{"id":"d1de16cc-c8ae-4081-8650-a96ddc65fcd0","arxiv_id":"2604.05805","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Infinitely many embedded tori in T^4#(S^2×S^2) share a geometric dual and are homotopic and diffeomorphic, but pairwise non-isotopic even after arbitrary external stabilizations.","lead":"The authors construct infinitely many tori in a fixed 4-manifold that are homotopic and diffeomorphic yet not isotopic, even after any number of external stabilizations. This shows that surface isotopy in 4-manifolds can stay exotic after stabilization.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The relative 2-handle homotopy classes may not remain a well-defined, stabilization-invariant separator of the surfaces built from non-homotopic tubing arcs.","rationale":"The reader’s weakest assumption is precisely the load-bearing point: the abstract’s final sentence asserts that relative 2-handle homotopy classes distinguish the surfaces and remain invariant under external stabilizations, but without the full handle calculations this cannot be checked. No stronger internal inconsistency is visible from the abstract alone, and the construction (Norman trick + non-homotopic tubing arcs) is standard enough that the only critical gap is the claimed invariance. Hence the UNVERDICTED status with low confidence is appropriate and needs no adjustment.","tokens_in":1875,"tokens_out":455,"duration_ms":10618,"concrete_test":"Obtain the full text and re-derive the effect of one external stabilization on the handle decomposition: recompute the fundamental group (or relative homotopy classes) of the complement of the 0- and 1-handles after connected sum with S^{2}\times S^{2}; if previously distinct 2-handle classes become homotopic, the claimed invariant is not stabilization-invariant and the infinite family does not survive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the homotopy classes of the 2-handles (relative to the boundary), taken in the complement of the image of the 0- and 1-handles, form an isotopy invariant of the Norman-trick surfaces that (i) separates those obtained from non-homotopic tubing arcs and (ii) is unchanged by arbitrary external stabilizations. The abstract asserts both properties, yet the invariance under external connected sum with S^{2}\times S^{2} is the least secure step: external stabilization alters the ambient 4-manifold and the complement of the 0- and 1-handles, and it is not immediate that the relative homotopy classes of the 2-handles survive or continue to distinguish the family. If that invariance fails, the infinite non-isotopic family collapses after sufficiently many stabilizations, falsifying the strongest claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript claims that there exist infinitely many embedded tori in T^4#(S^2×S^2) sharing a common geometric dual that are pairwise homotopic and diffeomorphic, yet pairwise non-isotopic even after arbitrarily many external stabilizations. The surfaces are produced by applying the Norman trick to a fixed immersed surface along non-homotopic tubing arcs; their isotopy classes are asserted to be separated by the relative homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the 0- and 1-handles.","tokens_in":2042,"tokens_out":765,"duration_ms":15175,"significance":"If the construction and the claimed stabilization-invariant distinction both hold, the result would be a meaningful contribution to the study of exotic surfaces in 4-manifolds: an infinite family of homotopic, diffeomorphic, non-isotopic tori with a common geometric dual that survive arbitrary external stabilizations. Distinguishing such surfaces by relative 2-handle homotopy classes is a natural and potentially reusable technique in the area. The claim sits outside the most common stabilization-collapse phenomena and would therefore be of genuine interest if fully verified.","major_comments":[{"comment":"The abstract's strongest claim—that the surfaces remain pairwise non-isotopic after arbitrarily many external stabilizations—rests on the assertion that the relative homotopy classes of the 2-handles (in the complement of the 0- and 1-handles) form a well-defined isotopy invariant that continues to separate the family after external connected sum with S^2×S^2. External stabilization changes both the ambient 4-manifold and that complement; without a careful argument that the relative classes survive and remain distinguishing, the infinite non-isotopic family may collapse after sufficiently many stabilizations. This invariance step is load-bearing for the central claim and must be checked in full detail.","section":null},{"comment":"The abstract states that the surfaces arise from a fixed immersed surface via the Norman trick along non-homotopic tubing arcs, and that the resulting relative 2-handle classes separate them. It is essential that the manuscript verify that distinct tubing-arc homotopy classes produce genuinely distinct relative 2-handle classes (rather than classes that become equivalent after the Norman tubing or after handle cancellation), and that this distinction is independent of choices in the handle decomposition. Absent that verification, the infinite family may not be infinite up to isotopy.","section":null}],"minor_comments":[{"comment":"Only the abstract was available for this review. A full assessment of the handle calculations, embedding arguments, and the claimed stabilization invariance requires the complete manuscript (including any figures of the immersed surface, tubing arcs, and handle decompositions).","section":null},{"comment":"The abstract would benefit from a brief indication of the ambient fundamental group or the precise sense in which the tori are 'homotopic' (as maps, or as surfaces with fixed dual), to orient the reader before the technical sections.","section":null}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review; the full text of arXiv:2604.05805 was not provided. The logical outline is standard for the area and internally coherent as stated, but the load-bearing stabilization-invariance of the relative 2-handle classes cannot be verified from the abstract alone. I recommend obtaining the full manuscript before a definitive editorial decision. If the full paper carefully establishes that invariance, the result would likely warrant minor or major revision rather than rejection; if the invariance is only sketched, major revision would be appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: this abstract claims an infinite family of embedded tori in T^4#(S^2\times S^2) that share a geometric dual, are homotopic and diffeomorphic, yet remain pairwise non-isotopic after any number of external stabilizations. That is a clean, useful existence result inside smooth 4-manifold topology if the details hold.\n\nWhat looks new is the concrete construction: fix one immersed surface, run the Norman trick along tubing arcs that are non-homotopic, and separate the resulting surfaces by relative homotopy classes of the 2-handles (taken in the complement of the 0- and 1-handles). The Norman trick itself is classical, and non-isotopic surfaces after stabilization have been studied before, but an infinite family that keeps a common geometric dual and survives arbitrary external stabilizations is a solid incremental contribution. The abstract states the claim without circularity or free parameters; the logical skeleton is coherent for the area.\n\nThe soft spot is exactly the one the stress-test flags, and it is load-bearing rather than minor. The distinguishing invariant must both separate the surfaces coming from non-homotopic arcs and remain unchanged under external connected sum with S^2\times S^2. External stabilization changes the ambient manifold and the complement of the 0- and 1-handles, so it is not automatic that those relative 2-handle classes survive or continue to distinguish. Without the full text we cannot check the handle calculations, the non-homotopy of the arcs, or the invariance argument. That is why confidence stays low and the paper is still unverdicted.\n\nWho it is for: people who work on surfaces in 4-manifolds, isotopy after stabilization, and geometric duals. A serious referee should see the full paper; the claim is important enough inside the subfield and the outline is standard enough that desk rejection would be premature. I would not cite it yet (abstract only), but I would bring it to a reading group once the PDF appears, mainly to pressure-test the stabilization step. Send it to peer review.","headline":"Abstract-only existence claim of infinite non-isotopic tori in T^4#(S^2\times S^2) surviving external stabilizations; standard Norman-trick outline, but the stabilization-invariant separator is unverified.","tokens_in":2671,"tokens_out":538,"would_cite":false,"duration_ms":5617,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K40","57R52","57R40"],"pacs":[],"model":"grok-4.5","headline":"Infinitely many tori in T^4#(S^2×S^2) share a geometric dual yet remain non-isotopic after any number of external stabilizations.","keywords":["4-manifolds","embedded tori","isotopy","external stabilization","Norman trick","geometric dual","T^4#(S^2×S^2)","handle homotopy"],"falsifier":"An explicit ambient isotopy (possibly after a finite number of external stabilizations) that carries one of the constructed tori onto another built from a non-homotopic tubing arc, or a calculation showing that the relative 2-handle homotopy classes become equivalent after some stabilization.","tokens_in":2746,"feed_emoji":"🧵","tokens_out":903,"duration_ms":31327,"temperature":0.7,"pith_summary":"This paper constructs infinitely many embedded tori inside the 4-manifold T^4#(S^2×S^2) that all share one common geometric dual sphere. The tori are homotopic to one another and diffeomorphic as abstract surfaces, yet they lie in distinct isotopy classes, and those classes stay distinct even after any number of external stabilizations. The surfaces are built from a single fixed immersed surface by the Norman trick, tubing along different arcs that are not homotopic to each other. The authors show that the resulting isotopy classes can be told apart by looking at the homotopy classes of the 2-handles (relative to the boundary) that appear once the 0- and 1-handles of a handle decomposition have been removed. For a reader interested in 4-dimensional topology, the result supplies a concrete infinite family of surfaces that cannot be made isotopic by the usual stabilization moves.","feed_headline":"Infinitely many non-isotopic tori share one dual in T^4#(S^2×S^2)","feed_subtitle":"Homotopic, diffeomorphic surfaces stay distinct after any number of external stabilizations","key_machinery":"The Norman trick applied to a fixed immersed surface via non-homotopic tubing arcs, together with the resulting relative homotopy classes of 2-handles in the complement of the image of the 0- and 1-handles, which serve as an isotopy invariant that survives arbitrary external stabilization.","core_discovery":"There exist infinitely many embedded tori in T^4#(S^2×S^2) that possess a common geometric dual, are mutually homotopic and diffeomorphic, yet are pairwise non-isotopic, and remain non-isotopic after arbitrarily many external stabilizations; they arise by applying the Norman trick to one fixed immersed surface along non-homotopic tubing arcs, and their isotopy classes are distinguished by the relative homotopy classes of the 2-handles in the complement of the 0- and 1-handles.","pith_inferences":["Similar non-isotopic families may appear in other simply-connected 4-manifolds that admit a geometric dual sphere and enough room for the Norman trick.","The same relative 2-handle invariant could be used to detect non-isotopy for higher-genus surfaces or for knotted spheres with duals.","If the invariant remains non-trivial after stabilization, it suggests that external stabilization alone cannot erase all tubing-arc data."],"forward_implications":["Infinitely many distinct isotopy classes of tori share a single geometric dual sphere inside T^4#(S^2×S^2).","Homotopy and diffeomorphism type of an embedded surface, even together with a geometric dual, do not determine its isotopy class after external stabilization.","The relative homotopy type of 2-handles after removing 0- and 1-handles can serve as a practical invariant for distinguishing stabilized surfaces.","The Norman trick with non-homotopic arcs systematically produces infinite non-isotopic families from a single immersion."],"fun_headline_variants":["Infinitely many non-isotopic tori share one dual in T^4#(S^2×S^2)","Homotopic diffeomorphic tori stay non-isotopic past all external stabilizations","Norman trick on fixed immersion yields infinite non-isotopic tori via distinct tubes","Non-homotopic arcs give infinitely many tori with shared dual yet distinct isotopy","Tori distinguished by 2-handle homotopy classes remain non-isotopic after stabilizations"],"cache_read_input_tokens":128,"weakest_assumption_plain":"That the relative homotopy classes of the 2-handles in the complement of the 0- and 1-handles form a well-defined isotopy invariant of the constructed surfaces that is unchanged by external stabilizations and separates surfaces coming from non-homotopic tubing arcs.","fun_headline_variants_meta":{"raw":{"variants":["Infinitely many non-isotopic tori share one dual in T^4#(S^2×S^2)","Homotopic diffeomorphic tori stay non-isotopic past all external stabilizations","Norman trick on fixed immersion yields infinite non-isotopic tori via distinct tubes","Non-homotopic arcs give infinitely many tori with shared dual yet distinct isotopy","Tori distinguished by 2-handle homotopy classes remain non-isotopic after stabilizations"]},"model":"grok-4.5","effort":"low","cost_usd":0.003198,"raw_usage":{"total_tokens":1057,"prompt_tokens":690,"num_sources_used":0,"completion_tokens":119,"cost_in_usd_ticks":31980000,"prompt_tokens_details":{"text_tokens":690,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":248,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":690,"tokens_out":119,"duration_ms":7753,"temperature":1.0,"reasoning_tokens":248,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T09:12:55.437575+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"An explicit ambient isotopy (possibly after a finite number of external stabilizations) that carries one of the constructed tori onto another built from a non-homotopic tubing arc, or a calculation showing that the relative 2-handle homotopy classes become equivalent after some stabilization.","supporting_citations":[],"review_version":2}