{"id":"2095dc32-db0a-4f60-8d8a-6c336dd2e434","arxiv_id":"2412.00183","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper derives an inversion formula that defines de Sitter boundary operators as integrals of bulk fields against the bulk-to-boundary propagator, reproducing known two-point functions and perturbation theory.","lead":"This paper builds a boundary theory for quantum fields in an accelerating universe, using a continuous family of operators instead of the discrete list familiar from anti-de Sitter space. It then gives a formula that turns any bulk field into boundary operators, which could be used in non-perturbative cosmological calculations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The inversion formula rests on the de Sitter completeness relation (4.14), imported from AdS by analytic continuation without an independent dS derivation; this is the most load-bearing assumption.","rationale":"The reader's weakest assumption matches the most load-bearing point: the completeness relation (4.14) is the only ingredient that is imported from AdS via analytic continuation without a direct dS derivation, and it is exactly the relation that makes the inversion formula invert the expansion. The paper's own warning that the distribution must be understood through analytic continuation underscores that this is a real gap rather than a stylistic choice. I considered the incomplete convergence proof in §3.3, which the author also acknowledges; that affects the regime where the expansion can be inserted into correlators, but even formal operator-level inversion already requires (4.14). The inversion formula (4.10) is the strongest claim, and it hinges on (4.14). A direct numerical or analytic check in dS coordinates, as proposed, would settle the issue. The verdict should remain CONDITIONAL: the construction is coherent and internally consistent, but this key identity needs independent verification. Since the reader already assigned CONDITIONAL, no verdict change is needed; hence UNCHANGED.","tokens_in":37934,"tokens_out":4082,"duration_ms":40984,"concrete_test":"Verify (4.14) directly in de Sitter planar coordinates without AdS analytic continuation. Fix d=3 (and repeat for d=2) and take a smooth, compactly supported test function f(Y2) on the dS patch. Numerically evaluate I(Y1) = ∫ dλ (1/(2πξ±_Δ)) ∫ dY2 Π_Δ(σ±) f(Y2) using the explicit Π_Δ(σ) in (2.30), the coefficient from (2.40), and the σ± prescription; compare I(Y1) with f(Y1) on a grid of Y1 values. If the match fails beyond numerical tolerance, or if the λ-integral requires additional contact terms, then Eq. (4.10) does not invert (3.1) and the central construction is not established. Alternatively, perform the same check analytically for f(Y2) a dS plane wave mode, reducing the composition to a known Kontorovich-Lebedev integral and checking the delta-function normalization exactly in dS coordinates.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that (4.10) inverts the bulk-to-boundary expansion (3.1), i.e. that every bulk scalar with a Källén-Lehmann representation yields boundary primaries O_Δ(y). The decisive step is the completeness relation (4.14), used in §4.2 to go from (4.13) back to (3.1). This relation is derived in Appendix D by Wick-rotating the AdS harmonic-analysis completeness relation (D.7). The paper itself notes that (4.14) must be understood through analytic continuation to AdS and provides no direct de Sitter proof. The Wick rotation passes through several delicate steps: the measure factor (D.11), the phase relation (D.10), and the replacement of the AdS delta function δ(X1,X2) with the dS delta δ(Y1,Y2). Each step is plausible, but none is checked directly in dS. If (4.14) is not a genuine identity at the distributional level, or if the analytic continuation picks up extra local terms, then (4.10) fails to invert (3.1), and the boundary operators defined by (4.10) do not reproduce the bulk field in the expansion. The subsequent consistency checks (§4.3) partly mask this because they use additional identities (split representation, V-diagram, broken leg) that also ultimately rely on the same analytic continuation. Thus a direct, coordinate-space verification of (4.14) is the single most decisive test of the paper's construction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a non-perturbative description of the late-time conformal boundary of QFT in de Sitter spacetime. It introduces a continuous family of boundary primary operators O_Δ(y) living on the principal series, together with a bulk-to-boundary expansion in which a bulk scalar is written as an integral over these operators with kernel determined by the Källén-Lehmann spectral density. The main result is an inversion formula, Eq. (4.10), that constructs the boundary operators from the bulk field by integrating against the bulk-to-boundary propagator. The paper also defines a boundary-to-bulk connector operator, studies the large-Δ behaviour of the spectral density, and verifies the inversion formula by recovering boundary two-point functions and by reproducing perturbative correlators, including an explicit three-point example. Several appendices provide supporting identities, including a completeness relation imported from AdS harmonic analysis.","tokens_in":38249,"tokens_out":10124,"duration_ms":90140,"significance":"If the construction is valid, it gives a concrete, symmetry-based definition of boundary operators for de Sitter that is adapted to the continuous principal-series spectrum, going beyond the formal discrete AdS-like expansion whose operators generically lie off the dS unitary representations. The proposed inversion formula and the boundary-to-bulk connector are potentially useful technical tools for cosmological bootstrap computations. The paper is also commendable for its explicit consistency checks, including the two-point function with contact terms, the V-diagram and broken-leg identities, and a three-point example. However, the central inversion step depends on a completeness relation that is obtained only by analytic continuation from AdS, and the convergence of the expansion inside correlation functions is explicitly left open; these are load-bearing gaps that need to be addressed before the construction can be regarded as fully established.","major_comments":[{"comment":"The claim that Eq. (4.10) is the inverse of the bulk-to-boundary expansion (3.1) relies on the de Sitter completeness relation (4.14). The derivation in Appendix D is exclusively by Wick rotation from the AdS harmonic-analysis completeness relation (D.7), and the paper itself states that (4.14) must be understood through analytic continuation to AdS. The later consistency checks in §4.3.1 and §4.3.2 use the split representation, the V-diagram and the broken-leg identity, all of which are derived from the same analytic continuation, so they do not provide an independent test of (4.14). I ask for a direct de Sitter derivation or a coordinate-space distributional check of (4.14) — for example, by acting with both sides on the mode-function basis of §2.1 — because if (4.14) fails as an identity, Eq. (4.10) does not invert Eq. (3.1).","section":"§4.2, Eq. (4.14), and Appendix D"},{"comment":"The convergence of the bulk-to-boundary expansion inside correlation functions is not established. Equation (3.31) controls only the large-Δ behaviour of the bulk-to-boundary coefficient a_Δ; the matrix elements ⟨ψ_i|O_Δ|ψ_j⟩ in Eq. (3.13) also enter the convergence analysis and their large-Δ asymptotics are not determined. The paper's own conclusion states this explicitly. Since the non-perturbative construction is intended to justify inserting (3.1) into correlators, this missing bound is a load-bearing gap. Please either supply the needed estimate or state clearly in the abstract and introduction that the expansion is formal and that only the large-Δ behaviour of a_Δ is analysed.","section":"§3.3 and concluding bullet in §5"},{"comment":"The construction covers only the principal-series part of the de Sitter spectrum. For a generic scalar whose Källén-Lehmann density has complementary-series or exceptional-series contributions, Eq. (3.1) is not an equality and Eq. (4.10) defines only a principal-series component of the boundary theory. The title and abstract claim a construction of the de Sitter late-time boundary without this restriction. Please qualify the claims and state explicitly that the paper constructs the principal-series sector, with other representations left to future work.","section":"§3.1, Eq. (3.1), and §5"},{"comment":"The recovery of the power-law coefficient of the boundary two-point function is not an independent check. With the normalization α_Δ=1, Eq. (3.10) fixes a_Δ = sqrt(2 g_Δ ρ(Δ)), and substituting this into Eq. (4.21) makes α_Δ equal to 1 by construction. The independent content of the two-point check is the contact-term coefficient β_Δ, which is not fixed by the normalization convention. The paper should present the two-point check in this way and should highlight that γ_Δ in Eq. (3.12) is independent of the spectral density, since this is a non-trivial prediction of the construction.","section":"§4.3.1, Eqs. (4.20)–(4.21)"}],"minor_comments":[{"comment":"Combining (3.14) and (3.30) appears to give the exponent δ - d/4 - 1/2 rather than the displayed δ - (d-2)/4; please check the exponent and the surrounding constants.","section":"Eq. (3.31)"},{"comment":"The notation for the continuous operators O_Δ and the discrete operators O_Δ is easy to confuse, especially since both appear in the same equations; please use a more distinct typesetting consistently from the first occurrence.","section":"Throughout"},{"comment":"There is a truncated grant reference in the acknowledgements ('European Union funding (ERC, , 101118787)'); the missing grant number should be filled in.","section":"Acknowledgements"},{"comment":"The coefficients ζ_± are constrained by hermiticity only up to their real part; the paper should state explicitly that the imaginary part of ζ_± is left undetermined by the present arguments.","section":"§4.4, Eqs. (4.39)–(4.44)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid and technically detailed contribution that fits the journal's scope. The stress-test concern about Eq. (4.14) is justified: the inverse direction of the construction rests on a completeness relation imported from AdS by analytic continuation, and the stated caveat in Appendix D is not sufficient for a claim as central as ‘inversion formula’. I would encourage the editor to seek a revised version that either supplies a direct de Sitter verification of (4.14) or substantially qualifies the construction's domain and convergence status. The paper's own limitation statements in §3.3 and §5 should be moved into the abstract-level claims if no proof is added."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Frankly: this is the most concrete proposal I have seen for a non-perturbative dS boundary, and the inversion formula (4.10) appears new. The paper clearly explains why a naive AdS-like discrete expansion fails, then builds a continuous principal-series bulk-to-boundary expansion, derives the inversion formula, and checks it by recovering the boundary two-point function, contact terms, and perturbative correlators. Those checks are real and non-trivial; the diagrammatic identities in Appendix E are useful on their own. The large-Δ analysis of the spectral density is a reasonable first step on convergence, though the author admits the full convergence proof is missing because boundary OPE coefficients are not bounded.\n\nThe stress-test concern is on target: the completeness relation (4.14) is the load-bearing step, and it is imported from AdS harmonic analysis by analytic continuation. The author notes that it must be understood distributionally through the AdS continuation, but there is no independent derivation in dS. If extra local terms appear in the analytic continuation, the inversion formula fails to invert the expansion. I do not see a concrete error here; it is a gap that should be closeable, and the referee should ask for a direct coordinate-space check of (4.14) or a derivation that does not rely solely on Wick rotation.\n\nThe paper leans on the author's own prior work on the Källén-Lehmann representation and contact terms. That is not a flaw by itself, but one should read [4,22] to evaluate the starting assumptions. The self-citation is transparent, and the new result is the inversion formula and the boundary-to-bulk connector, which go beyond those papers.\n\nWho is this for? People working on dS bootstrap, dS holography, and cosmological correlators. It gives a new tool and a research program, but the completeness relation is the part to scrutinize before building on it. My recommendation: send it to peer review, and ask for a direct verification of (4.14) in de Sitter, or at least a discussion of its distributional status that goes beyond analytic continuation.","headline":"A serious, genuinely new proposal for a continuous dS boundary, but its central completeness relation is imported from AdS and needs direct scrutiny.","tokens_in":38829,"tokens_out":2241,"would_cite":true,"duration_ms":23815,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81T20","83C47"],"pacs":["04.62.+v"],"model":"deepseek-v4-flash","headline":"Every bulk scalar in a Bunch-Davies de Sitter theory with a Källén-Lehmann representation has a well-defined late-time conformal boundary built from principal-series primary operators.","keywords":["de Sitter","late-time boundary","bulk-to-boundary expansion","principal series","inversion formula","Källén-Lehmann spectral density","boundary-to-bulk connector","cosmological correlators"],"falsifier":"Evaluate the left side of the de Sitter completeness relation, $\\int d\\lambda\\,(1/2\\pi\\xi^\\pm_\\Delta)\\,\\Pi^\\pm_\\Delta(\\sigma)$, against a smooth compactly supported test function of two bulk points in planar coordinates, without Wick rotation; if the result is not $\\delta(Y_1,Y_2)$ on the full manifold, the inversion formula fails.","tokens_in":37684,"feed_emoji":"🌌","tokens_out":15263,"duration_ms":108617,"temperature":0.7,"pith_summary":"The paper tries to fix a gap in the non-perturbative understanding of de Sitter spacetime: the late-time boundary should be conformal, but the natural AdS-style expansion fails because it would produce boundary operators with dimensions off the unitary irreducible representations of the de Sitter group. The proposal is a continuous bulk-to-boundary expansion: instead of summing over discrete primaries, integrate over the principal series $\\Delta=d/2+i\\lambda$. The central technical result is an inversion formula that constructs the boundary primaries as integrals of the bulk field against a bulk-to-boundary propagator, with normalization fixed by the Källén-Lehmann spectral density. This gives a well-defined boundary operator content for interacting bulk theories and reproduces the known perturbative results in the free-field limit. A sympathetic reader should care because it turns the late-time de Sitter boundary into a concrete object that could support a conformal bootstrap for cosmological correlators.","feed_headline":"Inversion formula builds de Sitter boundary operators from bulk fields","feed_subtitle":"Continuous family of principal-series primaries with contact terms; perturbation theory recovered.","key_machinery":"The central object is the boundary-to-bulk connector $\\hat D_\\Delta(\\eta,y)=(-\\eta)^\\Delta\\,{}_0F_1(\\Delta-d/2+1,\\tfrac14\\eta^2\\partial_y^2)$, a differential operator that lifts a boundary two-point function to a bulk-to-boundary propagator and a free bulk-to-bulk propagator; together with the inversion kernel $K^\\pm_\\Delta$ and the completeness relation $\\int d\\lambda\\,(1/2\\pi\\xi^\\pm_\\Delta)\\,\\Pi^\\pm_\\Delta(\\sigma)=\\delta(Y_1,Y_2)$ obtained by Wick rotation from AdS. The connector packages descendants automatically and reduces de Sitter diagram identities (split representation, V diagram, broken leg) to algebraic steps. The inversion formula is the inverse of the expansion only insofar as this completeness relation holds.","core_discovery":"On its own terms, the central discovery is equation (4.10): the boundary operator $O_\\Delta(y)$, with $\\Delta=d/2+i\\lambda$ on the principal series, is obtained from a bulk field $\\phi$ by $O_\\Delta(y)=\\frac{1}{N_\\Delta}\\int_{\\text{bulk}} K^\\pm_\\Delta(\\eta',y';y)\\,\\phi(\\eta',y')$, where $K^\\pm_\\Delta$ is the bulk-to-boundary propagator and $N_\\Delta=2\\pi\\xi^\\pm_\\Delta a_\\Delta$ is fixed by the Källén-Lehmann spectral density. This inverts the bulk-to-boundary expansion $\\phi(\\eta,y)=\\int d\\lambda\\,a_\\Delta\\,\\hat D_\\Delta(\\eta,y)O_\\Delta(y)$, so the boundary theory is constructible directly from bulk data. The resulting boundary operators satisfy conformal Ward identities, have two-point functions of the form $\\delta_{\\lambda_1\\lambda_2}/y_{12}^{2\\Delta_1}+\\gamma_{\\Delta_1}\\delta_{\\lambda_1,-\\lambda_2}\\delta(y_1-y_2)$, and reproduce perturbative cosmological correlators through the broken-leg identity.","pith_inferences":["If the completeness relation can be proven directly in de Sitter without the AdS continuation, the same inversion logic should extend to complementary and exceptional series, turning the suspected discrete sums into a theorem.","The non-commutativity of the boundary operators suggests the late-time boundary is not an ordinary Euclidean CFT; any bootstrap based on these operators will need to treat contact terms and operator ordering as part of the crossing equations.","The exponential suppression of $a_\\Delta$ at large $\\Delta$ implies the principal-series integral can be truncated in numerical studies of de Sitter correlators, offering a concrete path to finite-coupling checks in models such as the $O(N)$ model.","A spinning generalization of the inversion formula, once exceptional-series representations are included, would provide the de Sitter analogue of a boundary stress tensor and a handle on massless spin-2 physics in de Sitter."],"forward_implications":["Any scalar bulk two-point function admitting a Källén-Lehmann representation defines a family of boundary primaries with principal-series dimensions, so the late-time boundary exists without assuming a discrete spectrum.","Boundary two-point functions necessarily contain local contact terms in addition to the CFT power law, and the boundary operators fail to commute; these are direct consequences of bulk unitarity and canonical commutation relations.","Perturbative de Sitter in-in correlators are recovered from the inversion formula, and the boundary-to-bulk connector reorganizes standard diagrammatic identities (split representation, V diagram, broken leg), simplifying actual calculations.","The bulk-to-boundary coefficient decays exponentially at large scaling dimension for theories that flow to a CFT in the UV, making the continuous expansion well-behaved inside correlation functions."],"supporting_citations":[{"why":"Supplies the de Sitter Hilbert-space resolution into unitary irreducible representations, the spectral decomposition, and the unitarity argument for boundary contact terms that the continuous expansion builds on.","marker":"[4]"},{"why":"Provides the Källén-Lehmann inversion formulas and spectral-density results, including late-time contact terms and large-dimension limits used for the normalization and convergence analysis.","marker":"[22]"},{"why":"Supplies the AdS boundary-operator/bulk-state map and discrete bulk-to-boundary expansion that the paper contrasts and adapts to the continuous de Sitter spectrum.","marker":"[18]"},{"why":"Provides the analytic continuation to Euclidean AdS and the inversion formula for de Sitter two-point functions used in the Wick-rotated completeness relation.","marker":"[21]"},{"why":"Supplies the analyticity and unitarity framework for cosmological correlators and the perturbative in-in setup whose boundary correlators the inversion formula must reproduce.","marker":"[5]"},{"why":"Supplies the de Sitter-to-AdS dictionary, split representation, and $i\\epsilon$ prescriptions used throughout the diagrammatic identities.","marker":"[12]"},{"why":"Provides the V-diagram and broken-leg integral identities used to verify the inversion formula against perturbation theory and three-point structures.","marker":"[45]"},{"why":"Supplies the orthogonality relation for the Bessel-function transform used in the direct derivation of the inversion formula.","marker":"[48]"}],"fun_headline_variants":["Inversion formula builds de Sitter boundary operators from bulk fields","Non-perturbative de Sitter boundary from bulk spectra","Principal-series primaries via bulk-to-boundary inversion","From bulk fields to de Sitter boundary operators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on a distributional completeness relation in de Sitter that the paper derives only by analytic continuation from anti-de Sitter harmonic analysis; if that identity does not hold as a genuine de Sitter distribution, the inversion formula does not invert the expansion.","fun_headline_variants_meta":{"raw":{"variants":["Inversion formula builds de Sitter boundary operators from bulk fields","Non-perturbative de Sitter boundary from bulk spectra","Principal-series primaries via bulk-to-boundary inversion","From bulk fields to de Sitter boundary operators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1708,"prompt_tokens":997,"completion_tokens":711,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":645}},"tokens_in":613,"tokens_out":711,"duration_ms":6088,"temperature":1.0,"reasoning_tokens":645,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:38:32.986621+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the left side of the de Sitter completeness relation, $\\int d\\lambda\\,(1/2\\pi\\xi^\\pm_\\Delta)\\,\\Pi^\\pm_\\Delta(\\sigma)$, against a smooth compactly supported test function of two bulk points in planar coordinates, without Wick rotation; if the result is not $\\delta(Y_1,Y_2)$ on the full manifold, the inversion formula fails.","supporting_citations":[{"cited_title":"Jones,The Theory of Electromagnetism, Pergamon Press (1964)","cited_arxiv_id":null,"evidence_quote":"Supplies the orthogonality relation for the Bessel-function transform used in the direct derivation of the inversion formula."}],"review_version":1}