{"id":"15d72743-e2e5-4efc-9450-1b4dc2c92a52","arxiv_id":"2508.19719","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Fitting S2's orbit with a Schwarzschild-constrained neural network yields Λ ≤ 5.67×10^-40 m^-2, but the quoted bound is a residual of the fit rather than an independent measurement.","lead":"This paper uses a physics-informed neural network to fit the orbit of the S2 star around the Milky Way's central black hole and derives an upper limit on the cosmological constant from the small difference between the fitted precession and the general-relativistic prediction. The claimed bound is two orders of magnitude tighter than a previous machine-learning estimate, but it is still far weaker than cosmological probes such as Planck.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Residual δφReg−δφSP is a self-consistency error of the PINN, not an independent Λ signal; a zero-Λ mock test is needed.","rationale":"The reader's weakest assumption correctly identifies the circularity: the PINN is trained with a physics loss that enforces Schwarzschild dynamics, so δφReg is not an independent measurement of total precession; the residual δφReg−δφSP is a self-consistency error. This is the most load-bearing concern because the entire Λ constraint rests on Eq. (10) and the interpretation of this residual. No part of the analysis includes Λ in the governing equations, so the PINN cannot represent a Λ-induced precession; any such effect would be absorbed into the fitted orbital elements or the residual. The other issues (missing extended mass, unexplained σReg, unavailable data) are secondary. A zero-Λ mock test would directly settle whether the method produces a spurious positive bound, which is the core of the claim. The paper is honest about S1/S9 limitations, but the S2 result inherits the same circularity. Therefore I agree with the reader's REJECT verdict; no adjustment is needed.","tokens_in":12385,"tokens_out":6495,"duration_ms":69338,"concrete_test":"Generate mock S2 astrometric data from a pure Schwarzschild orbit (Λ=0) using the same observation epochs, noise level, and phase sampling as the real Gillessen et al. (2017) data. Run the full iPINN pipeline exactly as in §§3–4, including the w-selection procedure and the δφReg−δφSP residual extraction. If the pipeline yields a positive upper bound on Λ of order 10⁻⁴⁰ m⁻² (or any nonzero bound of comparable magnitude) from the Λ=0 mocks, the residual is dominated by noise and model self-consistency rather than by Λ, and the central claim is unsupported. Repeat for multiple noise realizations to assess the distribution of derived bounds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline bound (eq. 21) follows from Eq. (10), δφReg = δφSP + δφΛ, where δφReg is extracted from the PINN's predicted orbit and δφSP from the inferred semi-latus rectum p. However, the PINN is trained with a physics loss (eq. 20) enforcing the Schwarzschild equations (5)–(7), which contain no Λ term. The network is therefore strongly constrained to produce a Schwarzschild-like orbit; the periapsis shift in the fit is essentially the Schwarzschild precession for the fitted parameters. Hence δφReg and δφSP are not independent: δφReg is derived from the same physics-constrained model that produced the fitted p. The residual δφReg−δφSP is a measure of the physics-loss residual (model inconsistency), not a cosmological signal. Its small value (≈6×10⁻⁴ arcmin) is within the reported σReg (4.1×10⁻⁴ arcmin), and σReg itself is unexplained. Other precession sources (extended mass distribution, black-hole spin) are ignored. Without demonstrating that the pipeline recovers zero Λ on mock Schwarzschild data, the residual cannot be interpreted as a Λ-induced precession.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies a physics-informed neural network (iPINN) to astrometric data of the S-stars around Sgr A*, focusing on S2, and infers the orbital elements and the total periapsis precession δφReg. The Schwarzschild precession δφSP is computed from the inferred semi-latus rectum, and the difference δφReg − δφSP is attributed to the cosmological precession. Using the Kerr et al. (2003) formula, the paper derives an upper bound Λ ≤ 5.67×10^-40 m^-2, about two orders of magnitude tighter than the previous PINN-based estimate of Galikyan et al. (2023). The same procedure is applied to S1 and S9, yielding numerically tighter but less reliable bounds; the authors conclude that the S2 constraint is the most robust.","tokens_in":12645,"tokens_out":11477,"duration_ms":127325,"significance":"If the derivation were valid, the paper would present a novel, local dynamical constraint on the cosmological constant and demonstrate an interesting use of PINNs for extracting relativistic precession from sparse astrometric data. The exploration of the loss-weight hyperparameter and the honest discussion of partial-orbit coverage for S1/S9 are useful contributions, and the sub-percent recovery of the S2 orbital elements is a positive result for the machine-learning approach. However, the central constraint is not supported: the residual used to isolate Λ is an internal self-consistency error of a Schwarzschild-constrained fit rather than a measured non-Schwarzschild precession, no mock-injection or zero-Λ validation is provided, the quoted uncertainties have no documented derivation, and important astrophysical precession sources are omitted. The headline bound therefore cannot be accepted as a physical constraint.","major_comments":[{"comment":"The decomposition δφReg = δφSP + δφΛ is circular as implemented. The PINN output u(φ) is trained by minimizing Eq. (20), whose physics loss enforces the Schwarzschild equations (5)–(7), which contain no Λ term. The 'total precession' δφReg is read off from that same fitted orbit, while δφSP is computed from the same inferred p via Eq. (9). To the extent the physics loss is minimized, δφReg is driven toward the Schwarzschild precession for the fitted (e, p), so δφReg − δφSP is a self-consistency error of the fit, not an independent measurement of a non-Schwarzschild precession. The reported residual ≈6×10^-4 arcmin is within 1.5σ of σReg = 4.1×10^-4 arcmin and is statistically indistinguishable from zero. The paper needs a mock-injection test, including Λ = 0, demonstrating that the residual tracks an injected cosmological precession; no such validation is presented.","section":"§2.3, Eq. (10); §3.2, Eq. (20); §4.2"},{"comment":"The uncertainties σReg and σSP are not derived anywhere in the manuscript. No ensemble size, bootstrap procedure, or error-propagation formula is given. In particular, p is an inferred iPINN parameter, but no uncertainty on p is reported, so the formal σSP ≈ 1.6×10^-9 arcmin appears to assume a delta-function p. The 3σ upper-limit construction in §4.2 therefore lacks a validated statistical basis. The authors should specify exactly how σReg was obtained (e.g., from retraining with different seeds), how the Gaussian assumption is justified, and how hyperparameter choice enters the quoted uncertainty.","section":"§4.2, Table 1"},{"comment":"Equation (10) omits known physical contributions to the S2 periastron shift. An extended mass distribution in the Galactic center and black-hole spin (Lense–Thirring) produce precessions that are not included in the model. The paper cites Galikyan et al. (2024) for a cluster-density constraint but does not incorporate such a term. At the claimed precision of ~10^-4 arcmin, a quantitative bound on these contaminants is required before the residual can be attributed to Λ. Without it, the extracted δφΛ is degenerate with unmodeled Newtonian/post-Newtonian effects.","section":"§2.3, Eq. (10); §4.2"}],"minor_comments":[{"comment":"The text states that inferred orbital parameters agree with observational values to within 0.3%, but for p the deviation is (228 − 225.25)/228 ≈ 1.2%. Please correct this statement.","section":"§4.2, Table 1"},{"comment":"The cosmological precession formula is typeset ambiguously. Please write it as a clear fraction, verify dimensional consistency, and confirm the sign/factor against Kerr et al. (2003), since a factor error here would directly rescale the bound.","section":"Eq. (12)"},{"comment":"The variable φ is defined as the true anomaly in §2.1 but is later called the 'declination of the elliptical orbit' in §3.2. Please standardize the notation.","section":"§3.1–§3.2"},{"comment":"The text says w = 10^-3 is adopted for both S1 and S9, while Fig. 7 reports optimal w values that minimize LossPhys. Please clarify which w values were actually used and how they were selected.","section":"§4.3, Fig. 7"},{"comment":"The loss curves are said to be normalized by their values in the first epoch. This normalization is not defined in the captions or in the main text, making it difficult to interpret the minima. Please specify the procedure and the number of training epochs.","section":"Figs. 3 and 4"},{"comment":"The data-availability statement says data will be shared on reasonable request, but no statement is made about code availability. Given the central role of the PINN implementation, releasing code would substantially improve reproducibility.","section":"Data Availability"}],"recommendation":"reject","confidential_remarks":"The central result is not supported as it stands, and the required fix is not a small revision: the Λ term must enter the equations used in the physics loss, and the inference must be validated on mock data with known injected Λ (including zero). The S1/S9 discussion is honest but those constraints are already acknowledged to be unreliable. I recommend rejection; a future version with a Λ-inclusive physics loss, mock tests, and a real error analysis could be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline number—Λ ≤ 5.67×10⁻⁴⁰ m⁻²—is probably not a real constraint on the cosmological constant. The paper builds on Galikyan et al. (2023) and improves the bound two orders of magnitude by tuning the loss weight w, but the logic that converts the fitted orbit into a Λ measurement has a circular step. The PINN is trained with a physics loss that enforces the Schwarzschild equations (5)–(7). So the inferred periapsis shift δφReg and the Schwarzschild precession δφSP, computed from the same fitted semi-latus rectum p, are not independent. Their difference is a self-consistency error of the fit, not a cosmological signal. The reported residual of ~6×10⁻⁴ arcmin is within the quoted (unexplained) σReg of 4.1×10⁻⁴ arcmin, and no mock test with Λ=0 is shown. Other precession sources (extended stellar mass, black hole spin) are not modeled. These are not minor caveats; they break the interpretation of eq. (21).\n\nThat said, the paper has real merits. The PINN recovers S2's orbital parameters e and p to sub-percent accuracy, the loss-weight survey is systematic, and the authors are honest about the S1/S9 results being unreliable due to limited phase coverage. They even warn that tighter bounds from S1/S9 should not be taken at face value. That is clear thinking and good scientific hygiene.\n\nThe work is not useless: it demonstrates that PINNs can reconstruct a precessing orbit from sparse astrometric data, and it highlights the danger of subtracting two outputs of the same fitted model. But as a constraint on Λ, it is not competitive with Planck+BAO by tens of orders of magnitude, and the current derivation does not establish an independent measurement.\n\nWho should read it? Anyone applying PINNs to orbital dynamics, and anyone interested in inference pitfalls in ML-based astronomy. It would be a good reading-group case study. My recommendation: if it comes to your desk, send it to peer review only with the explicit requirement of a zero-Λ injection test (fit a mock Schwarzschild orbit and show the pipeline yields δφΛ consistent with zero) and a derivation of σReg. Without that, the central claim remains unsupported. I would not cite the bound.","headline":"Likely a fit residual, not a Λ measurement—honest PINN work that needs a zero-Λ mock test before the bound can be taken seriously.","tokens_in":13182,"tokens_out":3502,"would_cite":false,"duration_ms":38520,"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":"A physics-informed neural network trained on S2's astrometry infers the orbit's total precession, subtracts the Schwarzschild contribution, and attributes the residual to the cosmological constant, obtaining Λ ≤ 5.67 × 10^-40 m^-2.","keywords":["cosmological constant","stellar orbits","Sgr A*","orbital precession","Schwarzschild precession","physics-informed neural networks","inverse PINN","Galactic Center"],"falsifier":"Fit the same S2 astrometric data with a conventional orbit model that adds the extended cluster mass and black-hole spin as free parameters alongside Λ; if the best-fitting Λ is inconsistent with zero or with the quoted 5.67 × 10^-40 m^-2 upper bound at the 3σ level, the residual is not measuring Λ. A second independent check is to recompute δφ_Reg with a network ensemble and a bootstrap of the data: if the scatter between realizations exceeds the quoted σ_Reg, the bound is an artifact of a single trained network.","tokens_in":12209,"feed_emoji":"🕳️","tokens_out":10018,"duration_ms":94533,"temperature":0.7,"pith_summary":"The paper sets out to show that the cosmological constant Λ can be constrained from the orbit of a single star near the Milky Way's central black hole, without relying on cosmology-wide data. Using 25 years of S2 astrometric data, a physics-informed neural network infers the orbital parameters and the total precession of the orbit, then subtracts the known Schwarzschild precession. The leftover is assigned to the cosmological precession caused by Λ, yielding an upper bound Λ ≤ 5.67 × 10^-40 m^-2, roughly two orders of magnitude tighter than the previous machine-learning-based estimate. The paper argues the bound holds for S2 because S2 is the only star in the sample with a fully observed orbit; extending the same analysis to two longer-period stars, S1 and S9, gives numerically tighter bounds, but the paper concludes those are unreliable because only part of each orbit has been observed.","feed_headline":"The S2 star's orbit caps Λ at 5.7e-40","feed_subtitle":"Neural net analysis of S2's orbit tightens the cosmological constant bound by ~100×.","key_machinery":"Key machinery is the inverse physics-informed neural network (iPINN). A fully connected network maps the orbital phase angle φ to the inverse radius u = 1/r, trained by minimizing a weighted sum of a regression loss against astrometric data and a physics loss enforcing the Schwarzschild-metric equations in Darwin variables (eqs. 5–7). The physical model's unknown parameters e and p are optimized along with the network weights, acting as an inverse solver. The trained network is then evaluated over two full orbital periods, the two minima of u(φ) locate successive periapsis passages, and their angular separation defines the total precession δφ_Reg. Subtracting the analytic Schwarzschild prece","core_discovery":"The central discovery the paper argues for is that stellar orbits around Sgr A* can act as a local probe of the cosmological constant. With the inverse PINN framework, the authors recover the eccentricity and semi-latus rectum of the S2 orbit to within about 0.3% of the values reported from direct orbital fits, and read off a total relativistic precession of about 12.149 arcminutes per orbit. Subtracting the Schwarzschild prediction leaves a tiny residual, which the analytic formula for cosmological precession converts into the upper bound Λ ≤ 5.67 × 10^-40 m^-2. The authors present this as roughly a hundredfold tightening of the previous PINN-based limit, and argue that the S2-based constra","pith_inferences":["Because the residual is attributed to Λ after assuming no other precession sources, the number can also be read as an upper bound on any unmodeled precession such as an extended stellar cluster, black-hole spin, or modified gravity; a dedicated multi-parameter fit with those terms free would be needed to tell which interpretation survives.","The subtraction is not fully independent: δφ_Reg and δφ_SP both come from the same trained network and the same fitted e and p, so the quoted error bars on δφ_Λ likely understate systematic correlations. A bootstrap or ensemble-of-networks estimate would test this.","A similar residual analysis applied to future stars with complete orbits should produce a sequence of bounds that scale with the semimajor axis and eccentricity as the cosmological-precession formula predicts; if they do not, the attribution of the residual to Λ is suspect.","The apparently tighter S1 and S9 bounds provide a cautionary template: any machine-learning constraint whose nominal precision improves with less complete data should be re-examined for overfitting before being quoted."],"forward_implications":["If the bound is right, a single well-observed stellar orbit around a supermassive black hole can constrain Λ to 10^-40 m^-2, a regime between cosmological and Solar System limits that no other single probe covers.","The method's sub-percent recovery of e and p from noisy, unevenly sampled astrometry suggests PINN-based inference can extract dynamical parameters from sparse data, which is directly relevant for future S-star monitoring.","The S1 and S9 tests imply that apparently tighter bounds from longer-period stars should not be trusted until their orbits are phase-complete; full orbital coverage, not just more data points, is the deciding factor.","Tuning the loss-weight w is identified as a necessary step: the paper finds w = 6.0 × 10^-5 minimizes the physics loss and argues that earlier PINN analyses, by not scanning w, left the inferred bound partly uncontrolled.","The framework extends to future ELT/VLT astrometry to tighten the Λ bound and to test alternative gravity theories by searching for precession residuals beyond Schwarzschild plus Λ."],"supporting_citations":[{"why":"Supplies the analytic cosmological-precession formula used to convert the residual precession into a bound on Λ.","marker":"Kerr et al. (2003)"},{"why":"Provides the 25-year astrometric data set of S-stars and the adopted Sgr A* mass and S2 reference orbital parameters.","marker":"Gillessen et al. (2017)"},{"why":"Supplies the Darwin-variable formulation of Schwarzschild geodesics encoded in the PINN physics loss.","marker":"Chandrasekhar (1998)"},{"why":"Earlier PINN-based analysis of S2 whose Λ bound this paper tightens by about two orders of magnitude.","marker":"Galikyan et al. (2023)"},{"why":"Reports the observed S2 periapsis advance of about 12 arcminutes per orbit that grounds the expected Schwarzschild precession.","marker":"Abuter et al. (2020)"},{"why":"Introduces the physics-informed neural network loss formulation on which the regression-plus-physics training is based.","marker":"Raissi et al. (2017a)"},{"why":"Provides the Thiele-Innes transformation used to project observed RA/Dec positions onto the orbital plane.","marker":"Becerra-Vergara, E. A. et al. (2020)"}],"fun_headline_variants":["S2 orbit caps cosmological constant at 5.7e-40 m^-2","AI tightens Λ bound 100× using S2 star's precession","Star orbit around Sgr A* yields tightest Λ limit yet","Neural network extracts Λ from S2's orbital wobble","S2's precession gives new upper bound on cosmic constant"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the precession the trained network measures is exactly the sum of Schwarzschild and cosmological precession, with no other source—extended stellar mass, black-hole spin, or alternative gravity—contributing, and that the residual after subtracting Schwarzschild is wholly due to Λ.","fun_headline_variants_meta":{"raw":{"variants":["S2 orbit caps cosmological constant at 5.7e-40 m^-2","AI tightens Λ bound 100× using S2 star's precession","Star orbit around Sgr A* yields tightest Λ limit yet","Neural network extracts Λ from S2's orbital wobble","S2's precession gives new upper bound on cosmic constant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000705,"raw_usage":{"total_tokens":3041,"prompt_tokens":794,"completion_tokens":2247,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":2162}},"tokens_in":538,"tokens_out":2247,"duration_ms":16835,"temperature":1.0,"reasoning_tokens":2162,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:32:18.495477+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fit the same S2 astrometric data with a conventional orbit model that adds the extended cluster mass and black-hole spin as free parameters alongside Λ; if the best-fitting Λ is inconsistent with zero or with the quoted 5.67 × 10^-40 m^-2 upper bound at the 3σ level, the residual is not measuring Λ. A second independent check is to recompute δφ_Reg with a network ensemble and a bootstrap of the data: if the scatter between realizations exceeds the quoted σ_Reg, the bound is an artifact of a single trained network.","supporting_citations":[{"cited_title":"W., Hauck J","cited_arxiv_id":null,"evidence_quote":"Supplies the analytic cosmological-precession formula used to convert the residual precession into a bound on Λ."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the 25-year astrometric data set of S-stars and the adopted Sgr A* mass and S2 reference orbital parameters."},{"cited_title":"International series of monographs on physics, Clarendon Press, https://books.google.co.jp/books?id=LBOVcrzFfhsC","cited_arxiv_id":null,"evidence_quote":"Supplies the Darwin-variable formulation of Schwarzschild geodesics encoded in the PINN physics loss."},{"cited_title":"A., Gurzadyan V","cited_arxiv_id":null,"evidence_quote":"Earlier PINN-based analysis of S2 whose Λ bound this paper tightens by about two orders of magnitude."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the observed S2 periapsis advance of about 12 arcminutes per orbit that grounds the expected Schwarzschild precession."}],"review_version":1}