{"id":"c009ed04-d992-49fe-863f-ac94870bc05a","arxiv_id":"1908.06610","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The authors argue their trapped-state extrapolation remains a plausible indicator of possible 3n and 4n resonances, but they concede the width and pole structure are unknown.","lead":"This reply defends the authors' earlier claim that trapped three- and four-neutron states may extrapolate to resonances near zero energy. It adds two-body benchmark checks but concedes it cannot distinguish a resonance from a virtual state.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Trap-extrapolation benchmark lacks a virtual-state control, and the reply concedes it cannot distinguish resonance from virtual state; few-neutron 'resonance indications' remain conditional.","rationale":"The reader's weakest assumption matches my reading: the reply's Fig. 2 benchmarks are real resonances, but no control is provided for the admitted resonance/virtual-state ambiguity. I do not see the reply as internally inconsistent; it is explicitly exploratory and acknowledges that widths are not quantified and that ACCC is a different procedure. It also adds value by showing the 4n trapped energy is stable over long imaginary time and by reporting the E4n<E2n subset check. However, the absence of a virtual-state control means the central inference from trap extrapolation to 'resonance indications' is conditional, not established. Since the author-side evidence is honestly circumscribed and the reader already assigned CONDITIONAL, I would not change the verdict; the concern confirms the condition rather than deepening it to rejection.","tokens_in":3056,"tokens_out":8431,"duration_ms":94654,"concrete_test":"Take the same two-Gaussian S-wave interaction used in Fig. 2 and retune its parameters to produce a shallow virtual state (or no pole at all) in the untrapped two-body system. Apply the identical linear V0-to-zero extrapolation to the trapped energies for the same three Woods-Saxon radii. If the extrapolated intercept is positive and statistically comparable to the resonance cases, the method is shown not to identify resonances specifically; the 3n/4n intercepts should then be reported as pole-agnostic, and the claim of 'indications' of resonances would need explicit S-matrix pole extraction (e.g., ACCC or complex scaling) to stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reply's central defense is that trapping plus linear extrapolation in V0 continues to indicate possible few-neutron resonances. The load-bearing step is the analytic continuation of a trapped bound-state energy to V0=0, and the reply's own final paragraph concedes that this step cannot distinguish a resonance from a virtual state. The two-body validation in Fig. 2 does not remove that concession: it tests only two cases known to be resonances, so it shows the method can reproduce a known pole when one exists, but gives no control case for a virtual state or for a very broad resonance. The imaginary-time stability shown in Fig. 1 is likewise computed inside a finite Woods-Saxon well; in that setting the state is confined by the trap, and the 'no decay' observation pertains to the trapped Hamiltonian, not to the untrapped continuum. The reply also defines 'true bound states' as states with compact support in the trap, which conflates confinement by the external potential with intrinsic binding. Thus the strongest assertion supported by the reply is only that the extrapolated intercept is stable across geometries; whether that intercept is a resonance pole, a virtual state, or a finite-volume artifact remains unresolved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript is a reply to the Comment by Deltuva and Lazauskas on the authors' earlier Letter (PRL 118, 232501 (2017)) concerning possible trineutron and tetraneutron resonances. The reply defends the use of an external trapping potential with a linear extrapolation to zero trap depth as a way to indicate possible few-neutron resonances. It presents new imaginary-time evolution results for four neutrons in a Woods-Saxon well, argues that the absence of decay over long imaginary time distinguishes the 4n state from a simple pair of dineutrons, and reports additional two-body benchmark calculations for two S-wave resonances. The authors also argue that the analytic continuation in the coupling constant (ACCC) approach used by the Commenters is not equivalent to applying an external trap. In the final paragraph they acknowledge that their current extrapolation cannot distinguish between a resonance and a virtual state, and they do not quantify resonance widths.","tokens_in":3273,"tokens_out":3467,"duration_ms":35449,"significance":"If the benchmarking and stability checks in this reply are accepted, they provide useful evidence that the trapped-state extrapolation is a reproducible diagnostic for identifying resonance-like structures in specific model cases. The two-body benchmarks in Fig. 2 are a concrete and honest test, and the authors explicitly narrow their original claim by admitting that resonance versus virtual-state discrimination is not possible with the current method. The reply therefore strengthens the methodological side of the original Letter but does not resolve the central ambiguity raised by the Comment, namely whether the extrapolated few-neutron intercepts correspond to S-matrix poles, virtual states, or finite-volume artifacts.","major_comments":[{"comment":"The benchmark in Fig. 2 tests only two cases that are known to be resonances, and it contains no control case with a virtual state or a very broad resonance. Since the authors state in the final paragraph that 'our current extrapolation cannot distinguish between a resonance and a virtual state,' the benchmark does not establish that the few-neutron extrapolated intercepts are resonance poles rather than virtual states. A virtual-state control case would be needed before the extrapolation can be claimed to indicate possible few-neutron resonances.","section":"Fig. 2 and final paragraph"},{"comment":"The imaginary-time stability of the 4n energy in Fig. 1 is computed inside a finite Woods-Saxon well with V0 = -1.25 MeV and RWS = 6.0 fm. For a Hamiltonian with a confining external well, the lowest eigenstate is necessarily long-lived in imaginary time, regardless of whether the untrapped system has a resonance, a virtual state, or no singularity at all. The comparison with the 8Be inset is not made on equal footing because the 8Be evolution is not performed with the same external confinement. Therefore the absence of decay in Fig. 1 does not support the statement that 'this 4n state is more complex than a pair of dineutrons.'","section":"Fig. 1"},{"comment":"The reply asserts that 'Bound states are states whose wave functions have compact support' and uses this to conclude that all calculated 3n and 4n states in the trap are bound. This is not a standard physics definition: bound states are normalizable eigenstates with energy below the continuum threshold, and wave functions in a finite-range external well generally do not have compact support. More importantly, this definition makes 'bound' equivalent to 'confined by the external trap,' which is exactly the point raised by the Commenters. The argument therefore does not rebut the distinction between trapped states and intrinsic bound states.","section":"First paragraph, definition of bound states"}],"minor_comments":[{"comment":"The symbol '□/2' appears to be a font substitution for Γ/2; please use a proper Gamma in the figure and caption.","section":"Fig. 2 and caption"},{"comment":"The sentence 'including in the extrapolation only the points where E4n < E2n for RWS = 7.5 fm still identifies the potential 4n resonance at approximately 2.5 MeV' would benefit from stating the V0 range used and the uncertainty of the fitted intercept.","section":"Paragraph 2 (RWS = 7.5 fm extrapolation)"},{"comment":"Reference [10] is listed as 'to be published,' and reference [12] is cited as already containing the arguments of the Comment; in a formal reply, a published or preprint identifier should be provided where available.","section":"References"},{"comment":"The parameters of the two-Gaussian S-wave potential used for the Fig. 2 benchmarks are not given in the reply; a reader should not need to consult the original Letter to reproduce the benchmark.","section":"Fig. 2 benchmark"}],"recommendation":"major_revision","confidential_remarks":"The reply is honest and the benchmarks are useful, but the central ambiguity between a resonance and a virtual state is explicitly conceded by the authors. For a journal reply, I would want to see either a virtual-state control case in the benchmark or a clear reformulation of the conclusion that avoids claiming resonance identification. With that change, the reply could be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This reply does what a reply should: it owns the missed references, avoids overclaiming, and tosses in two new checks that actually test the method. The two-body benchmark in Fig. 2 is clean evidence that the linear trap extrapolation can locate a known resonance when one exists. The imaginary-time stability run in Fig. 1 is a reasonable sanity check against the trivially dissociating dineutron picture. I also give them credit for explicitly conceding that the extrapolation cannot distinguish a resonance from a virtual state, and for not repeating the stronger language from the original Letter.\n\nThe soft spot is exactly that concession. The new benchmarks include no virtual-state control, so we still do not know whether the same linear procedure would produce a similar V0=0 intercept for a virtual state or a broad continuum artifact. The two-body tests show the method works in the cases where a pole exists, but they say nothing about how often it might produce a false positive. The 8Be comparison is also not as clean as it looks: the 8Be decay is computed for the untrapped Hamiltonian, while the 4n stability is computed inside a Woods-Saxon well. Confinement by the trap is not intrinsic binding, and the \"compact support\" definition of a bound state is simply wrong for a finite-range potential. That passage is the weakest in the reply.\n\nSo what is actually established? Only that the original trapped-state extrapolations are stable across geometries and can reproduce known two-body resonances. The central question—whether the few-neutron signals are S-matrix poles or virtual-state/continuum artifacts—remains open, as the authors themselves admit in the final paragraph. They are not evading anything; they just do not have the tools to close the gap.\n\nThis is a useful reply for anyone working on few-neutron physics or trapped-continuum extrapolation methods. It deserves a serious referee, because the two-body benchmarks and the stability check are honest evidence that a contested method is not vacuous. I would send it out, with the expectation that the referee asks for a virtual-state control or a clearer statement about what the intercept actually means. It should not be used as evidence that the resonances exist.","headline":"Honest reply with useful sanity checks, but the resonance-vs-virtual-state gap survives; fine as a reply, not a resolution.","tokens_in":3794,"tokens_out":1888,"would_cite":false,"duration_ms":22725,"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 reply maintains that extrapolating trapped few-neutron energies to zero trap depth can still indicate possible resonances, while conceding it cannot distinguish a resonance from a virtual state.","keywords":["trineutron","tetraneutron","few-neutron resonances","trap extrapolation","auxiliary-field diffusion Monte Carlo","Woods-Saxon potential","virtual state","imaginary-time evolution"],"falsifier":"Take a two-body model whose untrapped spectrum is known to contain a virtual state and no resonance. If the zero-depth extrapolation returns a positive intercept near that virtual-state energy, the positive intercepts reported for three and four neutrons cannot be taken as resonance evidence. Alternatively, evolve the trapped four-neutron state in the $E_{4n}<E_{2n}$ regime for substantially longer imaginary time; a downward bend in the energy would contradict the claimed stability and the conclusion that the state is more complex than separated dineutrons.","tokens_in":2836,"feed_emoji":"⚛️","tokens_out":6973,"duration_ms":66586,"temperature":0.7,"pith_summary":"This reply to a Comment defends the claim that linearly extrapolating the energy of three- or four-neutron states held in an external trap to zero trap depth can indicate possible few-neutron resonances. The authors argue that this trap-based procedure differs from analytic continuation in the coupling constant, and that the states they compute are bound in the conventional sense inside the trap. They add two-body test calculations in which the same extrapolation locates two known resonances within the fit uncertainties, and they report that the four-neutron energy stays flat over long imaginary time in a regime where the dineutron is bound, which they read as evidence against a simple pair-of-dineutrons structure. They explicitly concede that the extrapolation cannot distinguish a resonance from a virtual state and says nothing about widths. The question matters because the existence of few-neutron resonances remains experimentally unsettled.","feed_headline":"Reply keeps trap-extrapolation path to few-neutron resonances open","feed_subtitle":"New two-body tests match resonance energies; 4n state stays stable in imaginary time.","key_machinery":"The load-bearing object is the linear extrapolation of the trapped ground-state energy to zero trap depth, with the depth $V_0$ of an external Woods-Saxon well as the control parameter. The reply's support comes from two-body checks: two S-wave two-Gaussian interactions tuned to resonances at 0.78 MeV and 4.40 MeV, whose extrapolated intercepts match the known positions within the fit uncertainty. The imaginary-time evolution of the energy serves as a decay diagnostic, with a quickly decaying unbound $^8$Be-like $4^+$ state as the counterexample and the flat four-neutron energy as the claimed stable case.","core_discovery":"The central claim of this reply is that the trap-extrapolation analysis of the original Letter remains a valid indicator of possible few-neutron resonances. When the energy of a few-body state in a Woods-Saxon trap is extrapolated linearly to zero well depth, the intercept reproduces the known energies of two S-wave two-body resonances, and the same procedure applied to three- and four-neutron states yields a common positive energy scale. For the four-neutron system, the energy shows no decay over very long imaginary-time evolution in a regime where $E_{4n}<E_{2n}$, which the authors take to mean the state is more complex than two dineutrons or a dineutron plus two neutrons; restricting the extrapolation to points with $E_{4n}<E_{2n}$ still places a possible tetraneutron resonance near 2.5 MeV. The authors do not claim that these resonances definitely exist, do not claim to know their widths, and explicitly state that the extrapolation cannot distinguish a resonance from a virtual state.","pith_inferences":["A testable extension the authors do not pursue: run the same zero-depth extrapolation on a two-body model known to produce only a virtual state; a matching positive intercept would mean the method locates near-threshold features generally, not resonances specifically.","The imaginary-time stability criterion could be turned into a quantitative diagnostic by comparing trapped three- and four-neutron decay curves against a ladder of known unbound cases in the same Monte Carlo setup, which would give a time threshold beyond which non-decay is meaningful.","If a future experiment sees a narrow structure near the extrapolated energy, it would retrospectively validate the method; if it sees only a smooth threshold rise, the virtual-state interpretation becomes the more natural reading of the same calculation."],"forward_implications":["If the extrapolation is accepted, the common positive energy scale found for trapped three- and four-neutron states remains an indication that few-neutron resonances may exist, but with no width information attached.","The two-body benchmark implies that, within this interaction class, the zero-depth intercept carries information about where a resonance sits, not merely about the trap itself.","The long flat four-neutron energy curve implies that a trapped four-neutron state in the $E_{4n}<E_{2n}$ regime is not trivially a dineutron pair or a dineutron plus two free neutrons, so a realistic explanation must involve a more collective structure.","Because the method cannot separate resonances from virtual states, any claimed 3n or 4n resonance derived from trap extrapolation should be phrased as a localized near-threshold feature until scattering data decide."],"supporting_citations":[{"why":"Original Letter whose trapped three- and four-neutron energies are defended here.","marker":"[1]"},{"why":"Supplies the zero-trap-depth extrapolation procedure used to interpret trapped few-body energies.","marker":"[2]"},{"why":"Experimental tetraneutron resonance candidate that the extrapolated energies are compared with.","marker":"[3]"},{"why":"Calculation suggesting the resonance may be very broad, used to temper claims about observable effects.","marker":"[9]"},{"why":"Earlier work that already raised objections similar to the Comment, acknowledged in the reply.","marker":"[12]"},{"why":"The Comment being answered, whose analytic-continuation-based objection the reply addresses.","marker":"[13]"},{"why":"Unbound-state imaginary-time decay example used as a comparison showing the four-neutron energy remains flat.","marker":"[14]"}],"fun_headline_variants":["Trap extrapolation still points to few-neutron resonance energies","Reply defends extrapolation: tetraneutron hint at 2.5 MeV","No decay in imaginary time: 4n state more than dineutrons","Extrapolation reproduces two-body resonances, hints tetraneutron","Reply: trap method keeps few-neutron resonance question open"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the energy of a bound state in a finite external trap, extrapolated linearly to zero trap depth, lands on a resonance or virtual state of the untrapped Hamiltonian; the authors test this on two-body examples but do not prove it, and they concede the extrapolation cannot tell a resonance from a virtual state.","fun_headline_variants_meta":{"raw":{"variants":["Trap extrapolation still points to few-neutron resonance energies","Reply defends extrapolation: tetraneutron hint at 2.5 MeV","No decay in imaginary time: 4n state more than dineutrons","Extrapolation reproduces two-body resonances, hints tetraneutron","Reply: trap method keeps few-neutron resonance question open"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":1090,"prompt_tokens":808,"completion_tokens":282,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":185}},"tokens_in":424,"tokens_out":282,"duration_ms":3036,"temperature":1.0,"reasoning_tokens":185,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:39:03.999419+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-body model whose untrapped spectrum is known to contain a virtual state and no resonance. If the zero-depth extrapolation returns a positive intercept near that virtual-state energy, the positive intercepts reported for three and four neutrons cannot be taken as resonance evidence. Alternatively, evolve the trapped four-neutron state in the $E_{4n}<E_{2n}$ regime for substantially longer imaginary time; a downward bend in the energy would contradict the claimed stability and the conclusion that the state is more complex than separated dineutrons.","supporting_citations":[{"cited_title":"Gandolfi, H","cited_arxiv_id":null,"evidence_quote":"Original Letter whose trapped three- and four-neutron energies are defended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the zero-trap-depth extrapolation procedure used to interpret trapped few-body energies."},{"cited_title":"Kisamori et al","cited_arxiv_id":null,"evidence_quote":"Experimental tetraneutron resonance candidate that the extrapolated energies are compared with."},{"cited_title":"Fossez, J","cited_arxiv_id":null,"evidence_quote":"Calculation suggesting the resonance may be very broad, used to temper claims about observable effects."},{"cited_title":"Deltuva, Phys","cited_arxiv_id":null,"evidence_quote":"Earlier work that already raised objections similar to the Comment, acknowledged in the reply."},{"cited_title":"Deltuva and R","cited_arxiv_id":null,"evidence_quote":"The Comment being answered, whose analytic-continuation-based objection the reply addresses."},{"cited_title":"Pastore, R","cited_arxiv_id":null,"evidence_quote":"Unbound-state imaginary-time decay example used as a comparison showing the four-neutron energy remains flat."}],"review_version":1}