{"id":"0fe63af6-58d9-4213-a793-4446f828a65e","arxiv_id":"2608.08384","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"A c→0 contraction of the Klein-Gordon equation yields three Carrollian quantum sectors, and the time-like sector shows temporal tunneling with |T|²=1+|R|².","lead":"This paper derives quantum wave equations for the Carrollian limit where the speed of light goes to zero, producing three sectors with different spatial dynamics. It finds a time-like sector where particles cannot move but can tunnel across time barriers, which may inform holography, null-surface physics, and fracton models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The hybrid 'contraction' (3.7) does not yield Eq. (6.1): substituting the stated scalings into the Klein-Gordon equation gives a massless wave equation at leading order, with neither m0² nor the χ∂t compensating term; the claimed three-sector derivation is therefore not demonstrated.","rationale":"I focused on the contraction itself because the abstract and Section 3.2 explicitly promise a systematic derivation. If two of the three sectors are not limits of the Klein-Gordon equation, the paper's taxonomy becomes a construction rather than a derivation, and the advertised novelty is unsupported. The time-like sector and its Bogoliubov relation |T|²=1+|R|² are internally consistent, and I do not dispute the reader's positive assessment of those calculations. The reader's rationale already notes that the limiting steps are not actually performed and that the compensating field is inserted manually; my check sharpens this into a concrete algebraic failure for the hybrid scaling, which provably produces a massless wave equation rather than Eq. (6.1). This is load-bearing because the central claim depends on all three sectors emerging from controlled contractions. Consequently the paper is best read as an exploration of a known time-like equation plus two ad hoc Carroll-invariant constructions. The no-propagation obstruction raised by the reader is related but secondary: even if the constructed equations are physically meaningful, they are not derived from the announced limit. The verdict should remain REJECT.","tokens_in":27507,"tokens_out":9401,"duration_ms":103882,"concrete_test":"Re-derive the hybrid limit with explicit bookkeeping: substitute c=ϵc̄, t unchanged, x=ϵX, m0=m̄/ϵ, and ψ=ϵ²Φ into Eq. (2.1), write the equation at each order in ϵ, and attempt to recover Eq. (6.1) with finite m̄² and a finite χ∂T term by any allowed normalization. Report the resulting leading-order equation. If the only reachable options are the massless wave equation ∂T²−∂X²=0 or the time-like equation ∂T²+m̄²/ℏ²=0, then Eq. (6.1) is not a contraction limit. Repeat the same bookkeeping for the space-like scaling (3.6) and check whether any first-order time term is produced by the contraction or must be inserted as a separate compensating structure.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's central claim is that three inequivalent Carrollian sectors arise from systematic c→0 contractions of the Klein-Gordon equation (2.1). The time-like limit is checkable: with c→ϵc and m→m/ϵ², the term (ℏ²∂t²+m²)ψ survives. The hybrid limit, however, fails this check. Apply the stated scaling (3.7) directly: set c=ϵc̄, keep t, put x=ϵX and m0=m̄/ϵ in (2.1). The equation becomes ϵ⁻²[(1/c̄²)∂t² − ∂X² + ϵ² m̄²/ℏ²]Φ=0. After multiplying by ϵ², the leading-order equation is (1/c̄²)∂t²Φ − ∂X²Φ=0, i.e. a massless wave equation, not Eq. (6.1). If one instead multiplies to keep the mass term, the ∂X² term is suppressed and the equation collapses to the time-like sector. In neither normalization does a first-order χ∂t term appear. Equation (6.1) is therefore obtained by adding the compensating field χ by hand, not by the announced contraction. The same structural problem afflicts the space-like 'Approach 1': no local rescaling of a second-order Klein-Gordon equation can produce a first-order time derivative, so Eq. (5.4) is an Ansatz introduced after the limit rather than a limit of (2.1). Since the space-like and hybrid sectors are constructed rather than derived, the central claim of a systematic three-sector split is not established. A consistent probabilistic interpretation of these ad hoc equations cannot compensate for the missing derivation. This concern is distinct from, and more basic than, the no-propagation obstruction cited from ref. [40]: even if the final equations are Carroll-invariant, they are not obtained from the ultra-relativistic contraction that the title and abstract promise.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to develop Carrollian quantum mechanics by taking c→0 contractions of the Klein-Gordon equation (2.1). Three sectors are announced: time-like, space-like, and hybrid. For the time-like sector the paper derives the Marsot equation, a continuity equation with vanishing current, the temporal box spectrum, and temporal barrier scattering with the Bogoliubov relation |T|²=1+|R|². The space-like sector is obtained by adding a compensating field χ (or an additive χ̃) and is claimed to describe tachyonic null states with spatial current. The hybrid sector combines temporal and spatial second derivatives with a first-order χ∂t term and is analyzed in damped/amplified regimes. The paper also claims a fracton/time-like Carroll duality and presents a classical hybrid particle action in Appendix A.","tokens_in":27853,"tokens_out":17040,"duration_ms":171906,"significance":"If the three-sector derivation were valid, the paper would provide a useful catalogue of Carrollian quantum-mechanical models with exact box and barrier solutions. The time-like part is genuinely valuable: the temporal tunneling computation is algebraically consistent and the identity |E|²−|F|²=1 follows correctly from the indefinite Klein-Gordon norm. However, the central claim is not established: the space-like and hybrid equations are not obtained from the announced contractions, and the space-like probabilistic interpretation has an inconsistency between the current and the physical flux. The no-propagation obstruction cited as ref. [40] is not engaged. These issues affect the core of the paper rather than presentation.","major_comments":[{"comment":"The announced contractions do not produce the equations used in Sections 5 and 6. For the hybrid scaling (3.7), substitute c=ϵc̄, x=ϵX, m0=M/ϵ into Eq. (2.1). After multiplying by ϵ², the leading-order equation is (1/c̄²)∂t²Φ−∂X²Φ=0, a massless wave equation, not Eq. (6.1); no ℏ²χ∂t term is generated. If c is instead kept fixed, (3.7) leads to −∂X²Φ+M²Φ=0 at leading order, still without the χ∂t term. The same structural objection applies to the space-like sector: no rescaling of the second-order Klein-Gordon equation can produce the first-order derivative in Eq. (5.4). Equations (5.4) and (6.1) are therefore introduced by hand after the limit, and the abstract's claim of a systematic three-sector derivation from c→0 contractions is unsupported.","section":"Section 3.2, Eqs. (3.5)–(3.7)"},{"comment":"The continuity and probability interpretation in the space-like sector are inconsistent. The continuity equation is derived for J=ℏ²(ψ*∂xψ−ψ∂xψ*), which is purely imaginary for plane-wave modes. The paper then defines a 'physical flux' J≡−iJ, but with this substitution the conserved equation becomes ∂tρ+i∂xJ=0, not ∂tρ+∂xJ=0. Thus the real flux is not the current appearing in the conservation law, and the claimed probabilistic interpretation with a real current is not demonstrated.","section":"Section 5.3, Eqs. (5.14)–(5.16)"},{"comment":"The paper cites ref. [40], which states that single Carrollian scalars cannot propagate, but it does not explain why the space-like and hybrid sectors circumvent this obstruction. The space-like sector is explicitly said to retain spatial propagation, yet its energy eigenstates have purely imaginary energies and hence real exponential time dependence, so the solutions do not exhibit oscillatory propagation. If the χ coupling is meant to evade the no-go result, an explicit argument is required; without it, the propagation claim in Section 5.1 is unsupported.","section":"Section 5.2 and ref. [40]"},{"comment":"The claimed fracton/time-like Carroll duality contains a sign error. The Lagrangian L=|∂tϕ|²+m0²|ϕ|² given in Eq. (4.45) yields the Euler-Lagrange equation ℏ²∂t²ϕ−m0²ϕ=0, not ℏ²∂t²ϕ+m0²ϕ=0 as written in Eq. (4.46). The equation with the plus sign is not the oscillator equation of this Lagrangian, so the duality as stated is incorrect unless the Lagrangian is also changed.","section":"Section 4.5, Eq. (4.46)"}],"minor_comments":[{"comment":"Section 3.1 specifies c→ϵc, but the scalings (3.5)–(3.7) never list the c-scaling. This ambiguity is directly related to the contraction failure identified above and should be clarified.","section":"Section 3.1 vs. Section 3.2"},{"comment":"The equation is described as Hermitian for complex ψ, but with χ=iχ′ the operator −ℏ²χ′∂t is anti-Hermitian. The terminology should be corrected or the Hermiticity claim justified with respect to a specific inner product.","section":"Section 5.1, Eq. (5.4)"},{"comment":"The classical compensating field χ of Eq. (A.5) transforms as δχ=−e p·β, whereas the quantum compensating field in Eq. (5.3) transforms as δχ=β²∂t+2β·∂x. The paper does not connect these two mechanisms or explain why the same symbol χ is used for both.","section":"Appendix A"},{"comment":"References [18] and [19] are identical duplicates and should be merged.","section":"References"},{"comment":"There are numerous typos and stylistic slips (e.g., 'factories' instead of 'factorizes' in Section 4.4.1, 'thistemporal' in Section 4.4). A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The time-like sector is a good, self-contained contribution, and a revised paper focused on that sector (or reframed as a study of particular Carroll-invariant equations) could be viable. As submitted, however, the advertised three-sector derivation is the paper's central claim, and it fails for the space-like and hybrid cases; the space-like probabilistic interpretation also has a real inconsistency. These are not local presentation issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I've read this paper through carefully. The headline is that the time-like Carroll sector is worked out well, but the central claim—that three sectors emerge from systematic c→0 contractions of the Klein-Gordon equation—does not hold up. The hybrid limit (3.7) fails: substituting x→ϵx, m→m/ϵ into (2.1) gives a massless wave equation, not Eq. (6.1). And no rescaling of a second-order equation can produce a first-order time derivative, so the space-like equation (5.4) is not a limit of (2.1) but an Ansatz with a compensating field χ. The paper would be honest if it presented the space-like and hybrid equations as constructions, not derivations.\n\nWhat is genuinely new and useful: the temporal tunneling calculation in Sec. 4.4. The matching is clean, and the Bogoliubov relation |T|²=1+|R|² follows correctly from the indefinite Klein-Gordon norm. The temporal box problem is a nice exercise—essentially a harmonic oscillator in time, with mass quantization in the zero-potential case. The fracton duality is a clean dimensional reduction, though it repeats ref [15], as the author acknowledges.\n\nThe space-like and hybrid sections contain a lot of work—box spectra, damped oscillator regimes, scattering amplitudes—and the algebra is mostly consistent as a set of exercises. But the physical interpretation is shaky: states with vanishing density are called 'null' and are declared physical without a clear argument, and the hybrid density involves an ad hoc exponential weight e^{-χ't} that makes the continuity equation work but is not derived. The paper also calls dissipative equations 'Hermitian' without qualification, and the hybrid temporal tunneling amplitudes are never actually computed—the author says they follow by analogy and omits them.\n\nFor a reader: this is for people working on Carrollian mechanics or fractons who want a catalog of exact solutions. The time-like tunneling result is worth a look. But as a unified framework, the paper overreaches. I would not cite it as a source for the three-sector derivation, but it might be cited for the time-like tunneling formula.\n\nRecommendation: I would not desk-reject; the time-like section is solid and the paper engages seriously with the literature. But it needs major revision before any acceptance. The author should either prove the contractions actually produce the claimed equations, or reframe the paper as a study of Carroll-invariant equations with compensating fields. A referee who checks Sec. 3.2 will quickly find the hybrid problem.","headline":"The time-like Carroll sector is solid, but the claimed three-sector derivation from the c→0 contraction is not; the hybrid limit gives a massless wave equation and the space-like equation is an Ansatz.","tokens_in":28456,"tokens_out":6459,"would_cite":false,"duration_ms":60107,"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":"By contracting the Klein-Gordon equation as the speed of light approaches zero, this paper derives three Carroll-invariant quantum theories and shows that the time-like one exhibits temporal tunneling with the Bogoliubov relation $\\lvert…","keywords":["Carroll symmetry","Klein-Gordon equation","ultra-relativistic limit","temporal tunneling","Bogoliubov transformation","fracton duality","tachyonic modes","null states"],"falsifier":"A direct check is to apply a non-zero Carroll boost to the space-like equation, keep all terms, and verify that the $\\chi$-dependent terms cancel the Laplacian's extra terms for every smooth $\\psi$; if no local choice of $\\chi$ closes the algebra, or if the canonical equal-time commutator of the zero-norm states yields no measurable flux, the central claim fails.","tokens_in":27206,"feed_emoji":"⏳","tokens_out":9580,"duration_ms":96993,"temperature":0.7,"pith_summary":"Taking the speed of light to zero is not a single limit: the paper shows that a systematic contraction $c\\to0$ of the Klein-Gordon equation splits into three Carroll-invariant quantum theories—time-like, space-like, and hybrid—each with its own continuity equation, probability density, current, and exact solutions. The time-like sector is immobile in space but supports temporal tunneling: a time-dependent barrier mixes positive and negative frequencies, yielding the Bogoliubov relation $\\lvert T\\rvert^2=1+\\lvert R\\rvert^2$ and emergent antiparticle-like modes. The space-like sector requires a non-dynamical compensating field to restore Carroll boosts; its propagating energy eigenstates are null states with vanishing density, their physical content carried by the spatial flux. The hybrid sector interpolates between the two, giving a damped or amplified wave equation whose box solutions divide into underdamped, critically damped, and overdamped regimes. The framework matters because Carroll symmetry is the ultra-relativistic counterpart of Galilean symmetry and controls the dynamics of null surfaces and immobile ('fractonic') excitations.","feed_headline":"Zero-speed-of-light limit yields three quantum theories","feed_subtitle":"A temporal barrier in one of them scrambles positive and negative frequencies into antiparticle-like modes.","key_machinery":"The machinery is the controlled Carrollian contraction: replacing $c$ by $\\epsilon c$ and sending $\\epsilon\\to0$ under three different scalings of the coordinates, mass, and wavefunction. The space-like and hybrid sectors are carried by a non-dynamical Stückelberg-like compensating field $\\chi$ (an auxiliary field with no kinetic term) whose Carroll boost variation is a differential operator, e.g. $\\delta_C\\chi=\\beta^2\\partial_t+2\\beta\\cdot\\partial_x$; it cancels the non-invariance of the spatial Laplacian. The time-like sector is carried by the indefinite Klein-Gordon norm, which turns temporal scattering into a Bogoliubov transformation preserving $\\lvert E\\rvert^2-\\lvert F\\rvert^2=1$, i.e. $\\lvert T\\rvert^2=1+\\lvert R\\rvert^2$. These two objects—the contraction scalings and the compensating field—fix which sectors exist and which states are physical.","core_discovery":"The central claim is that the ultra-relativistic limit of relativistic spin-0 quantum mechanics is not a single equation but a three-way branching. Depending on how coordinates, mass, and wavefunction are rescaled before $c\\to0$, the Klein-Gordon equation contracts to (i) the time-like Carroll equation $(\\hbar^2\\partial_t^2+m_0^2)\\psi=0$, which is second-order in time and has no spatial derivatives; (ii) a space-like sector, first-order in time with a compensating field $\\chi$ whose boost variation is $\\delta_C\\chi=\\beta^2\\partial_t+2\\beta\\cdot\\partial_x$, giving purely imaginary energies and zero-norm propagating states; and (iii) a hybrid sector containing both $\\partial_t^2$ and $\\partial_x^2$ plus a first-order $\\chi\\partial_t$ term, which is a damped/amplified oscillator for real wavefunctions and tachyonic for complex ones. For each sector the paper constructs the continuity equation, the probability density and current, and exact particle-in-a-box and rectangular-barrier solutions. The signature result is temporal tunneling: a rectangular potential step in time converts part of an incoming positive-frequency mode into a negative-frequency mode, with amplitudes satisfying $\\lvert T\\rvert^2=1+\\lvert R\\rvert^2$, the indefinite-norm analogue of $\\lvert T\\rvert^2+\\lvert R\\rvert^2=1$.","pith_inferences":["If the null states of the space-like sector are taken literally, an immediate test is to construct interference experiments where two such fluxes meet; the density would remain zero everywhere while the current pattern should still show interference fringes—something impossible in standard non-relativistic quantum mechanics.","The same contraction recipe should produce an exactly parallel trio of Galilean sectors in the $c\\to\\infty$ limit, with a 'spatial tunneling' analogue of temporal tunneling; the paper only sketches this direction.","The relation $\\lvert T\\rvert^2=1+\\lvert R\\rvert^2$ is identical in form to particle creation by time-dependent backgrounds in quantum field theory; if the analogy holds, the temporal barrier should also produce a calculable entanglement entropy between the positive- and negative-frequency sectors.","One could look for an experimental realization in ultracold atom or photonic simulators where an engineered time-dependent dispersion mimics the Carrollian slow light, checking whether the predicted zero-density propagating mode appears as a dark state that still carries flux."],"forward_implications":["Temporal barriers act as tunable beam splitters in the time-like sector: a pure positive-frequency state emerges as a superposition of positive- and negative-frequency parts with probabilities set by the barrier height and duration.","A particle confined in a temporal box has quantized energy levels; for a zero interior potential the rest mass itself is quantized in units of $\\pi\\hbar/T$.","Propagating space-like Carroll states are null in the sense of zero probability density, so measurements in that sector should be organized around the spatial flux rather than local density.","Hybrid Carrollian particles tunnel through a static spatial barrier with the same transmission and reflection probabilities as an ordinary non-relativistic particle, while the overall amplitude is multiplied by a global damping or amplification factor.","Time-like Carroll quantum mechanics coincides with the 0+1-dimensional reduction of a dipole-symmetric scalar field theory, providing a quantum-mechanical realization of isolated fractonic monopoles."],"supporting_citations":[{"why":"Establishes the Carroll limit as the c→0 contraction of the Poincaré group, the foundational symmetry used throughout.","marker":"[1]"},{"why":"Provides the alternative analogue of the Galilei group on which the Carroll group construction rests.","marker":"[2]"},{"why":"Supplies the classical time-like and space-like Carroll particles whose quantum counterparts the paper derives, and the classical Carroll/fracton duality.","marker":"[13]"},{"why":"Gives the quantum Carroll/fracton particle correspondence that the time-like sector is matched against.","marker":"[15]"},{"why":"Introduced the time-like Carroll quantum equation that the paper extends to three sectors with full probabilistic structure.","marker":"[32]"},{"why":"Source for the compensating-field idea used to restore Carroll boost invariance in the space-like and hybrid sectors.","marker":"[36]"},{"why":"States the known no-propagation result for single Carrollian scalars that the paper's χ-dressed construction must overcome.","marker":"[40]"},{"why":"Provides the Klein-Gordon probability density and its indefinite norm, which underlies the |T|²=1+|R|² relation.","marker":"[24]"}],"fun_headline_variants":["Carrollian quantum mechanics splits into three distinct sectors","Temporal tunneling appears in zero-speed-of-light limit","Three quantum worlds emerge when light stops","Zero-light limit gives quantum theory a triple identity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The space-like and hybrid sectors survive only if adding a non-dynamical auxiliary field that shifts under Carroll boosts lets a single scalar field carry spatial flux, in contrast to the known result the paper cites that a bare Carrollian scalar cannot propagate.","fun_headline_variants_meta":{"raw":{"variants":["Carrollian quantum mechanics splits into three distinct sectors","Temporal tunneling appears in zero-speed-of-light limit","Three quantum worlds emerge when light stops","Zero-light limit gives quantum theory a triple identity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000486,"raw_usage":{"total_tokens":2455,"prompt_tokens":1061,"completion_tokens":1394,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":677,"completion_tokens_details":{"reasoning_tokens":1335}},"tokens_in":677,"tokens_out":1394,"duration_ms":12001,"temperature":1.0,"reasoning_tokens":1335,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:37:49.526357+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to apply a non-zero Carroll boost to the space-like equation, keep all terms, and verify that the $\\chi$-dependent terms cancel the Laplacian's extra terms for every smooth $\\psi$; if no local choice of $\\chi$ closes the algebra, or if the canonical equal-time commutator of the zero-norm states yields no measurable flux, the central claim fails.","supporting_citations":[{"cited_title":"Une nouvelle limite non-relativiste du groupe de poincaré,","cited_arxiv_id":null,"evidence_quote":"Establishes the Carroll limit as the c→0 contraction of the Poincaré group, the foundational symmetry used throughout."},{"cited_title":"On an analogue of the galilei group,","cited_arxiv_id":null,"evidence_quote":"Provides the alternative analogue of the Galilei group on which the Carroll group construction rests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Klein-Gordon probability density and its indefinite norm, which underlies the |T|²=1+|R|² relation."}],"review_version":1}