{"id":"e4f86efe-7cb6-4414-a679-79e591bf1c7f","arxiv_id":"2412.12889","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Analytical obstructions to weak approximation by smooth maps exist for every integer Sobolev exponent p≥2, and for spheres S^{2n} when p=4n-1.","lead":"For every integer p≥2, the authors build a compact target manifold so that some p-integrable Sobolev map is not a weak limit of smooth maps. The result settles the weak approximation problem for all integer exponents, extending the single known counterexample at p=3.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4 is stated in the abstract and introduction, but Section 5 explicitly says 'we omit the details' for its proof; the higher-order sphere extension is therefore unsupported, leaving the paper's full central claim conditional.","rationale":"Good-faith reading: the paper's main construction for integer p is detailed and appears internally consistent. I checked the key estimates in Proposition 2.1: the parameter choices (2.13)-(2.24) are consistent, the use of Proposition 2.2 on overlapping balls is handled by merging, and the application of the VMO homotopy criterion is standard. I therefore do not share a specific technical objection to Proposition 2.1; it is load-bearing but not evidently wrong. The decisive issue is completeness: the abstract promises higher-order extensions, and Theorem 1.4 is explicitly left without proof. The reader's rationale already flags this red flag, though the weakest_assumption field points to Proposition 2.1; my concern is the omitted proof. A conditional verdict is appropriate: the main theorem may well be correct, but the full claim as stated is not established. If the authors supply the missing proof or explicitly relegate Theorem 1.4 to a remark, the verdict could move to ACCEPT for Theorems 1.1-1.3.","tokens_in":37231,"tokens_out":42260,"duration_ms":385502,"concrete_test":"Write out the complete proof of Theorem 1.4: construct the smooth periodic map u_sm (the sphere analogue of Lemma 5.1) and verify that the Hopf-degree transport lower bound of Corollary 4.3 holds for every weakly converging sequence in W^{s,p}(Q_ell,S^{2n}), using (5.2) to bound the first-order energy. If the proof fails for non-integer s or for s close to 1, or if the Hopf-invariant defect cannot be controlled on L^{4n-1}, then the theorem should be downgraded to a conjecture or restricted to s=1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is not just Theorems 1.1-1.2 but also the advertised extension to all W^{s,p} with sp in N\\{0,1} (Theorems 1.3-1.4). Section 5 gives a genuine proof sketch for Theorem 1.3 via Lemma 5.1, the Gagliardo-Nirenberg inequality (5.2), and the higher-order uniform boundedness principle. For Theorem 1.4, however, the text states: 'Since the procedure is actually simpler than for Theorem 1.1... we omit the details.' This is a missing proof for a stated theorem, not an internal inconsistency. The transfer is not automatic: it requires a smooth periodic approximant of the Whitehead-product map that controls all derivatives near the singularity set, and it requires the Hopf-degree transport argument of Proposition 4.2/Corollary 4.3 to survive under the Gagliardo-Nirenberg energy comparison. Neither step is written. The reader's weakest_assumption, Proposition 2.1, is intricate but I found no concrete error in it after checking the parameter choices and the merging-balls step; the load-bearing gap in the manuscript as submitted is the unproved Theorem 1.4. This does not affect the validity of Theorem 1.1 or the plausibility of Theorem 1.2, but it means the paper overclaims its full scope.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for every integer p≥2 there exists a compact Riemannian manifold N such that for every M with dim M>p, the sequential weak closure of smooth maps in W^{1,p}(M,N) is a proper subset of W^{1,p}(M,N). The proof is quantitative: it constructs a periodic singular retraction onto the p-skeleton of the (p+1)-torus and shows its relaxed energy on cubes of side ℓ grows like ℓ^{p+1} ln ℓ while its Sobolev energy grows like ℓ^{p+1}, then applies a nonlinear uniform boundedness principle. For p=4n−1 the target can be S^{2n}, via a Whitehead-product construction with Hopf invariant 2. The paper also states extensions to higher-order spaces W^{s,p} with sp∈N\\{0,1} (Theorems 1.3 and 1.4) and provides a proof sketch in Section 5.","tokens_in":37498,"tokens_out":8028,"duration_ms":74485,"significance":"The first-order results are a major advance: they settle the weak approximation problem in the negative for every integer exponent p≥2, generalizing Bethuel's p=3 counterexample and showing that analytical obstructions are not a sporadic phenomenon. The detailed estimates for the relaxed energy, the bubbling proposition (Proposition 2.1), and the use of a uniform boundedness principle are valuable tools. The paper is carefully written and the proofs of Theorems 1.1 and 1.2 appear sound. However, the advertised higher-order results are not proved at the same level: Theorem 1.4 is explicitly not proved, and Theorem 1.3 is only sketched. The paper as submitted therefore substantiates the first-order claims but overclaims the full scope.","major_comments":[{"comment":"Theorem 1.4 is stated in the abstract and in the introduction as one of the paper's main outcomes, but its proof is reduced to the sentence “we omit the details.” This is not an automatic consequence of the first-order sphere construction: it requires a smooth periodic approximant of the Whitehead product map with derivative bounds near the singularity set (an analogue of Lemma 5.1 for the sphere construction), and it requires the Hopf-degree transport statement of Proposition 4.2 and Corollary 4.3 to remain valid under the Gagliardo–Nirenberg energy comparison (5.2). Neither step is written. Because the theorem is advertised as proved, the manuscript as submitted overclaims its scope.","section":"Section 5, Theorem 1.4"},{"comment":"The proof of Theorem 1.3 is a sketch rather than a complete proof. In particular, the claim that Proposition 3.5 “remains valid” when condition (3.13) is weakened to u(Q_{ℓ,α})⊂Q_{ℓ,α}+B_{C_1 ε} is not demonstrated, and this weakened lower bound is exactly what transfers the relaxed-energy growth from v to v_lip. While the statement is plausible for sufficiently small ε, the verification should be supplied if Theorem 1.3 is to be counted among the paper's theorems; otherwise the result should be presented as conditional.","section":"Section 5, proof of Theorem 1.3"},{"comment":"The topological description N_λ ≃ (T^n\\ B^n)×S^{m−1} ∪_∂ S^{n−1}×B^m and the connected-sum description for m=1 are introduced with the phrase “without giving detailed arguments.” These descriptions are not needed for the proof of the counterexample, but they are used in the introduction to advertise the structure of the target manifold; the authors should either provide the missing details or clearly label these assertions as heuristic.","section":"Section 3.5, Remark 3.10"}],"minor_comments":[{"comment":"There is a typo: “strong appproximation” should be “strong approximation.”","section":"Section 1.1"},{"comment":"In the proof of Proposition 4.1, “Whithead product” should be “Whitehead product.”","section":"Section 4.1"},{"comment":"The formula for V(x) should be parenthesized, e.g. V(x):=∑_{j=1}^n (1+x_{2j-1})/2 + ∑_{j=1}^m |x_{2n+j}|^2, to avoid ambiguity in the scope of the product.","section":"Section 3.5, definition of V"},{"comment":"The notation B_{√λ/2}(0) should specify that this is the ball in the R^m factor of T^n × R^m, since the ambient Euclidean space has dimension 2n+m.","section":"Section 3.5, Remark 3.10"},{"comment":"The sentence “the case s∉N might be somehow more subtle” is followed by a brief additivity argument; since the paper otherwise gives precise estimates, a slightly more explicit treatment of the fractional Gagliardo–Nirenberg step would improve readability.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper whose first-order results, Theorems 1.1 and 1.2, are likely correct and important. The main obstacle to acceptance is the discrepancy between the abstract and introduction, which advertise Theorems 1.3 and 1.4 as proved, and Section 5, which provides only a sketch and explicitly omits the proof of Theorem 1.4. I recommend major revision: either supply a complete proof of Theorem 1.4 (and fill the unstated verification in Theorem 1.3), or restate the higher-order results as conditional and remove them from the paper's central claims. The concern is about completeness, not correctness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What to know: The paper's core theorems 1.1 and 1.2 are a real advance. For every integer p≥2 they construct a compact target N (built from the p-skeleton of the (p+1)-torus) for which weak approximation fails whenever dim M>p, and for p=4n-1 the target can be S^{2n}. That replaces Bethuel's isolated p=3 counterexample with a general theorem and an infinite sphere family. The proof is detailed and the mechanism is clear: a periodic singular retraction has relaxed energy growing like ℓ^{p+1} ln ℓ, versus Sobolev energy ℓ^{p+1}; the bubbling proposition 2.1, the conical joint degree estimate, and the branched transport lower bound are all written out and hang together.\n\nCredit where due: I checked the parameter choices in Prop 2.1 and the merging-balls step; nothing is obviously wrong. The discussion of why the minimal-connection strategy cannot work for this target is honest and useful.\n\nNow the soft spot, and it's a real one: Theorem 1.4, the higher-order sphere case, is stated in the abstract and intro but Section 5 says 'we omit the details.' That is not a cosmetic omission. The transfer from W^{1,p} to W^{s,p} needs a smooth periodic approximant controlling all derivatives near the singular set, and the Hopf-degree transport of Prop 4.2–Cor 4.3 to survive the Gagliardo–Nirenberg comparison. Neither step is written. So the paper as submitted overclaims its full scope. Theorem 1.3 has a genuine proof sketch via Lemma 5.1 and appears plausible; Theorem 1.4 is unsupported. There are also a few topological statements in Prop 3.9/Remark 3.10 given without full detail, but those are less central and can be checked directly.\n\nWho is it for: specialists in Sobolev mappings and calculus of variations. It deserves a serious referee: the main results are important and the proof is substantial. I'd send it to review with the request that the authors either prove Theorem 1.4 or explicitly downgrade it to a conjecture.","headline":"Strong paper proving analytical obstructions for every integer p≥2, but the higher-order sphere extension (Thm 1.4) is stated without proof and should be either supplied or downgraded.","tokens_in":38020,"tokens_out":2545,"would_cite":true,"duration_ms":23311,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58D15","46E35","46T10","58C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every integer $p\\ge 2$ there is a compact manifold target into which some Sobolev map cannot be weakly approximated by smooth maps.","keywords":["weak approximation of Sobolev mappings","relaxed energy","analytical obstruction","Brouwer degree estimate","Whitehead product","Hopf invariant","bubbling of Sobolev maps","higher-order Sobolev spaces"],"falsifier":"Compute or bound from above the relaxed energy of the periodic singular retraction for the smallest case $p=2$, $n=3$: if one can exhibit, for every $\\ell$, a sequence of smooth maps $u_k$ into $N_0$ (or into the compact manifold $N_\\lambda$) with $E(u_k,Q_\\ell)\\le C\\ell^3$ and $u_k\\to u$ a.e., then $E^{\\mathrm{rel}}(u,Q_\\ell)$ would be $O(\\ell^3)$ and the claimed $\\ell^3\\ln\\ell$ growth\\u2014and with it the main theorem\\u2014would be refuted. Equivalently, produce two maps $u,v$ satisfying the hypotheses of Proposition 2.1 for which no finite family of disjoint balls with the stated energy and homotopy properties exists; that would break the load-bearing bubbling premise.","tokens_in":37026,"feed_emoji":"🌐","tokens_out":14082,"duration_ms":114706,"temperature":0.7,"pith_summary":"The paper establishes that weak approximation by smooth maps fails for every integer exponent $p\\ge 2$. Specifically, for each such $p$ it constructs a compact manifold $N$ (retracting onto the $p$-skeleton of a $(p+1)$-torus, and for a parameter choice just a connected sum of two $(p+1)$-tori) such that whenever the domain dimension exceeds $p$, there is a finite-energy map $M\\to N$ that is not the weak limit of any sequence of smooth maps. The obstruction is analytical, not topological: it is produced by a periodic singular retraction whose relaxed energy grows like $\\ell^{p+1}\\ln\\ell$ on cubes of side $\\ell$, while its ordinary Sobolev energy only grows like $\\ell^{p+1}$. For $p=4n-1$ the same phenomenon occurs with the sphere $S^{2n}$ as the target, generalizing the previously known $p=3$ case. The same results are obtained for higher-order Sobolev spaces $W^{s,p}$ with $sp\\in\\mathbb{N}$ and $sp\\ge 2$. A sympathetic reader would care because this closes the weak-approximation problem for integer exponents: density of smooth maps in the weak topology can no longer be expected once the relevant homotopy condition fails, even on a ball.","feed_headline":"Weak smooth approximation fails for every integer p≥2","feed_subtitle":"A periodic retraction pays an extra logarithmic energy cost that no smooth approximant can avoid.","key_machinery":"The central object is the periodic singular retraction $u:\\mathbb{R}^{p+1}\\setminus\\Sigma\\to\\tilde N_0$ into the $p$-skeleton of the standard cubical decomposition, defined on each unit cube centered at $\\sigma\\in\\Sigma$ by $u(x)=\\sigma+(x-\\sigma)/(2|x-\\sigma|_\\infty)$, where $\\Sigma=(\\mathbb{Z}+1/2)^{p+1}$. The paper's proof that the relaxed energy cannot match the Sobolev energy rests on two structural ingredients. The first is a finite-scale bubbling statement (Proposition 2.1): two continuous Sobolev maps with bounded energy that are close in average distance are homotopic outside finitely many small balls, and the energy inside those balls controls the homotopy gap; this localizes the difference between an approximating sequence and the singular limit. The second is a conical joint estimate on Brouwer degrees (Proposition 3.2), which bounds the total degree with respect to a lattice of singularities by an energy integral over cones; applied to the boundary of cubes in Proposition 3.5, it yields the defect lower bound (3.14) involving the difference $|Dv|^p-|D(\\Theta_{\\ell,\\alpha}\\circ v)|^p$, where $\\Theta_{\\ell,\\alpha}$ is the retraction onto the cube $Q_{\\ell,\\alpha}$. Iterating the inequality $E^{\\mathrm{rel}}(Q_{5\\ell})\\ge 5^{p+1}E^{\\mathrm{rel}}(Q_\\ell)+c\\ell^{p+1}$ produces the log-factor growth. For the sphere target, the analogous mechanism is the Whitehead product construction of a periodic map with Hopf invariant $2$ per singularity, controlled by an integral estimate on the Hopf degree.","core_discovery":"On the paper's own terms, the central discovery is that analytical obstructions to weak density of smooth maps are a general phenomenon, occurring for every integer exponent $p\\ge 2$. The proof identifies the mechanism: for the singular retraction $u$ of $\\mathbb{R}^{p+1}$ onto the $p$-skeleton $\\tilde N_0$ of the standard cubical lattice, the relaxed energy satisfies $\\liminf_{\\ell\\to\\infty} E^{1,p}_{\\mathrm{rel}}(u,Q_\\ell)/\\ell^{p+1}=\\infty$, although the Dirichlet energy $E^{1,p}(u,Q_\\ell)$ is comparable to $\\ell^{p+1}$. The superlinear growth is obtained by iterating a strengthened self-similarity estimate $E^{1,p}_{\\mathrm{rel}}(u,Q_{5\\ell})\\ge 5^{p+1}E^{1,p}_{\\mathrm{rel}}(u,Q_\\ell)+c\\ell^{p+1}$, whose proof uses a finite-scale bubbling proposition for Sobolev maps into uniform Lipschitz neighborhood retracts and a conical joint estimate on Brouwer degrees. The relaxed-energy gap is transferred from the noncompact skeleton to a compact manifold $N_\\lambda$ (a level set of an explicit function on $T^{p+1}\\times\\mathbb{R}^m$) and then, via a nonlinear uniform boundedness principle, to the existence of a map in $W^{1,p}(M,N)$ that no sequence of smooth maps can weakly approximate. In the sphere case, the singular map is built by a Whitehead product with Hopf invariant $2$ for each lattice singularity, and an integral estimate on the Hopf invariant supplies the corresponding degree control on bubbles.","pith_inferences":["The $\\ell^{p+1}\\ln\\ell$ growth of the relaxed energy suggests a logarithmic \\u201centropy cost\\u201d for smoothing a periodic lattice of singularities: the cost is proportional to the boundary area of the domain times the logarithm of the system size. If the mechanism is robust, for $p=2$ one would expect the minimal energy of smooth approximants on a cube of side $\\ell$ to exceed the singular map","Because the constructed target retracts onto the $p$-skeleton of a $(p+1)$-torus, the same mechanism may appear for any target whose universal cover has a $p$-skeleton that is not uniformly retractible; nilmanifolds with torsion-free nilpotent fundamental group are natural next candidates to check.","The finite-scale bubbling proposition is stated for uniform Lipschitz neighborhood retracts, so it may have independent uses in quantitative homotopy problems for variational functionals, for instance in estimating minimal connection energies in Ginzburg\\u2013Landau type models, provided the constants can be made effective."],"forward_implications":["For every integer $p\\ge 2$ there is a compact target manifold $N$ (e.g. $T^{p+1}\\#T^{p+1}$) such that weak approximation by smooth maps fails as soon as $\\dim M>p$; this shows that the positive results known under the hypothesis $\\pi_1(N)=\\cdots=\\pi_{p-1}(N)=0$ cannot be extended to arbitrary targets.","When $p=4n-1$, the target can be taken to be the sphere $S^{2n}$; this gives an infinite family of counterexamples, including the earlier $S^2$ case, and shows that the earlier restriction to $p=3$ was an artifact of the available Hopf-invariant-one maps, since the new construction uses Hopf invariant $2$.","The obstruction already appears for continuous Sobolev maps, so the failure is not due to a lack of regularity of approximating sequences; the non-weakly-approximable map can be chosen with values in a compact manifold, not merely in a cell complex or a noncompact skeleton.","The same conclusion holds in higher-order Sobolev spaces $W^{s,p}$ whenever $sp\\in\\mathbb{N}$ and $sp\\ge 2$ with $\\dim M>sp$, so the analytical obstruction persists across all Sobolev regularities for which the question is open.","The proof shows why a natural strategy for weak density\\u2014passing to the universal cover to trivialize $\\pi_1$\\u2014cannot work in general: the homotopy between an approximant and the singular limit would require eliminating arbitrarily large cycles in the cover, and the quantitative control needed for that elimination is exactly what the bubbling analysis rules out."],"supporting_citations":[{"why":"The previously known $p=3$ counterexample into $S^2$ that the present construction extends to every integer $p\\ge 2$.","marker":"[8]"},{"why":"The density criterion requiring the first $p-1$ homotopy groups of the target to vanish, which the new construction shows is essential.","marker":"[30]"},{"why":"Supplies the nonlinear uniform boundedness principle that converts the relaxed-energy gap into the existence of a non-weakly-approximable map.","marker":"[33]"},{"why":"Provides the uniform boundedness principle used for the higher-order Sobolev spaces.","marker":"[47]"},{"why":"The integral estimate on the Hopf invariant, used to control the degree contributions of bubbles in the sphere case.","marker":"[53]"},{"why":"The Whitehead product construction, used to build the periodic sphere-valued map with Hopf invariant $2$.","marker":"[63]"},{"why":"The classification of elements of Hopf invariant one, which explains why the sphere construction needs Hopf invariant $2$.","marker":"[1]"},{"why":"Establishes the strong and weak approximation framework and the topological obstructions against which the new analytical obstructions are contrasted.","marker":"[6]"},{"why":"Strong density results for higher-order Sobolev spaces, used to frame and prove the higher-order counterparts.","marker":"[14]"}],"fun_headline_variants":["Weak smooth approximation fails for all p≥2 in maps to manifolds","Every p yields a Sobolev map with no smooth weak limit","Sphere targets also obstruct weak smooth limits for p=4n-1","Energy gap behind missing smooth weak limits for all p","Analytical obstruction: smooth maps never weakly dense for p≥2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument leans on a finite-scale bubbling statement: if two Sobolev maps into the target have bounded energy and are very close on average, then away from finitely many tiny balls they are homotopic, with the homotopy gap charged by the energy inside those balls; if that statement fails at the scales chosen in the proof, the extra log of relaxed energy disappears and the obstruction would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Weak smooth approximation fails for all p≥2 in maps to manifolds","Every p yields a Sobolev map with no smooth weak limit","Sphere targets also obstruct weak smooth limits for p=4n-1","Energy gap behind missing smooth weak limits for all p","Analytical obstruction: smooth maps never weakly dense for p≥2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000466,"raw_usage":{"total_tokens":2389,"prompt_tokens":1076,"completion_tokens":1313,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":692,"completion_tokens_details":{"reasoning_tokens":1223}},"tokens_in":692,"tokens_out":1313,"duration_ms":12169,"temperature":1.0,"reasoning_tokens":1223,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:38:18.575926+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute or bound from above the relaxed energy of the periodic singular retraction for the smallest case $p=2$, $n=3$: if one can exhibit, for every $\\ell$, a sequence of smooth maps $u_k$ into $N_0$ (or into the compact manifold $N_\\lambda$) with $E(u_k,Q_\\ell)\\le C\\ell^3$ and $u_k\\to u$ a.e., then $E^{\\mathrm{rel}}(u,Q_\\ell)$ would be $O(\\ell^3)$ and the claimed $\\ell^3\\ln\\ell$ growth\\u2014and with it the main theorem\\u2014would be refuted. Equivalently, produce two maps $u,v$ satisfying the hypotheses of Proposition 2.1 for which no finite family of disjoint balls with the stated energy and homotopy properties exists; that would break the load-bearing bubbling premise.","supporting_citations":[{"cited_title":"Math.219 (2020), no","cited_arxiv_id":null,"evidence_quote":"The previously known $p=3$ counterexample into $S^2$ that the present construction extends to every integer $p\\ge 2$."},{"cited_title":"Hajłasz, Approximation of Sobolev mappings, Nonlinear Anal","cited_arxiv_id":null,"evidence_quote":"The density criterion requiring the first $p-1$ homotopy groups of the target to vanish, which the new construction shows is essential."},{"cited_title":"III, Comm","cited_arxiv_id":null,"evidence_quote":"Supplies the nonlinear uniform boundedness principle that converts the relaxed-energy gap into the existence of a non-weakly-approximable map."},{"cited_title":"Monteil and J","cited_arxiv_id":null,"evidence_quote":"Provides the uniform boundedness principle used for the higher-order Sobolev spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Whitehead product construction, used to build the periodic sphere-valued map with Hopf invariant $2$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classification of elements of Hopf invariant one, which explains why the sphere construction needs Hopf invariant $2$."},{"cited_title":"Bethuel,The approximation problem for Sobolev maps between two manifolds, Acta Math.167 (1991), no","cited_arxiv_id":null,"evidence_quote":"Establishes the strong and weak approximation framework and the topological obstructions against which the new analytical obstructions are contrasted."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Strong density results for higher-order Sobolev spaces, used to frame and prove the higher-order counterparts."}],"review_version":1}