{"id":"0305e26e-956d-4355-8dd3-9dcb2c2a9105","arxiv_id":"2507.21292","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In four-dimensional Lambda-vacuum spacetimes, algebraic specialness plus local conformal flatness of null infinity forces the asymptotic Weyl data into the Kerr-de Sitter-like class.","lead":"This paper develops a new decomposition of the Weyl tensor into pieces tied to a family of hypersurfaces, then uses it to show that in four dimensions, algebraically special Einstein spacetimes with a conformally flat null infinity must have the same boundary data as Kerr-de Sitter-like solutions. The result gives a geometric handle on which Lambda-vacuum spacetimes are related to rotating black holes.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2's automatic λ>0 extendability of the multiple WAND (Lemma 6) fails for spacetimes with horizons, e.g. the ingoing principal null direction of Kerr–de Sitter; the theorem needs an explicit extendability hypothesis.","rationale":"The paper's algebraic decomposition, Fefferman–Graham expansion, and reduction of the type II condition to equations (79)–(80) are carefully executed, and the derivation of the conformal Killing vector ξ from (82) is internally consistent. For the conformally flat branch, the appeal to the previously classified Kerr–de Sitter-like data in [38], together with the uniqueness of the asymptotic initial data in Theorem 1, largely supports the claimed coincidence; the general-γ sufficiency question explicitly flagged in Section 5.1 is not the decisive gap for the conformally flat statement. The single most load-bearing weakness is the automatic extendability of k for λ>0 claimed by Lemma 6. The lemma's global-hyperbolicity argument works only inside D^-(I+), but in the intended black-hole examples this domain does not contain a neighbourhood of I+; the ingoing repeated PND of Kerr–de Sitter is a concrete counterexample to the lemma's generality. This does not necessarily destroy the classification, because one may select an extending WAND, but it means Theorem 2 as stated is not established for all algebraically special spacetimes with conformally flat I. The reader's CONDITIONAL verdict is therefore appropriate: the hypotheses of Theorem 2 should be restated, and the headline classification should be scoped to spacetimes admitting a multiple WAND that extends transversally to I. I see no reason to move beyond the reader's conditional assessment.","tokens_in":32417,"tokens_out":17081,"duration_ms":222800,"concrete_test":"Take Schwarzschild–de Sitter with Λ>0, a Petrov type D metric whose I+ is locally conformally flat. In a Fefferman–Graham chart adapted to I+, write the two repeated principal null directions k_+ and k_- and compute whether the ingoing k_- extends smoothly to I+ as a nonvanishing g-null vector field. If it does not extend, as expected because its integral curves cross the black-hole horizon and never reach I+, then Lemma 6 is false as stated and Theorem 2's λ>0 case must carry an explicit extendability hypothesis. A complementary check is to verify directly that points arbitrarily close to I+ admit future-directed ingoing null geodesics that do not intersect I+.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central classification rests on Theorem 2, whose λ>0 branch is justified by Lemma 6. Before Lemma 6, the paper assumes k extends transversally to I for λ<0 and claims Lemma 6 makes this unnecessary for λ>0. The lemma's proof restricts to M=D^-(I+) and uses that every future-inextendible causal geodesic from M\\I+ meets I+ exactly once. That is a property of the chosen domain, not of the vector field k. In black-hole spacetimes in the intended class, such as Schwarzschild–de Sitter or Kerr–de Sitter (Petrov type D, locally conformally flat I), points arbitrarily close to I+ admit future-directed ingoing repeated principal null geodesics that cross the horizon and end at the singularity, never meeting I+. Hence those points are not in D^-(I+), and D^-(I+) contains no collar neighbourhood of I+. Consequently Lemma 6 cannot establish that the ingoing PND extends to I+ as a nonvanishing vector field. Since Theorem 2 is stated for an arbitrary geodesic multiple WAND k, its λ>0 branch is not proven for all algebraically special spacetimes with conformally flat I. The theorem should either assume transversal extendability for both signs of λ or explicitly restrict to a multiple WAND whose integral curves are future-complete to I+, and the coincidence claim should be scoped accordingly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a covariant algebraic decomposition of Riemann-type and Weyl-type tensors with respect to a non-null vector, derives Gauss, Codazzi, and Ricci-type identities for the Weyl tensor, and combines these with Fefferman-Graham expansions to compute the leading and subleading fall-off of the Weyl tensor components near conformal infinity in Lambda-vacuum spacetimes. In four dimensions, the authors specialize the algebraic type II (multiple WAND) conditions and obtain two asymptotic equations, (79) and (80), relating the boundary metric, the rescaled electric Weyl tensor W^⊥, and an auxiliary unit vector y. For locally conformally flat I they integrate these equations and conclude that W^⊥ has the form (89) with ξ a conformal Killing vector of the boundary metric. The paper then invokes the existing classification of Mars, Paetz, Senovilla, and Simon to argue that this coincides with the Kerr-de Sitter-like class with conformally flat I.","tokens_in":32696,"tokens_out":10812,"duration_ms":128363,"significance":"If the main theorem is established, the paper provides a clean geometric characterization of the Kerr-de Sitter-like class in four dimensions and demonstrates that local conformal flatness of I is a robust criterion for algebraic specialness. The algebraic decomposition and the explicit asymptotic formulas (60)-(62) are useful and appear to be derived carefully; the honesty about the open sufficiency question for equations (79)-(80) is commendable. However, the lambda>0 branch of Theorem 2 rests on Lemma 6, whose proof is not valid for black-hole spacetimes of the intended class, and the 'exact match' statement in the abstract is stronger than what is proven for lambda<0. With an explicit extendability hypothesis and a scoped statement of the classification, the paper would be a solid contribution.","major_comments":[{"comment":"The proof of Lemma 6 restricts to M = D^-(I^+) and uses the property that every future-inextendible causal geodesic starting in M\\I^+ intersects I^+ exactly once. This is a property of the chosen domain of dependence, not of the vector field k. In spacetimes with horizons in the intended class, such as Kerr-de Sitter, the future-directed ingoing repeated principal null geodesic from points arbitrarily close to I^+ crosses the event horizon and never reaches I^+; those points are not in D^-(I^+), so D^-(I^+) contains no collar neighbourhood of I^+. Consequently the map phi in Lemma 6 need not cover a neighbourhood of I^+, and the lemma does not prove that an arbitrary geodesic multiple WAND k extends to I for lambda>0. Since Theorem 2 is stated for an arbitrary such k with no lambda>0 extendability assumption, its lambda>0 branch is not established as written. The theorem should either assume transversal extendability of k for both signs of lambda, or explicitly allow the choice of a multiple WAND whose integral curves are future-complete to I^+, and the subsequent classification statement should be scoped accordingly.","section":"Section 5, Lemma 6 and Theorem 2"},{"comment":"The abstract and the introduction claim that four-dimensional algebraically special spacetimes with locally conformally flat I 'match exactly' the Kerr-de Sitter-like class. What the body proves is the necessary statement of Theorem 2 for lambda != 0 (with the extendability caveat above) and, for lambda>0, the converse by invoking the classification of [38] and the fact that Kerr-Schild metrics are of algebraic type at least II. For lambda<0 no converse is proved: the paper does not show that every set of data of the form (89), with xi a boundary conformal Killing vector, arises from an algebraically special bulk spacetime, nor that the Kerr-de Sitter-like class with conformally flat I exhausts those data in the negative-lambda case. The text after Eq. (80) itself states that sufficiency of (79)-(80) for the bulk type II property is open. The 'exact match' claim should therefore be restricted to lambda>0, or the missing converse for lambda<0 should be proved.","section":"Abstract and Section 5.1"}],"minor_comments":[{"comment":"The symbol W^⊥ is used both for the rescaled leading coefficient Omega^{-1} Weyl^⊥|_I in Eq. (59) and for the limit L_{\\partial\\Omega} W^⊥|_I in Eq. (90). The two are consistent, but the overloaded notation should be flagged explicitly to avoid confusion.","section":"Section 5, notation around Eq. (59) and Appendix A"},{"comment":"There are numerous typographical errors and inconsistencies in names (for example 'admiting', 'sastisfies', 'Moverover', 'Frierdrich', and 'Storminger' in the references). A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The case W^⊥ = 0 is included by allowing kappa = 0, but for kappa = 0 the expression with xi is singular and xi itself is not defined. It would be cleaner to state the zero case separately, as is done just before Eq. (81).","section":"Equation (89)"},{"comment":"The statement that Eq. (77) follows from Eq. (76) and the tracelessness of the Weyl tensor is used to discard one of the algebraic equations; a short derivation or a reference for that four-dimensional identity would make the reduction easier to verify.","section":"Section 5.1, Eq. (77)"}],"recommendation":"major_revision","confidential_remarks":"The central calculation in Sections 2-4 and Appendix A appears careful, and the paper is transparent about the open sufficiency of the asymptotic equations. The main concern is Lemma 6: for Lambda>0 black-hole spacetimes such as Kerr-de Sitter, the past domain of dependence of I^+ does not contain a collar of I^+, so the proof cannot establish extendability of an arbitrary multiple WAND. I believe this is fixable by adding an explicit extendability hypothesis for both signs of Lambda and by carefully scoping the 'exact match' claim to cases where the converse is actually established. The paper is otherwise within the scope of a serious relativity journal, and the algebraic decomposition plus asymptotic expansions are likely to be useful independently of the classification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on arXiv:2507.21292. The paper builds a genuinely useful asymptotic toolbox and uses it to give a new characterization of the Kerr-de Sitter-like class, but the main theorem's λ>0 branch has a real gap: it relies on Lemma 6, which claims automatic extendability of the multiple WAND to I, and that proof does not hold in spacetimes with horizons. The statement needs to be re-scoped.\n\nWhat is actually new: the covariant decomposition of Weyl-type tensors relative to a non-null vector u, together with the derived Gauss-Codazzi-Ricci identities for the Weyl tensor, is clean and reusable. The Fefferman-Graham expansion of the Weyl tensor components near I, with explicit leading terms, is also new and well executed—especially the 4D result W⊥ = -(3|λ|/2) g(3), which gives a gauge-free characterization of asymptotic data. Theorem 2, which forces the boundary electric data into the Kerr-de Sitter-like form when conformally flat I and type II are imposed, is novel and supplies the missing four-dimensional analogue of [7]. The paper is also honest: it states that sufficiency of (79)-(80) for bulk type II is open, and the algebraic derivations in the appendix are detailed enough to check.\n\nThe soft spot is Lemma 6. The lemma is supposed to show that for λ>0, any geodesic multiple WAND extends transversally to I, making an extendability hypothesis unnecessary. But the proof restricts to M = D^-(I+) and uses the fact that every future-inextendible causal geodesic from M\\I+ intersects I+. That is true for points in the domain of dependence, not for points near I+ that lie beyond a horizon. In Kerr-de Sitter, the ingoing principal null direction is a multiple geodesic WAND whose integral curves cross the horizon and hit the singularity, never reaching I+. Such points are not in D^-(I+), so the lemma does not apply. Consequently Theorem 2 as stated is too strong: for λ>0, extendability of the chosen WAND to I should be added as a hypothesis, or the theorem should be restricted to WANDs with future-complete integral curves to I. This is a clean restatement, and the classification likely survives for the physically intended case where the outgoing PND is chosen, but the advertised 'coincide with the Kerr-de Sitter-like class' goes beyond what is proven.\n\nThere are no serious citation problems. The heavy self-citation is to results the authors previously established, and those are the relevant ones. No circularity detected.\n\nWho this is for: mathematical relativists, especially those working on conformal infinity, asymptotic data with Λ, and algebraic classification. It deserves a serious referee. I would recommend major revision, asking the authors to fix the extendability assumption and recalibrate the abstract and title to match. The core machinery and the conditional statement are likely right.","headline":"Strong, reusable asymptotic toolbox and a plausible new characterization of the Kerr-de Sitter-like class, but the λ>0 branch of the main theorem rests on a WAND-extendability lemma that does not cover black-hole spacetimes.","tokens_in":33208,"tokens_out":9542,"would_cite":true,"duration_ms":117203,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C30","83C20"],"pacs":["04.20.-q","04.20.Ha","04.20.Jb"],"model":"deepseek-v4-flash","headline":"Four-dimensional Λ-vacuum spacetimes that are algebraically special with a locally conformally flat null infinity coincide with the Kerr–de Sitter-like class; the paper's main theorem pins their electric Weyl data at infinity to a single…","keywords":["Weyl tensor","algebraically special spacetimes","Kerr–de Sitter spacetime","conformally flat null infinity","Fefferman–Graham expansion","cosmological constant","asymptotic initial data","algebraic classification of the Weyl tensor"],"falsifier":"Two concrete checks would settle the claim. First, follow the ingoing principal null direction of the Kerr–de Sitter metric backward from the horizon: if a future-inextendible null geodesic with that tangent fails to intersect $\\mathscr I^+$ exactly once, the hypothesis behind the $\\lambda>0$ extension lemma fails on exactly the spacetimes the theorem targets, and the classification would need a stricter assumption. Second, compute the electric Weyl part at $\\mathscr I$ for a known Kerr–de Sitter-like metric, for instance from its Kerr–Schild form, and verify that it reproduces (89) with $\\xi$ a conformal Killing vector; any algebraically special $\\Lambda$-vacuum spacetime with conformally flat $\\mathscr I$ whose $W^\\perp$ at $\\mathscr I$ violates this form would refute the classification.","tokens_in":32213,"feed_emoji":"🌌","tokens_out":24454,"duration_ms":216608,"temperature":0.7,"pith_summary":"Four-dimensional vacuum spacetimes with non-zero cosmological constant that are algebraically special and have a locally conformally flat null infinity are claimed to be exactly the Kerr–de Sitter-like class. The paper reaches the claim by developing a covariant decomposition of Weyl-type tensors with respect to a non-null vector, which turns the Gauss, Codazzi and Ricci identities into identities for the Weyl tensor whose terms live entirely on the hypersurfaces orthogonal to that vector. Paired with the Fefferman–Graham expansion of the metric at null infinity, these identities deliver the first two orders of the Weyl tensor's fall-off, and the algebraic type-II condition becomes two intrinsic equations at the boundary. When the boundary metric is locally conformally flat, those equations integrate to a single algebraic form for the electric part of the Weyl tensor at $\\mathscr I$, namely $W^\\perp_{\\alpha\\beta} = \\frac{\\kappa}{|\\xi|^5_\\gamma}\\left(\\xi_\\alpha\\xi_\\beta + \\frac{\\epsilon|\\xi|^2_\\gamma}{3}\\gamma_{\\alpha\\beta}\\right)$ with $\\kappa\\in\\{0,\\pm1\\}$ and $\\xi$ a conformal Killing vector of the boundary metric — precisely the data that define the Kerr–de Sitter-like class. Since metrics of that class are themselves algebraically special, the paper concludes that the two families coincide, giving a local geometric characterization of a family that contains Kerr–de Sitter.","feed_headline":"4D: algebraically special + flat null infinity = Kerr–de Sitter class","feed_subtitle":"A Weyl-tensor expansion at null infinity collapses the search to one known family: Kerr–de Sitter-like.","key_machinery":"The engine is a covariant decomposition of any Riemann-type or Weyl-type tensor with respect to a unit non-null vector $u$, $T = T^{\\parallel u} + \\epsilon\\, u \\circledast T^{\\perp\\parallel} + T^{\\perp} \\mathbin{?} \\eta$ (Lemma 1), built so that the Weyl tensor's components relate to the intrinsic geometry of the hypersurfaces orthogonal to $u$ through Gauss, Codazzi and Ricci-type identities (Proposition 2). Near null infinity, $u$ is taken to be the normalized gradient of the conformal factor, and the identities express the Weyl tensor entirely in terms of the boundary metric $\\gamma$ and its Fefferman–Graham coefficients. In four dimensions the tangential part collapses to $Weyl^{\\parallel}_g = -\\epsilon\\, Weyl^{\\perp}_g \\mathbin{?} \\gamma$ (here $?$ denotes the Kulkarni–Nomizu product), so the whole conformal curvature near $\\mathscr I$ is governed by the electric part $W^{\\perp}$. Feeding the first two orders of the Fefferman–Graham expansion into the algebraic type-II condition yields the boundary system (79)–(80); when $\\gamma$ is locally conformally flat the Cotton tensor of $\\gamma$ vanishes, and the system integrates to the single form (89) with $\\xi = f y$ satisfying the conformal Killing equation of $\\gamma$.","core_discovery":"On the paper's own terms the result is Theorem 2: let $(\\widetilde{M},\\tilde g)$ be a four-dimensional Einstein manifold with $\\lambda\\neq 0$ admitting a locally conformally flat null infinity $\\mathscr I$ and whose Weyl tensor has algebraic type at least II with a geodesic multiple WAND (Weyl Aligned Null Direction) $k$, with $k$ assumed to extend transversally to $\\mathscr I$ when $\\lambda<0$. Then the electric part of the Weyl tensor at $\\mathscr I$ takes the form $W^\\perp_{\\alpha\\beta} = \\frac{\\kappa}{|\\xi|^5_\\gamma}\\left(\\xi_\\alpha\\xi_\\beta + \\frac{\\epsilon|\\xi|^2_\\gamma}{3}\\gamma_{\\alpha\\beta}\\right)$ with $\\kappa\\in\\{0,\\pm1\\}$ and $\\xi$ a conformal Killing vector of the boundary metric $\\gamma$. Because this is exactly the asymptotic-data form that defines the Kerr–de Sitter-like class with conformally flat $\\mathscr I$, and because that class is known to consist of algebraically special (type at least II) metrics, the paper concludes that the two families coincide: four-dimensional algebraically special $\\Lambda$-vacuum spacetimes with conformally flat $\\mathscr I$ are exactly the Kerr–de Sitter-like class.","pith_inferences":["The transversal-extension hypothesis could probably be traded for a horizon-free (asymptotic simplicity) condition, which would split the family into spacetimes whose multiple WAND reaches $\\mathscr I$ and horizon-bound spacetimes that escape the theorem's assumptions.","The collapse to a single field is special to four dimensions, where conformal flatness is the only curvature condition on the three-dimensional boundary; in higher dimensions the boundary Weyl tensor enters independently, so the expected classification will be a family of data sets rather than one algebraic form.","If the equivalence holds, 'is this spacetime Kerr–de Sitter-like?' becomes a local question answerable at infinity — check algebraic type II and the vanishing of the boundary Cotton tensor — which is much cheaper than the global uniqueness arguments used elsewhere.","The computed fall-offs imply that the information carried by the electric part at $\\mathscr I$ is dimension-dependent — $O(\\Omega)$ in four dimensions, $O(\\Omega^2)$ in higher dimensions, with a $\\Omega^2\\log\\Omega$ anomaly in five — which suggests that proposals for gravitational radiation based on $W^\\perp$ transfer across dimensions only with care."],"forward_implications":["For $\\lambda>0$, where $(\\gamma, W^\\perp)$ are the free asymptotic data of the Cauchy problem at $\\mathscr I$, Theorem 2 says the data of every algebraically special spacetime with conformally flat $\\mathscr I$ are exhausted, up to the constants $(\\kappa, f, \\xi)$, by a single conformal Killing vector field of the boundary metric.","The case $W^\\perp=0$ yields data locally diffeomorphic to de Sitter near $\\mathscr I$, so de Sitter appears in the classified family as the trivial member.","The equivalence connects three previously separate descriptions of the same family: data of the form (89), the alignment of the bulk Weyl tensor with a Killing vector plus conformally flat $\\mathscr I$, and Kerr–Schild metrics sharing null infinity with a de Sitter background.","The general expansion of the Weyl tensor near $\\mathscr I$ is worked out in all dimensions and for both signs of $\\Lambda$, with the first terms given explicitly; the paper announces that the same machinery yields the higher-dimensional analogue of the classification.","The boundary system (79)–(80) gives an integration recipe around any locally conformally flat $\\gamma$: solve (80) for the pair $(f,y)$, form $W^\\perp$ through (79), and read off admissible asymptotic data for an algebraically special spacetime; whether satisfying the system asymptotically is sufficient for type II in a neighbourhood of $\\mathscr I$ is left open."],"supporting_citations":[{"why":"Defines and classifies the Kerr–de Sitter-like class with conformally flat infinity; the boundary-data classification that Theorem 2's conclusion plugs into.","marker":"[38]"},{"why":"Extends the Kerr–de Sitter-like class to all dimensions and characterizes it as Kerr–Schild metrics sharing null infinity with a de Sitter background, which closes the equivalence with algebraic specialty.","marker":"[40]"},{"why":"Shows that Kerr–Schild metrics with an (anti-)de Sitter background are of algebraic type at least II, supplying the known direction that the classified spacetimes are algebraically special.","marker":"[35]"},{"why":"Source of the Fefferman–Graham expansion and its coefficients, used to compute the explicit first terms of the Weyl-tensor fall-off near null infinity.","marker":"[17]"},{"why":"The higher-dimensional algebraic classification of the Weyl tensor (types, WANDs, type II) that Section 5 rewrites as the boundary equations (79)–(80).","marker":"[11]"},{"why":"Classical existence and structure result for past asymptotically simple solutions with positive cosmological constant, the backdrop for the $\\lambda>0$ extension of the multiple WAND to the boundary.","marker":"[24]"},{"why":"Well-posed asymptotic initial value problem for asymptotically de Sitter spacetimes in all dimensions, used to justify extending the multiple WAND to null infinity for $\\lambda>0$.","marker":"[27]"},{"why":"A multiple WAND implies a geodesic multiple WAND, which sets up the frame used near the boundary in the type-II conditions.","marker":"[15]"}],"fun_headline_variants":["4D algebraically special + flat null infinity = Kerr–de Sitter class","Weyl expansion shows: algebraically special + flat I+ are Kerr–de Sitter","Flat null infinity forces algebraically special 4D vacua into Kerr–de Sitter class","Exact match: algebraically special 4D with flat I+ are Kerr–de Sitter-like","Kerr–de Sitter-like class fully characterized by Weyl algebra and flat I+"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the multiple WAND $k$, the privileged null direction singled out by the algebraic condition, reaches null infinity and meets it transversally: this is assumed outright for $\\lambda<0$, while for $\\lambda>0$ it is derived from a global-hyperbolicity argument whose key property — that every future-inextendible null geodesic from the interior hits $\\mathscr I^+$ exactly once — fails when an event horizon is present and is open for the Kerr–de Sitter-like metrics the theorem is meant to classify.","fun_headline_variants_meta":{"raw":{"variants":["4D algebraically special + flat null infinity = Kerr–de Sitter class","Weyl expansion shows: algebraically special + flat I+ are Kerr–de Sitter","Flat null infinity forces algebraically special 4D vacua into Kerr–de Sitter class","Exact match: algebraically special 4D with flat I+ are Kerr–de Sitter-like","Kerr–de Sitter-like class fully characterized by Weyl algebra and flat I+"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001258,"raw_usage":{"total_tokens":5191,"prompt_tokens":1020,"completion_tokens":4171,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":4052}},"tokens_in":636,"tokens_out":4171,"duration_ms":29633,"temperature":1.0,"reasoning_tokens":4052,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:58:27.459993+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Two concrete checks would settle the claim. First, follow the ingoing principal null direction of the Kerr–de Sitter metric backward from the horizon: if a future-inextendible null geodesic with that tangent fails to intersect $\\mathscr I^+$ exactly once, the hypothesis behind the $\\lambda>0$ extension lemma fails on exactly the spacetimes the theorem targets, and the classification would need a stricter assumption. Second, compute the electric Weyl part at $\\mathscr I$ for a known Kerr–de Sitter-like metric, for instance from its Kerr–Schild form, and verify that it reproduces (89) with $\\xi$ a conformal Killing vector; any algebraically special $\\Lambda$-vacuum spacetime with conformally flat $\\mathscr I$ whose $W^\\perp$ at $\\mathscr I$ violates this form would refute the classification.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines and classifies the Kerr–de Sitter-like class with conformally flat infinity; the boundary-data classification that Theorem 2's conclusion plugs into."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the Kerr–de Sitter-like class to all dimensions and characterizes it as Kerr–Schild metrics sharing null infinity with a de Sitter background, which closes the equivalence with algebraic specialty."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that Kerr–Schild metrics with an (anti-)de Sitter background are of algebraic type at least II, supplying the known direction that the classified spacetimes are algebraically special."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the Fefferman–Graham expansion and its coefficients, used to compute the explicit first terms of the Weyl-tensor fall-off near null infinity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The higher-dimensional algebraic classification of the Weyl tensor (types, WANDs, type II) that Section 5 rewrites as the boundary equations (79)–(80)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classical existence and structure result for past asymptotically simple solutions with positive cosmological constant, the backdrop for the $\\lambda>0$ extension of the multiple WAND to the boundary."},{"cited_title":"Asymptotically de Sitter metrics from scattering data in all dimensions","cited_arxiv_id":"2311.02739","evidence_quote":"Well-posed asymptotic initial value problem for asymptotically de Sitter spacetimes in all dimensions, used to justify extending the multiple WAND to null infinity for $\\lambda>0$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"A multiple WAND implies a geodesic multiple WAND, which sets up the frame used near the boundary in the type-II conditions."}],"review_version":1}