{"id":"7b3cb00d-88e9-4ed6-8834-5a3a223cfceb","arxiv_id":"2608.11945","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Massive one-particle states are encoded on null infinity by projecting their momenta onto null frames, giving an isometric map into the complete Carrollian representation.","lead":"A new dictionary maps massive particles in flat spacetime onto boundary states at null infinity, the arena usually reserved for massless radiation. It puts massive and massless scattering amplitudes in a common Carrollian language and extends soft photon and graviton theorems to massive external legs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The embedding and Källén–Lehmann matching rest on the companion-paper representation [25]; if its κ>0 orbit or two-point function is wrong, the central claim has no target, so this dependency is the load-bearing gap.","rationale":"The reader's conditional verdict is appropriate. The paper's core dictionary is a formal integral construction whose target Hilbert space, inner product, and two-point function structure are all inherited from the companion preprint [25]. That is a genuine load-bearing dependency: the central 'isometric embedding' statement is only meaningful if S_O^{>0} is a positive-definite Hilbert space with the stated representation content, and the Källén–Lehmann application only works if the generic non-contact two-point function from [25] has the exact form assumed in Section 3. The paper does not provide the relevant derivations, so the correctness risk is medium rather than low. At the same time, I found no internal inconsistency in the explicit formulas: the null-frame parametrization, the momentum reconstruction, and the structure of the normalization identity are coherent, and a direct evaluation of (2.13) appears consistent with the claimed (2π)^2 δ^4 normalization. The concern is therefore about unverified foundations and omitted derivations, not about a demonstrated error. The verdict should remain CONDITIONAL, pending independent verification of [25] and an explicit derivation of the intertwining and normalization identities.","tokens_in":13681,"tokens_out":29568,"duration_ms":330059,"concrete_test":"Verify the two load-bearing ingredients imported from [25]: (i) the κ>0 orbit of the complete Carrollian representation carries a positive-definite inner product and the stated non-contact two-point function with exactly one undetermined f(μ^2,L); (ii) substituting (2.5) into the Poincaré intertwining equations of [25] for P_0 and J_{03} reproduces the |κ|^{1−Δ} factor and the 2m normalization. As a check independent of [25], evaluate the integral (2.13) explicitly using the null-frame completeness relation (A.4); if the d²z Jacobian produces anything other than (2π)^2 δ^4(p−p′), the normalization fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The construction's central claim—that (2.4)/(2.5) with reconstruction (2.8)/(2.9) is an isometric embedding into S_O^{>0}—is not self-contained. Section 2 imports from the companion paper [25] the existence and Hilbert-space structure of the 'complete Carrollian representation': the κ>0 and κ<0 orbits with nonzero quadratic Casimir, the state space S_O^{>0}, and the generic scalar non-contact two-point function used in Section 3 with undetermined coefficient f(μ^2,L). None of these ingredients is re-derived here. If the inner product on S_O^{>0} is not positive definite, or if the Ward-identity solution in [25] has additional L dependence or contact terms, then (2.14) does not describe an embedding into a Hilbert space and the fixing of f in (3.4) is void. The text also asserts, without displaying the calculation, that 'matching the action' of the Poincaré generators fixes the kernel (2.5) up to normalization, and that applying ⟨p| to the reconstruction gives (2.13). Those assertions can only be checked against [25]'s generator action and inner product. This is an external-dependency gap rather than an observed inconsistency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a Carrollian dictionary for massive one-particle states in 4D Minkowski spacetime, associating to each bulk momentum eigenstate |p> a set of boundary states |O_X> labeled by Carrollian coordinates and representation data. The dictionary kernel (2.5) encodes the bulk momentum through its projections onto a null frame at each celestial point, and the reconstruction kernel (2.9) is claimed to be its Hermitian adjoint. The central claim is that the two maps satisfy G^dagger G = I and G G^dagger = Pi, an isometric embedding of the massive one-particle Hilbert space into the state space S_O^{>0} of the 'complete Carrollian representation'. The paper then applies this dictionary to the Källén-Lehmann representation to fix the coefficient function of a non-contact Carrollian two-point function, and to soft photon and graviton theorems to derive hard actions for massive legs, distinguishing global from local symmetry actions. Appendices give spin matrix factors and a 3D analogue.","tokens_in":13858,"tokens_out":21987,"duration_ms":218311,"significance":"If the central isometry claim is correct, this is a notable step: it provides a concrete, normalized map between massive bulk states and Carrollian boundary data at null infinity, an issue that has remained open despite work at timelike infinity. The construction is explicit, with closed-form kernels, spin matrix factors, and a 3D counterpart, and it yields falsifiable statements, notably the fixing of f(mu^2,L) in terms of the bulk spectral density and the distinction between global and local soft-mode actions. However, the paper is not self-contained: the target state space, its inner product, and the generic two-point function are imported from the author's companion paper [25]. The manuscript also does not display the derivations of the completeness identity (2.13) or the Källén-Lehmann correlator (3.2), so the central claims cannot currently be verified from the text alone. The significance is therefore conditional on filling these gaps.","major_comments":[{"comment":"The central completeness identity is asserted without a derivation. The text states that applying 0 and kappa<0 orbits with nonzero quadratic Casimir, the state space S_O^{>0}, and the generic scalar non-contact two-point function used in Section 3. None of these ingredients is re-derived or even summarized in sufficient detail for the present claims. If the inner product on S_O^{>0} is not positive definite, or if the Ward-identity solution in [25] has additional L dependence or contact terms, then Eq. (2.14) does not describe an embedding into a Hilbert space and the fixing of f(mu^2,L) in Eq. (3.4) is void. The paper should either include the relevant results from [25] as an appendix or state clearly which properties of the representation are assumed.","section":"Section 2, second paragraph; Section 3, Eq. (3.4)"},{"comment":"The definitions of the dictionary and reconstruction maps are notationally ambiguous. Eq. (2.4) writes |O_X> = G|p> as an integral over bulk momentum states |p>, while Eq. (2.8) writes |p> = K|O_X> as an integral over boundary states |O_X>. If |O_X> are elements of the boundary Hilbert space S_O^{>0} and |p> are elements of the bulk Hilbert space, these equalities cannot hold as literal identities in a single Hilbert space. The reader cannot tell whether |O_X> are states in S, elements of H defined by the integral, or smeared operators. This ambiguity makes it impossible to verify the adjointness relation K = G^dagger and the composition identities (2.14). The maps should be defined precisely, including their domains, targets, and inner products.","section":"Section 2, Eqs. (2.4) and (2.8)"},{"comment":"The derivation of the Carrollian correlator from the Källén-Lehmann representation is not shown. The text states that 'applying the massive dictionary to both bulk legs gives' Eq. (3.2), but the evaluation of the delta-function constraints, the conversion of the positivity condition theta(p_1^0) to theta(kappa_1), and the treatment of the kappa_2<0 requirement are all omitted. Since the matching to the generic two-point function and the fixing of f(mu^2,L) in Eq. (3.4) are among the main physical results, this computation should be displayed at least in an appendix. Without it, the claimed relation between the bulk spectral density and the Carrollian two-point function cannot be checked.","section":"Section 3, Eq. (3.2)"}],"minor_comments":[{"comment":"The notation for q_3 is unclear: it is written as (1/8) partial^2 q_0, but the derivative symbol and the factor of 1/8 are not defined. Please specify the second derivative with respect to the complex coordinate or z^a.","section":"Section 2, Eq. (2.3)"},{"comment":"The variable rho is introduced in the footnote as rho = (m^2 + beta^2)/kappa = -2 p·q_3, but rho is not defined in the main text before Eq. (2.7). Its role as the Fourier conjugate to u should be stated in the main body.","section":"Section 2, footnote 1"},{"comment":"The statement that the kappa>0 and kappa<0 orbits 'have the nonzero quadratic Casimir' should be made more precise: the quadratic Casimir of the representation is a label of the orbit, while the relation C_2 = kappa rho - beta^2 = m^2 in footnote 1 refers to the reconstructed bulk momentum. The distinction between the representation-theoretic Casimir and the bulk mass parameter should be clarified.","section":"Section 2, second paragraph"},{"comment":"The denominator |beta_1^2 - beta_2^2| in the delta function should be discussed: it vanishes for configurations with beta_1^2 = beta_2^2, and the distributional interpretation of the expression in that limit should be given.","section":"Section 3, Eq. (3.2)"},{"comment":"There are several typographical and consistency issues: the abstract contains 'K¨all´en' with a diacritic, the text uses inconsistent spacing in 'K all´ en–Lehmann', and in Eq. (2.3) the expression 'z^a z_a' appears as 'z aza'. These should be corrected during revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central claims depend on the author's companion paper [25], which is not available to the reader and is cited for the target representation, its inner product, and the generic two-point function. For a fair evaluation, the editor should require that [25] be posted or that its relevant results be summarized within the present paper. The notation in Eqs. (2.4) and (2.8) is confusing and should be fixed; as written, it obscures the domain and target of the dictionary map. If the normalization issue in Eq. (2.13) is not resolved, the central isometry claim may require a different prefactor, so the revision should include a full derivation of the completeness identity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper fills a real gap: no one had put massive one-particle states on null infinity in the Carrollian dictionary. The idea of encoding the timelike momentum by its projections onto null frames at each celestial point is clean, and the reconstruction formula (2.7) ties it together nicely. The Källén–Lehmann application fixing the non-contact two-point coefficient, and the soft-theorem result that local and global modes act differently on massive legs, are both worth having. The spin matrix construction is explicit enough that it looks right, and the 3D appendix suggests the mechanism is robust.\n\nThe soft spots are real but are mostly presentation gaps rather than obvious errors. The completeness identity (2.13)–(2.14) is the heart of the dictionary, and it is simply asserted. The 2m normalization is said to be \"fixed\" without a displayed Jacobian. A referee will need to see that calculation, and it should be straightforward. The second concern is the dependence on the companion paper [25]: the state space S_O^{>0}, the κ>0 orbit structure, and the Ward-identity two-point function are all imported without re-derivation. If any of those is wrong, the dictionary has no valid target and the spectral matching in Section 3 is void. That is a genuine external-dependency gap, not a minor inconvenience. Finally, the generic-spin unitarity R R† = 1 is stated from formulas in Appendix B without proof; low-spin checks are given, but the general statement is asserted.\n\nI have not redone the algebra, but nothing in the paper makes me think the central claim is false. The setup is standard enough that the missing Jacobian probably works, and the author seems to know exactly what is being claimed. The paper is a letter, so some omission is expected, but the omitted items are load-bearing rather than decorative.\n\nThis is a paper for Carrollian and flat-space holographers, and it is worth a serious referee. The right referee will ask for a derivation of (2.13), explicit reconciliation with [25], and a proof of the generic-spin unitarity. If those come through, the paper will be a solid addition. It deserves peer review, not desk rejection.","headline":"A plausible and genuinely novel massive dictionary on null infinity, but the load-bearing isometry check and the target-space structure from the companion paper are asserted rather than shown.","tokens_in":14442,"tokens_out":2501,"would_cite":true,"duration_ms":30475,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a normalized Carrollian dictionary mapping massive one-particle states in 4D Minkowski space onto boundary states at null infinity, via momentum projections on a null frame at every celestial point.","keywords":["Carrollian holography","null infinity","massive particles","null frame projection","isometric embedding","soft photon theorem","soft graviton theorem","spectral density"],"falsifier":"Compute the completeness integral (2.13), $\\int_{\\kappa>0} du\\,d^2z\\,d^2\\beta\\,d\\kappa\\, K(X;p')G(p;X)$, for a generic massive momentum pair and check whether it equals $(2\\pi)^2\\delta^{(4)}(p-p')$; failure would disprove the isometric embedding. A second direct check is to compute the boundary two-point function (3.2) from an explicit free massive scalar and compare with the bulk spectral form; a mismatch would disprove the claimed fixing of $f(\\mu^2,L)$.","tokens_in":13363,"feed_emoji":"🌌","tokens_out":17946,"duration_ms":152026,"temperature":0.7,"pith_summary":"Massive particles travel along timelike worldlines and never arrive at null infinity, which has left their Carrollian boundary description open. This paper constructs a dictionary that puts massive one-particle states onto null infinity anyway: the bulk momentum is encoded by its projections onto a null frame attached to every celestial point, so no endpoint on $\\mathscr{I}^\\pm$ is needed. Matching the Poincaré action on both sides fixes the dictionary kernel, and the reconstruction map is its Hermitian adjoint, making the dictionary an isometric embedding of the bulk Hilbert space into an overcomplete boundary state space. The paper applies this to the spectral representation of the two-point function, fixing the coefficient of the non-contact Carrollian correlator in terms of the bulk spectral density, and to soft photon and graviton theorems, where global boundary modes reproduce ordinary charge and Poincaré conservation while local modes act by integrals over the celestial sphere.","feed_headline":"Massive particles gain a Carrollian description at null infinity","feed_subtitle":"Each massive momentum is encoded as projections on every celestial point, linking bulk to boundary.","key_machinery":"The load-bearing object is the null frame $\\{q_0^\\mu(z), q_1^\\mu(z), q_2^\\mu(z), q_3^\\mu(z)\\}$ attached to each celestial point $z^a$, together with the complete Carrollian representation whose $\\kappa>0$ and $\\kappa<0$ orbits carry nonzero quadratic Casimir. The dictionary kernel uses delta functions to read off $\\beta^a$ and $\\kappa$ as the frame projections of the bulk momentum, while the reconstruction formula inverts those projections; the spin matrix factor $R^{(j)}$ carries all spin dependence and is unitary on the positive-energy mass shell. The work this machinery does is to turn a single timelike momentum into a distribution of data over the whole celestial sphere, so that bulk and boundary Poincaré actions can be matched without sending the massive worldline to null infinity.","core_discovery":"The central discovery is that a normalized map from the massive one-particle Hilbert space $H_j^{(+)}$ to the state space $S_O^{>0}$ of the complete Carrollian representation is fixed by requiring the Poincaré generators to act identically on both sides. The scalar dictionary kernel is $G(p;X)=2m\\,e^{2iu\\,p\\cdot q_3(z)}|\\kappa|^{1-\\Delta}\\delta^{(2)}(\\beta^a+p\\cdot q_a(z))\\delta(\\kappa+p\\cdot q_0(z))$, where $q_0,q_1,q_2,q_3$ are the null frame at celestial point $z^a$; the delta functions set $\\beta^a=-p\\cdot q_a$ and $\\kappa=-p\\cdot q_0$, so $\\beta^a$ and $\\kappa$ are the projections of the massive momentum onto that frame. The momentum is reconstructed from these labels by $p^\\mu=\\frac{m^2+\\vec\\beta^2}{2\\kappa}q_0^\\mu-\\beta^a q_a^\\mu+\\kappa q_3^\\mu$. On the unitary line $\\Delta=2+i\\nu$, the reconstruction kernel is the complex conjugate of the dictionary kernel, hence the adjoint, and the two compositions obey $\\hat G^\\dagger \\hat G = I_{H_j^{(+)}}$ and $\\hat G \\hat G^\\dagger = \\Pi_{S_O^{>0}}$ with $\\Pi$ a nontrivial projector; the boundary states therefore form a Parseval continuous frame, an overcomplete basis for the bulk Hilbert space. The paper further claims the spectral representation fixes $f(\\mu^2,L)=\\frac{\\mu^2}{2\\pi^2}\\rho_{KL}(\\mu^2)$, and that soft photon and graviton Ward identities imply local large gauge transformations and supertranslations act on massive legs as integrals over the celestial sphere while global modes reproduce the ordinary $U(1)$ and Poincaré actions.","pith_inferences":["If the embedding is correct, every massive $S$-matrix element can in principle be rewritten as an integral of Carrollian boundary correlators over $\\kappa>0$, so massive scattering becomes boundary data even though massive worldlines never touch null infinity.","The projector $\\Pi_{S_O^{>0}}$ suggests a selection rule for any holographic dual: only the image of the dictionary map corresponds to physical massive bulk states, so the boundary Hilbert space is effectively a quotient of the overcomplete Carrollian space.","The same projection mechanism should extend to subleading soft orders and spinning hard legs: local subleading superrotations are expected to act nonlocally on massive scalars, with global modes still reducing to the Lorentz action.","Because the same bulk Hilbert space also admits Carrollian descriptions at timelike infinity, the null-infinity dictionary and the timelike-infinity construction should be related by an integral transform; the paper leaves this connection open."],"forward_implications":["Massive and massless external states can be described in a common Carrollian boundary framework on $\\mathscr{I}^\\pm$, with the standard radiative dictionary covering massless legs and the new dictionary covering massive legs.","The complete Carrollian representation is necessary for massive states because its $\\kappa\\neq 0$ orbits have nonzero quadratic Casimir; the sector usually used for massless radiation cannot carry them.","Bulk dynamics enters the boundary two-point function only through the spectral density, fixing $f(\\mu^2,L)=\\frac{\\mu^2}{2\\pi^2}\\rho_{KL}(\\mu^2)$; unitary bulk theories then require $f\\geq 0$ for $\\mu^2>0$, with no independent $L$-dependence.","Soft photon and graviton theorems for massive legs imply that local large gauge transformations and supertranslations act by nonlocal integrals over the celestial sphere, while global modes reproduce charge conservation, momentum conservation, and Lorentz invariance.","The construction extends to arbitrary spin through a unitary spin matrix factor and to a 3D bulk with a 2D boundary, preserving the same isometry and projection identities."],"supporting_citations":[{"why":"Introduces the complete Carrollian representation whose $\\kappa>0$ and $\\kappa<0$ orbits carry nonzero Casimir, and supplies the target state space $S_O^{>0}$ and the generic non-contact two-point function used in the spectral matching.","marker":"[25]"},{"why":"Supplies the momentum-space spectral representation of the bulk Wightman two-point function with spectral density $\\rho_{KL}$, from which the boundary correlator is derived.","marker":"[32–36]"},{"why":"Establishes the interpretation of soft theorems as Ward identities of asymptotic symmetries, the basis for extracting the hard actions on massive legs.","marker":"[3, 24, 37–39]"},{"why":"Provides the soft photon theorem and its large-gauge-transformation Ward identity that the massive photon hard action is tested against.","marker":"[4, 40, 41]"},{"why":"Provides the leading and subleading soft graviton theorems and their translation and Lorentz Ward identities used for supertranslation and superrotation actions.","marker":"[42–45]"},{"why":"Defines the conformally soft photon and graviton currents as the $\\Delta=1$ residue used in constructing the soft charges.","marker":"[46]"}],"fun_headline_variants":["Massive particles gain Carrollian description via celestial projections","Carrollian dictionary maps massive momentum to null-frame projections","Overcomplete Carrollian basis links massive states to null infinity","Soft theorems framed by Carrollian massive states at null infinity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The dictionary presumes that the complete Carrollian representation defined in the companion paper [25] actually exists with the stated $\\kappa>0$ and $\\kappa<0$ orbits, target state space, and non-contact two-point function; if that representation is flawed or cannot be extended to null infinity, the massive dictionary has no valid target space and the spectral matching collapses.","fun_headline_variants_meta":{"raw":{"variants":["Massive particles gain Carrollian description via celestial projections","Carrollian dictionary maps massive momentum to null-frame projections","Overcomplete Carrollian basis links massive states to null infinity","Soft theorems framed by Carrollian massive states at null infinity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1532,"prompt_tokens":1087,"completion_tokens":445,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":703,"completion_tokens_details":{"reasoning_tokens":379}},"tokens_in":703,"tokens_out":445,"duration_ms":4585,"temperature":1.0,"reasoning_tokens":379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:21:40.292407+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the completeness integral (2.13), $\\int_{\\kappa>0} du\\,d^2z\\,d^2\\beta\\,d\\kappa\\, K(X;p')G(p;X)$, for a generic massive momentum pair and check whether it equals $(2\\pi)^2\\delta^{(4)}(p-p')$; failure would disprove the isometric embedding. A second direct check is to compute the boundary two-point function (3.2) from an explicit free massive scalar and compare with the bulk spectral form; a mismatch would disprove the claimed fixing of $f(\\mu^2,L)$.","supporting_citations":[{"cited_title":"Missing Descendants in the Carrollian Conformal Family","cited_arxiv_id":"2607.28400","evidence_quote":"Introduces the complete Carrollian representation whose $\\kappa>0$ and $\\kappa<0$ orbits carry nonzero Casimir, and supplies the target state space $S_O^{>0}$ and the generic non-contact two-point function used in the spectral matching."}],"review_version":1}