{"id":"0e60f799-d55f-47db-83ce-6ad05a72f768","arxiv_id":"2505.10199","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"GWTC-3 gravitational wave data exclude pseudo-complex GR deviations at 1PN and, for the first time, at 1.5PN that would remove black hole horizons.","lead":"Using gravitational wave data from the LIGO-Virgo-KAGRA third catalog, the authors constrain a modified gravity theory, pseudo-complex general relativity, in which black hole horizons can disappear for a large enough deviation parameter. They report that deviations at 1PN and 1.5PN order, sufficient to remove the horizon, are excluded, the 1.5PN case for the first time.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed 1.5PN exclusion depends on comparing point estimates to the General 90% bounds; the Restricted (common-deviation) bound quoted in the same table overlaps the predictions, and no posterior propagation is performed. This may not hold under a full hierarchical analysis.","rationale":"The reader's weakest_assumption (final-spin to inspiral mapping) is a real gap, but it is likely conservative in direction if the final spin is indeed the largest spin in the system. The more acute issue is that the claimed 1.5PN exclusion is not supported by the statistical comparison as presented. The paper compares point estimates (or approximate ranges) of the predicted δφ to the 90% General bounds, without building a full posterior predictive distribution for δφ from the joint GWTC-3 posteriors. For the decisive χ=0.7 group the upper end of the quoted δφ range is -3.7%, only 0.7 percentage points outside the General bound; a proper propagation of the correlations among q and spins could move nontrivial posterior mass inside. Moreover, the paper's own Table I lists the Restricted (common-deviation) 1.5PN bound as (-5%,-1%), which overlaps the predicted values; since pcGR predicts a universal b, the correct treatment is a hierarchical comparison with b as a common unknown, not an ad hoc choice of the General bound. The abstract states 'rule out 1.5PN' without these qualifications, while the body limits the claim to the General bounds. The public code and data are a positive feature, and the formula checks (e.g., b_c for n=1 and n=1.5, standard PN coefficients like PNGR=6.44 and -50.26) are correct, but the central exclusion claim is not yet established. A full reanalysis with posterior samples and a hierarchical likelihood is required before the paper can be accepted; hence UNVERDICTED rather than CONDITIONAL, since the current evidence is insufficient rather than merely in need of minor revision.","tokens_in":9442,"tokens_out":25437,"duration_ms":261948,"concrete_test":"Re-analyze the nine O3b events using the public GWTC-3 posterior samples. For each posterior draw, compute b_c from Eq. (5) using the relevant spin (separately: χf, max(χ1,χ2), and an effective inspiral spin), then compute δφ_1.5 from Eqs. (7)-(10), obtaining the full posterior predictive distribution. Compute (i) the 90% credible interval of the predicted δφ_1.5 and its overlap with the GWTC-3 General and Restricted bounds; and (ii) a hierarchical likelihood with b as a common unknown parameter, comparing pcGR to GR via a Bayes factor. If the predictive interval overlaps either bound, or if the Bayes factor does not strongly favor b > b_c, the asserted exclusion is not supported.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central claim is 'for the first time rule out 1.5PN' horizonless pcGR. In Sec. IV this rests on Table I: predicted δφ = PNpcGR/PNGR (Eq. 10) at b = b_c (Eq. 5) is compared to the GWTC-3 'General' 90% bounds. Two problems make this comparison load-bearing. First, the prediction is evaluated at point estimates (median q and χ) and the quoted δφ ranges are not full posterior predictive intervals; they are obtained by varying input parameters individually, ignoring correlations. For n=1.5 and χ=0.7, the upper end is -3.7%, only 0.7 percentage points outside the General bound (-3%,3%). With the full joint posterior of q, χ1, χ2, χf from GWTC-3, a non-negligible fraction of the predictive distribution can fall inside the bound, so the 'rule out' at 90% confidence is not established. Second, the choice of the General bound is not self-evident: because b is a universal parameter of pcGR, the deviation is common across events; the Restricted (common deviation) bound quoted in the same table is (-5%,-1%) for 1.5PN, and the predicted upper values (-4.9% for GW200115, -3.7% for the χ=0.7 group) lie inside it. The paper does not perform a hierarchical likelihood comparison with b as an unknown, which is the only way to decide which bound applies. The Sec. III choice of the final spin for b_c is likewise defended as conservative only if the final spin exceeds every component spin; this is asserted, not checked, and if violated the predicted δφ is inflated. These gaps affect exactly the new 1.5PN result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper updates constraints on the pseudo-complex General Relativity (pcGR) and dirty-black-hole metric parameter b, for post-Newtonian orders n = 1, 1.5, 2, 3, 3.5, using the inspiral phases of the nine GWTC-3 O3b events that passed the LVK parameterized-test selection. The authors evaluate the horizonless threshold b_c of a single spinning object (Eq. 5) at the averaged final-spin posterior of each event, compute the pcGR-to-GR phase-coefficient ratio delta_phi = PN_pcGR/PN_GR (Eq. 10, with PN_pcGR from Eq. 9 at b = b_c), and compare delta_phi with the 90% 'General' and 'Restricted' bounds of the LVK parameterized tests. They conclude that 1PN horizonless pcGR is excluded (extending earlier work), that 1.5PN horizonless pcGR is excluded for the first time, and that 2PN-3.5PN remain allowed. The analysis code is publicly available.","tokens_in":9792,"tokens_out":39163,"duration_ms":351538,"significance":"A robust 1.5PN exclusion would be a genuinely new constraint on an exotic-metric family, and the paper's mechanics are partly commendable: the analytic b_c construction is transparent; the quoted GR phase coefficients at 1PN (6.44) and 1.5PN (-16*pi) match the standard TaylorF2 values, confirming the mapping; the code is public; and the use of GWTC-3 rather than O1 events is a clear step forward. The 1PN confirmation, whose median predictions lie far outside both quoted bounds, is credible. However, the headline 1.5PN claim is not established by the analysis as presented, for the reasons detailed below; the result becomes publishable only after a proper statistical treatment of the comparison.","major_comments":[{"comment":"The claim that 1.5PN horizonless pcGR is 'rule[d] out' rests on comparing the predicted delta_phi with the 'General' 90% bound (-3%, 3%) only. The same table quotes the 'Restricted' (common-deviation) 90% bound (-5%, -1%) for 1.5PN, and the predicted values in Table I overlap that interval: -3.7% to -6.5% for the chi=0.7 group and -4.9% to -9.1% for GW200115. Because b is a universal parameter of pcGR (the per-event variation enters only through the q-dependent factor Q in Eq. 8), the common-deviation comparison is the pertinent one, and a defensible exclusion requires a joint/hierarchical likelihood over b across the nine events, or at minimum a comparison against the Restricted bound. As it stands, the 1.5PN exclusion is not established against the appropriate bound.","section":"Sec. IV / Table I / Fig. 3"},{"comment":"The quantity delta_phi is evaluated at point estimates (averaged posteriors) of q and chi, and the reported ranges are obtained by substituting the 90% bounds of the inputs one parameter at a time, ignoring correlations in the joint posterior. For the n=1.5, chi=0.7 case, the upper end of the quoted range (-3.7%) lies only 0.7 percentage points inside the General bound (-3%, 3%); under a full posterior predictive computation using the joint q, chi_1, chi_2, chi_f posteriors, a non-negligible fraction of the predictive mass of delta_phi will fall inside the bound, so the 90%-confidence exclusion language is not supported. The authors should compute the full predictive distribution of delta_phi (or of b directly), for example by reweighting the public LVK posterior samples of the parameterized-test parameters, and report the posterior mass outside the relevant bound.","section":"Sec. III / Table I"},{"comment":"The horizonless threshold b_c is evaluated with the remnant (final) spin and then applied to a two-body inspiral phase coefficient. The justification given, namely that the final spin exceeds every component spin for all events considered and thus provides a more restrictive bar, is asserted but not demonstrated for the nine events, and no derivation is provided for mapping a single-object horizon condition to the inspiral PN coefficient. The authors should either derive the appropriate effective spin entering the inspiral horizonless condition or demonstrate robustness with a sensitivity analysis (e.g., using effective inspiral spin or the component spins in Eq. 5) and quantify the resulting change in delta_phi. I note that if the final-spin ordering failed, b_c and |delta_phi| would increase and the exclusion would be strengthened; nevertheless, the physical basis of the chosen threshold must be part of the argument.","section":"Sec. III / Eq. (5)"},{"comment":"Evaluating Eq. (9) at the tabulated b_c for the chi=0, q=1 rows gives PN_pcGR values larger than those tabulated by a systematic factor of roughly 2^(n-1) (n=1.5: 6.14 vs 4.33; n=2: 23.7 vs 11.8; n=3: -112 vs -28.1; n=3.5: -172 vs -30.4), while the n=1 row agrees. Because delta_phi is the central quantity of the paper, the authors must reconcile Eq. (9) with the numbers actually used in Table I, give the exact convention connecting B(f) in Eq. (7) to the LVK delta_phi-hat_n parameters (including any powers of 2 arising from the (M 2*pi*f)^(2n/3) factor), and confirm that the GitHub code implements the convention printed in the paper.","section":"Sec. II / Eq. (9) vs Table I"}],"minor_comments":[{"comment":"Equation (6) prints the SPA prefactor as (pi*G*M*f)^(5/3); the standard stationary-phase expression has (pi*G*M*f)^(-5/3). The exponent should be corrected; the ratio delta_phi is unaffected, but the printed formula is wrong as it stands.","section":"Sec. II / Eq. (6)"},{"comment":"The sentence 'pcGR converges with GR predictions in the weak-field regime for larger (or more precisely (M/r)^n)' is garbled; it should say that (M/r)^n tends to zero at large r.","section":"Sec. II / first paragraph"},{"comment":"As quoted in Table I, the Restricted 1.5PN interval (-5%, -1%) excludes delta_phi=0 at 90%, a statement that would itself be notable; the authors should verify this number against the source (arXiv:2112.06861) and comment on its meaning, and in general should cite the specific table or figure from which each General and Restricted bound is taken.","section":"Table I / quoted bounds"},{"comment":"Reference [7] spells the author name 'Birnholz' whereas the current paper and reference [6] use 'Birnholtz'; the typo should be fixed.","section":"References"},{"comment":"The acknowledgment thanks the 'annonymous referee'; this should read 'anonymous referee'.","section":"Acknowledgments"},{"comment":"Table I mixes input parameters (q, chi with their 90% bounds) with derived quantities (b_c, PN_pcGR, PN_GR, delta_phi); splitting the table or adding explicit column labels for medians and upper/lower 90% values would greatly aid reproducibility, and the caption should state that the chi=0, q=1 rows are not GWTC-3 events but a comparison with [6].","section":"Table I / presentation"},{"comment":"The statement that 'The 1PN deviation lies outside both the General and Restricted bounds' is not strictly true for the full quoted range: for the chi=0.7 group the lower end of the quoted delta_phi range is 13%, inside the General upper bound of 14%. The claim should either be restricted to medians or be supported by the full comparison.","section":"Sec. IV / 1PN claim"},{"comment":"Grouping the eight non-GW200115 events at a fixed 'averaged' chi=0.7 discards the per-event posterior information shown in Fig. 2; using each event's own posterior would be more principled and would connect directly to the hierarchical comparison requested in the major comments.","section":"Sec. III / event grouping"},{"comment":"Section V's 'effectively ruled out' is weaker than the abstract's 'for the first time rule out 1.5PN'; the abstract, main text, and summary should make the same, appropriately qualified claim.","section":"Abstract vs Sec. V"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short research note whose headline claim ('for the first time rule out 1.5PN') considerably exceeds what the analysis supports as written, because the comparison is essentially a point-estimate check against published 90% intervals. The systematic factor of roughly 2^(n-1) between Eq. (9) and Table I should be resolved before the paper is reviewed again, as it suggests a convention mismatch between the printed formulas, the code, and the LVK parameterization. I would also ask the authors to double-check the provenance of all quoted GWTC-3 bounds, since some quoted intervals (e.g., Restricted 1.5PN, (-5%, -1%)) have properties worth explaining. The paper may fit the journal as a research note provided the claims are appropriately qualified and the statistical treatment is upgraded."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — this short paper does one genuinely new thing: it uses GWTC-3 events to attempt the first exclusion of 1.5PN pseudo-complex GR deviations that would remove black hole horizons. The 1PN exclusion is confirmed and looks robust; the 1.5PN claim is the interesting and fragile part.\n\nWhat the paper does well: it applies the authors' prior parameterized framework cleanly to the latest LVK catalog, computes b_c analytically, and presents the results in easy-to-read tables and figures. The code is public, the 9 O3b events match the LVK PN-test sample, and the body text is careful to say that the 1.5PN deviation lies outside the General bounds rather than claiming a blanket exclusion.\n\nThe main soft spot is statistical. The predicted δφ at b_c is computed at posterior medians or means of q and χ, and the quoted ranges come from varying input parameters one at a time rather than from the joint posterior. For the χ≈0.7 group at 1.5PN, the upper end is -3.7% against a General bound of (-3%, 3%) — a 0.7 percentage point margin. Propagating the full joint posterior could move a non-negligible fraction of the predictive distribution inside the bound, so the 90% exclusion is not actually established. A proper hierarchical analysis with b as a common unknown would settle whether the General or Restricted bound is the right comparison; the paper's argument that the deviation is event-dependent has merit, but without a joint likelihood it remains an assertion.\n\nThe final-spin choice for b_c is physically motivated but asserted rather than checked; that is minor, though if the final spin is not the maximum spin for some event, the predicted δφ would be inflated.\n\nThe abstract does overstate slightly: 'rule out 1.5PN' omits the 'General bound' qualifier that the body carefully includes.\n\nVerdict: the paper deserves serious refereeing. The 1PN update is solid, and the 1.5PN claim is a plausible first pass that needs a fuller statistical treatment. I would accept for peer review and ask for a revision with a hierarchical likelihood and posterior predictive intervals.","headline":"A genuinely new 1.5PN exclusion attempt with a marginal statistical foot; worth refereeing, but the headline claim needs a proper hierarchical treatment.","tokens_in":10352,"tokens_out":3719,"would_cite":true,"duration_ms":38340,"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":"Using nine merging black-hole signals from the third gravitational-wave transient catalog, this paper reports that horizonless pseudo-complex general relativity and dirty-black-hole solutions are excluded at 1PN and, for the first time…","keywords":["pseudo-complex general relativity","dirty black holes","horizonless solutions","post-Newtonian phase","gravitational-wave astronomy","parametrized tests of general relativity","black hole horizons"],"falsifier":"Recompute $\\delta\\phi$ for all nine events using the mass-weighted effective inspiral spin (or the component spins) in place of the remnant spin in Eq. (5), and check whether the 1PN and 1.5PN ratios still fall outside the catalog's 90% bounds; if they do not, the exclusion of horizonless pcGR solutions at these orders does not follow from the current data.","tokens_in":9215,"feed_emoji":"🕳️","tokens_out":13425,"duration_ms":124515,"temperature":0.7,"pith_summary":"This paper asks whether a family of modified black-hole metrics that can remove the horizon entirely—pseudo-complex general relativity (pcGR) and dirty-black-hole models with a running mass function—is still compatible with the gravitational waves observed from merging black holes. Using nine events from the third gravitational-wave transient catalog, it computes, for each event, the smallest deviation strength $b_c$ that would make the remnant horizonless, then evaluates the post-Newtonian phase shift such a deviation would leave in the inspiral waveform and compares it with the catalog's measured bounds on deviations from general relativity. The central result is that at $b_c$ the predicted 1PN phase ratio $\\delta\\phi$ lies outside both the general and restricted bounds, and the 1.5PN ratio, tested here for the first time, lies outside the general bounds. At 2PN, 3PN, and 3.5PN the predicted ratios remain inside the bounds, so horizonless solutions at those orders are not excluded. If the result holds, any horizonless pcGR or dirty-black-hole description of these binaries must place its leading deviation at 2PN or higher.","feed_headline":"1.5PN horizonless pcGR models excluded by gravitational waves","feed_subtitle":"Nine black-hole mergers put horizonless pcGR solutions outside observed 1PN and 1.5PN bounds.","key_machinery":"The load-bearing object is the family of modified Kerr metrics with a mass function $m(r)=M[1-b(M/r)^n]$, representing a dirty black hole in general relativity and a vacuum solution in pcGR. The horizonless threshold is the critical value $b_c$ at which $\\Delta = r^2 + a^2 - 2m(r)r$ has no positive real roots, so no Killing horizon exists; the paper evaluates $b_c$ from the remnant spin $\\chi$ via Eq. (5). The observable imprint is the pcGR post-Newtonian phase coefficient $PN^{pcGR}_n = \\frac{20}{(n-4)(2n-5)}\\left(\\frac{(n+2)(n+1)}{3}Q+\\rho\\right)b$, derived through the stationary-phase approximation, and the argument proceeds by comparing the ratio $\\delta\\phi = PN^{pcGR}_n / PN^{GR}_n$ with the catalog's general and restricted bounds at each order. This ratio converts an abstract no-horizon condition into a measurable waveform-phase prediction.","core_discovery":"The paper's central claim is that horizonless pcGR/dirty-black-hole solutions are observationally excluded at two post-Newtonian orders. For each of the nine events, setting the deviation parameter $b$ to the critical horizonless value $b_c$ (Eq. 5, evaluated with the remnant spin $\\chi$) produces a phase ratio $\\delta\\phi = PN^{pcGR}_n / PN^{GR}_n$ that exceeds the measured bounds at 1PN and 1.5PN: at 1PN the ratio lies outside both the General bounds (which allow deviation coefficients to vary across events) and the Restricted bounds (which require a common deviation), while at 1.5PN—tested here for the first time—it lies outside the General bounds. GW200115, the low-spin outlier, provides the most stringent 1PN and 1.5PN exclusions and comes closest to excluding 2PN. The paper concludes that the earlier 1PN exclusion is confirmed and that 1.5PN horizon-removing deviations are newly ruled out, while 2PN, 3PN, and 3.5PN horizonless solutions remain within the observational bounds.","pith_inferences":["Because $b_c$ grows as spin decreases, the exclusion power of this method is spin-dependent; a catalog with more low-spin mergers, or a different choice of which spin enters Eq. (5), could either sharpen or erode the stated bounds, a direction the paper does not quantify.","If horizonless deviations are forced to 2PN or higher, their inspiral-phase signatures shrink, so the cleanest future tests would move to merger/ringdown observables such as the photon ring and quasinormal-mode spectra, which the paper flags as its planned next step.","The same 'horizonless threshold to inspiral phase ratio' mapping should transfer to any Kerr-like metric with a similar mass-function form, letting the method constrain other parametrized deviations without building new waveform models."],"forward_implications":["Horizonless pcGR and dirty-black-hole solutions with 1PN or 1.5PN leading deviations are incompatible with the nine events considered, so any surviving horizonless model must have $b < b_c$ at those orders.","The 2PN, 3PN, and 3.5PN parameter regions remain observationally open, so horizonless solutions whose deviations first appear at those orders are not ruled out by this dataset.","The low-spin event GW200115 yields the most stringent 1PN and 1.5PN bounds and comes closest to excluding 2PN, indicating that low-spin mergers carry the most exclusion power per event.","Compared with earlier single-event bounds, the updated constraints are narrower at 1PN and have shifted the allowed 2PN and 3PN intervals in a direction more accommodating of pcGR horizonless objects.","Applying the same ratio test to the next observing run's catalog should refine these exclusions and may decide whether 2PN-order horizonless deviations become excluded as sensitivity improves."],"supporting_citations":[{"why":"Gives the pseudo-complex Kerr line element and the mass-function ansatz $m(r)=M[1-b(M/r)^n]$ used throughout.","marker":"[5]"},{"why":"Derives the critical horizonless $b_c$ formula (Eq. 5) and introduces the post-Newtonian phase-deviation method.","marker":"[6]"},{"why":"Extends the method to dirty black holes and supplies the form of the pcGR post-Newtonian coefficients compared here.","marker":"[7]"},{"why":"Provides the catalog of compact-binary coalescences from the second part of the third observing run, from which the nine events are drawn.","marker":"[13]"},{"why":"Supplies the General and Restricted 90% bounds on post-Newtonian deviations used to decide which orders are excluded.","marker":"[14]"},{"why":"Gives the stationary-phase approximation used to write the waveform phase and hence the deviation term.","marker":"[19]"}],"fun_headline_variants":["GWTC-3 rules out horizonless pcGR at 1.5PN","First 1.5PN exclusion of horizonless dirty black holes","Nine mergers exclude horizonless pcGR at 1PN and 1.5PN","Gravitational waves tighten bounds on horizonless pcGR","Horizonless pcGR excluded at 1PN and 1.5PN by GWTC-3"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The exclusion rests on assuming that the deviation strength needed to erase the horizon, computed from the merged remnant's final spin, also sets the size of the phase deviation during the early inspiral; if the relevant spins are instead the inspiral component spins or a mass-dependent $b_c$, the predicted phase shifts change and the claimed exclusion could fail.","fun_headline_variants_meta":{"raw":{"variants":["GWTC-3 rules out horizonless pcGR at 1.5PN","First 1.5PN exclusion of horizonless dirty black holes","Nine mergers exclude horizonless pcGR at 1PN and 1.5PN","Gravitational waves tighten bounds on horizonless pcGR","Horizonless pcGR excluded at 1PN and 1.5PN by GWTC-3"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000937,"raw_usage":{"total_tokens":4011,"prompt_tokens":951,"completion_tokens":3060,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":2956}},"tokens_in":567,"tokens_out":3060,"duration_ms":20748,"temperature":1.0,"reasoning_tokens":2956,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:14:41.791499+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute $\\delta\\phi$ for all nine events using the mass-weighted effective inspiral spin (or the component spins) in place of the remnant spin in Eq. (5), and check whether the 1PN and 1.5PN ratios still fall outside the catalog's 90% bounds; if they do not, the exclusion of horizonless pcGR solutions at these orders does not follow from the current data.","supporting_citations":[{"cited_title":"Caspar, T","cited_arxiv_id":null,"evidence_quote":"Gives the pseudo-complex Kerr line element and the mass-function ansatz $m(r)=M[1-b(M/r)^n]$ used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the critical horizonless $b_c$ formula (Eq. 5) and introduces the post-Newtonian phase-deviation method."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the method to dirty black holes and supplies the form of the pcGR post-Newtonian coefficients compared here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the catalog of compact-binary coalescences from the second part of the third observing run, from which the nine events are drawn."},{"cited_title":"Cutler and E","cited_arxiv_id":null,"evidence_quote":"Gives the stationary-phase approximation used to write the waveform phase and hence the deviation term."}],"review_version":1}