{"id":"5e10dfbc-3f30-4f7a-9341-af0a511ab520","arxiv_id":"2610.00454","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":8,"one_line_summary":"Selection-corrected inference on the 27 Gaia neutron star candidates yields an intrinsic local density near 120 kpc^-3, a decreasing period distribution, and a forecast of 300 to 650 detections in Gaia DR4.","lead":"Using the 27 known Gaia neutron star candidates and a detailed model of how Gaia detects them, this paper infers how many such systems truly exist and what they look like. It concludes that about 97% remain undetected and predicts Gaia's next data release will find 300 to 650 more.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The injection campaign applies the M_NS>1.25 cut to astrometric masses while the real sample used joint Gaia+RV masses; since AMRF masses are lower limits, beta_eff may be biased low and N_2kpc/n0 biased high, scaling the ~3%-discovered headline.","rationale":"The paper's most distinctive quantitative claims are the intrinsic space density n ≈ 120 kpc^-3, the implied ~900 systems within 2 kpc and ~3% discovery fraction (Fig. 4), and the DR4 forecast, all tied to the modeled selection fraction beta_eff through N_2kpc times beta_eff ≈ 27. I examined the candidate-selection steps that define beta_eff (Table 1) for a systematic, rather than statistical, error. The strongest candidate is the documented mismatch in Section 2.2: the real sample applied M_NS > 1.25 to joint Gaia+RV masses, while injections apply it to astrometric masses. Since the AMRF astrometric mass is a lower limit, the injection cut is systematically stricter, and because the inferred intrinsic mass distribution rises steeply toward 1.25 Msun, the mass boundary carries the most population weight. A small offset between astrometric and joint masses translates directly into a beta_eff bias that can exceed the statistical error, propagating into N_2kpc, n0, the undiscovered fraction, the Galactic birth-rate normalization, and the DR4 yield. The paper is fully honest about calling this a best approximation, but honesty about a systematic does not bound its size; the concrete test above would bound it. This is exactly the reader's weakest assumption, so I agree. A secondary sensitivity I considered but do not elevate: alpha_P = -0.75 rests on three short-period systems and softens to -0.13/-0.37 for the M_NS > 1.35 subset (Appendix C); this is a numeric caveat on one headline parameter rather than a threat to the population inference as a whole. I give credit for the sound importance-reweighting likelihood (Eqs. 2–4), the passing posterior predictive check (mu = 26.6 vs N = 27), the parameterization robustness checks in Appendix B, and the honest scaling guidance for possible WD contamination. Nothing I found is internally inconsistent or circular. The CONDITIONAL verdict stands: the framework is sound and the qualitative picture robust, but the absolute normalization is hostage to an untested selection-function calibration.","tokens_in":53800,"tokens_out":10290,"duration_ms":123499,"concrete_test":"For the 27 real systems, recompute the astrometric-only AMRF masses using the same machinery gaiamock applies to injections, and measure the offset distribution relative to the published joint Gaia+RV masses. Then re-run the selection-function calculation with the M_NS > 1.25 cut replaced by a joint-mass-equivalent cut (threshold shifted by the median offset, or per-injection offsets drawn from the measured distribution) and recompute beta_eff. If beta_eff changes by more than its quoted ~0.007–0.009 statistical error, the intrinsic count, space density, and DR4 yield scale inversely and the headline numbers need revision; if beta_eff is stable, the best-approximation selection function is validated for the normalization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The space-density and discovery-fraction claims scale inversely with the modeled selection fraction beta_eff, since N_2kpc times beta_eff ≈ 27 (Fig. 4). Section 2.2 documents two mismatches between the injection selection and the real sample: the M_NS > 1.25 cut was applied to joint Gaia+RV masses for the published sample but to astrometrically inferred masses in the injection campaign, and part of the observed sample came from earlier compact-object searches whose selection steps are not reproduced. The first mismatch has a definite direction: the AMRF astrometric mass is a lower limit on the companion mass, so requiring astrometric M_NS > 1.25 in the injections is stricter than the real joint-mass cut. The inferred intrinsic M_NS distribution rises steeply toward the 1.25 Msun boundary (mu ≈ 1.30, sigma ≈ 0.18, with a posterior that decreases with mass; Section 3.1), so the boundary region carries large population weight. Even a ~0.05–0.1 Msun offset between astrometric and joint masses can shift the recovered fraction passing the mass cut by more than the quoted statistical uncertainty (beta_eff = 0.029 with errors 0.007–0.009). Calling the selection function a best approximation is honest, but it leaves the absolute normalization — N_2kpc ≈ 900, n ≈ 120 kpc^-3, the ~97% undiscovered claim, the DR4 yield, and the Section 5 birth rates — uncertain at a level the quoted 1-sigma intervals do not capture. The qualitative picture of a rare, short-period-weighted population detected at a few percent survives; the headline magnitudes are conditional on an untested mass-threshold calibration.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper presents the first selection-corrected inference of the intrinsic demographics of the 27 Gaia-detected neutron star (NS) candidates in au-scale orbits with solar-type companions (El-Badry et al. 2024b, 2026). The authors characterize the Gaia DR3 selection function with a 140,000-injection gaiamock forward-model campaign that replicates the astrometric NSS pipeline plus additional NS-candidate cuts, and fit the 27-object sample with an inhomogeneous Poisson hierarchical Bayesian model (Eqs. 2-4). They infer a decreasing orbital-period power law (alpha_P = -0.75 ± 0.35), an NS mass distribution peaked near 1.30 M_sun (sigma ≈ 0.18), a visible-star mass peak near 0.85 M_sun, an eccentricity distribution centered near e ≈ 0.54, and a normalization of N_2kpc ≈ 900 (n ≈ 120 kpc^-3), implying the sample is ~3% complete. They then compare to the au-scale WD-MS population (Yamaguchi et al. 2026), forecast Gaia DR4 yields of ~300-650 systems under four long-period assumptions, and place the Gaia NS birth rate in a broader synthesis of Galactic NS populations (Section 5, Table 3).","tokens_in":54002,"tokens_out":8343,"duration_ms":50206,"significance":"If the normalization is correct, this is a high-impact result: the inference that Gaia has detected only ~3% of the local Gaia NS population, with ~900 systems within 2 kpc and an order-of-magnitude DR4 forecast, would reframe both formation-channel studies and future survey planning. The methodological core is sound: the importance-reweighted Poisson likelihood (Eqs. 3-4) follows established selection-corrected population inference, the posterior predictive check reproduces N=27 (mu = 26.6^+5.5_-4.9), the injection campaign is large and public, and the authors honestly state their main limitations (the astrometric-vs-joint mass-cut mismatch, the heterogeneous parent samples, and the WD/ultramassive-WD contamination). The Section 5 synthesis (birth rates/lifetimes across isolated NSs, HMXBs, LMXBs, spiders, PSR+WD, DNS, LVK mergers) is a useful, clearly caveated compilation.","major_comments":[{"comment":"The normalization results — N_2kpc ≈ 900, n ≈ 120 kpc^-3, the '~97% undiscovered' claim, the DR4 yield, and the Section 5 birth rates — all scale inversely with the modeled selection fraction beta_eff (Eq. 3; Fig. 4 shows N_2kpc × beta_eff ≈ 27). The paper acknowledges (Section 2.2) that the M_NS > 1.25 cut was applied to astrometrically inferred masses in the injection campaign but to joint Gaia+RV masses for the published sample, and Section 2.1 states that the astrometric mass is a lower limit. The direction of the resulting bias is therefore definite: the injection cut is stricter than the real cut, so beta_eff = 0.029 is biased low and N_2kpc and n_0 are biased high. Because the intrinsic M_NS distribution rises steeply toward the 1.25 M_sun boundary (mu ≈ 1.30, sigma ≈ 0.18; Section 3.1), even a 0.05-0.1 M_sun offset between astrometric and joint masses can shift the mass-cut fraction by more than the quoted statistical error on beta_eff (0.007-0.009). This systematic is not reflected in any of the reported 1-sigma intervals. Please add a quantitative sensitivity analysis (e.g., recompute beta_eff and the derived quantities under an effective joint-mass threshold shifted by 0.05-0.1 M_sun, or re-run the injection with joint-mass cuts) and state the resulting systematic on N_2kpc, n, the discovery fraction, and the DR4 forecast.","section":"Sections 2.2 and 3.2","authors_acknowledge":"In addition, the published MNS >1.25 M⊙ threshold was applied to the final joint Gaia+RV companion masses, whereas our synthetic systems only consider astrometrically inferred masses.","authors_acknowledge_p":0.64},{"comment":"The headline decreasing period slope alpha_P = -0.75 ± 0.35 is, per Section 3.1, 'mainly driven by the fact that 3 observed Gaia NSs reside at P < 400 d despite poor sensitivity there.' Appendix C's robustness check is in direct tension with the paper's characterization of this as a robust result: the M_NS > 1.35 subset (19 systems, the cleanest with respect to ultramassive-WD contamination) gives alpha_P = -0.13 ± 0.51 (fiducial) and -0.37 ± 0.26 (broken power law), both consistent with a flat distribution. This conflicts with Appendix B's statement that 'the declining period distribution' is treated 'as a robust result.' Since alpha_P feeds the DR4 forecast (Eq. 13) and the WD-MS comparison (alpha_P ≈ -0.445; Section 3.3), the claim should be reframed in the abstract and conclusions (e.g., as favored but not established, with the cleanest subsample consistent with a flat slope), and the Appendix C tension should be discussed explicitly in the main text rather than only in the appendix.","section":"Section 3.1, Appendix B, Appendix C","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"There is a typo in the table header: 'T able 1' should read 'Table 1'.","section":"Page 5, Table 1 header","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The measurement-uncertainty Monte Carlo uses 96 fixed samples per object drawn once and held fixed through the MCMC; given the small quoted uncertainties this is likely adequate, but a brief statement on the convergence of this fixed-draw choice (and its effect on the posterior widths) would be helpful.","section":"Section 2.3","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The phrase '1 in every 103 – 104 NSs' renders the superscripted exponents ambiguously; please write 10^3 - 10^4 in running text.","section":"Section 5.1.2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'The quoted uncertainties on DR4 yields include both uncertainties from the posteriors and in the selection-function model' does not specify what the selection-function-model uncertainty is; clarify whether this is the injection Poisson uncertainty or an estimate that includes the mass-cut systematic discussed in Section 2.2.","section":"Section 3.4","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The reliance on tools developed within the same group (gaiamock; the Yamaguchi et al. 2026 hierarchical framework) is not, in my view, a circularity problem: both are independently built, published (or preprinted) methods applied here to a distinct sample. The blocking issues are the two load-bearing systematics above. The mass-cut mismatch has a definite direction that inflates the inferred normalization, and the alpha_P robustness claim is internally contradicted by the paper's own Appendix C. Both are fixable within the manuscript's scope through a sensitivity analysis and a reframing of the period-slope claim; I would not accept before those are addressed. The Section 5 synthesis is a strength and should survive the revision largely unchanged."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Two things to know before you read this. First, it's the first intrinsic, selection-corrected census of the Gaia neutron star population — a real new result, not a repackaging. Second, the headline numbers ride on the selection function in ways the authors partly flag but don't fully resolve: the ~120 kpc^-3 density (and the 3% discovered fraction that comes with it) scales inversely with the modeled detection fraction beta_eff, and the -0.75 period slope is driven by three systems at P<400 days in a low-sensitivity region.\n\nWhat it does well. The machinery — gaiamock forward model plus inhomogeneous Poisson hierarchical inference — is established and independently published; applying it to the 27-candidate NS sample is a clean, legitimate extension, and the heavy self-citation is tool citation, not circularity. The posterior predictive check reproduces N=27 (mu=26.6), the alternative parameterizations give consistent selected counts, and the authors explicitly test the M_NS>1.35 subset. The birth-rate synthesis (Table 3, Figure 8) is broad, honestly bracketed, and labelled as plausible ranges rather than measurements. The WD-MS comparison is a genuinely useful addition.\n\nWhere it's soft. The stress-test concern lands. The injection campaign applies the M_NS>1.25 cut to astrometric masses, which are lower limits, while the published sample used joint Gaia+RV masses. That makes the injection cut stricter than the real one, biasing beta_eff low and N_2kpc/n0 high. The authors acknowledge the mismatch in Section 2.2 [b103] and call the selection function a best approximation, but the acknowledgment doesn't bound the bias. Since the inferred mass distribution rises steeply toward the 1.25 M_sun boundary (mu=1.30, sigma=0.18), even a 0.05–0.1 M_sun offset moves the fraction passing the mass cut by more than the quoted statistical error. So N≈900, n≈120, the ~97% undiscovered claim, and the DR4 forecast all carry a systematic the 1-sigma intervals don't capture. The period slope is honest but fragile: it shifts to -0.13 to -0.37 for the high-mass subset (Appendix C), and the separable model misses the observed P-MNS correlation (rho=0.58). The DR4 yield is dominated by the unconstrained long-period tail, which they acknowledge.\n\nBottom line: the qualitative picture — a rare, short-period-weighted population detected at a few percent — survives. The quantitative normalization is conditional on an untested mass-threshold calibration. That's a revision request, not a rejection.\n\nFor anyone working on compact-object binaries, NS demographics, or Gaia astrometry selection, this is the reference. Worth a serious referee; my recommendation is to accept for peer review, conditional on tightening the mass-cut calibration and presenting the systematic on beta_eff explicitly.","headline":"First intrinsic census of the Gaia NS population — real new result with honest execution, but the density, birth-rate, and DR4-yield normalizations ride on a selection-function mass-cut calibration the quoted uncertainties don't cover.","tokens_in":54686,"tokens_out":2573,"would_cite":true,"duration_ms":17958,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"Only ~3 percent of the neutron stars in wide Sun-like binaries near Earth have been found: modeling Gaia's astrometric selection, this paper infers about 900 such systems within 2 kpc and forecasts 300–650 detections in Gaia DR4.","keywords":["Gaia","neutron stars","binary stars","astrometry","selection function","hierarchical Bayesian inference","space density","birth rates"],"falsifier":"Run the same selection-corrected machinery on Gaia DR4: whether the actual yield lands in the forecast 300–650 range, and how many candidates appear below 400 days and beyond 1000 days, directly tests the modeled detection fraction $\\beta_{\\rm eff} \\approx 0.029$, since the inferred population size is pinned by $N_{2\\,{\\rm kpc}} \\times \\beta_{\\rm eff} \\approx 27$. A targeted radial-velocity survey of the predicted $\\sim$870 undiscovered systems within 2 kpc would independently settle the space density.","tokens_in":55170,"feed_emoji":"🛰️","tokens_out":14783,"duration_ms":162136,"temperature":0.7,"pith_summary":"The paper tries to establish what the Gaia neutron stars — Sun-like stars in roughly Earth-orbit-sized (about one astronomical unit) binaries with invisible companions, presumably neutron stars — are like once Gaia's detection bias is removed. The 27 candidates found in Gaia DR3 are a strongly biased sample, so the authors forward-model Gaia's astrometry pipeline and fold the resulting selection function into a hierarchical Bayesian fit to infer the intrinsic population. They conclude that within 2 kpc there are about 900 such systems, a space density near $120~\\mathrm{kpc}^{-3}$, only $\\sim$3% of which have been discovered. The inferred population favors short orbital periods (power-law slope $\\alpha_P = -0.75 \\pm 0.35$), neutron star masses peaking near $1.3~M_\\odot$ with a high-mass tail, companions of about $0.85~M_\\odot$, and moderate-to-large eccentricities. If this is right, Gaia's fourth data release should multiply the known sample to roughly 300–650 candidates and open the long-period tail that distinguishes formation scenarios.","feed_headline":"Gaia finds only 3% of nearby neutron-star binaries","feed_subtitle":"Selection-corrected census predicts ~900 systems within 2 kpc and 300–650 new detections in Gaia DR4.","key_machinery":"The load-bearing object is gaiamock, a forward model that simulates Gaia epoch astrometry with the real scanning law and replicates the DR3 non-single-star pipeline to decide whether a synthetic binary would be detected and pass the neutron-star candidate cuts. A campaign of 140,000 injected binaries run through that pipeline defines the per-system detection probability, and an importance-reweighting estimator converts those injections into the selection fraction $\\beta(\\Lambda)$ that enters a hierarchical Bayesian Poisson model over orbital period, visible-star mass, neutron-star mass, and eccentricity. The identity $N_{2\\,{\\rm kpc}} \\times \\beta_{\\rm eff} \\approx 27$ ties the inferred population size to the 27 observed systems, so the machinery transforms 'what Gaia happened to see' into 'what is actually out there,' with the selection model as the quantity that carries the entire argument.","core_discovery":"The central claim is that the 27 Gaia DR3 neutron-star candidates are the bright tip of a larger intrinsic population whose properties differ from what the raw sample shows. After running 140,000 injected binaries through the modeled astrometric selection and fitting a selection-corrected hierarchical Bayesian model, the paper finds an intrinsic orbital-period distribution that declines toward long periods with slope $\\alpha_P = -0.75 \\pm 0.35$; visible companions peaking near $M_1 \\approx 0.85~M_\\odot$; neutron-star masses peaking near $M_{\\rm NS} \\approx 1.3~M_\\odot$ with a high-mass tail; and eccentricities clustered near 0.5 rather than near zero. The same fit yields an intrinsic count $N \\approx 900$ within 2 kpc, a local space density $n \\approx 120~\\mathrm{kpc}^{-3}$, a discovery fraction of only $\\sim$3%, and — extrapolating the period law to the longer DR4 baseline — a forecast of 300–650 new detections. Placing this in context, the authors estimate that only 0.01–1% of neutron stars are born into these wide, Sun-companion orbits, giving a birth rate 10–100 times below that of high-mass X-ray binaries and broadly comparable to symbiotic X-ray binaries.","pith_inferences":["My inference: the forecast short-period excess rests on only three systems detected at $P_{\\rm orb} < 400$ d in the least sensitive part of the selection function, so Gaia DR4's actual short-period yield is the sharpest available test of the $\\alpha_P = -0.75$ slope.","My inference: since the model pins the population by $N_{2\\,{\\rm kpc}} \\times \\beta_{\\rm eff} \\approx 27$, any independent calibration of the detection efficiency — the realized DR4 count, or a targeted survey of predicted systems — rescales the space density and birth rate in direct proportion.","My inference: applying the same selection-corrected machinery uniformly to Gaia black holes, white-dwarf binaries, and neutron stars would yield a self-consistent census of dark companions at au-scale separations, with the WD–MS comparison already begun here.","My inference: the residual $P_{\\rm orb}$–$M_{\\rm NS}$ correlation ($\\rho_S = 0.58$) that the separable model cannot reproduce hints that mass and period are physically linked, for example through kick-dependent binary survival, and the several-hundred-object DR4 sample is the natural place to test it."],"forward_implications":["The roughly 900 systems inferred within 2 kpc imply that about 870 Gaia NS binaries still await discovery in the solar neighborhood, detectable through astrometric orbital solutions and follow-up radial velocities.","Gaia DR4 is forecast to enlarge the sample by an order of magnitude, to 300–650 candidates, most at periods beyond 1000 days, giving direct access to the long-period tail that tests binary-evolution and natal-kick models.","Because these neutron stars have not accreted, their selection-corrected mass distribution (peak near $1.3~M_\\odot$, tail above $1.6~M_\\odot$) approximates the NS birth-mass distribution, linking the population to core-collapse and kick physics.","Gaia NSs are about 250 times rarer than au-scale WD–MS binaries while sharing a declining period distribution and near-solar-mass companions, so the two populations together separate the effects of ordinary mass transfer from those of a supernova.","Only 0.01–1% of neutron stars are born into these wide orbits, and the inferred Gaia NS birth rate is 10–100 times below that of high-mass X-ray binaries and comparable to symbiotic X-ray binaries, onto which Gaia NSs should eventually evolve."],"supporting_citations":[{"why":"Supplies gaiamock, the forward model of Gaia epoch astrometry and the DR3 astrometric-orbit pipeline through which all injected binaries are filtered.","marker":"El-Badry et al. 2024a"},{"why":"Provides the original Gaia NS candidate sample and the selection cuts (G < 15, joint Gaia+RV companion masses) that define the observed dataset.","marker":"El-Badry et al. 2024b"},{"why":"Extends the candidate catalog to the full 27 systems, including the longer-period and lower-mass companions that motivated the additional injections.","marker":"El-Badry et al. 2026"},{"why":"Supplies the selection-corrected hierarchical Bayesian likelihood with importance-reweighted detection fractions that the inference is built on.","marker":"Essick et al. 2025"},{"why":"Provides the hierarchical Bayesian implementation being adapted, and the WD–MS binary demographics used as the direct comparison population.","marker":"Yamaguchi et al. 2026"},{"why":"Defines the astrometric mass-ratio function boundary and Thiele–Innes precision cuts that decide whether an injected system passes the NS candidate selection.","marker":"Shahaf et al. 2024"},{"why":"Sets the DR4 forecasting procedure (same-cuts versus less-stringent definitions) that the DR4 yield forecast follows.","marker":"Nagarajan et al. 2025"},{"why":"Provides the empirically inferred NS birth-mass distribution that the bias-corrected Gaia NS masses are compared against.","marker":"You et al. 2025"}],"fun_headline_variants":["Gaia census: 27 neutron-star finds are just 3% of hidden population","Most Gaia neutron stars remain hidden: 900 predicted within 2 kpc","Gaia sees 3% of neutron-star binaries, forecast 300-650 more","Rare birth: only 1% of neutron stars form in Sun-like orbits","Gaia's selection bias masks 97% of nearby neutron-star binaries"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that gaiamock's modeled detection fraction, $\\beta_{\\rm eff} \\approx 0.029$, faithfully represents how Gaia DR3 actually selected the 27 real candidates; the inferred $\\sim$900 systems, the $\\sim$120 $\\mathrm{kpc}^{-3}$ density, and all derived birth rates scale inversely with this number, and the authors themselves call the selection function a 'best approximation' because the real sample's mass threshold used joint Gaia+RV masses while the injections used astrometric masses.","fun_headline_variants_meta":{"raw":{"variants":["Gaia census: 27 neutron-star finds are just 3% of hidden population","Most Gaia neutron stars remain hidden: 900 predicted within 2 kpc","Gaia sees 3% of neutron-star binaries, forecast 300-650 more","Rare birth: only 1% of neutron stars form in Sun-like orbits","Gaia's selection bias masks 97% of nearby neutron-star binaries"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"cost":0.000224028,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000224028,"upstream_inference_prompt_cost":0.000140588,"upstream_inference_completions_cost":8.344e-05},"total_tokens":1507,"prompt_tokens":1209,"completion_tokens":298,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":206}},"tokens_in":1209,"tokens_out":298,"cost_source":"openrouter_usage","duration_ms":2621,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":206,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-03T03:23:26.274315+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Run the same selection-corrected machinery on Gaia DR4: whether the actual yield lands in the forecast 300–650 range, and how many candidates appear below 400 days and beyond 1000 days, directly tests the modeled detection fraction $\\beta_{\\rm eff} \\approx 0.029$, since the inferred population size is pinned by $N_{2\\,{\\rm kpc}} \\times \\beta_{\\rm eff} \\approx 27$. A targeted radial-velocity survey of the predicted $\\sim$870 undiscovered systems within 2 kpc would independently settle the space density.","supporting_citations":[],"review_version":1}