{"id":"c5abc108-b044-4ca8-ad15-7ee41c3bb53a","arxiv_id":"2512.10008","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For several scalar-field profiles, extremal Reissner-Nordström black holes can be glued from collapse only above a profile-dependent threshold in eM, and only for scalar masses below about m/e≈0.28.","lead":"Numerical experiments map when the characteristic-gluing construction of Kehle and Unger can form an exactly extremal Reissner-Nordström black hole from charged scalar collapse. The required size of the final black hole depends strongly on the scalar-field profile, and grows when the scalar has mass or the cosmological constant is negative.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"C^2 'minimum' eM values are not established as true minima: §3.4 admits missed solutions, so Table 1 thresholds may be upper bounds pending independent global search.","rationale":"The reader's conditional verdict already identifies the incompleteness of the C^2 root search as a key assumption. My stress-test converges on that same point: the paper's own text in §3.4 concedes that solutions may have been missed, and the numerical method is not guaranteed to find all roots. Because the central quantitative claim depends on those minima, independent verification is needed before the thresholds can be trusted. However, the qualitative conclusion—that gluing to extremality is possible only for sufficiently large eM and that the required eM depends strongly on the Ansatz—is likely robust, and the C^0 threshold has an analytic condition supporting it. Therefore the appropriate verdict remains CONDITIONAL, with no change from the reader's assessment.","tokens_in":25402,"tokens_out":15906,"duration_ms":157347,"concrete_test":"Independently reimplement the C^2 even-Ansatz gluing at eM = 3000, 4000, and 4417 using a global root-finder (e.g. multi-start Newton with 10^4 random initial guesses, or homotopy/deflation) and high-accuracy ODE integration (step <= 1e-4 with convergence checks). If any solution with q=1, ∂_U Φ(1)=0 and ∂_U^2 Φ(1)=0 is found for eM < 4417, the claimed minimum is wrong; if none is found, the Table 1 C^2 entry gains support. Repeating the same test for the C^1 branches at the quoted minima would further settle whether the 'only if' thresholds are genuine.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline quantitative claim is that for each Ansatz gluing is possible only if eM exceeds a stated minimum. The weakest load-bearing point is the C^2 search: §3.4 explicitly says 'our numerical algorithm is sometimes unstable, and thus there may well exist solutions we have not found,' and the branch plot in Fig. 12 is described as likely incomplete. Since the C^2 entries in Table 1 (e.g. (eM)_min=4417) are presented as minimum values, any undiscovered branch reaching q=1 at smaller eM would invalidate those entries. Appendix A also notes Broyden's method 'is not stable, and is not guaranteed to converge.' The same concern applies, less acutely, to C^1 where uniqueness of the solution branch is asserted only as 'we have not found any examples.' Thus the Table 1 thresholds are best interpreted as upper bounds on the true minima unless a more exhaustive search is performed. The C^0 results are on firmer ground because I_max is obtained from a one-dimensional scan and the condition (23) is analytic, but the exact digits still lack independent verification.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a numerical study of characteristic gluing for constructing spherically symmetric spacetimes that form exactly extremal Reissner–Nordström black holes in finite time, following the existence proof of Kehle and Unger. For several scalar-field Ansätze (even and odd bump functions, polynomials, and a modified even family), the authors solve the constraint/transport equations along the gluing null cone to accuracy C^0, C^1, and C^2, and determine, for each Ansatz and regularity class, the smallest value of eM for which gluing to an extremal RN horizon appears possible, together with the largest scalar mass-to-charge ratio m/e for which such gluing is found. They also study the effect of a cosmological constant. The results are summarized in Table 1, and the paper additionally reports that the dynamical horizon can be temporarily superextremal during the gluing.","tokens_in":25753,"tokens_out":3224,"duration_ms":37645,"significance":"If the numerics are reliable, this is a valuable quantitative companion to Kehle–Unger: it shows that the proof's large-eM regime is not necessary in practice, that the threshold depends strongly on the Ansatz and on regularity, and that mass-to-charge thresholds are well below the rigorous bound m/e < 1. The paper is transparent about the limitations of the C^2 search, and it gives concrete profiles that could seed future explicit spacetime constructions. The main scientific value is in the C^0 and C^1 results; the C^2 entries currently have the status of upper/lower bounds rather than established extrema. No code or data are shipped, and no residual tolerances or error bars are reported, which limits reproducibility for a numerical paper.","major_comments":[{"comment":"The C^2 entries in Table 1 — (eM)_min = 4417, 2076 and max m/e = 0.00992 — are presented as established extrema, but the text explicitly states that 'there may well exist solutions we have not found' and that Fig. 12 likely omits branches. These values are therefore only bounds obtained from the branches the solver happened to find: for (eM)_min they are upper bounds on the true infimum, and for max m/e they are lower bounds on the true supremum. The abstract and Table 1 should be reworded accordingly, or an independent global search (e.g. continuation/random multistart with documented success rates) should be added.","section":"§3.4 and Table 1"},{"comment":"The claim that 'we have not found any examples' of multiple C^1 solutions is used to infer uniqueness of the branch and hence to interpret the C^1 minima as true minima. This is a numerical observation, not a proof, and the C^2 experience shows that extra branches can appear at higher k. The C^1 (eM)_min values (720, 2909, 78.5, 239) should similarly be stated as 'the smallest value found' unless the search over the parameter space is made exhaustive or its completeness is quantified. The distinction matters because the paper's central claim is that gluing is possible only above these thresholds.","section":"§3.3.2 and §3.4"},{"comment":"The numerical accuracy of the thresholds is not quantified. The text reports step size 0.001 and 'regularly checked . . . convergence tests', but no residual norms, no tolerance for the Broyden solves, no estimated error bars for (eM)_min, Imax, or the critical m/e values, and no data/code are provided. Since the paper's contribution is quantitative, at least the final thresholds should be accompanied by an estimate of discretization and root-finding error. This is not fatal to the qualitative conclusions, but it prevents the reader from assessing how many significant digits in Table 1 are meaningful.","section":"Appendix A and §3.1"}],"minor_comments":[{"comment":"The abstract says 'gluing is possible only if the final black hole mass is large enough.' Given the numerical, Ansatz-dependent nature of the evidence, 'only if' is too strong; 'for each of the Ansätze studied, gluing was found to be possible only when eM exceeded a certain value' would be more accurate. The same overstatement appears in the Discussion.","section":"Abstract and §1"},{"comment":"The notation K = 2k+1 for the number of basis functions is easily confused with the regularity order k. Since the paper uses both k and K in close proximity, a different symbol for the number of parameters would improve readability.","section":"Eq. (18)–(20)"},{"comment":"The figure relies on colors (black/blue/green) to label branches. The colors will not survive grayscale printing. Please add line styles or explicit labels to all branch plots.","section":"Fig. 12"},{"comment":"The equations are written in a slightly unusual mixed notation, with ∂_V A_U rather than ∂_V A_U after Eq. (5e). Throughout, the gauge choice A_V = 0 should be stated before Eq. (5) is used, not later in §2.3.","section":"§2.2, Eq. (5)"},{"comment":"The discussion of superextremal horizons is interesting but would benefit from a short explanation of why the quasilocal definition is gauge-invariant, since the renormalised mass ϖ is defined via the induced metric on the sphere. Currently the paper only states the formula (25).","section":"§3.2.5"},{"comment":"Typo in the Introduction: 'produce spactimes' should be 'produce spacetimes'.","section":"Spelling"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the topic is timely. The main issue is interpretational: Table 1 and the abstract present incomplete numerical searches as definitive thresholds. The authors can fix this by reframing the C^2 (and partly C^1) entries as bounds and by adding error/residual reporting. I do not see a fundamental flaw in the approach; the revision is feasible without new conceptual work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful numerical companion to Kehle and Unger. The new quantitative output — minimum eM for C0/C1/C2 gluing, maximum m/e for massive scalars, and the Λ trends — gives a concrete sense of how far the proven \"eM ≫ 1\" regime is from the actual threshold. The C0 results are on firm ground: I_max comes from a one-dimensional scan, condition (23) is analytic, and the Appendix B extension of sup∂_U r < 0 to nonzero Λ is clean. I would trust C0 as solid, C1 as indicative, and C2 as provisional. The soft spots are real but mostly acknowledged. No code or data are shipped, and there are no residual tolerances or error bars. The numerical method is Broyden with a finite-difference Jacobian, which the authors themselves describe as unstable and not guaranteed to converge. The bigger issue is the C2 search: Section 3.4 states explicitly that there may well be solutions they have not found, and Figure 12 is labeled as likely incomplete. So the C2 entries in Table 1 — (eM)_min ≈ 4417, max m/e ≈ 0.00992 — are upper bounds on the true minima, not established values. I would push that point a bit harder than the reader's summary did. The same caveat applies weakly to C1, where uniqueness is only asserted as \"we have not found any examples.\" The paper is honest about these limitations and does not oversell. The modified-even Ansatz result is a nice touch: it shows the thresholds are genuinely Ansatz-dependent, so there is no universal minimum eM over all profiles, only a lower bound for the families tried. The central qualitative claims — gluing requires sufficiently large eM, massive scalars work only below an m/e threshold well below 1, and positive Λ lowers the required eM — are consistent with the proved results, including the m/e ≥ 1 obstruction from [12]. The circularity concern does not land: the thresholds are outputs of shooting, not parameters tuned to a known outcome. Who is this for: anyone working on third-law violations, characteristic gluing, or charged scalar collapse. The exact digits in Table 1 shouldn't be quoted without the caveats, but the paper deserves a serious referee. I would accept it for review and ask for code/data and a reframing of the C2 entries as upper bounds rather than minima.","headline":"Useful numerical companion to Kehle-Unger: the C0 thresholds look solid and the qualitative picture holds, but the headline C2 numbers in Table 1 should be read as upper bounds, not proven minima, and the paper deserves a serious referee.","tokens_in":696,"tokens_out":803,"would_cite":true,"duration_ms":32764,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C22","83C75"],"pacs":[],"model":"deepseek-v4-flash","headline":"For each scalar-field profile tested, gluing to an extremal Reissner-Nordstrom black hole works only above a minimum black-hole mass and, for massive scalars, below a maximum mass-to-charge ratio.","keywords":["extremal black holes","characteristic gluing","third law of black hole mechanics","charged scalar field","Reissner-Nordstrom","numerical relativity","mass-to-charge ratio","cosmological constant"],"falsifier":"Run the same Ansatz and ODE integration below the quoted (eM)_min, say below 15.96 for the C0 even profile, and find any parameter set with Q(1)=qM, the required transverse derivatives of Phi vanishing, inf r>0, and sup partial_U r<0: that would refute the threshold. Alternatively, find a valid extremal gluing with m/e above the quoted maximum for that Ansatz.","tokens_in":1573,"feed_emoji":"","tokens_out":7883,"duration_ms":116441,"temperature":0.7,"pith_summary":"This paper numerically implements the characteristic gluing construction that was used to prove that an extremal Reissner-Nordstrom black hole can form in finite time from gravitational collapse of a charged scalar field. It asks how large the final black hole must be and how light the scalar must be for the construction to work. It finds that, for each scalar-field profile, extremal gluing requires a minimum dimensionless mass eM, ranging from about 10 to 2900 depending on the profile and the required smoothness. It also finds that adding a scalar mass allows gluing only up to a finite mass-to-charge ratio, always below 1, with the maximum varying strongly with the Ansatz. If these thresholds are correct, they turn the existence proof into concrete quantitative conditions and show that violating the third law of black hole mechanics is possible but requires substantial, though finite, fine-tuning.","feed_headline":"Extremal black hole gluing needs a minimum mass","feed_subtitle":"Numerical scans set the mass and mass-to-charge thresholds for finite-time extremal black hole formation.","key_machinery":"The central object is the characteristic gluing along an outgoing null cone C, with free scalar profile rho(V) e^{-iV}. Charge balance reduces to Q/(e r_+^2) = I(alpha), where I(alpha)=int xi^2 Im(Phi partial_V Phi) dV and xi=r/r_+ solves the null Raychaudhuri equation. Because I depends only on the profile parameters, its maximum over the region where r>0 sets the minimum mass via (eM)_min = 1/I_max at q=1. For C^k gluing, 2k transverse derivatives of Phi must vanish at the horizon, producing a multi-parameter shooting problem solved numerically.","core_discovery":"The paper performs characteristic gluing numerically for several scalar-field profiles on a null cone, connecting a flat or (A)dS sphere to a Reissner-Nordstrom horizon. It finds that extremal gluing (q=1) works only above a minimum dimensionless mass eM that depends on the profile and the desired smoothness: e.g., 15.96 for the C0 even profile at Lambda=0, 720 for the C1 even profile, and 4417 for the C2 even profile; other profiles give different values. A scalar mass m permits gluing only up to a maximum m/e, always well below 1, with values such as 0.2059 (C0 even) and 0.0272 (C1 even) at Lambda=0. A positive cosmological constant lowers the required mass, a negative one raises it. Gluin","pith_inferences":["If these thresholds are true minima, optimizing the scalar profile could push eM_min considerably lower, and it is an open numerical question whether it can reach zero to form arbitrarily small extremal black holes.","The strong Ansatz dependence of the threshold, paired with a universal final extremal state, evokes critical collapse and suggests possible scaling relations between profile families and minimal mass.","The temporary superextremal phase indicates gluing data routinely overshoot the final charge-to-mass ratio, which invites numerical evolution to test whether such horizons are stable or seed instabilities.","The m/e maxima remain well below 1, so the known no-go bound at m/e=1 might be sharpenable; varying the phase or adding many parameters could move toward it."],"forward_implications":["For each profile tested, extremal gluing has a finite minimal eM; below it no solution satisfies the required r>0 and partial_U r<0 conditions.","A positive cosmological constant lowers the minimal mass while a negative one raises it.","Including a scalar mass gives a maximum m/e for which q=1 can be reached; beyond it no extremal gluing exists even at very large eM.","Higher smoothness classes demand substantially larger masses and can exhibit multiple solution branches, as seen in C^2.","Some gluing solutions pass through a temporarily superextremal horizon before reaching the final stationary black hole."],"fun_headline_variants":["Extremal black hole gluing needs a minimum mass","Minimum mass varies with scalar profile for extremal gluing","Numerics pin down mass thresholds for extremal black holes","Mass-to-charge cap found for scalar field in black hole gluing","Cosmological constant shifts required mass for extremal gluing"],"cache_read_input_tokens":27520,"weakest_assumption_plain":"The thresholds rely on numerical searches that assumed the root-finder captured all gluing solutions, that the maximum of I occurs where it was computed, and that gluing data can be extended to a full spacetime by existing theorems.","fun_headline_variants_meta":{"raw":{"variants":["Extremal black hole gluing needs a minimum mass","Minimum mass varies with scalar profile for extremal gluing","Numerics pin down mass thresholds for extremal black holes","Mass-to-charge cap found for scalar field in black hole gluing","Cosmological constant shifts required mass for extremal gluing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1387,"prompt_tokens":710,"completion_tokens":677,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":592}},"tokens_in":454,"tokens_out":677,"duration_ms":6855,"temperature":1.0,"reasoning_tokens":592,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T17:16:35.024718+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same Ansatz and ODE integration below the quoted (eM)_min, say below 15.96 for the C0 even profile, and find any parameter set with Q(1)=qM, the required transverse derivatives of Phi vanishing, inf r>0, and sup partial_U r<0: that would refute the threshold. Alternatively, find a valid extremal gluing with m/e above the quoted maximum for that Ansatz.","supporting_citations":[],"review_version":1}