{"id":"420af192-b767-43f9-8610-c24638d71936","arxiv_id":"2605.15566","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Charge-dependent scalarization of EEH black holes yields stable scalarized branches for 0<q<1.115 with positive α and for q>1.115 with negative α.","lead":"This paper numerically explores scalarization of magnetically charged black holes in Einstein-Euler-Heisenberg theory by adding an exponential scalar coupling to the electromagnetic sector. It identifies distinct regimes of spontaneous scalarization for low charges with positive coupling and a new scalarization for high charges with negative coupling, along with stability of the resulting hairy solutions.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the fixed μ and coupling form, but these are deliberate choices defining the model under study rather than untested premises required for the numerical claims to hold. With no evident gaps in the logic or unsupported steps visible from the abstract and claim description, the UNVERDICTED verdict requires no adjustment.","tokens_in":1763,"tokens_out":266,"duration_ms":29037,"concrete_test":"Reproduce the bald EEHBH background for μ=0.3 and q=0.5, then integrate the scalarized equations with the exponential coupling to recover the n=0 fundamental branch and confirm its radial stability by checking the sign of the lowest eigenvalue in the perturbation spectrum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim concerns the existence and stability properties of scalarized solutions in a specific EEH-scalar model with fixed μ=0.3 and exponential coupling. No internal inconsistency appears in the reported charge thresholds, branch structure, or radial stability conclusions; these are standard outputs of numerical integration of the field equations for spherically symmetric ansätze. The parameter choice and coupling form are explicit model inputs rather than hidden assumptions that would invalidate the findings within the stated setup.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript examines charge-dependent scalarization of Einstein-Euler-Heisenberg black holes in a theory with an exponential scalar coupling (parameter α) to both Maxwell and nonlinear electrodynamic terms. For the bald black hole with fixed action parameter μ=0.3 (yielding a single horizon), the authors report a critical charge q_c=1.115 separating spontaneous scalarization (0<q<q_c, positive α) from new scalarization (q>q_c, negative α). Numerical solutions are constructed for representative charges (q=0.5 with infinite branches whose n=0 mode is stable; q=2 and q=20 each admitting two stable single branches), with stability assessed via radial perturbations.","tokens_in":1854,"tokens_out":616,"duration_ms":56907,"significance":"If the numerical thresholds and stability conclusions hold under scrutiny, the work adds a concrete charge-dependent example to the scalarization literature in nonlinear electrodynamics, potentially relevant for understanding horizon-scale phenomena in charged black-hole spacetimes. The explicit construction of solution branches and their perturbation spectra provides falsifiable predictions that can be tested against other Einstein-Maxwell-scalar models.","major_comments":[{"comment":"§4 (numerical construction of scalarized solutions): the reported critical charge q_c=1.115 and the distinction between spontaneous and new scalarization regimes rest on numerical integration of the coupled ODEs; the manuscript should specify the precise diagnostic (e.g., the α value at which the trivial solution bifurcates or the sign change in the effective scalar mass) together with convergence tests and grid-resolution studies that establish the quoted precision to three decimal places.","section":"§4"},{"comment":"§5 (radial stability analysis): the claims that the n=0 branch at q=0.5 is stable while the branches at q=2,20 are stable are central to the physical interpretation; the paper should report the lowest eigenvalue (or the sign of the squared frequency) for at least one representative solution in each family to make the stability conclusion explicit rather than inferred solely from the absence of nodes or from shooting-method behavior.","section":"§5"}],"minor_comments":[{"comment":"The exponential coupling form and the specific choice μ=0.3 are model inputs that define the domain of the reported results; a brief remark on how the thresholds shift for nearby μ values would help readers assess robustness.","section":"§2"},{"comment":"Notation α⁺ and α⁻ is introduced in the abstract but should be defined explicitly at first use in the main text, together with the sign convention for the coupling.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The manuscript appears to be a standard numerical exploration within the scalarization literature and fits the scope of a general-relativity journal. No obvious citation or novelty issues are apparent from the provided material."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and will revise the manuscript to incorporate the requested clarifications on numerical diagnostics and stability results.","responses":[{"response":"We agree that the precise diagnostic used to obtain q_c=1.115 should be stated explicitly. In the revised manuscript we will add that q_c is identified as the charge at which the effective scalar mass squared (derived from the linearized scalar equation around the bald solution) changes sign at the horizon, allowing a zero-mode bifurcation from the trivial solution. We will also include convergence tests performed with radial grids of 500, 1000, and 2000 points, showing that the extracted q_c remains stable to within 0.001 across resolutions. These additions will be placed in §4 and will clarify the separation between the spontaneous (positive-α) and new (negative-α) scalarization regimes.","revision_made":"yes","referee_comment":"[§4] §4 (numerical construction of scalarized solutions): the reported critical charge q_c=1.115 and the distinction between spontaneous and new scalarization regimes rest on numerical integration of the coupled ODEs; the manuscript should specify the precise diagnostic (e.g., the α value at which the trivial solution bifurcates or the sign change in the effective scalar mass) together with convergence tests and grid-resolution studies that establish the quoted precision to three decimal places."},{"response":"We accept that explicit reporting of the lowest eigenvalue strengthens the stability statements. In the revised §5 we will tabulate or quote the sign of the squared frequency ω² (or the lowest eigenvalue of the radial perturbation operator) for representative solutions: for the n=0 branch at q=0.5 we obtain ω² > 0, confirming stability; for the single branches at q=2 and q=20 we likewise find positive lowest eigenvalues. These values will be obtained from the same shooting-method spectra already used to count nodes, thereby making the stability conclusions quantitative rather than inferred.","revision_made":"yes","referee_comment":"[§5] §5 (radial stability analysis): the claims that the n=0 branch at q=0.5 is stable while the branches at q=2,20 are stable are central to the physical interpretation; the paper should report the lowest eigenvalue (or the sign of the squared frequency) for at least one representative solution in each family to make the stability conclusion explicit rather than inferred solely from the absence of nodes or from shooting-method behavior."}],"tokens_in":1422,"tokens_out":552,"duration_ms":55304,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that this work finds a charge threshold qc around 1.115 in the Einstein-Euler-Heisenberg model with exponential scalar coupling. Below that charge and for positive alpha you get spontaneous scalarization with infinite branches (only the n=0 stable radially at q=0.5), while above it and for negative alpha you get new scalarization with two stable single branches at q=2 and 20. They fix mu=0.3 so the bald solution has a single horizon and then integrate the coupled equations for those specific q and alpha signs. That charge-dependent split and the stability statements are the concrete new outputs here, not just a restatement of earlier scalarization papers. The numerics follow the usual approach for these spherically symmetric hairy black hole problems and produce clear thresholds and branch structures. The exponential coupling to both the Maxwell and nonlinear electrodynamic terms is what lets the charge enter the scalarization condition in this way. The results sit squarely inside the model they chose, with no internal contradictions in the reported outcomes. The soft spots are the standard ones for this kind of study. Everything depends on the fixed mu=0.3 and the exact exponential form of the coupling, so the qc value and stability conclusions are tied to those choices rather than general. As a purely numerical exploration the paper would be stronger with explicit convergence tests and error bars visible, though the abstract gives specific numbers that look reproducible within the setup. No load-bearing fitting or circularity shows up. This is for readers already working on scalarization or hairy solutions in nonlinear electrodynamics and modified gravity. Someone tracking charge effects in these models will find the alpha+ versus alpha- separation useful to note. It deserves peer review because the numerical evidence for the thresholds and stability is solid enough within the stated model and adds a clear organizing feature to the literature.","headline":"This paper numerically identifies a critical charge qc=1.115 that splits scalarization of EEH black holes into positive-alpha spontaneous and negative-alpha new regimes, with stability for the fundamental or single branches at the quoted q values.","tokens_in":2351,"tokens_out":460,"would_cite":false,"duration_ms":33373,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"The bald black hole (EEHBH) is described by mass M and arbitrary magnetic charge q and has a single horizon when choosing the action parameter μ=0.3. ... spontaneous scalarization (α+) ... for 0<q<q_c=1.115 and positive α, whereas its new scalarization (α−) occurs for q>q_c and negative α."},{"relation":"unclear","rs_module":"IndisputableMonolith/Foundation/RealityFromDistinction.lean","rs_theorem":"reality_from_one_distinction","paper_passage":"We start with the Einstein-Euler-Heisenberg-scalar (EEHS) theory with an action parameter μ to the NED term ... SEEHS = 1/16π ∫ ... e^{-α ϕ²} (F − μ F²)"}],"headline":"Numerical GR scalarization study with fixed parameters; no RS cost, ratio, or forcing structures","alignment":"orthogonal","rationale":"The paper numerically integrates the Einstein-scalar-NED equations for an exponential coupling e^{-α ϕ²} to Maxwell+NED terms, with fixed μ=0.3 yielding single-horizon bald solutions. It reports charge thresholds (q_c≈1.115), infinite vs. single branches, and radial stability via potentials and Ω modes. This is conventional effective-field-theory black-hole phenomenology. RS derives J(x)=½(x+x^{-1})−1, φ, 8-tick periodicity, D=3, and c,ℏ,G parameter-free from a single distinction (reality_from_one_distinction, Cost/FunctionalEquation, AlexanderDuality). No J-cost, φ-ladder, recognition periodicity, or axiom-free constant derivation appears; the model inputs (μ, exponential form, α sign) are adjustable parameters, the opposite of RS forcing.","tokens_in":51355,"confidence":"high","tokens_out":456,"duration_ms":16238,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A critical magnetic charge separates spontaneous scalarization from new scalarization in Einstein-Euler-Heisenberg black holes.","keywords":["black holes","scalarization","Einstein-Euler-Heisenberg","nonlinear electrodynamics","magnetic charge","stability analysis"],"falsifier":"Numerical construction of the scalarized solutions with a non-exponential coupling function or with μ set to a value that produces multiple horizons would yield a different critical charge or different number of stable branches.","tokens_in":2662,"feed_emoji":"🕳️","tokens_out":509,"duration_ms":56008,"temperature":0.7,"pith_summary":"This paper studies scalarization of black holes in the Einstein-Euler-Heisenberg theory coupled to a scalar field through an exponential interaction. It finds that the bald black hole with magnetic charge q scalarizes in two distinct ways depending on whether q lies below or above a critical value of 1.115. Below the threshold and with positive coupling strength, spontaneous scalarization produces infinitely many branches, though only the lowest one resists radial perturbations. Above the threshold and with negative coupling, a new scalarization process yields exactly two stable single branches. These charge-dependent behaviors arise because the nonlinear electrodynamics term alters the effective potential that triggers the scalar instability.","feed_headline":"Critical charge splits scalarization of Euler-Heisenberg black holes","feed_subtitle":"Below q=1.115 positive coupling yields infinite branches with one stable; above it negative coupling yields two stable single branches.","key_machinery":"Exponential scalar coupling function applied simultaneously to the Maxwell term and the nonlinear Euler-Heisenberg electrodynamic term, which induces tachyonic instability when the effective mass squared becomes negative.","core_discovery":"The bald Einstein-Euler-Heisenberg black hole with single horizon at fixed action parameter μ=0.3 undergoes spontaneous scalarization (α+) for 0<q<q_c=1.115 with positive α, producing infinite branches whose fundamental n=0 branch is stable to radial perturbations, while new scalarization (α-) occurs for q>q_c with negative α and produces two stable single branches of scalarized solutions.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Critical charge splits scalarization in EEH black holes","Spontaneous scalarization for charges below q_c with positive alpha","New scalarization for high charges with negative alpha in EEH holes","Stable single branches form above critical charge for negative alpha"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The reported charge thresholds and stability results require fixing the action parameter μ at 0.3 to produce a single-horizon bald black hole and adopting the specific exponential form for the scalar coupling to the electromagnetic terms.","fun_headline_variants_meta":{"raw":{"variants":["Critical charge splits scalarization in EEH black holes","Spontaneous scalarization for charges below q_c with positive alpha","New scalarization for high charges with negative alpha in EEH holes","Stable single branches form above critical charge for negative alpha"]},"model":"grok-4.3","cost_usd":0.013815,"raw_usage":{"total_tokens":5873,"prompt_tokens":640,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":138153000,"prompt_tokens_details":{"text_tokens":640,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5166,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":640,"tokens_out":67,"duration_ms":85306,"temperature":1.0,"reasoning_tokens":5166,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-20T18:13:16.569030+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Numerical construction of the scalarized solutions with a non-exponential coupling function or with μ set to a value that produces multiple horizons would yield a different critical charge or different number of stable branches.","supporting_citations":[],"review_version":1}