{"id":"f5915141-95f6-400a-b2e3-295d1240af4e","arxiv_id":"2607.27751","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Iterated rim surgery produces topologically isotopic genus-g surfaces whose smoothly extendable mapping classes lose one projective homology symmetry per step, ending in projective rigidity.","lead":"This paper shows that surfaces in 4-manifolds can be topologically identical yet progressively lose smooth ambient symmetries under carefully chosen rim surgeries. It also embeds a genus-40 surface in a hyperbolic 4-manifold whose smooth and topological symmetry groups are both trivial.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified for the central claim (Theorem B); the proof is internally sound. Main risk is the external [38, Lemma 5.1] dependency in Theorem C, which is secondary.","rationale":"The central claim, as captured by the reader's strongest_claim, is Theorem B. Its proof is constructive and depends only on the Fintushel–Stern rim-surgery formula, standard Lefschetz fibration theory, and self-contained convex-geometric lemmas. I checked the key steps: the rim Newton profile's invariance (Lemma 5), the stabilizer computation (Lemmas 7 and 15), the large-perturbation lemma (Lemma 8), and the full-monodromy fibrations (Lemma 11). No internal inconsistency or circularity emerged. The reader's conditional verdict is driven by two external dependencies: Pyronneau's unpublished preprint (used only for a remark about topological flexibility in Theorem A) and the self-embedding reading of [38, Lemma 5.1] (used for Theorem C). Neither affects the central quantitative filtration of Theorem B. Therefore the reader's weakest_assumption does not identify the load-bearing concern for the central claim; it identifies a valid but secondary concern. The verdict remains CONDITIONAL because the secondary Theorem C dependency is real and unverified, but no change to the reader's verdict is needed.","tokens_in":21421,"tokens_out":48460,"duration_ms":412141,"concrete_test":"As a verification of the central mechanism, explicitly compute the monodromy group of the twisted fiber sum in Lemma 11 for g=2: list the vanishing cycles in the factorization W_2 (ψ W_2 ψ^{-1}) and confirm their Dehn twists generate Mod(Σ_2). If they do, the full-monodromy input is secured; the rest of Theorem B follows from the algebraic lemmas.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I reviewed Theorem B's proof in detail. The rim Newton profile is well-defined and natural (Lemma 5); Lemma 7's basis construction is correct (the intersection graph is a path and distinct weights force line preservation); Lemma 8's perturbation argument is rigorous (bounded operator norms imply finite possibilities, then a limiting argument yields A(Z)=Z); Lemma 15's strict filtration via symplectic transvections works. The construction in Lemma 11 uses standard full-monodromy Lefschetz fibrations (E(2) for g=1, twisted fiber sum of hyperelliptic fibrations for g≥2), and the subsequent surgery argument in Theorem 16 produces rim Newton profiles consisting of a single translation class [2Z_i] of affine dimension i, yielding pairwise nondiffeomorphism and the upper bounds ρ(E∞(X_g,F_i))⊆P_i. I found no internal gap, circularity, or hidden assumption in the proof of Theorem B. The reader's weakest_assumption concerns Lemma 22 and Theorem C, which is a complementary result; even if that concern landed, Theorems A and B would stand. Hence no load-bearing objection to the central claim.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a convex-geometric refinement of the relative Seiberg-Witten invariant, the \"rim Newton profile,\" and uses it to study exotic knottings of surfaces in 4-manifolds through extendable mapping class groups. Theorem A states that, under hypotheses F^2≥0, π1(X\\νF)=1, and nonzero relative invariant, iterated rim surgery produces a topologically isotopic, projectively rigid exotic copy. Theorem B, the central result, constructs for each genus g a 4-manifold X_g and surfaces F_0,...,F_{2g} that are mutually topologically isotopic, topologically flexible, pairwise nondiffeomorphic, and with successively constrained projective homological symmetry groups P_0⊋...⊋P_{2g}={1}. Theorem C uses hyperbolic geometry to produce a genus-40 totally geodesic surface with trivial smooth and topological extendable mapping class groups. An appendix shows that the relative diffeomorphism or homeomorphism group determines the pair.","tokens_in":21624,"tokens_out":25820,"duration_ms":250069,"significance":"If the main results hold, this is a significant contribution: Theorem B gives a quantitative, genus-uniform filtration of exotic knottedness, where each rim surgery kills one projective homological symmetry while preserving topological flexibility. The proof is unusually checkable: Lemma 5 (naturality of the rim Newton profile), Lemma 7 (stabilizer of a weighted zonotope is ±I), Lemma 8 (large-scale detection of the added zonotope), and Lemma 15 (strict filtration by symplectic transvections) are all proved carefully from stated assumptions, with no fitted parameters or circularity. The rim-surgery formula is cited explicitly to Fintushel-Stern, and the initial flexible fiber inputs are supplied by full-monodromy Lefschetz fibrations. The main caveat is Theorem C, which rests on a strong external lemma that is not quoted; this does not affect the central Theorem B but must be resolved.","major_comments":[{"comment":"The proof of Theorem C depends on a specific reading of [38, Lemma 5.1] that is not reproduced. The lemma must embed the given arithmetic hyperbolic manifold itself (not merely a finite cover) totally geodesically in a closed arithmetic hyperbolic manifold of one higher dimension, preserving compactness, and must apply twice: once to the genus-40 surface and once to the resulting compact 3-manifold. This is load-bearing for Theorem C and Remark 23. Please quote the lemma, state its hypotheses, and verify them for both applications, including the compactness claim. If [38] only provides finite-cover embeddings or noncompact outputs, Theorem C and Remark 23 do not follow as written.","section":"Section 2.4, Theorem A and Remark 18"},{"comment":"The topological-flexibility clause in Theorem A ('If F is ordinary...') is deferred to Pyronneau's unpublished preprint [45, Theorem 4.3]. Since the clause is part of a theorem statement, it should not rest on an unavailable manuscript without at least a precise statement of the cited theorem. Either incorporate a proof, or state Theorem A without this clause and record topological flexibility as conditional on [45].","section":"Section 2.4, Theorem A and Remark 18"}],"minor_comments":[{"comment":"The displays labeled (2.5) and (3.3) appear inside proofs without being integrated into the global equation numbering; this is confusing for cross-referencing. Please renumber or use unnumbered displays.","section":"Proof of Theorem 9 and Theorem 16"},{"comment":"In the blow-up reduction, the verification that π1(\\tilde X\\ν\\tilde F)=1 after blowing up n points on F is implicit. It is true, but should be stated explicitly, since the rim-surgery formula and Boyer's theorem are applied to the proper transform.","section":"Section 2.4, F^2>0 case"},{"comment":"The notation 'W_g := h_g^2 = 1, h_g = ...' is compressed and potentially misleading. Spell out that the monodromy factorization is h_g^2=1 with h_g given by the displayed word.","section":"Section 3.1, Lemma 11"},{"comment":"Terms such as 'simplest type' and 'admissible ternary quadratic form' are used without definitions; please add precise references or definitions so the hypotheses of [38, Lemma 5.1] can be checked.","section":"Section 4, Lemma 22"}],"recommendation":"major_revision","confidential_remarks":"The central Theorem B appears sound and the authors should be credited for the detailed, checkable proofs of the Newton-polytope machinery. The revision should focus on the two external dependencies: quote and verify [38, Lemma 5.1] for Theorem C, and do not leave a theorem clause resting on an unpublished preprint. If these are resolved, the paper would be a strong fit for the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the central theorem is in good shape. The rim Newton profile is a genuine addition to the toolkit for exotic surface knottings, and the proof of Theorem B survives close reading. The weak spot is Theorem C, whose proof depends on a nonstandard reading of Martelli–Riolo–Slavich that the authors should be asked to justify.\n\nWhat's new: the profile turns relative Seiberg–Witten support into convex geometry, and the iteration with torus knots gives a quantitative filtration of knottedness. Lemma 7 does what it claims, and Lemma 15's strict chain is correct. I checked the proof of Theorem 16 (the heart of Theorem B): no hidden assumptions, no parameter fitting, no circularity. The claim that the surfaces are topologically flexible does not depend on Pyronneau's preprint, as you thought; Lemma 3 plus smooth flexibility of F_0 covers that. So the topological-flexibility dependency you flagged is actually not load-bearing for the main result.\n\nThe real soft spot is Lemma 22. It asserts that [38, Lemma 5.1] embeds the input manifold itself into a closed arithmetic hyperbolic 4-manifold, not merely a finite cover. The authors do not reproduce the lemma, and that reading is nonstandard for arithmetic embedding results. If the reading is wrong, the genus-40 rigid example is not established. This is secondary—Theorems A and B stand without it—but the abstract sells Theorem C as a headline, so the authors should be pushed to verify the lemma statement or supply a direct proof. They also do not flag this interpretive risk themselves, which is an oversight.\n\nThe blow-up reduction in Theorem A looks fine on close reading; isotoping the blow-up points on the surface is standard, and isotopy extension lifts it. The paper is honest about the non-explicit nature of the hyperbolic manifolds and about the coarse nature of the rim profile. Citation practice is reasonable; self-citations in [7] and [50] are used for peripheral facts, not to define the target construction.\n\nBottom line: this deserves serious peer review. The referee should check Lemma 22 carefully and, for the remark about topological flexibility when the starting surface is not flexible, will need Pyronneau's preprint or a replacement argument. For the main theorem, I see no obstruction.","headline":"Theorems A and B are solid and the rim Newton profile is a real new tool; Theorem C rests on an unverified reading of a cited lemma.","tokens_in":22221,"tokens_out":4873,"would_cite":true,"duration_ms":43789,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57K40","57R58"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that ambient symmetry loss can grade exotic knottedness of surfaces in 4-manifolds, with each successive rim surgery removing one more projective homological symmetry until none remain.","keywords":["exotic knotted surfaces","4-manifolds","extendable mapping class groups","rim surgery","relative Seiberg-Witten invariants","Newton polytopes","projective rigidity","hyperbolic 4-manifolds"],"falsifier":"Check the genus-1 model from Example 17: the final rectangle [−2a,2a]+[−4b,4b] must have stabilizer ±I in PSp(2,Z); if a smoothly extendable mapping class of the twice-rim-surgered torus acts by a symplectic matrix that does not preserve both primitive lines, the symmetry-breaking theorem fails. For the hyperbolic theorem, verify the quoted embedding lemma in a concrete case by exhibiting a totally geodesic embedding of the genus-40 surface into a closed hyperbolic 4-manifold; failure to produce such an embedding, or a nontrivial homeomorphism of the pair fixing the surface, would contradict T","tokens_in":21193,"feed_emoji":"🔗","tokens_out":9044,"duration_ms":93012,"temperature":0.7,"pith_summary":"This paper claims that exotic knottedness of surfaces in 4-manifolds can be quantified by the gradual loss of ambient symmetries. For every genus g, the authors construct 2g+1 embedded surfaces in a single simply connected 4-manifold that are all topologically isotopic and topologically flexible, yet each successive rim surgery removes one more projective homological symmetry from the smoothly extendable mapping class group, ending at a projectively rigid surface. A complementary hyperbolic construction yields a closed genus-40 totally geodesic surface whose smooth and topological extendable mapping class groups are both trivial. The method extracts convex-geometric data—Newton polytopes—from the relative Seiberg-Witten invariant and shows that the zonotopes added by rim surgery act as symmetry detectors. If correct, knottedness has finer gradations than 'knotted versus unknotted,' and ambient symmetry groups provide a quantitative probe of those gradations.","feed_headline":"Exotic surfaces lose ambient symmetries one at a time","feed_subtitle":"A new construction produces topologically identical surfaces whose smooth symmetries shrink stepwise until none remain.","key_machinery":"The engine is the rim Newton profile: for each orbit of the rim-torus lattice in the affine exponent set of the relative Seiberg-Witten invariant, take the Newton polytope of the restricted Laurent polynomial, up to translation. Rim surgery—cutting out a torus neighborhood of a curve on the surface and gluing in the complement of a knot—changes the profile by Minkowski-adding a segment determined by the knot's symmetrized Alexander polynomial, so iterated surgery builds centrally symmetric zonotopes. The sign symmetry of the knot polynomial makes these zonotopes centrally symmetric, which is exactly why the final stabilizer contains ±I but, with an integral basis and pairwise distinct weight","core_discovery":"The central claim is that relative Seiberg-Witten invariants, read through Newton polytopes of their supports, detect ambient symmetries of embedded surfaces after rim surgery. Rim surgery along a curve multiplies the relative invariant by a knot polynomial evaluated at the rim-torus variable; the paper packages the invariant's support into a finite multiset of translation classes of polytopes, the rim Newton profile, which is invariant under diffeomorphisms of pairs. Starting from a smoothly flexible fiber of a full-monodromy genus-g surface fibration, whose rim Newton profile consists only of points, each of 2g carefully chosen rim surgeries adds a centered segment to every polytope in the","pith_inferences":["The same profile technique should apply to any relative invariant with a product formula under a surgery operation: whenever an initial flexibility condition forces point polytopes, the stabilizer chain measures symmetry loss without needing the full invariant.","A natural test is whether the actual images ρ(E^∞(X_g,F_i)) are themselves nested, not merely the upper bounds P_i; finding an example where an extension realizes a transvection outside P_i would show whether the filtration is sharp.","Because the rim Newton profile forgets Laurent polynomial coefficients, it cannot distinguish a polytope from its negative; a coefficient-sensitive refinement could eliminate the residual ±I ambiguity and potentially produce smoothly rigid, topologically flexible surfaces, which the paper leaves open.","The hyperbolic construction suggests a broader source of rigid pairs: any closed arithmetic hyperbolic surface with trivial isometry group defined over a number field other than the rationals should embed totally geodesically into infinitely many closed hyperbolic 4-manifolds, yielding the same rigid surface in varying ambient manifolds."],"forward_implications":["For every genus g, there exist 2g+1 pairwise nondiffeomorphic surface pairs in a simply connected 4-manifold that are all topologically isotopic, so a single topological isotopy class can contain arbitrarily long finite chains of exotic surfaces.","Knottedness becomes graded: each rim surgery provably removes one more primitive homology line from the possible smooth ambient symmetries, giving a quantitative filtration of the smooth pair.","Any surface with nonnegative self-intersection, simply connected complement, and nonzero relative Seiberg-Witten invariant admits a projectively rigid exotic copy; in genus one, only the hyperelliptic involution can survive.","A closed hyperbolic 4-manifold can contain a totally geodesic surface with no nontrivial smoothly or topologically extendable mapping classes, and every topologically isotopic copy of that surface is likewise rigid in both categories.","Rim Newton profiles distinguish pairs by affine dimension and lattice-point count, providing an effective computable invariant for detecting exotic knotted surfaces."],"fun_headline_variants":["Exotic surfaces dim symmetries step by step","Topologically same surfaces, smooth symmetries shrink","New knotted surfaces lose smooth symmetries gradually","Symmetry loss revealed in exotic surface pairs","Rim surgery yields surfaces with fading symmetries"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The hyperbolic theorem rests on a quoted lemma, not reproduced in the paper, asserting that a compact arithmetic hyperbolic 3-manifold over a field different from the rationals embeds itself—not merely a finite cover—totally geodesically into a closed arithmetic hyperbolic 4-manifold; if that reading is wrong, only the genus-40 surface itself, not the rigid pair, is established.","fun_headline_variants_meta":{"raw":{"variants":["Exotic surfaces dim symmetries step by step","Topologically same surfaces, smooth symmetries shrink","New knotted surfaces lose smooth symmetries gradually","Symmetry loss revealed in exotic surface pairs","Rim surgery yields surfaces with fading symmetries"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1213,"prompt_tokens":710,"completion_tokens":503,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":433}},"tokens_in":454,"tokens_out":503,"duration_ms":5033,"temperature":1.0,"reasoning_tokens":433,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T01:57:03.290700+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the genus-1 model from Example 17: the final rectangle [−2a,2a]+[−4b,4b] must have stabilizer ±I in PSp(2,Z); if a smoothly extendable mapping class of the twice-rim-surgered torus acts by a symplectic matrix that does not preserve both primitive lines, the symmetry-breaking theorem fails. For the hyperbolic theorem, verify the quoted embedding lemma in a concrete case by exhibiting a totally geodesic embedding of the genus-40 surface into a closed hyperbolic 4-manifold; failure to produce such an embedding, or a nontrivial homeomorphism of the pair fixing the surface, would contradict T","supporting_citations":[],"review_version":1}