{"id":"10b1a980-44ca-45d6-ad56-8982cc311924","arxiv_id":"2603.04886","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Without prior assumptions, regional internal–external potential field separation is non-unique; under a source-free shell assumption it is unique but highly unstable via the spherical Hardy–Hodge decomposition.","lead":"Regional magnetic data cannot uniquely separate internal from external sources without extra assumptions; with a source-free shell assumption the split is unique but highly unstable. This explains why local geomagnetic surveys struggle with source separation.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified beyond abstract-only unavailability of the proofs.","rationale":"The reader correctly withholds a verdict for lack of proofs and correctly notes that uniqueness rests on the shell assumption. That assumption is, however, an explicit modeling hypothesis rather than an internal inconsistency or hidden gap; questioning its geophysical realism does not undermine the mathematical claim as stated. No further load-bearing technical concern can be extracted from the abstract alone, so the UNVERDICTED status and LOW confidence remain appropriate.","tokens_in":1922,"tokens_out":319,"duration_ms":14157,"concrete_test":"Obtain the full text and re-derive the uniqueness statement under the shell hypothesis from the spherical Hardy–Hodge decomposition alone (without extra regularity or boundary conditions); if a non-trivial null space appears for any non-empty open subdomain of the sphere, the uniqueness claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"With only the abstract, the central claims (regional non-uniqueness without assumptions; uniqueness plus high instability under the exterior-sources-above-source-free-shell hypothesis, via spherical Hardy–Hodge) are clearly scoped and appear internally consistent. The modeling premise is stated explicitly rather than hidden, and no equation, estimate, or function-space hypothesis is available to isolate a concrete soft spot in the argument itself. The reader’s flagged assumption is load-bearing for geophysical applicability, not for the mathematical claim as formulated.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies internal–external potential field separation for magnetic data given only on a regional subdomain of a spherical observation surface. It claims that, without prior assumptions, such a separation is not uniquely determined, in contrast to the classical global Gauss procedure on the full sphere. Under the additional geometric hypothesis that exterior sources lie above a source-free spherical shell, uniqueness is restored via the spherical Hardy–Hodge decomposition, but the resulting inverse problem is highly unstable. The work aims to explain the intrinsic mathematical difficulties of regional data-based separation in geomagnetism.","tokens_in":2024,"tokens_out":752,"duration_ms":13369,"significance":"If the non-uniqueness, conditional uniqueness, and instability statements hold rigorously, the paper would supply a clear functional-analytic foundation for a well-known practical obstacle in aeromagnetic and ground-based geomagnetism. Explicitly tying the restoration of uniqueness to a geophysically motivated source-location constraint, and documenting high instability of the resulting map, would be of genuine interest to both potential theory and applied geomagnetism. Credit is due for framing the problem against the classical global Gauss baseline and for invoking the spherical Hardy–Hodge decomposition as the natural tool; those strengths can only be fully assessed once the proofs and estimates are available.","major_comments":[{"comment":"Only the abstract is available for this review. The central claims—regional non-uniqueness without assumptions, uniqueness under the exterior-sources-above-source-free-shell hypothesis, and high instability of the resulting separation—rest on Hardy–Hodge arguments, function-space settings, and quantitative instability estimates that cannot be checked. A full technical assessment of correctness is therefore impossible from the material provided.","section":null},{"comment":"Abstract: the uniqueness claim is load-bearing and is conditioned on the geometric premise that exterior sources lie above a source-free spherical shell. The manuscript must state the precise function-space hypotheses (trace spaces, regularity of the subdomain boundary, admissible source classes) under which uniqueness holds, and must exhibit the argument that this premise is both necessary and sufficient for uniqueness in the regional setting.","section":null},{"comment":"Abstract: the claim that the conditional separation is “highly unstable” is load-bearing for the paper’s explanatory goal. Without a modulus of continuity, singular-value decay rates, or an explicit ill-posedness estimate derived from the spherical Hardy–Hodge decomposition, the instability assertion remains qualitative and cannot be verified or compared with the stable global Gauss case.","section":null}],"minor_comments":[{"comment":"Abstract: the phrase “geophysically reasonable assumption” is appropriate for motivation but should be paired, in the full text, with a precise mathematical formulation of the source-free shell so that the modeling premise is not left informal.","section":null},{"comment":"Abstract: a brief indication of the function spaces (e.g., Sobolev or Hardy-type spaces on the sphere and on the regional subdomain) would help the reader anticipate the technical setting before the full development.","section":null}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review: the full manuscript body was not supplied. Under those constraints a standard accept/revise/reject decision cannot be reached; the recommendation is therefore uncertain. Once the full text is available, the load-bearing points to check are the Hardy–Hodge uniqueness argument under the shell hypothesis and the quantitative instability estimates. The modeling premise is stated openly in the abstract and does not appear to be hidden circularity."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this is a clean theoretical diagnosis of a problem geomagnetists already feel in practice: on a full sphere Gauss separation is unique and stable; on a proper subdomain it is not unique without extra assumptions, and under the usual “exterior sources above a source-free spherical shell” hypothesis it becomes unique but highly unstable. That is the contribution, obtained via the spherical Hardy–Hodge decomposition.\n\nWhat is new is the regional non-uniqueness result and the uniqueness-plus-instability theorem under that geometric source constraint. The global case is classical; the paper’s value is making the regional failure modes precise and explaining why regional data-based separations are intrinsically hard. The abstract is well-scoped, the modeling premise is stated up front rather than smuggled in, and there is no data-fitting or free-parameter circularity. For potential theory and geomagnetism this is useful organization of a long-standing difficulty.\n\nSoft spots are almost entirely about what we cannot see. We have only the abstract, so the function-space settings, boundary regularity, and quantitative instability rates are unchecked. The load-bearing geophysical assumption (exterior sources above a source-free shell) is explicit; if it fails in a given application the uniqueness claim does not apply, but that is a modeling limitation, not a hidden flaw in the math as formulated. Nothing in the abstract looks incoherent or over-claimed.\n\nThis is for people who work on regional potential-field separation, aeromagnetic/ground surveys, or spherical Hardy–Hodge methods. A serious referee should see the full proofs and estimates. I would send it to peer review; it deserves that time even if the instability rates turn out milder or sharper than the abstract suggests. Bring it to reading group only if someone is already deep in regional geomagnetic inverse problems; otherwise wait for the full text.","headline":"Abstract-only: regional non-uniqueness of internal–external separation, and uniqueness-plus-high-instability under a source-free shell via spherical Hardy–Hodge — clear claims, proofs unchecked.","tokens_in":2608,"tokens_out":485,"would_cite":false,"duration_ms":4841,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["86A25","31B05","46F12"],"pacs":[],"model":"grok-4.5","headline":"Regional internal–external magnetic field separation is not unique without assumptions and becomes unique but highly unstable under a geophysically motivated spherical-shell constraint.","keywords":["internal-external field separation","spherical Hardy-Hodge decomposition","regional geomagnetic data","potential field uniqueness","source-free spherical shell","instability of inverse problems","geomagnetism"],"falsifier":"Construct a smooth magnetic field that is generated by currents both inside the Earth and below the assumed shell yet coincides, on a chosen open spherical cap, with a purely internal field; any numerical separation algorithm that claims uniqueness must then return the wrong exterior component.","tokens_in":2825,"feed_emoji":"🧲","tokens_out":591,"duration_ms":7481,"temperature":0.7,"pith_summary":"Internal–external field separation tries to split a magnetic field observed on a surface into the part generated inside that surface and the part generated outside it. On a complete sphere the split is classical, unique, and stable, but when measurements cover only a regional subdomain the same split is no longer uniquely determined by the data alone. The paper shows that uniqueness can be restored by the modeling assumption that all exterior sources lie above a source-free spherical shell; under that assumption the spherical Hardy–Hodge decomposition yields a unique separation. That separation is nevertheless severely unstable, which accounts for the practical difficulties that have long been observed when regional aeromagnetic or ground-based surveys attempt the same decomposition. The result therefore supplies both a rigorous uniqueness statement and a precise explanation of why regional separations remain fragile.","feed_headline":"Regional magnetic-field split is unique only under a shell assumption—and still unstable","feed_subtitle":"A spherical Hardy–Hodge argument shows why aeromagnetic surveys cannot stably separate internal from external sources.","key_machinery":"The spherical Hardy–Hodge decomposition, which splits a vector field on a sphere into an internal potential part, an external potential part, and a toroidal remainder; under the shell assumption the remainder vanishes and the two potential parts become uniquely recoverable from regional data.","core_discovery":"Without any prior geometric constraints an internal–external potential-field separation performed on a proper subdomain of a spherical observation surface is non-unique. Once exterior sources are required to lie above a source-free spherical shell, the spherical Hardy–Hodge decomposition produces a unique separation, yet the reconstruction operator is highly unstable.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Regional mag-field split nonunique without shell—and unstable even then","Hardy–Hodge: subdomain internal–external separation unique only under shell","Aeromagnetic surveys cannot stably isolate internal from external sources","Shell constraint yields unique yet highly unstable regional field separation","Without priors, spherical-subdomain potential split is non-unique"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The premise that all exterior sources sit above a source-free spherical shell; if that geometric constraint fails, uniqueness is lost.","fun_headline_variants_meta":{"raw":{"variants":["Regional mag-field split nonunique without shell—and unstable even then","Hardy–Hodge: subdomain internal–external separation unique only under shell","Aeromagnetic surveys cannot stably isolate internal from external sources","Shell constraint yields unique yet highly unstable regional field separation","Without priors, spherical-subdomain potential split is non-unique"]},"model":"grok-4.5","effort":"low","cost_usd":0.005074,"raw_usage":{"total_tokens":1381,"prompt_tokens":707,"num_sources_used":0,"completion_tokens":91,"cost_in_usd_ticks":50740000,"prompt_tokens_details":{"text_tokens":707,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":583,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":707,"tokens_out":91,"duration_ms":4864,"temperature":1.0,"reasoning_tokens":583,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T14:55:30.756762+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct a smooth magnetic field that is generated by currents both inside the Earth and below the assumed shell yet coincides, on a chosen open spherical cap, with a purely internal field; any numerical separation algorithm that claims uniqueness must then return the wrong exterior component.","supporting_citations":[],"review_version":1}