{"id":"66221d8f-5969-4457-9b28-5e1835415039","arxiv_id":"2501.13182","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Boundary hyperboloids formed by bulk lightcones satisfy a quadratic tangency condition equivalent to the null geodesic equation of an emergent bulk conformal metric.","lead":"The paper develops a Hamilton-Jacobi description of the boundary shapes left by bulk light cones and shows these shapes encode the interior geometry. It proves that, under a broad quadratic condition, the boundary curves are exactly the null geodesics of an emergent bulk metric.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of §4.2 is valid conditional on (4.16), but (4.16) is the equivalence principle itself, not a consequence of the causality inclusion property (4.1); so the broad claim that bulk geodesic motion is derived from boundary causality rests on an unproven postulate.","rationale":"The paper's own wording in §4.2 calls (4.16) 'the main physical assumption', so the conditional proof is not internally flawed. The concern is about the scope of the central claim: causality inclusion alone yields inequalities, not a quadratic null cone. The reader's weakest_assumption already identified exactly this gap, and the paper's conclusion explicitly defers the derivation of (4.16) to future work. I therefore see no reason to move the verdict: the result is a correct conditional reformulation, and the conditional framing is appropriate. The numerical check in §4.1 is stated without code or data, but that is a reproducibility gap, not a mathematical flaw, and it does not bear on the central derivation. I would keep the CONDITIONAL verdict unchanged.","tokens_in":15896,"tokens_out":9351,"duration_ms":99488,"concrete_test":"Construct, in the translation-invariant 2+1 example of §3.1, a smooth family x^μ(p;z)=X^μ+f^μ(p;z) satisfying the normal condition p_μ ∂_{p_ν} f^μ=0 and the inclusion inequality (4.1) for all future-directed timelike δX, but with saturation directions v(p)=(-∂_z f^0, -∂_z f^1, 1) tracing a non-quadric curve in RP^2, e.g. v(θ)=(cos θ, sin θ, 1+ε cos 3θ) for small ε. If such a family exists, then (4.16) is an independent postulate and the conditional verdict is mandatory. If no such family can satisfy (4.1), the gap closes and (4.16) may follow from causality.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem in §4.2 is internally consistent: given a family of boundary hyperboloids x^μ(p;X) and a non-degenerate g_{MN}(X) such that the kernel of ∂x^μ/∂X^M is null for every p (eq. 4.16), the argument via S=p·x correctly recovers the Hamilton-Jacobi equation and hence null geodesics. The load-bearing weakness is that (4.16) is not derived from the causality inclusion inequality (4.1). The inclusion property only implies a convex-hull statement: for every future-directed timelike or null δX, -p_ν(∂x^ν/∂X^M)δX^M ≥ 0 for all p, and saturation singles out some null directions. It does not imply that those directions are the zero set of a single quadratic form. In a (d+1)-dimensional tangent space a Lorentzian null cone is a quadric, and assuming that the boundary-selected directions form a quadric is exactly assuming the equivalence principle. The paper acknowledges this in §5 ('the equivalence principle') and lists the derivation of (4.16) from boundary dynamics as the outstanding question. But then the abstract's phrase 'from which the bulk geodesic equation can be derived under some assumptions' is doing substantial work: the assumption is as strong as the conclusion. The reverse logic establishes a reformulation, not a derivation of bulk geometry from boundary causality alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a Hamilton-Jacobi description of the boundary hyperboloids H±(X) formed by null geodesics emitted from a bulk point X and reaching an asymptotic AdS boundary. The authors first review how a bulk Hamilton-Jacobi function S(p;X) controls these surfaces and how, under a rank assumption, the Hamilton-Jacobi equation implies the bulk null geodesic equation. They then compute explicit hyperboloids for Poincaré AdS, planar black holes, and small AdS-Schwarzschild black holes, including photon-sphere effects. In Section 4, they translate bulk causality into a boundary inclusion property for light-cone cuts and use it to propose a method for measuring the bulk conformal metric from families of hyperboloids. Section 4.2 contains the paper's central converse: if a family of boundary hyperboloids satisfies the quadratic tangent condition (4.16) for a fixed nondegenerate metric g, then S = p·x satisfies the Hamilton-Jacobi equation and the curves are null geodesics of g. The paper is candid in Section 5 that the origin of the quadratic condition remains open and that it is tied to the equivalence principle.","tokens_in":53,"tokens_out":9072,"duration_ms":211395,"significance":"If taken as a conditional reformulation, the paper is a useful and concrete contribution. The derivation in §2.1.1 is sound, the examples in §3 are worked out in explicit parametric form, and the photon-sphere analysis in §3.2 is physically interesting. The numerical check in §4.1 that the metric reconstruction reproduces the AdS5 planar black hole metric adds concreteness. The main limitation is that the reverse derivation in §4.2 does not derive a common quadratic null cone from the causality inclusion property; instead, it assumes such a cone through (4.16). Since the paper acknowledges this in §5, the value lies in a precise equivalence statement and an algorithmic measurement protocol, not in a derivation of bulk geometry from boundary causality alone. With this caveat made central, the paper would be a reasonable and publishable contribution.","major_comments":[{"comment":"The central theorem is conditional on the quadratic tangent condition (4.16), but (4.16) is not derived from the causality inclusion property (4.1). Inequality (4.1) only constrains boundary displacements for timelike or null bulk displacements; saturation selects one null direction per boundary momentum p. That all these selected directions lie on the null cone of a single quadratic form g is an additional postulate, essentially the equivalence principle. The paper states this in §5, calling it 'the essential question,' so the derivation itself is not circular. However, the abstract's phrase 'from which the bulk geodesic equation can be derived' and the heading of §4.2 ('null geodesics in the bulk from causality') overstate the result. I recommend rephrasing the claim to state explicitly that (4.16) is an assumption and that the result is an equivalence: given a common quadratic null cone, boundary hyperboloids satisfying the tangent condition are exactly the null geodesic congruences of that cone. This is a substantive framing issue because it determines what the paper has actually proved.","section":"§4.2, Eq. (4.16); §5; Abstract"},{"comment":"The metric-measurement protocol also silently uses the bulk Lorentzian structure that it claims to measure. The forward direction is correct: if the bulk metric is known, then the saturation vectors of (4.1) are indeed null, and equal-slope displacements on nearby hyperboloids give null vectors. But the transition from 'saturation' to the existence of a single g in (4.3) uses the bulk notions of promptness and Lorentzian geometry. If the protocol is intended as a purely boundary reconstruction without prior bulk input, the existence of the quadratic form must be stated as an assumption rather than as a consequence of the inclusion property. Please make explicit in §4.1 that the measurement recovers the metric only after a quadratic null cone is assumed or supplied by the bulk theory.","section":"§4.1, Eqs. (4.1)–(4.3)"},{"comment":"The proof of the converse is valid under the stated maximal-rank assumptions, but the physical regime where these assumptions hold is not discussed. In particular, the matrix ∂δX^M/∂p^ν is required to have rank d, and ∂x^μ/∂X^M is required to have rank d. These conditions can fail at conjugate points, caustics, or where boundary hyperboloids develop cusps, as can occur near the photon sphere in the AdS-Schwarzschild example. The paper already acknowledges isolated singular points in §2.2, but a more precise statement of where the genericity assumptions hold, and what happens at their failure, would strengthen the claim that the construction applies to realistic bulk geometries.","section":"§4.2.1, rank assumptions"}],"minor_comments":[{"comment":"There is a typo: the expression 'δX^M δX^M' should read 'g_{MN}(X) δX^M δX^N'.","section":"Eq. (4.3)"},{"comment":"The index notation is inconsistent: the surface coordinate is x^μ(p;X), but (2.22) writes ∂x_{˙μ}/∂p_{˙ν}. This should be ∂x^{˙μ}/∂p_{˙ν} or equivalent, with the metric on the boundary used consistently.","section":"Eq. (2.22)"},{"comment":"The statement that 'd(d+3)/2 discrete values of p generically suffice' is plausible by counting symmetric matrices up to scale, but the nonlinear dependence of the null vector on p means a rank argument is needed. A brief justification or a reference would be helpful.","section":"Section 4.1, counting of evaluations"},{"comment":"Reference [13] is given as 'arXiv:25XX.XXXXX'. This placeholder should be updated or the reference should be marked as forthcoming with a working identifier.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and contains a useful, checkable reformulation of bulk null geodesics in terms of boundary hyperboloids. The main issue is the gap between the actual theorem, which assumes a common quadratic null cone, and the advertised 'derivation from causality.' I do not see this as fatal, because the authors are explicit about the open question in §5. I would be comfortable accepting after the abstract and §4.2 are reframed to state the assumption prominently, and after the technical clarifications in the major comments are addressed. The companion-paper placeholder in [13] should also be cleaned up."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper is worth reading. It gives a clean Hamilton-Jacobi description of boundary hyperboloids and works through explicit examples (planar and spherical black holes, photon sphere) that are genuinely useful. The main new result in section 4.2 is a valid conditional proof: given a family of boundary hyperboloids satisfying the quadratic tangency condition (4.16), the function S = p.x obeys the Hamilton-Jacobi equation and the boundary curves are null geodesics of the metric g. The logic is internally consistent, the rank assumptions are stated, and the examples check out.\n\nBut the soft spot is exactly where the abstract does the most work. The inclusion property (4.1) gives a convex-hull statement: for every timelike or null displacement, an infinite set of inequalities holds. It does not, by itself, imply that the saturating displacements are the zero set of a single quadratic form. Assuming that they are (4.16) is assuming the equivalence principle, that null directions form a quadric in each tangent space. The authors acknowledge this in section 5 and call it the outstanding question. So the paper is better read as a reformulation: if the boundary data satisfies a quadratic condition, then you get bulk geodesics. That is a useful sharpening, but it is not a derivation of bulk geometry from boundary causality alone.\n\nMinor points: the numerical check in section 4.1 is stated but no code or data is provided, so it is not reproducible. That is a small gap for a theory paper. The citation pattern looks fair; the Engelhardt-Horowitz inclusion property is properly credited, and the photon-sphere references are there.\n\nWho is this for? People working on holographic reconstruction, light-cone cuts, and bulk locality. They will get a clear framework and concrete examples. I would send it to a serious referee; it is not a desk reject. The referee should push on the status of (4.16) and ask for a sharper statement of what would count as deriving it. But the paper is honest about its limits, and the central conditional theorem is sound.\n\nRecommendation: engage with it.","headline":"Solid Hamilton-Jacobi framework for boundary hyperboloids, but the claimed derivation of bulk geodesics from causality assumes the quadratic null-cone structure it aims to derive.","tokens_in":16710,"tokens_out":2182,"would_cite":true,"duration_ms":22225,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["11.25.Tq","04.20.Cv"],"model":"deepseek-v4-flash","headline":"A single quadratic condition on boundary lightcone imprints is equivalent to the bulk Hamilton-Jacobi equation, so bulk geodesics can be derived from boundary causality.","keywords":["boundary hyperboloids","holographic lightcones","Hamilton-Jacobi formalism","null geodesics","light-cone cuts","bulk metric reconstruction","causal inclusion","AdS/CFT"],"falsifier":"Take a known bulk spacetime without symmetries, compute its boundary hyperboloids, and apply the reverse algorithm of section 4.2 to reconstruct the conformal metric; if the reconstructed metric differs from the original beyond a Weyl rescaling at any bulk point, the quadratic-cone assumption fails. Alternatively, search for a family of boundary hyperboloids whose tangent displacements respect causal inclusion but whose null directions cannot be fit by a single quadratic form, which would violate condition (4.16) and invalidate the geodesic derivation.","tokens_in":15620,"feed_emoji":"🔭","tokens_out":9615,"duration_ms":91005,"temperature":0.7,"pith_summary":"This paper studies how the lightcones of localized bulk events are seen from the boundary of a holographic spacetime, and what those boundary images can tell about the bulk. It argues that the boundary surfaces cut out by the future and past lightcones of a bulk point X—the hyperboloids $H^\\pm(X)$—encode the bulk conformal metric, with the boundary momentum doubling as the surface normal. Under one quadratic-cone assumption, the paper proves that the infinitesimal displacements of X that leave a hyperboloid tangent to itself are exactly the null directions of a bulk metric, which makes the quadratic condition equivalent to the bulk Hamilton-Jacobi equation. The upshot is a concrete recipe: measure nearby boundary hyperboloids, read off the conformal metric algebraically, and reconstruct bulk null geodesics from boundary data alone. If this picture holds, bulk geometry is not an input to the boundary description but an output of boundary causality.","feed_headline":"Bulk geodesics emerge from boundary lightcone shapes","feed_subtitle":"Under one quadratic tangency condition, boundary hyperboloids are exactly the null geodesic congruence of the reconstructed bulk metric.","key_machinery":"The load-bearing object is the Hamilton-Jacobi function $S(p;X)=p_\\mu x^\\mu(p;X)$, whose gradient in p gives the boundary hyperboloid and whose gradient in X gives the bulk momentum. The main identity is the quadratic tangent condition (4.16): the infinitesimal displacements $\\delta X(p;X)$ that keep the boundary hyperboloid tangent to itself are null with respect to a single metric g. The paper proves that this condition is equivalent to the Hamilton-Jacobi equation $g^{MN}(\\partial_M S)(\\partial_N S)=0$, and that the flow generated by $g^{MN}\\partial_N S$ satisfies the geodesic equation. This equivalence is what lets boundary curves, treated as given data, be reinterpreted as null geodesic congruences of the reconstructed conformal metric.","core_discovery":"The paper's central claim is that bulk causality leaves an imprint on the boundary strong enough to reconstruct the bulk metric and its null geodesics. For a fixed bulk point X, the future and past lightcones intersect the asymptotic boundary in codimension-one surfaces $x^\\mu(p;X)$, parametrized by the boundary momentum p, which is also the surface normal. The paper shows that if every infinitesimal bulk displacement $\\delta X$ that produces a tangent change of these surfaces satisfies a common quadratic equation $g_{MN}\\delta X^M\\delta X^N=0$ for some non-degenerate metric g, then that condition is equivalent to the bulk Hamilton-Jacobi equation. Consequently a family of boundary hyperboloids satisfying the quadratic tangent condition is exactly the null geodesic congruence of the reconstructed conformal metric, and the bulk geodesic equation follows from boundary data. The paper also gives an explicit algebraic procedure that recovers the conformal metric from nearby hyperboloids and verifies it on planar and spherical black hole examples, including points behind which orbiting geodesics create many branches.","pith_inferences":["One could test the reverse construction numerically in a non-symmetric spacetime by generating hyperboloids from a known metric and then running the section 4.2 algorithm to see whether the recovered conformal metric agrees with the original; agreement would confirm the equivalence beyond the symmetric examples.","The quadratic-cone assumption is essentially a statement of the equivalence principle; if an emergent bulk had momentum-dependent or non-Lorentzian null directions at some scale, the boundary data would show a p-dependent 'metric' and the reconstruction would break down, offering a diagnostic for Lorentz violation.","The same hyperboloid data may give access to gravitational focusing and geodesic deviation: the paper leaves open whether the boundary inclusion property implies convergence of nearby null geodesics, which would connect this kinematic construction to the emergence of gravity rather than just geometry.","A practical extension would be to feed the reconstructed conformal metric into a boundary-correlator search for bulk-point singularities, using the hyperboloid shapes as a prior for where to look for scattering events."],"forward_implications":["A holographic observer who can measure correlation-function singularities along boundary hyperboloids can determine the bulk conformal metric up to an overall Weyl factor, without solving any bulk equations of motion.","The inclusion property of hyperboloids for timelike-separated bulk points becomes a testable boundary signature of bulk causality; points close to black-hole horizons appear as time-translated copies of a common shape with logarithmically growing time delay.","Because the quadratic condition fixes the metric algebraically from finitely many tangent points, a finite set of boundary measurements over-determines the bulk metric and provides built-in consistency checks.","The same logic applies beyond asymptotically AdS boundaries, such as null infinity in flat space or artificial boundaries around local observers, making the construction a general way to read geometry from lightcone cuts."],"supporting_citations":[{"why":"introduces light-cone cuts $C^\\pm(X)$ and the program of reconstructing general bulk metrics from them; the paper's hyperboloids are closely related to these cuts.","marker":"[8]"},{"why":"shows that the causal inclusion property of light-cone cuts determines the spacetime conformal metric; section 4.1 turns this into an explicit algebraic measurement procedure.","marker":"[9]"},{"why":"describes the radar-camera setup in which boundary hyperboloids become observable through wavepacket correlators; the paper takes these surfaces as given boundary data.","marker":"[12]"},{"why":"provides the gravitational time-delay and causality theorems that justify the boundary inclusion property used to derive the metric from hyperboloids.","marker":"[23]"},{"why":"supplies the definitions of prompt null geodesics and transitive causality that separate light-cone cuts from full hyperboloids and underpin the inclusion inequalities.","marker":"[24]"}],"fun_headline_variants":["Bulk geodesics from boundary hyperboloid shapes","Boundary lightcones are the bulk geodesic imprint","Hyperboloids on boundary encode null geodesics","Boundary hyperboloids give bulk metric and geodesics","From boundary lightcone shapes to bulk geodesics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes there is a single, fixed notion of 'null' in the bulk: every infinitesimal shift of a bulk point that leaves a boundary hyperboloid tangent must lie on one common quadratic lightcone, and nothing in the causality-inclusion property alone forces that to be true.","fun_headline_variants_meta":{"raw":{"variants":["Bulk geodesics from boundary hyperboloid shapes","Boundary lightcones are the bulk geodesic imprint","Hyperboloids on boundary encode null geodesics","Boundary hyperboloids give bulk metric and geodesics","From boundary lightcone shapes to bulk geodesics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001309,"raw_usage":{"total_tokens":5272,"prompt_tokens":819,"completion_tokens":4453,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":435,"completion_tokens_details":{"reasoning_tokens":4378}},"tokens_in":435,"tokens_out":4453,"duration_ms":30746,"temperature":1.0,"reasoning_tokens":4378,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T16:23:36.865182+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a known bulk spacetime without symmetries, compute its boundary hyperboloids, and apply the reverse algorithm of section 4.2 to reconstruct the conformal metric; if the reconstructed metric differs from the original beyond a Weyl rescaling at any bulk point, the quadratic-cone assumption fails. Alternatively, search for a family of boundary hyperboloids whose tangent displacements respect causal inclusion but whose null directions cannot be fit by a single quadratic form, which would violate condition (4.16) and invalidate the geodesic derivation.","supporting_citations":[],"review_version":1}