{"id":"57ede5f0-f87f-4e8a-90f3-c7bbfbb10f64","arxiv_id":"2610.26692","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a static Palatini Kalb-Ramond black hole with vanishing cosmological constant, all observables fixed by the orbital mass ǞM keep Schwarzschild values, while radial distances, infall times, tidal forces with a σ−2 factor, photon-orbit instability, and axial gravitational quasinormal…","lead":"This paper works out what a black hole from a modified theory of gravity, Palatini Kalb-Ramond gravity, looks like to orbiting particles, light rays, shadows, and waves. It finds that many familiar black hole observables stay identical to Schwarzschild when compared at the same orbital mass, while distances, falling times, tidal stretches, and gravitational wave frequencies carry the theory's signature deformation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The fixed-orbital-mass invariance story rests entirely on the Eq. (10) metric and the Iyer–Wald mass identification, which are imported from the unpublished companion arXiv:2610.05799 and never independently checked; if that solution or charge identification is wrong, the σ-factorization…","rationale":"I read the paper in good faith as a phenomenological study whose core program is to exhibit which observables depend on the residual radial deformation σ after the orbital mass M̄ is held fixed. The geodesic, Doppler, tidal, shadow, and test-field calculations are mostly straightforward consequences of the stated metric, and the WKB versus time-domain agreement is internally consistent. The reader's strongest claim is accurately represented: the paper does show that several Schwarzschild-scale invariants survive at fixed M̄ while distances, times, azimuthal sweeps, tidal strengths, and the axial tensor spectrum scale with σ. The reader's weakest assumption is also the one I find most load-bearing. The paper never re-derives Eq. (10) and its mass normalization; it cites Ref. [30], a companion arXiv preprint, for the exact solution and for the Iyer–Wald mass. Because the entire edifice of fixed-M̄ comparisons is built on that metric, an error in the companion's field-equation solution or charge identification would invalidate not a single section but the organizing claim of the paper. I considered whether to elevate the asserted axial tensor reduction in Sec. IX.C instead; the derivation there is compressed, but the sector is explicitly restricted to δB_{μν} = 0, the auxiliary map (139)–(141) is stated, and the WKB/time-domain consistency of Table XIV supports the internal logic. It is therefore less fragile than the imported background. I also considered the lensing convention issue; it is a genuine inconsistency in how the conical reference is subtracted, but it affects the lensing observables and Fig. 26, not the fixed-M̄ invariance story at the core of the central claim. My recommendation is therefore unchanged from the reader's CONDITIONAL verdict: the paper's internal phenomenology is plausible, but its external anchor in the unpublished companion result needs to be checked or made available before the central claim can be accepted as fully independent.","tokens_in":49797,"tokens_out":4810,"duration_ms":41593,"concrete_test":"Independently substitute the ansatz (10)–(11) into the full Palatini Kalb–Ramond action, solve the independent-connection equation for Γ, and verify that √−h h^{μν} = √−g[(1 + ξ₁B²)g^{μν} + ξ₂B^{μρ}B_ρ^ν] indeed yields the auxiliary Schwarzschild geometry (141) with m_h = √c M̄. Then compute the Iyer–Wald conserved mass M for this solution directly from Wald's Noether-charge formula. If the metric does not satisfy the field equations identically, or if the resulting charge differs from ρM̄, the σ-factorizations in Eqs. (63)–(65), (77)–(80), and (144) lose their basis and the central claim must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that at fixed M̄ = M/ρ, the horizon, photon sphere, critical impact parameter, ISCO, and the tidal and QNM scalings follow from the line element (10) with F(r) = 1 − 2M̄/r, G(r) = σ²/F(r), σ² = 1 − q/2, and ρ = (1 − ℓ₁)σ. Every subsequent invariant, including Eqs. (63)–(65), (77)–(80), and (144), is a direct corollary of this metric rather than of a derivation performed in the paper. The solution, the parameter domain ℓ₁ < 1 and 1 − ℓ₁ − ℓ₂/2 > 0, and the identification of M as the Iyer–Wald conserved mass are all imported from Ref. [30], an arXiv preprint that is not yet a published or refereed source; Ref. [31] supplies only the general Iyer–Wald construction, not the application to this particular solution. If the companion derivation has an error in the field equations or in the charge computation, the factorization into σ and M̄ that organizes the whole paper is no longer grounded. This is not an internal inconsistency, but it is the most load-bearing external anchor of the paper: the abstract's invariance statements and the exact axial spectral map of Eq. (144) are only as secure as an unpublished companion result. The lensing convention mismatch between Eqs. (29)–(30) and Eqs. (97)–(101) is real and should be fixed, but it is confined to the lensing chapter and does not threaten the fixed-M̄ invariants as directly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper analyzes a static, spherically symmetric black hole in Palatini Kalb–Ramond gravity with zero cosmological constant, using the line element of Eq. (10) with F(r)=1−2Mbar/r, G(r)=σ²/F(r), Mbar=M/ρ, and σ²=1−q/2. At fixed orbital mass Mbar, it shows that the horizon, photon sphere, critical impact parameter, ISCO, and radial Doppler ratios coincide with their Schwarzschild values, while proper distances, infall times, azimuthal sweeps, tidal eigenvalues, and photon-orbit instability exponents retain a nontrivial dependence on σ. The paper also computes shadows, weak- and strong-field lensing, and scalar, electromagnetic, Dirac, and homogeneous axial gravitational quasinormal modes, reporting an exact spectral map for the axial gravitational sector, ωhat_T=ωhat_Sch/σ, and a WKB-to-time-domain agreement within 0.7% for the fundamental modes. The shadow comparison with Sgr A* and M87* is presented honestly as a q-degeneracy rather than a fit.","tokens_in":50201,"tokens_out":10399,"duration_ms":306629,"significance":"If the imported solution and mass identification are correct, the paper provides a clean and useful classification of which observables are invariant under the Kalb–Ramond deformation once the orbital mass is fixed and which are not. The work has several concrete strengths: the exact factorization relations are stated explicitly; the axial gravitational spectral map in Eq. (144) is a sharp, testable statement; the WKB calculation is supported by internal order-variation diagnostics, 70-digit working precision, and time-domain extraction with mesh-convergence checks; and no parameters are fitted to the EHT data, with the shadow degeneracy reported rather than hidden. These properties make the central fixed-Mbar statements credible and reproducible, provided the external input from the companion preprint is sound.","major_comments":[{"comment":"The central metric and the identification of M as the Iyer–Wald conserved mass are imported from the unpublished companion preprint Ref. [30]. Every fixed-Mbar invariant in the abstract and in Eqs. (63)–(65), (77)–(80), and (144) is a direct consequence of this imported input rather than of a derivation performed in the present paper. Since Ref. [30] is an arXiv preprint and not yet a refereed source, the correctness risk is load-bearing: if the companion derivation of the solution or of the mass charge is wrong, the σ-factorization that organizes the whole paper is no longer grounded. I recommend either adding an appendix that independently verifies the field equations and the mass identification for Eq. (10), or ensuring that Ref. [30] is available in published or otherwise fully validated form before acceptance.","section":"Sec. III, Sec. V, Sec. VII; Eq. (10)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There is an internal inconsistency in the weak-deflection convention. In Sec. IV.C the exact factorization gives Δφ=σΔφ_Sch, and the conical excess is defined as α_cone=Δφ−σπ=σ α_Sch, whose expansion is α_cone=σ[4Mbar/b+15πMbar²/(4b²)+O(Mbar³/b³)]. In Sec. VIII.A, however, the footnote defines δφ=Δφ−σπ, yet Eq. (97) reports δφ=π(σ−1)+4σMbar/b+15πσMbar²/(4b²). Setting Mbar=0 in Eq. (97) gives a nonzero deflection π(σ−1) in the massless conical spacetime, whereas Eq. (29) gives zero excess. Equation (101) in the Gauss–Bonnet subsection repeats Eq. (97), so the two weak-field methods agree with each other but not with the exact factorization. The π(σ−1) term must be removed, or the reference angle in Sec. VIII must be redefined consistently with Eq. (29).","section":"Sec. VIII.A, Eq. (97) vs. Sec. IV.C, Eqs. (28)–(30)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The strong-field deflection section contradicts the earlier exact result. Combining Eqs. (111) and (112) with the stated total Δφ=Δφ_D+Δφ_R−σπ gives +σ ln(b/b_c−1)+σ ln[216(7−4√3)]−σπ, whereas Eq. (31), obtained from the exact factorization Eq. (29), gives σ[−ln(b/b_c−1)+ln(216(7−4√3))−π]. The logarithmic divergence therefore has the wrong sign in Sec. VIII.C, and the strong-field total deflection is inconsistent with Eq. (31). Since the subsequent lens equation and observables in Eqs. (114)–(124) are built directly on this strong-field expansion, this error propagates into the reported image-separation and magnification quantities and must be corrected.","section":"Sec. VIII.C, Eqs. (111)–(112) vs. Sec. IV.C, Eq. (31)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The citation placeholder \"[?]\" in the strong-deflection expansion should be replaced with the appropriate reference, for example the Bozza or Tsukamoto papers already cited elsewhere.","section":"Eq. (31)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are unresolved equation references written as \"Eq. (??)\" near the definitions of the tortoise coordinate and the Dirac tetrad; these should be fixed.","section":"Sec. IX and Sec. IX.D","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Reference [30] and reference [84] are the same companion preprint and should be merged or cross-referenced consistently.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The text says that positive q places the relevant orbits at larger radii than Schwarzschild while negative q shifts them inward, but at fixed Mbar the photon sphere is q-independent. Please clarify whether this comparison is made at fixed conserved mass M or at fixed Mbar, and identify explicitly which 'orbits' are meant.","section":"Sec. VIII.B.1, Fig. 25","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The Einstein-ring angles for negative q are of order degrees or tens of degrees, which is not naturally a small-angle weak-field result. Since these values arise from the conical reference term π(σ−1), the text should state explicitly that this term is a topological/asymptotic contribution and discuss whether the weak-field lensing approximation remains meaningful in that regime.","section":"Sec. VIII.F, Table VI","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"There are numerous typographical errors, for example \"highlilthed\" in the conclusion, \"as we should experct\" in Sec. X.E, \"constrast\" in Sec. II, and \"In this manenr\" in Sec. V. A careful proofreading pass is needed.","section":"Throughout","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The main external risk is the companion preprint Ref. [30]: the entire fixed-Mbar story, including the exact axial spectral map, rests on a solution and mass identification that are not independently derived in this manuscript. If that preprint remains unpublished, I would view an appendix verifying the solution and charge as necessary. The lensing inconsistencies in Sec. VIII are concrete, clearly fixable, and confined to the lensing chapter, but they are significant enough that the paper should not be accepted in its current form. The geodesic, tidal, shadow, and quasinormal-mode parts of the paper appear internally consistent and well documented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know: this paper's real contribution is a catalogue of which observables of the Palatini Kalb-Ramond black hole retain their Schwarzschild values at fixed orbital mass M̄ = M/ρ and which carry the σ deformation. The organizing facts — horizon, photon sphere, critical impact parameter, and ISCO all at their Schwarzschild radii, while proper distances, infall times, azimuthal sweeps, tidal eigenvalues (σ⁻²), photon-sphere instability exponent (σ⁻¹), and axial gravitational QNMs (exact map ω̂_T = ω̂_Sch/σ) scale with σ — are real and follow cleanly from the metric F = 1 − 2M̄/r with radial coefficient σ²/F.\n\nWhat is genuinely good: the fixed-M̄ normalization is a sensible way to compare theories, and it is applied consistently across geodesics, Doppler, tidal, shadow, and wave sectors. The WKB/time-domain comparison is properly executed: second-order convergence, matrix-pencil extraction diagnostics, sub-0.7% agreement, and the constant 1.3% WKB order-variation in the tensor sector is exactly what the exact σ-scaling predicts. That internal consistency makes the numbers believable. The shadow comparison honestly reports a q-degeneracy rather than a fit, so there is no tuning.\n\nSoft spots, in proportion. The lensing chapter has a real convention inconsistency: Eqs. (29)–(30) define the conical excess without a π(σ−1) term, while Eqs. (97) and (101) reintroduce the σπ subtraction, and the \"negative deflection\" reading in Figure 26 is misleading — the angle is negative only because of the reference subtraction, not because gravity repels. This is confined to the lensing chapter and fixable, but it needs fixing. The axial gravitational spectral map in Sec. IX C rests on a reduction (the auxiliary metric becomes exactly Schwarzschild) that is asserted rather than derived; the Palatini field equations for the δB = 0 sector deserve to be written out. And the metric itself, the parameter domain, and the Iyer-Wald mass identification are imported from the unpublished companion arXiv:2610.05799. The paper is transparent about this, but it is load-bearing: if the companion's solution or charge identification has an error, the σ-factorization that organizes everything collapses. I would want the companion published or the key steps reproduced before fully trusting the framework.\n\nWho this is for: people designing observational tests of Lorentz-violating gravity, and anyone needing the fixed-mass comparison facts for this solution. It deserves a serious referee, with the request that the lensing convention be fixed and the tensor reduction spelled out.","headline":"A systematic and mostly sound catalogue of which Palatini Kalb-Ramond black hole observables stay Schwarzschild-degenerate at fixed orbital mass, held back by a localized lensing-convention mess and a load-bearing dependence on an unpublished companion paper.","tokens_in":50795,"tokens_out":2383,"would_cite":true,"duration_ms":22497,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek/deepseek-v4-flash","headline":"At fixed orbital mass, this black hole keeps Schwarzschild's horizon, photon sphere, and impact parameter; every Lorentz-violating effect lives in one radial factor $\\sigma$ that rescales distances, tides, and axial ringdown.","keywords":["Palatini Kalb-Ramond gravity","Lorentz symmetry violation","black hole shadow","quasinormal modes","gravitational lensing","tidal forces","geodesics","orbital mass"],"falsifier":"Substitute the ansatz of Eq. (10) with $F(r)=1-2\\bar{M}/r$, $\\sigma^2=1-q/2$, and $\\bar{M}=M/\\rho$ into the full Palatini Kalb-Ramond field equations for generic admissible $\\ell_1,\\ell_2$ and verify that the residual vanishes and that $M$ is indeed the conserved charge; a non-vanishing residual would remove the solution's footing and with it every scaling law. An independent observational check: in the axial gravitational sector, a measured mode that violates $\\hat{\\omega}_{\\mathrm{T}}=\\hat{\\omega}_{\\mathrm{Sch}}/\\sigma$ at fixed $\\bar{M}$ would falsify the spectral map directly, as would a proper-distance measurement contradicting the predicted $\\sigma$-scaling of the radial metric.","tokens_in":51172,"feed_emoji":"🕳️","tokens_out":18241,"duration_ms":125954,"temperature":0.7,"pith_summary":"This paper studies a static, spherically symmetric black hole in Palatini Kalb-Ramond gravity, a modified theory in which a background antisymmetric tensor field spontaneously breaks Lorentz symmetry and the metric and the affine connection are varied independently. Its central claim is that, once the orbital mass $\\bar{M}=M/\\rho$ that actually enters the Newtonian potential is held fixed, the horizon at $r_h=2\\bar{M}$, the photon sphere at $r_{\\mathrm{ph}}=3\\bar{M}$, the critical impact parameter $b_c=3\\sqrt{3}\\,\\bar{M}$, and the innermost stable circular orbit at $6\\bar{M}$ keep their Schwarzschild values, while a single radial factor $\\sigma^2=1-q/2$ (with $q=\\ell_2/(1-\\ell_1)$) carries the entire deformation. Every quantity that samples the radial metric — proper radial distances, infall times, accumulated azimuthal angles, tidal eigenvalues, and the photon-orbit instability exponent — rescales by an exact power of $\\sigma$, and in the homogeneous axial gravitational sector the quasinormal frequencies obey the exact relation $\\hat{\\omega}_{\\mathrm{T}}(q)=\\hat{\\omega}_{\\mathrm{Sch}}/\\sigma$. This matters because it separates observables that cannot distinguish the Lorentz-violating geometry from Schwarzschild, such as the shadow diameter once the dynamical mass-to-distance ratio is calibrated, from observables that can, such as tidal forces, strong-field lensing image separations, and gravitational ringdown rates.","feed_headline":"One factor σ carries every Kalb-Ramond deviation from Schwarzschild","feed_subtitle":"At fixed orbital mass the shadow matches Schwarzschild; distances, tides, axial ringdown scale by σ.","key_machinery":"The load-bearing object is the line element $ds^2=-F(r)\\,dt^2+(\\sigma^2/F(r))\\,dr^2+r^2\\,d\\Omega^2$ with $F(r)=1-2\\bar{M}/r$, a one-parameter radial deformation of Schwarzschild in which $\\sigma^2=1-q/2$ measures the Lorentz-violating deformation and $\\bar{M}=M/\\rho$ separates the conserved Iyer-Wald mass $M$ (the Noether-charge mass) from the orbital mass that enters the Newtonian potential. The argument is carried by the identity $\\psi=\\varphi/\\sigma$: substituting it into the geodesic equation puts the orbit problem in Schwarzschild form, so any quantity fixed by the effective potential $F(r)(\\epsilon+L^2/r^2)$ is shared with Schwarzschild, while any quantity that integrates the radial metric — proper length, proper time, azimuthal accumulation, the tortoise coordinate — acquires a power of $\\sigma$. In the gravitational sector the same factorization is realized by an auxiliary metric whose background is exactly Schwarzschild under rescaled time, radius, and mass, which is what produces the exact axial spectral map $\\hat{\\omega}_{\\mathrm{T}}=\\hat{\\omega}_{\\mathrm{Sch}}/\\sigma$.","core_discovery":"The paper's central claim, in its own normalization, is an exact factorization of the physics. With the orbital mass $\\bar{M}=M/\\rho$ defined so that the gravitational potential reads $F(r)=1-2\\bar{M}/r$, and with $\\sigma^2=1-q/2$, $q=\\ell_2/(1-\\ell_1)$, the quantities fixed by the circular-orbit effective potential — horizon, photon sphere, critical impact parameter, ISCO, marginally bound orbit, and the radial frequency ratios of the Doppler analysis — are exactly Schwarzschild, because that potential depends only on $\\bar{M}$. The deformation sits entirely in the radial metric component $G(r)=\\sigma^2/F(r)$, entering through the exact rescaling $\\psi=\\varphi/\\sigma$, which maps the geodesic equation to Schwarzschild form: at fixed $\\bar{M}$, energy, and angular momentum, $r(\\varphi)=r_{\\mathrm{Sch}}(\\varphi/\\sigma)$. Consequently proper distances, infall times, and accumulated azimuthal angles scale with $\\sigma$; the radial and transverse tidal eigenvalues for free fall gain a factor $\\sigma^{-2}$ while remaining traceless and in the ratio $-2$; and the photon-orbit instability exponent scales as $\\sigma^{-1}$ while the orbital frequency stays fixed. In the wave sector, scalar, electromagnetic, and massless Dirac fields propagate on the physical metric, whereas a homogeneous axial gravitational sector with $\\delta B_{\\mu\\nu}=0$ is carried by an auxiliary metric that becomes exactly Schwarzschild under the rescalings $T=\\sqrt{1-\\ell_1}\\,t$, $R=\\sqrt{1-\\ell_1-\\ell_2/2}\\,r$; this yields the exact spectral relation $\\hat{\\omega}_{\\mathrm{T}}(q)=\\hat{\\omega}_{\\mathrm{Sch}}/\\sigma$, so both oscillation and damping rates of these modes scale as $\\sigma^{-1}$ at fixed $\\bar{M}$, with WKB and time-domain evolution agreeing to within $0.7\\%$ for the reported fundamental modes.","pith_inferences":["The exact auxiliary-metric map suggests a stronger, unproven statement worth testing: the entire homogeneous axial sector may be physically indistinguishable from Schwarzschild with a rescaled time coordinate, which would make ringdown a clean null test of Lorentz violation in this theory class.","The paper's conversion formulas imply a concrete observational route it leaves implicit: a single independent measurement of proper radial distance (yielding $\\delta_{\\mathrm{prop}}=\\sigma-1$) or of the conserved mass would convert the currently degenerate shadow intervals of its Table III into a strip bound on $q$ — at $\\ell_1=0$, the Sgr A* VLTI interval alone would roughly bound $-0.9\\lesssim q","If the imported solution is confirmed, the split between conserved mass and orbital mass is likely generic in Palatini theories with a second metric scale: mass estimates from the Newtonian potential and from Noether charges need not coincide, and conflating them could mimic either a shadow anomaly or an accidental agreement with general relativity."],"forward_implications":["Shadow measurements of Sgr A* and M87* cannot by themselves pin down the Kalb-Ramond couplings: at fixed $\\bar{M}/D_A$ the model predicts exactly zero deviation in both published angular statistics, so the couplings remain free at that level of inference.","Strong-field lensing breaks that degeneracy: the limiting relativistic-image position $\\theta_\\infty$ grows with $q$ while the angular separation of the outermost image from the asymptotic image position shrinks, so image-separation measurements would probe $\\sigma$ directly.","The axial gravitational ringdown is fully predicted from Schwarzschild: at fixed $\\bar{M}$ every multipole obeys $\\hat{\\omega}_{\\mathrm{T}}=\\hat{\\omega}_{\\mathrm{Sch}}/\\sigma$, so measuring one axial mode together with an independent mass calibration determines $\\sigma$, while at fixed conserved mass $M$ the spectrum is independent of $\\ell_2$.","Test-field ringdown and photon-orbit instability respond differently: at leading eikonal order the scalar, electromagnetic, and Dirac oscillation frequencies track the physical photon orbit and stay fixed at fixed $\\bar{M}$, while their damping rates scale as $\\sigma^{-1}$, so the same shadow radius can accompany different relaxation times.","The choice of mass convention changes what is predicted: at fixed conserved mass $M$ rather than $\\bar{M}$, the horizon, photon sphere, and impact parameter all grow like $1/\\rho$, so astrophysical comparisons must state which mass is being calibrated."],"supporting_citations":[{"why":"Supplies the exact Palatini Kalb-Ramond solution, the definitions of $\\sigma$ and $\\rho$, and the identification of $M$ as the conserved charge; the paper's normalization and factorization stand on it.","marker":"[30]"},{"why":"Provides the Noether-charge construction used to identify $M$ as the conserved mass of the solution.","marker":"[31]"},{"why":"Establishes the geodesic-stability and Lyapunov-exponent connection to quasinormal modes used for the photon-orbit instability and eikonal damping formulas.","marker":"[54]"},{"why":"Supplies the M87* angular mass consistency statistic that the shadow comparison is tested against.","marker":"[1]"},{"why":"Supplies the Sgr A* shadow deviation statistics from two dynamical priors used in the shadow comparison.","marker":"[2]"},{"why":"Provides the strong-field limit expansion of the deflection angle used to derive the relativistic image observables.","marker":"[36]"},{"why":"Extends the strong-deflection expansion to general static spherical spacetimes, underpinning the lensing formulas in the strong-field sections.","marker":"[37]"},{"why":"Supplies the Gauss-Bonnet method used to compute the weak deflection angle as an independent check.","marker":"[39]"},{"why":"Provides the higher-order WKB formula used to compute the quasinormal frequencies.","marker":"[59]"},{"why":"Supplies the characteristic time-domain integration scheme used to extract frequencies and check the WKB estimates.","marker":"[60]"}],"fun_headline_variants":["One rescaling sigma maps Kalb-Ramond geodesics to Schwarzschild","Black hole shadow stays Schwarzschild while sigma sets distances and tides","Kalb-Ramond black hole: photon sphere unchanged, ringdown scales as 1/sigma","At fixed orbital mass, Kalb-Ramond shadow matches Schwarzschild exactly","Sigma alone carries every Kalb-Ramond deviation from Schwarzschild orbits"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The paper's scaling laws all rest on the imported claim, taken from a companion preprint and not re-derived here, that the line element with $F(r)=1-2\\bar{M}/r$ is an exact solution of the Palatini Kalb-Ramond field equations and that $M$ is its Iyer-Wald conserved mass; if that solution, mass identification, or positivity domain $1-\\ell_1>0$, $1-\\ell_1-\\ell_2/2>0$ fails, the factorization into $\\sigma$ and $\\bar{M}$ collapses.","fun_headline_variants_meta":{"raw":{"variants":["One rescaling sigma maps Kalb-Ramond geodesics to Schwarzschild","Black hole shadow stays Schwarzschild while sigma sets distances and tides","Kalb-Ramond black hole: photon sphere unchanged, ringdown scales as 1/sigma","At fixed orbital mass, Kalb-Ramond shadow matches Schwarzschild exactly","Sigma alone carries every Kalb-Ramond deviation from Schwarzschild orbits"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000107,"raw_usage":{"cost":0.0001069068,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001069068,"upstream_inference_prompt_cost":3.50168e-05,"upstream_inference_completions_cost":7.189e-05},"total_tokens":1731,"prompt_tokens":1336,"completion_tokens":395,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1189,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":304}},"tokens_in":1336,"tokens_out":395,"cost_source":"openrouter_usage","duration_ms":5257,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":304,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-20T07:06:47.566937+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Substitute the ansatz of Eq. (10) with $F(r)=1-2\\bar{M}/r$, $\\sigma^2=1-q/2$, and $\\bar{M}=M/\\rho$ into the full Palatini Kalb-Ramond field equations for generic admissible $\\ell_1,\\ell_2$ and verify that the residual vanishes and that $M$ is indeed the conserved charge; a non-vanishing residual would remove the solution's footing and with it every scaling law. An independent observational check: in the axial gravitational sector, a measured mode that violates $\\hat{\\omega}_{\\mathrm{T}}=\\hat{\\omega}_{\\mathrm{Sch}}/\\sigma$ at fixed $\\bar{M}$ would falsify the spectral map directly, as would a proper-distance measurement contradicting the predicted $\\sigma$-scaling of the radial metric.","supporting_citations":[{"cited_title":"An exact stationary axisymmetric vacuum solution within a metric--affine bumblebee gravity","cited_arxiv_id":"2402.13014","evidence_quote":"Supplies the exact Palatini Kalb-Ramond solution, the definitions of $\\sigma$ and $\\rho$, and the identification of $M$ as the conserved charge; the paper's normalization and factorization stand on it."},{"cited_title":"Quasinormal modes of black holes: from astrophysics to string theory","cited_arxiv_id":"1102.4014","evidence_quote":"Establishes the geodesic-stability and Lyapunov-exponent connection to quasinormal modes used for the photon-orbit instability and eikonal damping formulas."},{"cited_title":"First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole","cited_arxiv_id":"1906.11243","evidence_quote":"Supplies the M87* angular mass consistency statistic that the shadow comparison is tested against."},{"cited_title":"Strong field limit of black hole gravitational lensing","cited_arxiv_id":"gr-qc/0102068","evidence_quote":"Provides the strong-field limit expansion of the deflection angle used to derive the relativistic image observables."},{"cited_title":"Gravitational lensing in the strong field limit","cited_arxiv_id":"gr-qc/0208075","evidence_quote":"Extends the strong-deflection expansion to general static spherical spacetimes, underpinning the lensing formulas in the strong-field sections."},{"cited_title":"Gravitational Lensing by Black Holes","cited_arxiv_id":"0911.2187","evidence_quote":"Supplies the Gauss-Bonnet method used to compute the weak deflection angle as an independent check."},{"cited_title":"Higher order WKB formula for quasinormal modes and grey-body factors: recipes for quick and accurate calculations","cited_arxiv_id":"1904.10333","evidence_quote":"Supplies the characteristic time-domain integration scheme used to extract frequencies and check the WKB estimates."}],"review_version":1}