each figure caption and use distinct markers for even and odd m in Figure 15, since they obey different polynomials (A.22) and (A.27).","section":"Sec. 3.2, Figs. 14-15"},{"comment":"The definition of EoSP as min(S_AA') is imprecise: please state precisely what is minimized over (which boundary subregions A' and B', under which homology constraints) and how the condition that AA'BB' is the entire boundary is formalized in higher dimensions.","section":"Sec. 4.3, Eq. (4.16)"},{"comment":"The induction of the polynomials from values at l=1,2,3,4 is described clearly, but the statement that the pattern 'has been checked to apply to any non-integer l' is vague; please state the check explicitly (e.g., for which values of l and m it was performed).","section":"Appendix A"},{"comment":"There are typos in the Acknowledgement: 'Bart lomiej Czech' and 'theirhis' should be corrected.","section":"Acknowledgement"},{"comment":"The divergence table uses notation such as ∫ ∂G_AB log n_E which is not explained; please define ∂G_AB and G_ABC in the text or caption before the table is used in Section 6.","section":"Sec. 1 and Table 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript overlaps substantially with companion works [6], [7], and [58]; the editor may wish to verify that the new results are non-duplicative and that the reliance on [6] is properly delineated. The 'no Bell pairs' claim is very strong and likely to attract attention; if the paper is accepted, that claim should be framed as a statement about the leading-order holographic entropy model rather than as a universal property of all holographic states."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this paper genuinely advances holographic multipartite entanglement. It develops a concrete, usable formalism—MPT diagrams and generating polynomials—for identifying the E that maximizes CMI, and it extracts clean results: the divergence of -I3 with the number of intervals, the identity sup_E I(A:B|E) = 2EoSP(A:B), and the finite/infinite dichotomy for I4 between AdS3/CFT2 and higher dimensions. The EoSP quantity is a genuinely new and useful notion, and the asymptotic expansions for one-, two-, and four-gap cases are carefully done. Second, the central claim is more fragile than the paper's tone suggests. I agree with the stress-test note: the weaker disconnectivity condition of Sec. 4.1 is the load-bearing step, not the m<=5 MPT extrapolation. The proof there is diagrammatic. It splits an interval, enlarges the gap, and asserts CMI increases from sign patterns of RT surfaces, without an explicit inequality. It also does not track whether the entanglement wedge of ABE remains fully connected and E remains fully disconnected—both needed for formula (4.1) and the bound (4.15). If the monotonicity fails in some phase, a connected E could beat all disconnected candidates and 2EoSP would be false. That is a real gap; it should be proven or explicitly downgraded to a conjecture supported by numerics.\n\nThe MPT-pattern extrapolation is a lesser concern. It affects the finite-m rates, but the saturation to 2min(SA,SB) and the existence of divergence are already supported by the explicit construction in Sec. 4.2, which does not require zigzag maximality. The 'holds in practice' statement about the hill-climbing argument is a flag but not fatal.\n\nThe 'no Bell pairs' conclusion is overreach. The paper shows that at the maximizing configuration I(A:B)=I(A:E)=0 and I(A:BE)=2SA, strong entropic evidence for tripartite-global structure. But entropic data alone cannot exclude Bell-pair contributions in the full quantum state; the authors concede this in the introduction and then state the bold claim anyway. Rephrase as 'consistent with no Bell pairs at the level of entropy inequalities.'\n\nThe citation pattern is fine; self-citations are contextual, not load-bearing.\n\nThis paper deserves a serious referee. The methods are inventive, the results are plausible and partially verified, and the gaps are addressable. Send it to review with a request to tighten the disconnectivity proof and soften the no-Bell-pairs claim.","headline":"Creative, technically rich paper; the disconnectivity proof in Sec. 4.1 is the real weak spot, and the no-Bell-pairs claim needs softening.","tokens_in":46001,"tokens_out":3819,"would_cite":true,"duration_ms":49700,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper shows that by tuning the number and shape of a region E, holographic conditional mutual information saturates its quantum upper bound $2\\,\\mathrm{EoSP}(A:B)$, so distant small regions are fully tripartite entangled.","keywords":["holographic entanglement entropy","conditional mutual information","n-partite information","multipartite entanglement","Ryu-Takayanagi surface","entanglement of purification","AdS3/CFT2","Araki-Lieb inequality"],"falsifier":"Take AdS3/CFT2 with two fixed intervals A and B at a fixed cross ratio (say 1/3), and for m = 6, 7, 8 solve the full system of quadratic phase-transition equations for every MPT diagram allowed by the disconnectivity constraints, comparing $\\exp[I(A:B|E)/2]$ with the root of the generating polynomial $x(x-1)^m - (\\mathrm{CR}/4)[(\\sqrt{x}+1)^{m+1}-(\\sqrt{x}-1)^{m+1}]^2$ (plus the $(x-1)^m$ term when m is odd). If any non-zigzag diagram yields a larger value, the claimed divergence behavior fails. A complementary check in AdS4/CFT3 would measure $I_4$ for three small disks and a strip region E with m strips approaching all three gap regions; if $I_4$ fails to approach $2\\min(S_A,S_B,S_C)$ as m grows, the higher-dimensional saturation claim is false.","tokens_in":44934,"feed_emoji":"🔗","tokens_out":11689,"duration_ms":93842,"temperature":0.7,"pith_summary":"The paper claims that holographic n-partite information is much larger than previously appreciated: for fixed regions A and B, tuning the third region E can drive the conditional mutual information $I(A:B|E)$ all the way to its quantum-information-theoretic ceiling, and in asymptotically AdS3 this ceiling is $2\\min(S_A,S_B)$. The central identity is $\\sup_E I(A:B|E) = 2\\,\\mathrm{EoSP}(A:B)$, where EoSP is the entanglement of state-constrained purification, the minimal area of a surface that separates A from B while the purifying regions are constrained to be boundary subregions. Because the required E is a collection of intervals whose number m is allowed to grow without bound, $-I_3$ can diverge as $m\\to\\infty$ even though A and B are distant and have zero mutual information. A sympathetic reading of the argument is that any two small distant boundary regions are fully tripartite entangled with some third region, and that all bipartite entanglement in holography emerges from tripartite entanglement.","feed_headline":"Distant holographic patches can be fully tripartite entangled","feed_subtitle":"Tuning a third boundary region drives conditional mutual information to 2 min(S_A, S_B) as its intervals grow.","key_machinery":"The argument is carried by the multi-entanglement phase transition (MPT) rule and its diagrammatic bookkeeping, together with the disconnectivity condition; RT here refers to the Ryu-Takayanagi surfaces, the bulk minimal surfaces whose areas give boundary entanglement entropies. The MPT rule says that, for a fixed number m of intervals in E, the CMI maximum occurs when 2m entanglement phase transitions of RT surfaces happen simultaneously, which fixes the 2m endpoints of E; the 'zigzag' MPT diagram is the one that wins for every m tested (m up to 5). The disconnectivity condition states that at the maximum, $I(A:E)=I(B:E)=0$, and a weaker version is proved by splitting intervals of E and enlarging gaps: any connected configuration can be replaced by a disconnected one with no smaller CMI. With that condition, the CMI collapses to $S_A+S_B+\\sum_i S(E_i)-\\sum_i S(\\mathrm{Gap}_i)$, and requiring the RT surface of AE to stay disconnected yields inequalities whose tight form is $I(A:B|E) \\le 2S_D$ for any D separating A and B; taking the minimal such $S_D$ defines $\\mathrm{EoSP}(A:B)=\\min S(AA')$, the entanglement of state-constrained purification. Generating polynomials for the maximal $\\exp[I/2]$, such as $(1+\\mathrm{CR})\\,x(x-1)^m - \\mathrm{CR}(x+1)^{m+1}$ in the one-gap case, give the exact divergence rates in m.","core_discovery":"The discovery is that the upper bound of conditional mutual information is saturated by holographic configurations, and that the saturating configuration is the limit of an m-interval region at a multi-entanglement phase transition. In AdS3/CFT2 the bound is $I(A:B|E) \\le 2\\min(S_A,S_B)$, and with E chosen as the 'zigzag' MPT diagram with m intervals living in the gaps around A and B, $\\exp[I(A:B|E)/2]$ grows linearly, quadratically, or quartically in m depending on how many gap regions E occupies, so $-I_3$ diverges as $m\\to\\infty$ and the CMI approaches $2\\min(S_A,S_B)$. At the saturating configuration the disconnectivity conditions $I(A:E)=I(B:E)=0$ hold while $I(A:BE)=2S_A$, so A shares no bipartite correlation with B or E individually but is maximally entangled with their union; the paper takes this as a signature that $-I_3$ measures genuine tripartite global entanglement rather than classical correlations. The same method gives $I_4 \\le 2\\min(S_A,S_B,S_C)$ in higher dimensions with divergence as the number of strips grows, while in AdS3/CFT2 the upper bound of $I_4$ with three fixed regions is finite; for $-I_5$ the higher-dimensional upper bound reaches the information-theoretic value $2S_A$. From these the paper concludes that every bipartite entanglement emerges from tripartite entanglement and that no Bell pairs exist in holographic states.","pith_inferences":["The proof that the zigzag MPT diagram is globally maximal is verified explicitly for m up to 5 and extrapolated to all m; if a different diagram overtook it at some large m, the claimed polynomial divergence rates would likely change while the saturation bound $2\\,\\mathrm{EoSP}(A:B)$ might still hold.","Identifying the large-m cutoff with the UV scale maps the linear, quadratic, and quartic growth of $\\exp[I/2]$ into logarithmic divergences of $-I_3$ that match twice the entanglement-entropy divergence, suggesting the O(1) IR contributions also saturate, a point the paper argues directly via EoSP.","The same construction should apply to other holographic backgrounds (for example, higher-genus or multi-boundary wormholes), predicting that $\\sup_E I(A:B|E)=2\\,\\mathrm{EoSP}(A:B)$ holds whenever a separating minimal surface exists, with the throat replaced by the minimal cross-section of the entanglement wedge.","For n ≥ 6 the paper does not prove the upper bound but constructs configurations reaching $2\\min(S_A,\\dots)$; extending the disconnectivity proof to $I_n$ would either confirm the hierarchy that any n-1 distant regions are highly n-partite entangled or reveal a failure of the pattern."],"forward_implications":["Two fixed distant small regions in a holographic CFT can be fully tripartite entangled with a third region: CMI reaches $2\\min(S_A,S_B)$ as the number of intervals in E goes to infinity.","The saturating configuration has $I(A:E)=I(A:B)=0$ but $I(A:BE)=2S_A$, so A is purely quantum entangled with the union BE, making $-I_3$ a faithful measure of tripartite global entanglement for that state.","The general upper bound is $\\sup_E I(A:B|E) = 2\\,\\mathrm{EoSP}(A:B)$, where EoSP is the area of the minimal surface dividing the entanglement wedge of A and B; in the two-sided black hole this is twice the throat area, so the throat encodes tripartite rather than bipartite entanglement.","In AdS3/CFT2, $I_4$ with three fixed regions is finite no matter how complex E is, while in higher dimensions it diverges and can reach $2\\min(S_A,S_B,S_C)$; $-I_5$ likewise reaches $2S_A$ in higher dimensions, so the multipartite entanglement structure is qualitatively different in three bulk dimensions.","All bipartite entanglement between arbitrary intervals emerges from tripartite global entanglement of their subregions, so pure Bell-pair-like bipartite entanglement is absent in holographic states."],"supporting_citations":[{"why":"Supplies the holographic entanglement entropy formula $S_A = \\mathrm{Area}/4G_N$ used in every CMI calculation.","marker":"[2]"},{"why":"Establishes monogamy of mutual information, giving $-I_3 \\ge 0$ in holography and justifying $-I_3$ as an entanglement measure.","marker":"[29]"},{"why":"Defines the holographic entropy cone program that this paper contrasts with unbalanced inequalities arising at infinite m.","marker":"[37]"},{"why":"States the superbalance premise whose smooth-boundary assumption is violated when E has infinitely many intervals.","marker":"[42]"},{"why":"Introduces the entanglement of purification via the minimal surface that divides the entanglement wedge, which EoSP adapts by constraining purifications to boundary subregions.","marker":"[55]"},{"why":"Provides the connected-wedge and distillable-entanglement interpretation that gives EoSP its physical meaning.","marker":"[56]"},{"why":"Earlier evidence that CMI of distant small subregions probes bulk geometry, motivating the search for the maximizing region E.","marker":"[10]"},{"why":"Repackages holographic entropy relations in the I-basis, which the paper uses to identify which $I_n$ have nontrivial divergence and to organize the divergence table.","marker":"[40]"},{"why":"The companion paper whose upper-bound analysis of n-partite information this work extends to saturation and to $I_4$ and $-I_5$.","marker":"[6]"}],"fun_headline_variants":["Holographic tripartite info hits upper bound, diverges","No Bell pairs: holographic entanglement is inherently multipartite","Multipartite entanglement emerges from tripartite in holography","Tripartite foundation: holographic entanglement is global","Divergent CMI proves genuine triple entanglement in holography"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the claim that for every number m of intervals, the global maximum of CMI sits at the zigzag multi-entanglement phase transition diagram; this is verified explicitly only up to m = 5, and the hill-climbing argument used beyond that is not a rigorous proof of global maximality.","fun_headline_variants_meta":{"raw":{"variants":["Holographic tripartite info hits upper bound, diverges","No Bell pairs: holographic entanglement is inherently multipartite","Multipartite entanglement emerges from tripartite in holography","Tripartite foundation: holographic entanglement is global","Divergent CMI proves genuine triple entanglement in holography"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000175,"raw_usage":{"total_tokens":1350,"prompt_tokens":1071,"completion_tokens":279,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":687,"completion_tokens_details":{"reasoning_tokens":193}},"tokens_in":687,"tokens_out":279,"duration_ms":3894,"temperature":1.0,"reasoning_tokens":193,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T10:25:44.121819+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take AdS3/CFT2 with two fixed intervals A and B at a fixed cross ratio (say 1/3), and for m = 6, 7, 8 solve the full system of quadratic phase-transition equations for every MPT diagram allowed by the disconnectivity constraints, comparing $\\exp[I(A:B|E)/2]$ with the root of the generating polynomial $x(x-1)^m - (\\mathrm{CR}/4)[(\\sqrt{x}+1)^{m+1}-(\\sqrt{x}-1)^{m+1}]^2$ (plus the $(x-1)^m$ term when m is odd). If any non-zigzag diagram yields a larger value, the claimed divergence behavior fails. A complementary check in AdS4/CFT3 would measure $I_4$ for three small disks and a strip region E with m strips approaching all three gap regions; if $I_4$ fails to approach $2\\min(S_A,S_B,S_C)$ as m grows, the higher-dimensional saturation claim is false.","supporting_citations":[{"cited_title":"Ryu and T","cited_arxiv_id":null,"evidence_quote":"Supplies the holographic entanglement entropy formula $S_A = \\mathrm{Area}/4G_N$ used in every CMI calculation."},{"cited_title":"Hayden, M","cited_arxiv_id":null,"evidence_quote":"Establishes monogamy of mutual information, giving $-I_3 \\ge 0$ in holography and justifying $-I_3$ as an entanglement measure."},{"cited_title":"Entanglement structures from modified IR geometry","cited_arxiv_id":"2404.02737","evidence_quote":"Earlier evidence that CMI of distant small subregions probes bulk geometry, motivating the search for the maximizing region E."}],"review_version":1}