{"id":"e9b8be3d-da36-4fd0-97d6-5c2ba4099375","arxiv_id":"2512.01486","paper_version":4,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"The claimed smooth signature-changing BTZ solution fails its own consistency checks: the metric is not Euclidean inside and the geodesic analysis contains sign errors.","lead":"A proposed signature-changing BTZ black-hole metric, meant to turn the interior Euclidean and make infalling observers take infinite time to reach the horizon, does not support its own claims: the interior signature is (-,-,+), not (+,+,+), and the radial velocity is imaginary in the exterior. The paper's regularization relies on an invalid distributional identity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Radial velocity in Eq. (33) is imaginary in the exterior (r>r_h), so the infinite-proper-time claim is an artifact of invalid square-root handling.","rationale":"The paper's headline claim is singularity avoidance via infinite proper time to the horizon. Eq. (33) is the direct basis for that claim; if \\dot r is imaginary in the exterior, there is no real geodesic to integrate, and the claimed divergence at r_h is an artifact of complex branches. This alone invalidates the central result, independent of the distributional regularization details. The reader identified related symptoms (imaginary velocity, wrong interior signature), but the weakest_assumption focused on the interior Euclidean signature rather than the exterior imaginary radial velocity, which is an even more direct contradiction of the 'infalling observers' claim. My concern therefore reinforces the reader's REJECT verdict without changing it.","tokens_in":15639,"tokens_out":13193,"duration_ms":133707,"concrete_test":"Set M=\\ell=1, so r_h=1, and evaluate Eq. (33) at r=2 (exterior): \\dot r = -\\sqrt{1-4} = -i\\sqrt{3}. If this is the paper's radial velocity, the exterior infall is imaginary, so the proper-time integral \\int dr/\\dot r is not real; the infinite-proper-time result cannot describe a physical observer. Equivalently, evaluate \\dot r^2 from Eq. (31) with E=\\varepsilon^2 at r=2; it is negative, contradicting real radial motion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central atemporality claim rests on a real radial velocity that vanishes at the horizon. But the paper's own Eq. (33), \\dot r = -\\sqrt{\\varepsilon}\\sqrt{M-r^2/\\ell^2}, is imaginary in the exterior: for r>r_h, \\varepsilon=+1 and M-r^2/\\ell^2<0, so \\dot r = -i\\sqrt{r^2/\\ell^2-M}. The same sign error propagates through Eq. (5), the Painlevé-Gullstrand transformation (7)-(8) (where d\\Theta/dr contains \\sqrt{M-r^2/\\ell^2}), and the proper-time integral. Thus the 'infinite proper time to the horizon' is not a real quantity; it is an artifact of taking square roots of negative numbers without branch definitions. Additionally, Eq. (1) at r<r_h has g_tt = -\\varepsilon(-M+r^2/\\ell^2) = F<0 and g_rr=1/F<0, so the interior has signature (-,-,+), not the claimed Euclidean (+,+,+). Neither the exterior geodesics nor the Euclidean interior therefore exist as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a smooth signature-changing BTZ black-hole geometry in which the Lorentzian exterior transitions to a Euclidean interior at the horizon. It claims that the regularized metric is a genuine vacuum solution (R_{\\mu\\nu}=0), has bounded curvature invariants, makes the horizon unreachable in finite proper time ('atemporality'), is linearly stable, supports unitary scalar-field propagation, and preserves standard BTZ thermodynamics. The central mechanism is the smooth transition function replacing the discontinuous sign in the metric. I find that the central claims fail on elementary algebraic and distribution-theoretic grounds.","tokens_in":16059,"tokens_out":12042,"duration_ms":121471,"significance":"If the claims were correct, the paper would provide a self-contained classical mechanism for singularity avoidance and would repair a known distributional-consistency problem in earlier signature-changing black-hole models. The paper is systematic and works out many consequences—geodesics, perturbations, scalar fields, thermodynamics—and its explicit formulas are a strength because they can be checked directly. However, the check fails at several load-bearing points, so the significance is currently not realized.","major_comments":[{"comment":"For r<r_h, -M+r^2/\\ell^2<0 and \\varepsilon=-1, so g_{tt}=-\\varepsilon(-M+r^2/\\ell^2)<0 and g_{rr}=1/(-M+r^2/\\ell^2)<0. The interior signature is (-,-,+), not the claimed Euclidean (+,+,+). Moreover, the Painlevé-Gullstrand form in Eq. (8) and the radial velocity in Eq. (33) contain \\sqrt{\\varepsilon}\\sqrt{M-r^2/\\ell^2}, which is imaginary for r>r_h. The exterior radial velocity is therefore not real, and the 'infinite proper time to the horizon' conclusion is not a real-geometry result.","section":"§2.1, Eq. (1); §3.2, Eq. (33)"},{"comment":"The distributional regularization is imposed rather than derived. Equation (13) sets \\int \\delta(x)/|x|^n dx=0, but \\delta/|x|^n is not a well-defined distribution; Eq. (14) sets integrals of \\delta^2 to zero, although \\delta^2 is undefined in standard distribution theory; Eq. (15) is an integration-by-parts identity only if boundary terms are dropped, which is not justified on the non-compact domain. These prescriptions are what force the surface-layer terms to vanish, so the claim R_{\\mu\\nu}=0 is circular rather than a derived property.","section":"§2.2.1, Eqs. (13)–(15)"},{"comment":"The final 'regularized' Riemann components contradict the claimed regularity. For example, R^T_{rTr}=2M/(\\ell^2 r \\varepsilon) diverges at the horizon where \\varepsilon=0, and R^r_{rTr} and R^\\varphi_{T\\varphi r} contain \\sqrt{M-r^2/\\ell^2}/\\sqrt{\\varepsilon}, which is imaginary in the Lorentzian exterior. Thus the tensor is neither finite nor real where it is supposed to describe the physical vacuum.","section":"Appendix A, Eqs. (A19)–(A24)"},{"comment":"The Kretschmann scalar as written, K_{\\rm reg}=12M^2/(\\ell^4 r^4)(r^4+2r^2\\ell^2/3+\\ell^4/3), diverges as r\\to 0 like 4M^2/r^4. The text states that it 'remains bounded as r\\to 0' and uses this to claim that r=0 is not a curvature singularity. This is an internal contradiction. The comparison with a standard BTZ 'K\\sim 1/r^6' is also incorrect: BTZ is locally AdS with constant Kretschmann scalar.","section":"§5.1, Eq. (18)"},{"comment":"The proof of scalar-field unitarity relies on imposing Im(A^*B)=0 as a boundary condition. This is an additional restriction on the mode coefficients, not a consequence of the field equation or of the conserved current. For a generic solution with Im(A^*B)\\neq 0, the second term in j^T contains a factor \\sqrt{\\varepsilon_\\rho}/x, which is singular near the change surface. Unitarity for arbitrary initial data is therefore not established. In addition, Eq. (50) contains the factor \\sqrt{M-r^2/\\ell^2}, which is imaginary in the exterior, making the radial equation complex.","section":"§4.2.2, Eq. (56)"}],"minor_comments":[{"comment":"The abstract defines a smooth transition S_\\delta(r)=\\tanh[(r-r_h)/\\delta], but the body uses \\varepsilon_\\rho(x)=x^{1/(2\\kappa+1)}/(x^2+\\rho)^{1/2(2\\kappa+1)}. The relation between these two regularizations is never stated, and it is unclear which function enters the actual metric.","section":"Abstract and §2.2"},{"comment":"The conserved quantity is written as E=\\varepsilon^2(-M+r^2/\\ell^2)\\dot t, but \\varepsilon^2=1, so the factor is redundant. If E is intended to be the Killing energy -\\xi\\cdot u, the correct expression is E=\\varepsilon(-M+r^2/\\ell^2)\\dot t. Please clarify the sign and normalization.","section":"§2.1, Eq. (3)"},{"comment":"Several figures (Figs. 3, 4, 5, 7) are described as showing divergences and spectra, but the numerical QNM computation is not accompanied by code, data, or a reproducible algorithm. The reader cannot independently verify the claimed agreement with WKB.","section":"Figures and numerical claims"},{"comment":"The proof that U and V are linear in affine parameter does not by itself show that r=0 is unreachable in finite affine parameter; one must check whether the condition for r=0 can be satisfied at finite \\lambda. As written, the argument is incomplete.","section":"Appendix B, Theorem 5"}],"recommendation":"reject","confidential_remarks":"I concur with the stress-test concern. The errors are elementary and load-bearing: the interior signature is (-,-,+), the exterior radial velocity is imaginary, the Kretschmann scalar in Eq. (18) diverges at r=0, and the distributional regularization is ad hoc. These are not local presentation issues; the central atemporality and vacuum claims are falsified by the manuscript's own formulas. A substantially rewritten paper with corrected algebra and a genuinely distributional treatment might be considered in the future, but the present manuscript is not publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this applies the Capozziello et al. atemporality mechanism to BTZ and tries to fix the distributional treatment, but the central claims are invalidated by basic sign errors that the author's own equations make obvious. I wouldn't send it to a referee in its current form.\n\nWhat's new: the extension to BTZ and the proposal of a smooth transition function plus a modified Hadamard regularization. The author is honest about borrowing the atemporality idea from [10], and the paper is clearly organized, with detailed appendices and a nice list of open problems.\n\nWhat's wrong: the defining metric (1) does not have Euclidean signature in the interior. For r < r_h, both g_tt and g_rr are negative, so the signature is (-,-,+), not (+,+,+). More seriously, the radial velocity in Eq. (33), \\dot r = -\\sqrt{\\epsilon}\\sqrt{M-r^2/\\ell^2}, is imaginary in the exterior r > r_h, because M-r^2/\\ell^2 is negative there. That makes the whole geodesic analysis—including the claim of infinite proper time to the horizon—an artifact of taking square roots of negative numbers without branch definitions. The same problem appears in the Painlevé-Gullstrand transformation in Eqs. (7)-(8). Eq. (15), the integration-by-parts identity, is false as stated: the second derivative is not transferred in that way, and the conclusion that distributional terms vanish depends on the regularization scheme being contrived to produce zero.\n\nThe QFT section imposes Im(A*B)=0 as a boundary condition to make the probability current finite, which is a built-in assumption rather than a derived result. The numerical stability results are presented without code or data, so they can't be checked. And the abstract and body disagree: the abstract advertises a tanh transition function, while the body uses a different regulated function.\n\nWhat survives: the thermodynamics section, if the geometry were consistent, would likely reproduce BTZ thermodynamics because the exterior is unchanged—but that's not a new result. The paper is a serious attempt, but the sign errors are load-bearing rather than cosmetic.\n\nRecommendation: reject/desk-reject. The author should correct the signature function, redo the geodesics with proper branch cuts, and then reassess whether atemporality actually holds. If the geometry is fixed, the atemporality extension to BTZ might deserve a closer look. Right now, the paper's internal contradictions make it not ready for outside review.","headline":"The paper's central atemporality claim is undone by sign errors in its own metric and geodesic equations; the idea is not new, and the BTZ application needs a serious rewrite.","tokens_in":16422,"tokens_out":3088,"would_cite":false,"duration_ms":33683,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.60.Bc","04.70.Dy","04.20.Jb","11.10.Gh"],"model":"deepseek-v4-flash","headline":"Smooth signature change at the BTZ horizon yields a vacuum solution with bounded curvature and an unreachable singularity, the paper claims.","keywords":["signature change","BTZ black hole","singularity avoidance","atemporality","Hadamard regularization","distributional geometry","Painlevé-Gullstrand coordinates","Euclidean interior"],"falsifier":"Evaluate the metric (8) at any interior point r<rh using the regularized ερ from Eq. (12): the line element acquires a negative coefficient for dr² and an imaginary cross-term, so the signature is not (+,+,+). Checking this single point against the paper's claim of a Euclidean interior would settle whether the construction is what it states.","tokens_in":15576,"feed_emoji":"🕳️","tokens_out":6234,"duration_ms":64186,"temperature":0.7,"pith_summary":"This paper argues that replacing the discontinuous sign function in a signature-changing BTZ metric with a smooth transition function produces a globally smooth, real vacuum solution with no surface layers and bounded curvature everywhere. The author claims that radially infalling observers require infinite proper time to reach the horizon, implementing 'atemporality' as a classical singularity-avoidance mechanism. A sympathetic reader would care because the construction offers a concrete, computable model in which classical general relativity resolves its own central singularity without invoking quantum gravity. The paper also reports linear stability, unitary scalar-field propagation, and unchanged external thermodynamics as evidence that the geometry is physically viable.","feed_headline":"Signature change makes BTZ horizon require infinite proper time","feed_subtitle":"A smooth Lorentzian-to-Euclidean transition yields a vacuum black hole whose interior is causally sealed.","key_machinery":"The central object is the regularized signature function ερ(r), a smooth approximation to sign(−M+r²/ℓ²), together with Painlevé-Gullstrand coordinates that remain finite at the horizon. A modified Hadamard regularization treats the problematic second-derivative distributional terms ε″(r) consistently, making the distributional Ricci and Weyl tensors vanish and eliminating surface layers and impulsive waves. This machinery converts the horizon into an 'atemporality' surface where infalling observers' proper time diverges.","core_discovery":"The paper claims that a signature-changing BTZ metric, in which the signature smoothly transitions from Lorentzian (−,+,+) to Euclidean (+,+,+) at the horizon, satisfies the vacuum Einstein equations Rμν=0 identically once a modified Hadamard regularization is applied to the distributional curvature. Curvature invariants remain finite, and the horizon becomes a one-way causal membrane: geodesic and accelerated observers reach it only after infinite proper time. The central singularity at r=0 is softened into a topological boundary rather than a curvature singularity, so the spacetime is geodesically complete but metrically incomplete.","pith_inferences":["The vacuum claim rests on a cancellation that is not visible in the displayed regularized Ricci components (Sec. 3.1.1), which are nonzero; the text asserts Rμν=0 but does not show how the remaining terms vanish.","The abstract describes the transition function as tanh[(r−r_h)/δ], while the body and appendix use a different power-law function ερ(x)=x^{1/(2κ+1)}/(x²+ρ)^{1/(2(2κ+1))}; the two are inequivalent, and the paper's calculations use the latter.","The paper's own limitations section says the mechanism prevents access to the singularity rather than eliminating it; declaring the topological boundary at r=0 a 'resolution' is an interpretive step beyond the mathematics.","If the scheme generalizes to four dimensions, it would offer a classical alternative to quantum-gravity singularity resolution, but the paper identifies that generalization as technically challenging and does not attempt it."],"forward_implications":["If the construction is correct, the Lorentzian exterior is indistinguishable from standard BTZ up to the horizon, so known BTZ thermodynamics and observations are preserved.","Infalling observers, both geodesic and accelerated, require infinite proper time to reach the horizon, making the Euclidean interior causally sealed.","The r=0 curvature divergence is replaced by a finite Kretschmann invariant, with the origin reinterpreted as a topological boundary.","The geometry is claimed to be linearly stable: gravitational and scalar quasi-normal modes all have negative imaginary parts, indicating exponential decay.","Quantum scalar fields propagate unitarily across the change surface, with a finite conserved probability current."],"fun_headline_variants":["Smooth signature change seals BTZ black hole interior","Infinite proper time at horizon via smooth signature swap","Signature transition banishes BTZ singularity to topology","BTZ horizon becomes timeless with tanh signature change"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction assumes that the smooth function ερ(r) is a real signature function whose square root is meaningful in the metric; in the paper's own coordinates √ερ is imaginary for r<rh, so the interior is not a Euclidean (+,+,+) geometry as claimed, and the vacuum claim is further strained by the nonzero displayed Ricci components.","fun_headline_variants_meta":{"raw":{"variants":["Smooth signature change seals BTZ black hole interior","Infinite proper time at horizon via smooth signature swap","Signature transition banishes BTZ singularity to topology","BTZ horizon becomes timeless with tanh signature change"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000666,"raw_usage":{"total_tokens":2910,"prompt_tokens":810,"completion_tokens":2100,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":2038}},"tokens_in":554,"tokens_out":2100,"duration_ms":13649,"temperature":1.0,"reasoning_tokens":2038,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T19:11:16.197725+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the metric (8) at any interior point r<rh using the regularized ερ from Eq. (12): the line element acquires a negative coefficient for dr² and an imaginary cross-term, so the signature is not (+,+,+). Checking this single point against the paper's claim of a Euclidean interior would settle whether the construction is what it states.","supporting_citations":[],"review_version":1}