{"id":"d650509c-43ce-4d0d-a2a4-b014417652c2","arxiv_id":"2502.09894","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Stable minimax surfaces are shown to be HRT surfaces, the entanglement wedge is the smallest minimax homology region, and a cooperating time-sheet configuration would prove the equality of RT and HRT entropy cones.","lead":"This paper proves new properties of the minimax formula for holographic entanglement entropy, and shows how a conjectural 'cooperating' property of time-sheets would yield a graph model for time-dependent states. If that graph model exists, the entropy cones of static and time-dependent holographic states coincide, a major open question in holography.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6 rests on an unproven existence assumption: a smooth, everywhere timelike minimax time-sheet is assumed, not shown, and the proof explicitly invokes it in Lemma 2.2(i).","rationale":"The reader's weakest_assumption and my concern coincide: the variational existence assumptions in Section 2.2 are the least secure link in the proof of Theorem 2.6. I considered whether the stronger concern is the unproved cooperating conjecture (Conjecture 4.1) because the graph model and cone equality depend on it, but the paper explicitly presents that as a conjecture and even gives a potential obstruction; the reader already conditions the headline consequences on it. The more insidious gap is in the supposedly proven minimax=HRT theorem, where a smooth timelike minimax time-sheet is assumed rather than derived. This matters because Corollary 2.3 is the bridge from stability to extremality, and Lemma 2.2(i) is the bridge from the time-sheet to smoothness of the minimax surface. The paper's own text flags both the smooth-time-sheet assumption and the assumption that sup/inf are achieved, so this is not an artifact of my reading. The numerical and analytic tests in Section 4.2 support the cooperating conjecture in special configurations but do not test existence of smooth timelike minimax time-sheets, so they do not close the gap. A concrete construction in AdS3-Vaidya would settle whether the assumption is benign or false in a physically relevant class. Since the paper is honest and labels these issues, the appropriate verdict remains CONDITIONAL; I would not move it.","tokens_in":41273,"tokens_out":9601,"duration_ms":102343,"concrete_test":"Analytical check in a nontrivial dynamical spacetime: in AdS3-Vaidya (or a collapsing null shell) with a single boundary interval A, attempt an explicit construction of a smooth timelike time-sheet tau containing the known HRT surface gamma and homologous to D(A), with gamma maximal on tau. Concretely, take tau to be a small timelike deformation of the entanglement horizon, solve the constraint that tau is everywhere timelike and that the second variation of area on tau is negative definite at gamma, and verify the construction for intervals of varying boost.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem of the paper, Theorem 2.6 ('A stable minimax surface is an HRT surface'), is not self-contained: its proof of extremality passes through Lemma 2.2(i), which opens by asserting that the minimax surface lies on a fully timelike, smooth minimax time-sheet 'which, as explained at the end of subsection 2.2, can be always assumed to exist.' The cited explanation is not a proof; Section 2.2 instead states: 'In this paper, we make the assumption that one such everywhere timelike and smooth time-sheet exists,' and separately assumes 'that the sup and inf are achieved.' If the variational problem (2.2) has no minimizer in some NEC-obeying asymptotically AdS spacetime, or if every minimax time-sheet is forced to contain null pieces or seams, then Lemma 2.2(i) fails, stability cannot be shown to imply extremality, and Theorem 2.6—and with it the minimax identification with HRT and Theorem 3.1—is unsupported. The paper's 'floppiness' argument is heuristic and explicitly deferred to future work. This is the load-bearing gap because the graph-model discussion in Section 4, whatever its status, presupposes that the objects in Section 2 exist and have the claimed properties.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops the minimax reformulation of the HRT holographic entanglement entropy formula introduced in [23]. The authors define time-sheets and the minimax area (Eq. (2.2)), prove that a stable minimax surface is extremal (Corollary 2.3, Theorem 2.5) and hence an HRT surface (Theorem 2.6), establish the equivalence between the original and relaxed minimax prescriptions (Lemma 2.7, Theorem 2.8, Corollary 2.9), and prove several geometric properties: the entanglement wedge is the smallest minimax homology region (Theorem 3.1), causal holographic information upper-bounds entanglement entropy (Theorem 3.2), entanglement wedge nesting and spacelike separation of degenerate minimax surfaces hold (Theorem 3.3 and Corollary 3.4), and cooperating pairs of time-sheets exist for connected crossing regions (Theorem 3.5). On this basis the authors construct a candidate spacetime graph model and prove that, if a configuration of cooperating time-sheets exists, graph cuts reproduce HRT entropies (Theorem 4.1), which would imply equality of the RT and HRT entropy cones. They present analytic and numerical evidence for the cooperating conjecture in pure AdS3 and in spherically symmetric matter, and discuss a configuration in pure AdS3 for which cooperation appears difficult.","tokens_in":41575,"tokens_out":8220,"duration_ms":85207,"significance":"If the main theorems hold, minimax provides an independent covariant route to HRT and a spacetime-level tool for proving entropy inequalities, and the graph model would be a substantial step toward showing that the HRT and RT entropy cones coincide. The paper's strengths include the new relaxed-minimax equivalence, the direct proof that the entanglement wedge is the smallest minimax homology region, and a transparent numerical test of the cooperating conjecture. The authors also honestly flag the main unresolved points: the existence of smooth timelike minimax time-sheets, the disconnected-component case of Theorem 3.5, and the open status of the cooperating conjecture. Because the central results are conditional on these points, the contribution is significant but not yet fully established.","major_comments":[{"comment":"The variational problem (2.2) is the foundation of the paper, but the text immediately following it assumes that the sup and inf are achieved and then states: \"we make the assumption that one such everywhere timelike and smooth time-sheet exists.\" These existence assumptions are load-bearing: Lemma 2.2(i) invokes the smooth timelike time-sheet to conclude smoothness of the minimax surface, and Corollary 2.3 and Theorem 2.6 use that conclusion. No proof or sufficient condition is given, and the \"floppiness\" argument is explicitly deferred. Please either prove existence under the stated holographic assumptions (NEC, Einstein equations, AdS boundary conditions), or state Theorem 2.6 and its corollaries as conditional theorems with the assumptions made explicit.","section":"2.2, Eq. (2.2)"},{"comment":"Even granting the existence assumption, the treatment of non-smooth time-sheets is not justified. The proof asserts that if the maximal surface on a non-smooth time-sheet inherits a kink, \"a nearby time-sheet which resolves the kink would have a smaller-area maximal surface.\" This is a local variational statement, while minimax minimization is global over all time-sheets, so the assertion does not follow from minimality of the time-sheet. A rigorous argument is needed to rule out kinks on minimizing time-sheets; as written, the smoothness of the minimax surface is assumed rather than derived.","section":"Lemma 2.2(i)"},{"comment":"The theorem claims existence of cooperating time-sheet pairs for crossing regions, but the proof is a sketch and explicitly leaves out the case of disconnected minimax surfaces: \"we suspect that it should go through by applying the same procedure to each connected component individually.\" The proof also assumes without discussion that the minimum over configurations in (3.3) is attained (footnote 10). Since disconnected components are needed for the proof of SSA and for the general graph-model construction, this is a genuine gap in a central claim. Please either supply the full proof or clearly mark Theorem 3.5 as proven only for connected minimax time-sheets.","section":"Theorem 3.5"},{"comment":"In the equality case |γ+(α)_1| = |γ+(β)_2|, the proof says one can \"round off these corners\" to obtain a smaller-area maximal surface, but no construction is given and it is not shown that the rounded time-sheet remains in the correct homology class and remains a minimax time-sheet. Since this lemma is used to restrict cooperating configurations to minimally intersecting ones, it needs a rigorous proof or an explicit weakening.","section":"Theorem 3.6"}],"minor_comments":[{"comment":"There is a typo, \"codmiension\", which should be \"codimension\".","section":"2.1"},{"comment":"The term \"non-degeneracy\" is used in the proof but not defined; the argument that a small deformation produces a maximal surface arbitrarily close in area to the original one would benefit from a precise statement of the topology on the space of surfaces.","section":"2.3, Lemma 2.1"},{"comment":"The notation \"kmax = 8/13ϵ\" is unclear: it would help to specify the normalization of the affine parameter and to state explicitly that the subsequent time interval is of order ϵ.","section":"4.3"},{"comment":"In the graph-model definition, the weight of an edge is described as the area of the partial minimax surface on the shared partial time-sheet; if that partial surface has several connected components, the text should state explicitly that the weight is the sum of their areas.","section":"4.1"},{"comment":"The numerical statement that the second eigenvalue of the Hessian changes sign precisely at χ = χc is presented without an analytic expression for the eigenvalue, which makes the check harder to reproduce independently.","section":"4.2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is unusually transparent about its limitations. The main concern is not novelty or presentation, but that the central theorem is built on explicit existence assumptions and a sketched disconnected case. I would advise a major revision focused on making the assumptions precise and either proving or clearly conditioning Theorem 2.6 and Theorem 3.5."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper earns its keep. It proves real new properties of the minimax prescription: the stability–extremality equivalence (Theorems 2.5 and 2.6), the relaxed minimax equivalence (Theorem 2.8), the entanglement wedge as the smallest minimax homology region (Theorem 3.1), and the existence of cooperating pairs for two crossing regions (Theorem 3.5). Those are not restatements of earlier work. The graph model in Section 4 is a natural and potentially powerful construction, and the numerical checks in AdS3 with NEC-obeying matter are honest attempts to test the cooperating conjecture rather than post-hoc justification. The paper also clearly marks which claims are proven and which are conjectural, including scenarios where the cooperating conjecture may fail. That intellectual honesty is worth something.\n\nThe soft spot flagged by the stress-test is real and load-bearing. The proof of Theorem 2.6 passes through Lemma 2.2(i), which assumes a smooth, everywhere timelike minimax time-sheet exists. The paper's own Section 2.2 says only that one \"can be always assumed to exist,\" with a heuristic floppiness argument deferred to future work. The same holds for the assumption that the sup and inf are achieved. Without those existence results, stability cannot be shown to imply extremality, and the minimax=HRT identification—and everything downstream, including Theorem 3.1 and the graph model motivation—is unsupported. This is not a hidden flaw: the authors state it, but it is a genuine gap in the proof chain, not a minor technicality.\n\nTwo smaller issues. Theorem 3.5 is explicitly a sketch for disconnected components; that is flagged, but it matters for SSA. And the cooperating conjecture itself may be false, as the paper's own \"small triangle\" example suggests. That does not undermine the value of the paper—it asks the right question and gives partial evidence—but it does mean the headline consequence (RT = HRT cone) remains an open problem.\n\nWho should read this? Anyone working on holographic entanglement entropy, entropy cones, or covariant prescriptions. It will be cited, mainly for the minimax properties and the graph model proposal. It deserves a serious referee: the topic is important, the new results are non-trivial, and the gaps are stated rather than hidden. My own verdict is conditional, but that is a reason to send it to peer review with sharp requests, not to desk reject.\n\nRecommendation: send it to review, and ask referees to focus on the existence assumptions for minimax time-sheets and surfaces, and on whether the proof of Theorem 3.5 can be completed for disconnected components. If those can be patched or at least precisely formulated as conjectures with clear evidence, the paper will be a solid contribution.","headline":"Genuinely useful minimax properties plus a clearly conditional graph-model construction; the central theorem leans on an assumed, unproven existence claim, but the paper is honest about it and deserves peer review.","tokens_in":42045,"tokens_out":2109,"would_cite":true,"duration_ms":24409,"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 proves that a stable minimax surface is exactly the HRT surface, and that cooperating time-sheets would yield a graph model making the static and time-dependent holographic entropy cones coincide.","keywords":["holographic entanglement entropy","minimax surfaces","HRT formula","time-sheets","entanglement wedge","holographic entropy cone","graph models","null energy condition"],"falsifier":"A concrete check would be to exhibit an asymptotically AdS spacetime satisfying the null energy condition in which no smooth everywhere-timelike minimax time-sheet exists, or where a stable minimax surface has nonzero extrinsic curvature trace; either would break Theorem 2.6. For the cooperating conjecture, an explicit three-region configuration (for instance, a sharpened version of the \"small triangle\" setup in pure AdS$_3$) with a proof that no cooperating time-sheet configuration exists would falsify Conjecture 4.1 while leaving the cone-equality question open.","tokens_in":41110,"feed_emoji":"🧮","tokens_out":8090,"duration_ms":70176,"temperature":0.7,"pith_summary":"The paper establishes that the minimax prescription for holographic entanglement entropy—maximize the area of a surface on a timelike \"time-sheet,\" then minimize over time-sheets—produces exactly the same surface and area as the covariant HRT formula. It proves that a stable minimax surface is an HRT surface (Theorem 2.6), and that the entanglement wedge is the smallest spacetime homology region bounded by a minimax time-sheet. The paper then shows that if a collection of time-sheets \"cooperates\"—every partial HRT surface is maximal on its partial time-sheet—the time-sheets define a graph on which min cuts compute HRT entropies. That would imply the time-dependent (HRT) entropy cone equals the static (RT) cone; the cooperating property is proved for pairs but only conjectured for arbitrary collections, with explicit examples where it may fail.","feed_headline":"Minimax surfaces reproduce holographic entanglement entropy","feed_subtitle":"Stable minimax surfaces equal the HRT surface; cooperating time-sheets would unify the entropy cones.","key_machinery":"Time-sheets are piecewise timelike or null hypersurfaces homologous to $D(A)$ relative to the future and past conformal boundaries; the minimax surface is the maximal-area surface on a time-sheet, minimized over time-sheets. The load-bearing mechanism is the null focusing theorem (null energy condition plus Einstein equations): focusing makes HRT surfaces maximal on their entanglement horizons, so the infimum over time-sheets selects the least-area extremal surface. The cooperating property—each partial HRT surface is maximal on each partial time-sheet cut out by intersections—is the mechanism that turns a collection of time-sheets into a graph model: graph cuts correspond to unions of partial time-sheets, and cooperation prevents shortcut cuts from undercutting the true entropies.","core_discovery":"The central claim is that the minimax formula $S(A)=\\frac{1}{4G_N}\\inf_{\\tau}\\sup_{\\gamma\\in\\Gamma_\\tau}|\\gamma|$ — maximize area over surfaces $\\gamma$ on a time-sheet $\\tau$ homologous to the boundary domain of dependence $D(A)$, then minimize over time-sheets — computes the same entropy as the covariant HRT prescription. The paper proves that a stable minimax surface is an HRT surface: stability and extremality coincide for minimax surfaces, and the minimization over time-sheets selects the least-area extremal surface in the spacetime homology class. It also proves that the entanglement wedge is the smallest homology region bounded by a minimax time-sheet, that causal holographic information upper-bounds $S(A)$, and that entanglement wedges nest for nested boundary domains of dependence. For crossing regions, cooperating time-sheet configurations are introduced; a cooperating pair always exists for two crossing regions with connected time-sheets. For more regions the paper leaves the existence of cooperating configurations as a conjecture, noting that its graph-model and entropy-cone conclusions are conditional on it.","pith_inferences":["If the cooperating conjecture fails for three or more time-sheets, the graph-model route to cone equality is blocked, but cone equality could still hold: the paper's potential counterexample lives in a 2+1-dimensional spacetime where the RT inequalities are already known to hold.","A more flexible graph construction—for example allowing non-minimally intersecting configurations or using relaxed-minimax surfaces—might rescue a graph model even where strict cooperation seems impossible.","The minimax formulation suggests a quantum version, $S(A)=\\min_\\tau\\max_\\sigma(|\\gamma|+S(\\rho_{\\tau\\cap\\sigma}))$, which could extend the approach to quantum extremal surfaces and bulk entropy inequalities; the paper raises this as a direction rather than proving it.","If a spacetime graph model exists, tensor-network constructions for static states could carry over to time-dependent states, effectively integrating out bulk time inside the Wheeler-DeWitt patch; the paper leaves this as an application."],"forward_implications":["A stable minimax surface exists and equals the HRT surface, so holographic entanglement entropy can be computed without first assuming an extremal surface exists.","The entanglement wedge is the smallest spacetime homology region bounded by a minimax time-sheet, sharpening the role of the wedge in bulk reconstruction.","For any pair of crossing boundary regions, a cooperating pair of time-sheets exists, yielding a minimax-based proof of strong subadditivity that bypasses the maximin representative trick.","If the cooperating conjecture holds for all $N$, every time-dependent holographic state with $N$ specified regions admits a weighted graph whose min cuts give all HRT entropies; hence the HRT entropy cone coincides with the RT cone.","The relaxed minimax prescription—maximizing over surfaces achronal only within each time-sheet—agrees with the original minimax under the null energy condition, providing an equivalent formulation for entropy-inequality proofs."],"supporting_citations":[{"why":"Introduces the minimax prescription for covariant holographic entanglement entropy; this paper extends it and proves stable minimax equals HRT.","marker":"[23]"},{"why":"Establishes the maximin formulation, its equivalence to HRT, and proofs of SSA and MMI; used here for comparison and for the causal-bound argument.","marker":"[24]"},{"why":"The original covariant HRT proposal defining holographic entanglement entropy in time-dependent spacetimes, which minimax aims to reproduce.","marker":"[12]"},{"why":"Defines the holographic entropy cone and graph models via min cuts; the target structure for the time-dependent graph model.","marker":"[7]"},{"why":"Gives the static inclusion/exclusion proof of strong subadditivity that the minimax proof elevates to spacetime volumes.","marker":"[4]"},{"why":"Shows the maximin strategy cannot prove entropy inequalities beyond SSA and MMI, motivating the new minimax approach.","marker":"[15]"},{"why":"Proves the RT inequalities hold in 2+1 dimensions for topologically trivial time-dependent spacetimes, providing context for the small-triangle example.","marker":"[20]"}],"fun_headline_variants":["Minimax surfaces match covariant entropy","Time-dependent entropy via minimax time-sheets","Entanglement wedge from minimax principle","HRT entropy from stable minimax surfaces","Minimax approach to holographic entropy cone"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proofs assume that a smooth, everywhere timelike minimax time-sheet exists and that the supremum and infimum in the minimax formula are achieved; the graph-model conclusion additionally assumes that cooperating configurations of time-sheets exist for arbitrarily many regions.","fun_headline_variants_meta":{"raw":{"variants":["Minimax surfaces match covariant entropy","Time-dependent entropy via minimax time-sheets","Entanglement wedge from minimax principle","HRT entropy from stable minimax surfaces","Minimax approach to holographic entropy cone"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000265,"raw_usage":{"total_tokens":1606,"prompt_tokens":946,"completion_tokens":660,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":593}},"tokens_in":562,"tokens_out":660,"duration_ms":6535,"temperature":1.0,"reasoning_tokens":593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T20:08:10.398401+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check would be to exhibit an asymptotically AdS spacetime satisfying the null energy condition in which no smooth everywhere-timelike minimax time-sheet exists, or where a stable minimax surface has nonzero extrinsic curvature trace; either would break Theorem 2.6. For the cooperating conjecture, an explicit three-region configuration (for instance, a sharpened version of the \"small triangle\" setup in pure AdS$_3$) with a proof that no cooperating time-sheet configuration exists would falsify Conjecture 4.1 while leaving the cone-equality question open.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the maximin formulation, its equivalence to HRT, and proofs of SSA and MMI; used here for comparison and for the causal-bound argument."},{"cited_title":"Maximin is Not Enough","cited_arxiv_id":"1712.10004","evidence_quote":"Shows the maximin strategy cannot prove entropy inequalities beyond SSA and MMI, motivating the new minimax approach."},{"cited_title":"Holographic Entropy Cone with Time Dependence in Two Dimensions","cited_arxiv_id":"1905.03787","evidence_quote":"Proves the RT inequalities hold in 2+1 dimensions for topologically trivial time-dependent spacetimes, providing context for the small-triangle example."}],"review_version":1}