{"id":"1007e400-4203-4fe3-8272-9b5dff7c41b7","arxiv_id":"2412.00303","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors derive a black hole metric with a cosmological-constant-dependent effective mass and use EHT and VLBI data to put extremely weak bounds on a quantum parameter beta.","lead":"The paper combines a curvature-based uncertainty principle with a graviton-condensate model of black holes to produce a Schwarzschild-like metric whose mass depends on the cosmological constant. It then uses black hole shadow and light-bending observations to estimate, very weakly, a quantum gravity parameter.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central metric Eq. (10) is asserted, not derived: the mapping from AGEUP variances σ_x, σ_p to horizon radius and graviton momentum is never specified, and every subsequent observable inherits this gap.","rationale":"The reader's weakest_assumption is exactly the unshown translation from Eq. (7) to Eq. (10), and I agree that this is the most load-bearing concern. The entire paper's novelty is the Λ-dependent effective mass; if this mapping is non-unique or incorrect, the paper reduces to a standard EUP black-hole analysis with an ad hoc mass formula. I also note supporting internal inconsistencies: Eq. (13) with L* = sqrt(3/Λ) implies α = 1/(2π^2), not (3π)^{-1}; Eq. (20) contains an undefined n; and Eq. (42) has dimensions of inverse length rather than inverse time if M is a length, so the Lyapunov exponent is dimensionally suspect. These reinforce rejection but are secondary to the missing central derivation. A conditional acceptance would be conceivable if the authors supplied the missing algebra and corrected the internal errors; however, as written, the central claim is not established, so the reader's REJECT verdict stands unchanged.","tokens_in":19527,"tokens_out":5823,"duration_ms":54468,"concrete_test":"Re-derive Eq. (10) from Eq. (7) with explicit corpuscular identifications: set σ_x = 2M (or 2M_eff), σ_p = 1/(2M), N = R^2/l_Pl^2, and require saturation at the horizon; solve for M_eff and compare with Eq. (10). If no consistent assignment yields M_eff = M − 2ΛM^3/(3π^2) + β l_Pl^2/(2M), the central claim is unsupported. As a calibration, apply the same procedure to the standard EUP (1) with Mureika's method; the known result should be reproduced. This is an algebraic check with no free parameters.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III introduces Eq. (7), an uncertainty bound for variances σ_x, σ_p in dS/AdS, then states 'we determine the M_eff by using the corpuscular framework' and immediately writes M_eff = M − 2ΛM^3/(3π^2) + β l_Pl^2/(2M) as Eq. (10). No algebraic step connects the two. To obtain Eq. (10) one must (i) identify σ_x with a black-hole radius (most plausibly 2M or 2M_eff), (ii) identify σ_p with a graviton momentum (most plausibly 1/(2M) in ℏ=1 units), (iii) impose the corpuscular counting N∼R^2/l_Pl^2, and (iv) choose a saturation convention. The paper specifies none of these choices. The exact coefficients 2/(3π^2) and 1/2 are not derivable from the text, and alternative identifications give different forms; for instance, the β term stems from σ_p^2, and whether σ_p is the individual graviton momentum or the collective momentum changes the mass dependence. Since the metric (9), shadow (15), deflection (18), temperature (32), QNM (45), and strong-lensing coefficients (60)–(64) all inherit M_eff, this is the load-bearing pillar of the paper. The absence of the derivation, not the standard EUP/black-hole formalism that follows, is the decisive defect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript combines the Asymptotic Generalized Extended Uncertainty Principle (AGEUP) with the corpuscular graviton-condensate picture to propose a static, spherically symmetric metric with effective mass M_eff = M - (2ΛM^3)/(3π^2) + β l_Pl^2/(2M). It then applies this metric to black hole shadows, weak and strong deflection angles, Unruh-Hawking thermodynamics, eikonal quasinormal modes, and EHT/VLBI constraints on the quantum parameter β. The central claim is that the cosmological constant enters local black hole mass through the uncertainty relation, rather than only through a global dS/AdS background term.","tokens_in":19808,"tokens_out":10993,"duration_ms":102022,"significance":"If Eq. (10) were derived rather than asserted, the idea that large-scale curvature can enter the black hole mass through an uncertainty relation would be a genuinely interesting phenomenological contribution, and the paper covers a standard set of observational probes in a transparent way. The authors also cite the relevant corpuscular and AGEUP literature and present their numerical bounds explicitly. However, the load-bearing formula is currently introduced without derivation, and several quantitative results in Sections III, IV, and VI are internally inconsistent. These issues prevent the reader from assessing the validity of the claimed predictions, so the significance of the work cannot be established from the manuscript in its present form.","major_comments":[{"comment":"The step from Eq. (7) to Eq. (10) is not shown. The text states 'we determine the M_eff by using the corpuscular framework' and cites Refs. [65-69], but it does not specify how the uncertainty variables σ_x and σ_p are identified with the horizon radius and the graviton momentum, how the corpuscular counting N ~ R^2/l_Pl^2 is imposed, or what saturation convention is used. Without these identifications the coefficient 2/(3π^2) in the Λ-term and the coefficient 1/2 in the β-term cannot be checked. Since Eqs. (15), (18), (32), (45), and (60)-(64) all inherit M_eff, this missing derivation is the load-bearing pillar of the paper.","section":"III, Eq. (10)"},{"comment":"The derived value of α is inconsistent with the preceding formula. If L_* = sqrt(3/Λ), then Eq. (13), L_*^2 = 6π^2 α/Λ, gives 6π^2 α = 3 and hence α = 1/(2π^2) ≈ 0.0507. The paper instead states α = (3π)^(-1) ≈ 0.034. This arithmetic error affects the claimed derivation of the EUP modulation parameter from the cosmological horizon scale.","section":"III, Eq. (13) and following text"},{"comment":"The Lyapunov exponent in Eq. (42) is incorrect. Evaluating Eq. (41) for f(r) = 1 - 2M_eff/r at r_0 = 3M_eff gives λ = 1/(3√3 M_eff), not 1/(3√(2M_eff)). The printed expression has the wrong dimension, and the imaginary part of the eikonal QNM frequency in Eq. (44) should share the same 1/(3√3 M_eff) factor as the real part. This changes the expansion in Eq. (46) and the subsequent interpretation of damping rates.","section":"VI, Eqs. (41)-(42)"},{"comment":"The substitution of the EHT deviations is inconsistent. If δ in Eq. (17) is the shadow-length deviation, as implied by the quoted range -0.364 ≤ δ/M ≤ 0.987, then for Sgr A* the first term gives β ≈ 5.9 × 10^88, not the reported 9.24 × 10^78. If δ is instead taken as δ/M, the first term of Eq. (17) has the wrong dimension. In addition, these 'constraints' attribute the entire observed Schwarzschild deviation to the β-term; they are fits to the model, not predictions, and the choice of only the positive sign in the VLBI comparison in Eq. (20) further weakens the exclusion claim in Section VIII.","section":"IV, Eq. (17) and the numerical bounds"}],"minor_comments":[{"comment":"The regular-integral expression in Eq. (60) contains -6M^2π, while the coefficient k in Eq. (62) contains -6M^2π^2; one of these is a typographical error and the two equations should be made consistent.","section":"VII, Eqs. (60) and (62)"},{"comment":"The quantity n appearing in the second term of Eq. (20) is not defined, which makes the PPN expression difficult to reproduce.","section":"IV, Eq. (20)"},{"comment":"There are several typos and encoding artifacts, including 'Schrdinger' for Schrödinger and 'Godel' for Gödel, and Eq. (14) would be clearer as β_crit = 4ΛM^4/(3π^2 l_Pl^2) rather than β_crit/M^4 = ... .","section":"II and V"},{"comment":"The conclusion refers to a 'weak bound β < 10^120' and suggests the model may be observationally excluded, but the paper actually derives enormous positive β values by fitting deviations; the language should distinguish an upper bound on β from a fitted value under the model.","section":"VIII"}],"recommendation":"major_revision","confidential_remarks":"The central idea is appropriate for the journal, but the current version does not establish its main formula. I am not recommending rejection because the missing derivation could in principle be supplied, and the arithmetic errors in Sections III and IV are correctable. However, if the authors cannot provide a concrete derivation of Eq. (10) from Eq. (7) with the stated coefficients, or if the corrected numbers continue to undermine the observational claims, the paper should not proceed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis one is a reject, and for the reasons the reader gives. The central formula, the effective mass in Eq. (10), is simply asserted. The text says it is obtained 'by using the corpuscular framework' and then writes it down, but nothing in Section III shows how the AGEUP uncertainty relation (7) becomes M_eff = M - 2ΛM^3/(3π^2) + β l_Pl^2/(2M). You have to guess the identifications of σ_x and σ_p with horizon radius and graviton momentum, the graviton counting N∼R^2/l_Pl^2, and a saturation convention. Without those choices, the exact coefficients are unfalsifiable. Since everything else—shadow, deflection, QNMs, lensing—inherits this M_eff, the gap is load-bearing.\n\nWhat the paper does passably: it gives a clear review of AGEUP, and the observation that Mureika's EUP black hole length scale can be identified with the cosmological horizon is a reasonable short step. The shadow and weak-deflection calculations are standard, and the authors are honest that the resulting bounds on β are very weak and that the model might be observationally excluded. That honesty is worth some credit.\n\nThe soft spots are not just the derivation gap. The α value is internally inconsistent: with L* = √(3/Λ), Eq. (13) gives α = 1/(2π²), not the printed 1/(3π) or the ≈0.034. The Lyapunov exponent in Eq. (42) has the wrong dimension (1/√length instead of 1/length). Eq. (20) has an undefined n. And the β constraints from EHT and VLBI are fits to observed deviations, not predictions; with positive β selected by hand. The bounds, β ~ 10^72–10^95, are so weak that the paper's own conclusion questions whether the model is already excluded.\n\nWho gets value from this? Possibly someone cataloguing EUP/GUP black hole metrics, but they would have to redo the central derivation themselves. As is, I would not send it to a referee. The authors should supply the missing derivation and fix the internal errors; then the paper could be worth another look.\n\nRecommendation: desk reject.","headline":"The core effective-mass formula is asserted rather than derived, and internal errors in alpha, the Lyapunov exponent, and an undefined n make the paper unpublishable in its current form.","tokens_in":20453,"tokens_out":5487,"would_cite":false,"duration_ms":47716,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["95.30.Sf","04.70.-s","97.60.Lf","04.50.+h"],"model":"deepseek-v4-flash","headline":"The paper derives a Schwarzschild-like black hole metric whose effective mass is rescaled by the cosmological constant and the Planck length, making the universe's large-scale curvature a participant in local black hole physics.","keywords":["spacetime curvature","uncertainty principle","black holes","cosmological constant","corpuscular gravity","black hole shadow","quasinormal modes","gravitational deflection"],"falsifier":"The decisive check is a first-principles computation of the graviton-condensate ground-state energy in Schwarzschild--de Sitter spacetime: if the $\\Lambda$-dependent term in the effective mass is not $-2\\Lambda M^3/(3\\pi^2)$ with exactly that coefficient, then the metric of Eqs. (9)--(10) is refuted. Observationally, a shadow measurement of Sagittarius A* that pushes the allowed deviation $\\delta/M$ below the existing range of $-0.364$ to $0.987$ would tighten the bound on $\\beta$; if independent arguments force $\\beta$ to be of order one, the model would be excluded.","tokens_in":19263,"feed_emoji":"🕳️","tokens_out":14498,"duration_ms":117833,"temperature":0.7,"pith_summary":"This paper attempts to establish a concrete bridge between the largest scale in physics—the curvature of the universe—and the local quantum structure of a black hole. It does this by feeding the asymptotic generalized extended uncertainty principle (AGEUP), with the cosmological constant $\\Lambda$ built into the position uncertainty, into the corpuscular picture of a black hole as a Bose-Einstein condensate of gravitons. The output is a Schwarzschild-like metric $f(r) = 1 - 2M_{\\mathrm{eff}}/r$ whose effective mass $M_{\\mathrm{eff}} = M - \\frac{2\\Lambda M^3}{3\\pi^2} + \\frac{\\beta l_{\\mathrm{Pl}}^2}{2M}$ contains a $\\Lambda$-dependent term, so the cosmological constant would actively alter the black hole's mass rather than only bending spacetime at large distances. The paper then shows that this mass rescaling propagates into shadow radii, deflection angles, thermodynamics, and quasinormal ringing, and that current shadow and solar-system measurements can constrain the quantum modulation parameter $\\beta$. This matters because it is a concrete proposal in which large-scale curvature and local quantum-gravity corrections leave imprints in the same observable quantity, the black hole mass.","feed_headline":"Cosmological constant rewrites black hole mass in new metric","feed_subtitle":"If correct, large-scale curvature leaves imprints in black hole shadows and ringdown signals.","key_machinery":"The load-bearing machinery is the dS/AdS form of the AGEUP relation, Eq. (7), together with the corpuscular scaling $N \\sim R^2/l_{\\mathrm{Pl}}^2$. The relation supplies a curvature-induced quadratic correction in position uncertainty, while the corpuscular scaling converts uncertainty data into a mass scale: $\\sigma_x$ becomes the black hole radius and $\\sigma_p$ becomes the graviton momentum, turning Eq. (7) into the effective mass $M_{\\mathrm{eff}}$ of Eq. (10). Once $M_{\\mathrm{eff}}$ is defined, the paper routes it through standard formalisms—photon-sphere shadow calculation, finite-distance weak deflection, strong-deflection integrals, holographic-screen temperature, and eikonal quasinormal modes—so that every observable is expressed as a Schwarzschild result plus $\\Lambda$- and $\\beta$-dependent shifts.","core_discovery":"The paper's central claim is that the AGEUP uncertainty relation in de Sitter and anti-de Sitter backgrounds, $\\sigma_p \\sigma_x \\geq \\pi\\hbar\\left(1 - \\frac{\\Lambda}{6\\pi^2}\\sigma_x^2 + \\beta l_{\\mathrm{Pl}}^2 \\sigma_p^2\\right)$, can be combined with the corpuscular relation $N \\sim R^2/l_{\\mathrm{Pl}}^2$ for a graviton-condensate black hole to yield a Schwarzschild-like metric whose mass is rescaled to $M_{\\mathrm{eff}} = M - \\frac{2\\Lambda M^3}{3\\pi^2} + \\frac{\\beta l_{\\mathrm{Pl}}^2}{2M}$. In this construction the uncertainty in position is identified with the horizon radius and the uncertainty in momentum with the graviton momentum, so the curvature term in the uncertainty relation becomes a correction to the black hole mass. From this rescaled mass the paper derives a critical mass $M_{\\mathrm{crit}} = \\frac{\\pi}{2}\\sqrt{6/\\Lambda} \\approx 3.14\\times 10^{26}$ m at which the $\\Lambda$ correction cancels the classical mass; a value $\\alpha = 1/(3\\pi)$ for the EUP modulation factor if the large fundamental length is the cosmological horizon; and constraints on $\\beta$ from shadow radii, photon rings, and parametrized post-Newtonian light bending that range from about $10^{72}$ to $10^{96}$, far above laboratory bounds. The conceptual punch is that the cosmological constant is no longer a passive background but participates in the black hole's local gravitational field.","pith_inferences":["Editorial inference: the exact coefficients in $M_{\\mathrm{eff}}$ are not forced by the uncertainty relation alone; a first-principles derivation of Eq. (10) from the corpuscular Hamiltonian could change the coefficients while preserving the overall structure, so the metric should be regarded as a phenomenological template until that derivation exists.","Editorial inference: because the $\\Lambda$ term grows as $M^3$, the largest supermassive black holes should show the largest fractional mass rescaling; comparing shadow or ringdown observations across a wide mass range would separate the $\\Lambda$ effect from the $\\beta$ effect.","Editorial inference: applying the same AGEUP-plus-corpuscular replacement to rotating or charged black holes would introduce spin-charge degeneracies in the shadow; multi-frequency imaging would be needed to isolate $\\beta$.","Editorial inference: the framework suggests a testable relation between $\\alpha$, $\\Lambda$, and the cosmological horizon scale; an independent EUP experiment at cosmological distances could verify or break the relation $\\alpha = 1/(3\\pi)$."],"forward_implications":["If the central claim is right, the cosmological constant enters the black hole's local quantities: horizon radius, temperature, entropy, and quasinormal-mode frequencies all acquire shifts proportional to $\\Lambda M^2$, so the universe's large-scale curvature is not decoupled from the horizon.","There is a critical mass $M_{\\mathrm{crit}} \\approx 3.14\\times10^{26}$ m at which the $\\Lambda$ correction cancels the classical mass; for smaller masses the quantum term $\\beta l_{\\mathrm{Pl}}^2/(2M)$ can dominate, but the requirement $M_{\\mathrm{eff}}>0$ forces $\\Lambda>0$ and $\\beta>0$ in that regime.","For astrophysical black holes the predicted shadow and deflection shifts are tiny, so the model is consistent with current measurements; the existing shadow and solar-system bounds translate into very large values of $\\beta$ (approximately $10^{72}$ to $10^{96}$), so a positive detection would mean an unexpectedly strong quantum-gravity coupling.","In the strong-deflection limit the deflection coefficients take their Schwarzschild values ($\\bar{a}=1$), with $\\Lambda$ and $\\beta$ entering only through the regular integral and the critical impact parameter, making strong lensing a comparatively weak probe of these corrections.","The derived value $\\alpha = 1/(3\\pi)$ ties the EUP modulation factor to the cosmological horizon; if independently measured, this relation becomes a test of the identification of the large fundamental length with $\\sqrt{3/\\Lambda}$."],"supporting_citations":[{"why":"Supplies the AGEUP uncertainty relation with Ricci-scalar and Cartan-invariant corrections, including the dS/AdS form in Eq. (7) that is the paper's starting point.","marker":"[57]"},{"why":"Introduced the extended-uncertainty-principle black hole metric using the corpuscular framework; the present paper adapts that construction to AGEUP.","marker":"[44]"},{"why":"Provide the corpuscular graviton-condensate picture with N ~ R^2/l_Pl^2, the relation used to convert uncertainty data into the effective mass.","marker":"[65-69]"},{"why":"Gives the exact shadow-radius formula for a static spherically symmetric spacetime used to compute R_sh from M_eff.","marker":"[59]"},{"why":"Supplies the finite-distance Gauss-Bonnet weak deflection angle method used to derive Theta and the beta constraints.","marker":"[60]"},{"why":"Provides the Sgr A* shadow deviation bounds used to set the beta_EHT constraint.","marker":"[72]"},{"why":"Provides the M87* shadow deviation bounds used for the beta constraint.","marker":"[73]"},{"why":"Supplies the VLBI solar-system light-deflection measurement with PPN parameter used for the beta_PPN bound.","marker":"[75]"},{"why":"Provides the strong-deflection angle integration scheme used for the photon-ring constraints.","marker":"[84]"}],"fun_headline_variants":["Cosmological constant rewrites black hole mass in quantum metric","Curved spacetime alters black hole mass via uncertainty principle","Black hole mass gets cosmological correction from new quantum metric","AGEUP links cosmic curvature to black hole quantum gravity","Cosmos-scale uncertainty reshapes black hole mass and shadow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the uncalculated step from Eq. (7), an uncertainty relation in position and momentum, to Eq. (10), a mass formula: one must assume the position uncertainty is the horizon radius and the momentum uncertainty is the graviton momentum, with the exact coefficients presented, yet the paper asserts this corpuscular translation rather than deriving it.","fun_headline_variants_meta":{"raw":{"variants":["Cosmological constant rewrites black hole mass in quantum metric","Curved spacetime alters black hole mass via uncertainty principle","Black hole mass gets cosmological correction from new quantum metric","AGEUP links cosmic curvature to black hole quantum gravity","Cosmos-scale uncertainty reshapes black hole mass and shadow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000442,"raw_usage":{"total_tokens":2369,"prompt_tokens":1207,"completion_tokens":1162,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":823,"completion_tokens_details":{"reasoning_tokens":1083}},"tokens_in":823,"tokens_out":1162,"duration_ms":9412,"temperature":1.0,"reasoning_tokens":1083,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:32:18.171437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The decisive check is a first-principles computation of the graviton-condensate ground-state energy in Schwarzschild--de Sitter spacetime: if the $\\Lambda$-dependent term in the effective mass is not $-2\\Lambda M^3/(3\\pi^2)$ with exactly that coefficient, then the metric of Eqs. (9)--(10) is refuted. Observationally, a shadow measurement of Sagittarius A* that pushes the allowed deviation $\\delta/M$ below the existing range of $-0.364$ to $0.987$ would tighten the bound on $\\beta$; if independent arguments force $\\beta$ to be of order one, the model would be excluded.","supporting_citations":[{"cited_title":"Effects of Extended Uncertainty Principle on the Relativistic Coulomb Potential","cited_arxiv_id":"2008.03807","evidence_quote":"Introduced the extended-uncertainty-principle black hole metric using the corpuscular framework; the present paper adapts that construction to AGEUP."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the VLBI solar-system light-deflection measurement with PPN parameter used for the beta_PPN bound."}],"review_version":1}