{"id":"dbcdf612-78b2-48a8-a47d-b1e2266cd0c4","arxiv_id":"2412.19722","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"SpheriCo.jl enables longer semiclassical collapse simulations and yields hints of non-trivial correlations of the scalar field across a dynamically formed apparent horizon.","lead":"A new open-source Julia code, SpheriCo.jl, simulates the gravitational collapse of a spherical scalar field in both classical and semiclassical settings, using summation-by-parts methods that improve stability near the origin. Its first physics application produces two-point correlation maps of the quantum field that hint at non-trivial correlations between points inside and outside a forming apparent horizon.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The across-horizon correlation tongues are shown only at parameter choices where the PV regulator condition is violated and the fastest modes have lost convergence; no evidence is given that the signal survives increasing MPV or mode cutoff.","rationale":"The reader's weakest assumption identifies exactly the same issue: Pauli-Villars mass and mode truncation. I agree. The strongest claim is presented as a 'hint', and the paper is candid about limitations; but the hint is only credible if it survives a check that the numerical observable is converged. Here the paper's own diagnostic data undercut the required conditions. Section 6.1 shows the highest-frequency quantum modes lose their designed second-order accuracy at t≈7.9 for MPV=1, and section 7.1 states the PV mass violates the separation-of-scales criterion by a factor of ~3.6. Since the correlation function is defined through the same mode sum that enters the stress tensor, the loss of convergence in high-l ghost modes can directly corrupt C(t;r1,r2). The subcritical comparison is a good control for the presence of a horizon, but it cannot control for numerical artifacts that appear only in the supercritical spacetime (e.g. lapse collapse inside the horizon). A gauge-dependence test is also desirable; the paper itself flags it. However, the regulator/truncation issue is more elementary: without a convergence check of the correlator, the 'hint' may simply be numerical noise. I therefore keep the reader's CONDITIONAL verdict. On the positive side, the classical and semiclassical validation suite (convergence tests, robust stability, critical collapse echo, Minkowski-mode convergence) is substantial and gives independent support for treating SpheriCo.jl as a reliable tool; the caveat is specifically about the new correlator result.","tokens_in":39701,"tokens_out":5129,"duration_ms":52984,"concrete_test":"Recompute the supercritical equal-time correlator of figure 16 at t≈8 for three mode cutoffs, e.g. (kmax,lmax) = (20,60), (30,90), (40,120), at fixed MPV=1; and, if stable, at MPV=2 with (30,90). If the off-diagonal tongue amplitude, sign, or shape changes by more than the observed subcritical noise floor, the feature is not converged. Additionally, split (7.1) into the physical-field term (n=0) and the total ghost contribution (n=1..5): if the tongues are not dominated by the n=0 term, they are regulator artifacts rather than Hawking correlations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the equal-time correlator (7.1) in supercritical collapse develops off-diagonal 'tongues' connecting points inside and outside the apparent horizon, hinting at Hawking-pair correlations. For this to be a physical statement, the observable must be converged in the numerical and regulator parameters. Two conditions known to be violated from the paper's own tests: (i) section 7.1 admits MPV = 1 fails the required separation MPV >> 2π/λ ≈ 3.6, so the five PV ghost fields are not safely decoupled; (ii) section 6.1, figure 12 shows the fastest modes (l=60, n=5) at MPV=1 lose second-order convergence for t ≳ 7.9, which is inside the displayed time range of the tongues (figure 16, t=7.97). The correlator (7.1) is an alternating sum over the physical field and ghost fields; if high-l/high-n ghost modes are inaccurate at late times, their oscillatory errors can masquerade as physical correlations. The paper provides no convergence study of C(t;r1,r2) in kmax, lmax, or MPV, and the subcritical control only demonstrates that the feature is tied to horizon formation, not that it is robust to numerical resolution. The 'hint' therefore rests on an unverified assumption: that the tongue structure is independent of the PV mass and mode truncation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents SpheriCo.jl, an open-access Julia code for spherically symmetric gravitational collapse of a scalar field in classical and semiclassical (quantum-field-on-classical-background) settings, with the quantum stress tensor regularized by five Pauli-Villars ghost fields and WKB-derived counterterms for the cosmological constant and Planck mass. The numerical method combines summation-by-parts operators for the 1/r terms at the origin, a Z4-type constraint-damping option, Kreiss-Oliger dissipation, and an infalling-outer-boundary option, and it is validated extensively: second-order convergence for smooth and noisy data in super- and subcritical runs, reproduction of Choptuik-type echoing and universality, convergence to analytic Minkowski mode solutions, and a Minkowski-consistency test for backreaction. The central physics result is the equal-time two-point correlator (7.1) in collapse runs: supercritical runs develop off-diagonal 'tongues' correlating points inside and outside the apparent horizon (Fig. 16), which the authors present as a hint of Hawking-pair correlations, while subcritical runs show no persistent analogue. Sections 7.2 and 8 report that the correlator and horizon quantities are qualitatively similar with backreaction, within parameter ranges the authors themselves flag as limited.","tokens_in":40019,"tokens_out":18913,"duration_ms":169493,"significance":"If the correlator tongues are physical, this would be a notable advance: a four-dimensional, spherically symmetric numerical computation of across-horizon quantum correlations in a dynamically collapsing spacetime, going beyond earlier two-dimensional semiclassical collapse studies and complementing analogue-horizon observations. The manuscript has real strengths independent of the headline claim: the code ships with a public repository and archived data; the validation is machine-checkable and broad (analytic Minkowski modes, Choptuik universality, robust-stability tests); the WKB derivation of the Pauli-Villars counterterms (Appendix C) is a genuine methodological contribution; and the authors are unusually candid about their diagnostics, including the failure of their constraint-damping implementation (Section 5.3). The skeptic's concern lands: the paper's own Section 7.1 admits that the runs producing the tongues violate the PV-decoupling condition MPV ≫ 2π/λ̃, and Section 6.1 (Fig. 12) shows the fastest ghost modes losing second-order convergence within the displayed time range.","major_comments":[{"comment":"The central result — the across-horizon correlation tongues of Fig. 16 — is computed with MPV = 1, a value that violates the paper's own decoupling condition MPV ≫ 2π/λ̃ ≈ 3.6 stated in Section 4 and explicitly acknowledged in Section 7.1 ('we were not able to respect the condition ... Here we have 2π/λ̃ ≃ 3.6'). Because the correlator (7.1) is an alternating sum over the physical field and the five ghost fields with masses (3.12) of order MPV, the ghost sectors are not suppressed at the wavelengths of the physical pulse. No computation of C(t; r1, r2) for larger MPV — even in the no-backreaction case where the Section 6.2 backreaction instability does not apply — and no estimate of the ghost-sector fraction of the correlator is provided. The abstract's claim that the results 'hint at a non-trivial correlation across the horizon of Hawking quanta' therefore rests on an untested assumption about regulator robustness rather than on the paper's usual standard of evidence.","section":"§7.1 (final paragraph), §4 (third paragraph), Eq. (7.1)"},{"comment":"The fastest modes of the heaviest ghost field (l = 60, n = 5, µ5 = 2MPV) lose second-order convergence for t ≳ 7.9 (Fig. 12), which is inside the time range where the tongues are displayed in Fig. 16 (last panel, t = 7.96875), and the correlator runs use lmax = 90, beyond the l = 60 at which convergence was tested. No resolution study or kmax/lmax-study of the correlator itself is presented: the subcritical control in Fig. 16 only shows that the tonguelike feature is associated with horizon formation, not that the supercritical feature is converged. Since (7.1) sums over all modes up to the truncation, oscillatory errors in high-l, high-mass ghost modes at t ≳ 7.9 could contribute to the observed off-diagonal structure, so the late-time growth of the tongues is not currently established as a physical effect.","section":"§6.1, Fig. 12; §7.1, Fig. 16"},{"comment":"The foliation dependence of the claimed signal is asserted but not tested. The text states that 'we expect the details of the correlation tongues to depend on the choice of lapse function, but the qualitative structure of correlations outside and within the horizon to remain' and immediately afterwards that 'It is possible that our picture is greatly affected by the collapse of the lapse inside the apparent horizon', while Section 8 defers a wider parameter scan to future work. Because the slicing determines which points are related as 'inside' and 'outside' at equal time, the physical interpretation of the tongues requires at least one independent gauge computation (e.g., harmonic or polar slicing), which is not provided. In addition, the abstract's phrase 'correlation across the horizon of Hawking quanta' is stronger than the body's hedged statement (Section 7, opening) that the correlation 'may be interpreted as correlating pairs of Hawking quanta'; if a gauge test is not feasible, the abstract and Section 8 should be re-scoped accordingly.","section":"§7.1 (gauge paragraph), §8"},{"comment":"The counterterm expressions in the main text and in the WKB derivation do not agree: Eqs. (3.13)–(3.14) quote log arguments 39/1216 and 24/33 with coefficient 1/(12(2π)²) on the Gab term, whereas Eqs. (C.11)–(C.15) give 39/216 and 28/36 with coefficient 1/(24(2π)²). The effective Planck mass (3.15) constructed from these counterterms is used in the backreaction runs of Section 7.2 (Figs. 17–18), so the inconsistency leaves the reader unable to determine which values were implemented. Please reconcile the two presentations and state explicitly which counterterms were used in production.","section":"§3.2, Eqs. (3.13)–(3.15) vs Appendix C, Eqs. (C.11)–(C.15)"}],"minor_comments":[{"comment":"The domain is written as '0 ≥ t ≥ tmax, 0 ≥ r ≥ rmax'; the inequalities are reversed and should read 0 ≤ t ≤ tmax and 0 ≤ r ≤ rmax.","section":"§2.2, Eq. (2.17)"},{"comment":"In the definition of C(t1, t2; r1, r2), the second field factor is written as 'Φ̂(t, r2)' but should be 'Φ̂(t2, r2)'; as written the expression mixes t and t2 in the same formula.","section":"§7.1, Eq. (7.1)"},{"comment":"The three MPV panels use very different vertical scales (MPV = 0.1: up to 2e-5; MPV = 1: up to 1; MPV = 2: up to 6), which makes the several-orders-of-magnitude difference between the cases difficult to read; a common logarithmic scale would be clearer, and the caption should state that the MPV = 2 runs crash during the evolution.","section":"§6.2, Fig. 13"},{"comment":"Since the constraint-damping option is reported in Section 5.3 to be unstable or to increase constraint violation for many parameter choices, please state explicitly whether damping was enabled (and with which κ1, κ2) in the Section 7 correlator runs, so that the reader can assess the quality of the background geometry used in the mode evolution.","section":"§5.3, §7.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript header states 'Published in JHEP'; if this submission is intended for a different venue, the provenance should be clarified editorially. The paper is within the journal's scope and has strong methodological value: the code is open-access, the validation is broad and reproducible, and the authors are candid about diagnostics. The main risk is the gap between the abstract-level claim and the paper's own internal diagnostics (violated PV-decoupling condition, loss of convergence of fast modes within the displayed time range, and untested gauge dependence). If the authors can supply a correlator convergence study in MPV and kmax/lmax, or re-scope the headline to an explicitly gauge- and regulator-dependent numerical observation, the paper would be publishable; if the scans destroy the tongues, the physics conclusion changes but the methodological contribution remains. The counterterm inconsistency (M4) must be fixed regardless of the re-scoping decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"SpheriCo.jl is a serious numerical relativity code paper, and the correlation tongues are a suggestive hint, not a measured Hawking-pair signal. That distinction matters, because the central physics claim currently rests on parameter choices the paper itself flags as problematic.\n\nWhat is actually new and good: the paper extends the Pauli-Villars semiclassical collapse program to 4D spherical symmetry with angular modes, gets farther in time than earlier work via SBP operators, and computes the Planck-mass and cosmological-constant counterterms analytically. The validation suite is extensive and credible: second-order convergence for smooth and noisy data, reproduction of Choptuik echoing, agreement with analytic Minkowski modes, and a backreaction test that moves toward Minkowski as more modes are included. The code and data are public, and the authors are unusually candid about their constraint-damping scheme failing. That is real, reproducible work.\n\nThe soft spots are concentrated in the correlation claim. The paper states in section 7.1 that MPV = 1 violates the required condition MPV >> 2π/λ ≈ 3.6. Section 6.1 shows the fastest modes (l=60, n=5) lose second-order convergence around t ≈ 7.9, which is inside the time window of the displayed tongues (figure 16, t = 7.97). The correlator is an alternating sum over physical and ghost fields, so inaccurate high-modal ghost modes can in principle produce oscillatory artifacts. There is no convergence study of C(t; r1, r2) in kmax, lmax, or MPV. The subcritical control shows the feature appears only when a horizon forms, but not that it survives better regulation or higher resolution. The authors call it a hint and list the same concerns themselves, so this is not a hidden flaw, but it is a load-bearing gap for the physical interpretation.\n\nI disagree with any reading that dismisses the code or the paper on these grounds. The classical and semiclassical validation stands on its own, and even the correlation result is worth publishing as a preliminary observation if framed as such. The honest move is a referee asking for a convergence study of the correlator before the Hawking-pair language is accepted.\n\nWho this is for: numerical relativists working on semiclassical collapse or black-hole formation codes, and anyone interested in analogue-gravity correlation phenomenology. It deserves a serious referee, not a desk rejection. My recommendation: send it to peer review, with the expectation that the correlation claim gets conditional acceptance pending a demonstration that the tongues persist as MPV and the mode cutoff are pushed toward the regime where the regulator is legitimately decoupled.","headline":"Solid code paper with an honest but not-yet-established physics hint: the across-horizon correlator tongues are shown only where the paper's own regulator condition is violated and fast modes have lost convergence.","tokens_in":40575,"tokens_out":1265,"would_cite":true,"duration_ms":17259,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.25.dg","04.62.+v","04.70.Dy"],"model":"deepseek-v4-flash","headline":"SpheriCo.jl simulates stable semiclassical collapse and sees horizon-linking quantum correlation tongues.","keywords":["quantum fields in curved spacetime","gravitational collapse","Hawking radiation","numerical relativity","Pauli-Villars regularization","quantum correlations","apparent horizon","summation-by-parts"],"falsifier":"Re-run the supercritical correlator with the Pauli-Villars mass raised tenfold while keeping the same physical wavelength, and with more than 30 radial modes; if the horizon-linking tongues do not persist and converge, the reported correlation is a regularization or truncation artifact.","tokens_in":39467,"feed_emoji":"🕳️","tokens_out":5394,"duration_ms":51778,"temperature":0.7,"pith_summary":"SpheriCo.jl is a new open-access code that simulates the spherical gravitational collapse of a scalar field whose matter sector can be classical or a quantised field on a classical, dynamical background. The paper's central claim is that with summation-by-parts differentiation the semiclassical system stays stable long enough to compute field correlations after an apparent horizon forms. Running a supercritical pulse with Pauli-Villars-regularized quantum modes, the code sees two-point correlation tongues connecting points inside and outside the horizon, growing with time; subcritical runs show no persistent analogue. The authors present this as a hint of Hawking-pair correlations while explicitly noting that a more systematic parameter study is needed.","feed_headline":"Quantum correlations form across a black hole horizon","feed_subtitle":"A stable semiclassical collapse code finds correlation tongues between points inside and outside the apparent horizon.","key_machinery":"The argument is carried by the equal-time two-point correlation function from equation (7.1), evaluated from the evolved quantum modes, together with the Pauli-Villars ghost-field sector that cancels ultraviolet divergences and the summation-by-parts finite-difference operators of [21] that keep the $1/r$ terms stable near the origin. The SBP operators are the practical enabler: they let the code evolve faster-oscillating modes and reach times where the horizon has formed, while the Pauli-Villars fields make the stress-energy expectation value finite on the lattice. The correlator is then sliced in time: tongues appear off the diagonal when one point sits inside the apparent-horizon radius and the other outside.","core_discovery":"The central discovery the authors report is that the equal-time two-point correlator $C(t; r_1, r_2)$ of the quantum scalar, computed in the semiclassical approximation on a dynamically collapsing spacetime, develops non-trivial 'tongues' away from the main diagonal that correlate a point inside the apparent horizon with one outside, and this happens only in supercritical evolutions. This is presented as evidence hinting at correlation between pairs of Hawking quanta. The correlation persists when backreaction is included, with the same qualitative shape and somewhat stronger oscillations around the horizon.","pith_inferences":["If the tongues are physical, equal-time correlation across the horizon could become a practical numerical witness of Hawking-pair formation, though the paper stops short of claiming entanglement.","The tongues' angle may depend on the gauge slicing; computing the correlator on the double-null surfaces the paper already provides could give a slicing-invariant diagnostic.","Because the SBP treatment is generic, the same pipeline could be applied to other spherically symmetric semiclassical settings, such as regular black holes or dynamical horizons, without new regularization work.","The reported failure of Z4 constraint damping for this variable set suggests that damping schemes should be re-tuned or extended to reduction constraints before being used in long runs."],"forward_implications":["Semiclassical collapse simulations can now run past apparent-horizon formation with stable evolution near the origin, allowing correlators to be evaluated after the horizon exists.","The classical module reproduces Choptuik critical-collapse behavior with second-order convergence, so the code can be used as a testbed for critical phenomena.","In supercritical runs the equal-time correlator develops off-diagonal tongues linking points inside and outside the apparent horizon, while subcritical runs relax back to a Minkowski-like profile.","When backreaction is included, the apparent-horizon location, area, and mass track the classical run within ten percent, and the correlation picture persists with stronger oscillations around the horizon."],"supporting_citations":[{"why":"Supplies the Pauli-Villars ghost-field regularization scheme with five massive ghost fields used to render the stress-energy expectation value finite.","marker":"[18]"},{"why":"Provides the summation-by-parts finite-difference operators that stabilize the $1/r$ terms and the stiff high-$l$ quantum modes near the origin.","marker":"[21]"},{"why":"Provides the regularized first-order variables, including the auxiliary variable $\\lambda$, that remove coordinate singularities in the spherically symmetric Einstein equations.","marker":"[29]"},{"why":"Provides the Z4 constraint-damping formulation adapted here to control Hamiltonian and momentum constraint violation during evolution.","marker":"[32]"},{"why":"Earlier semiclassical collapse work that established the coherent-state quantum treatment and the PV regularization approach this paper extends.","marker":"[17]"},{"why":"Analogue-horizon numerical study that defined the 'tongue' correlation signature that this paper compares with its horizon-linking correlator.","marker":"[25]"},{"why":"Hawking's particle-creation result that motivates interpreting the horizon-crossing correlation as correlation between pairs of Hawking quanta.","marker":"[22]"}],"fun_headline_variants":["Gravitational collapse reveals quantum links across horizon","Simulation shows quantum correlations across black hole horizon","Horizon-spanning quantum correlations emerge in collapse simulation","Quantum scalar field correlates inside and outside horizon","Collapse simulation links quantum fields across horizon"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The simulations use a Pauli-Villars ghost mass that is not much larger than the shortest wavelength resolved, so the correlation tongues could be a regularization artifact rather than a sign of Hawking radiation.","fun_headline_variants_meta":{"raw":{"variants":["Gravitational collapse reveals quantum links across horizon","Simulation shows quantum correlations across black hole horizon","Horizon-spanning quantum correlations emerge in collapse simulation","Quantum scalar field correlates inside and outside horizon","Collapse simulation links quantum fields across horizon"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000752,"raw_usage":{"total_tokens":3260,"prompt_tokens":775,"completion_tokens":2485,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":391,"completion_tokens_details":{"reasoning_tokens":2415}},"tokens_in":391,"tokens_out":2485,"duration_ms":16783,"temperature":1.0,"reasoning_tokens":2415,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:56:38.427896+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the supercritical correlator with the Pauli-Villars mass raised tenfold while keeping the same physical wavelength, and with more than 30 radial modes; if the horizon-linking tongues do not persist and converge, the reported correlation is a regularization or truncation artifact.","supporting_citations":[{"cited_title":"Regularization of spherically symmetric evolution codes in numerical relativity","cited_arxiv_id":"gr-qc/0401113","evidence_quote":"Provides the regularized first-order variables, including the auxiliary variable $\\lambda$, that remove coordinate singularities in the spherically symmetric Einstein equations."}],"review_version":1}