{"id":"c8a7a99e-f92b-45cc-9631-a3ba29008d28","arxiv_id":"2506.20226","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Limits of vectorial fractional Allen-Cahn solutions are stationary nonlocal minimal partitions, and minimizing 3-partitions are smooth outside a small singular set, even under a reversed triangle inequality for s close to 1/2.","lead":"This paper proves that solutions of a vectorial fractional Allen-Cahn equation converge, in the sharp interface limit, to stationary nonlocal minimal partitions of the domain. It also establishes partial regularity for minimizers, including a surprising case where the surface tension triangle inequality is reversed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2's vectorial regularity proof is omitted ('line to line' from [51]); every compactness and stationarity step in Theorem 1.1 rests on it.","rationale":"I read the paper in good faith. The main theorems are precisely stated and the proofs are lengthy; apart from the omitted proof I found no internal inconsistency, no fitted parameters, and no circularity. Theorem 1.6 is careful to state s > s*(sigma,n), and the competitor analysis in Lemma 7.13 is explicit. The serious gap is structural: all compactness and stationarity results depend on Theorem 3.2. The authors' statement that the vectorial proof is \"line to line\" may well be correct, but this is precisely the kind of assertion that must be checked; it is the linchpin of the asymptotic result. Since the reader already made this the basis for CONDITIONAL, my read does not move the verdict. I would not reject the paper: the concern is a missing technical proof, not a demonstrated error. The concrete check above would settle whether the concern lands; if the proof transfers, the central claim appears sound.","tokens_in":75863,"tokens_out":6880,"duration_ms":76517,"concrete_test":"Re-derive Theorem 3.2 for d >= 2 by writing out the proof of [51, Theorem 3.3] with vector-valued v, tracking every use of [13, Lemma 4.5] and of maximum/comparison principles. The decisive point is the uniform C^{0,alpha} estimate used in Lemma 3.10 Step 1 to extract a subsequential limit: check whether it follows componentwise from boundedness of grad W(v) (then the extension is routine) or whether it needs a scalar comparison principle (then a vectorial substitute must be provided). Also verify that the stationarity identity in Corollary 3.3 can be obtained without additional regularity of grad W(v) beyond (H1)-(H3). If this cannot be completed, Theorem 1.1 is not currently supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing weakness is exactly the one the Reader identifies, and it is flagged by the paper itself. Section 3.1 says of Theorem 3.2: \"Since its proof follows from line to line the proof of [51, Theorem 3.3], we shall omit it.\" Theorem 3.2 is the regularity/stationarity statement for the degenerate Allen-Cahn boundary-reaction system (3.9): bounded weak solutions are C^infinity in B_R^+, C^{0,alpha} up to D_R, with z^a d_z v Holder continuous. From it the paper derives Corollary 3.3 (stationarity), Corollary 3.4 (monotonicity), Lemma 3.10 (clearing-out), and ultimately Theorem 4.5, which is the engine of Theorem 1.1. The transfer from [51] is not automatic: [51, Thm 3.3] is scalar, while (3.9) is a system coupled through grad W(v). The vectorial maximum principle supplied in Corollary 3.8 controls |u| rather than individual components, and the scalar boundary estimates of [13, Lemma 4.5] cannot be assumed to apply verbatim to systems. If any step in the omitted proof uses a scalar comparison principle, then \"line to line\" is not an argument. In that case the clearing-out property, the compactness of critical points, and the stationarity of the limiting nonlocal partition all collapse.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the vectorial fractional Allen-Cahn system for s in (0,1/2) with multiple wells and proves that bounded-energy critical points converge, as the characteristic parameter tends to zero, to stationary nonlocal partitions for the energy P_sigma^{2s}, where sigma_ij = |a_i - a_j|^2. Minimizers converge to minimizing partitions. The paper then develops a regularity theory for stationary and minimizing nonlocal partitions: stationary partitions have open chambers and an interface of Minkowski codimension one, and minimizing partitions are C^∞ outside a singular set of Hausdorff dimension at most n-2 under a nearly homogeneous coefficient condition, or for 3-partitions under a strict triangle inequality. A headline result is the same partial regularity under a strict inverse triangle inequality for s close to 1/2, together with the statement that in that case the i0-j0 interface is contained in the singular set.","tokens_in":76136,"tokens_out":15593,"duration_ms":154212,"significance":"If the main theorems are correct, this is a substantial advance: it extends the scalar critical-point compactness of Millot--Sire--Wang to arbitrary vector-valued multi-well systems, and it initiates a regularity theory for nonlocal partitions with surface-tension coefficients that need not satisfy the triangle inequality, a phenomenon that has no counterpart in the local theory. The paper has real strengths: the coefficient matrix is derived parameter-free from the wells, the quantitative stratification arguments are laid out in detail, the non-infiltration propositions are proved, and many auxiliary estimates are given in full. The central caveat is that Theorem 3.2, on which the compactness, monotonicity, clearing-out, and stationarity steps all rest, is stated without proof and deferred to a scalar predecessor.","major_comments":[{"comment":"The proof of Theorem 3.2 is omitted with the sentence \"Since its proof follows from line to line the proof of [51, Theorem 3.3], we shall omit it.\" This is load-bearing: Corollary 3.3 (stationarity), Corollary 3.4 (monotonicity), Lemma 3.10 (clearing-out), Proposition 4.6, and ultimately Theorem 4.5 all use Theorem 3.2. The transfer from [51] is not automatic because [51] treats the scalar case d=1, whereas (3.9) is a system coupled through the boundary reaction grad W(v); scalar comparison and barrier arguments do not have an obvious vector analogue. Please supply a complete proof of Theorem 3.2, or a detailed step-by-step verification that identifies each scalar estimate in the proof of [51, Theorem 3.3] and explains why it applies componentwise to v_epsilon, and in particular why the boundary Holder estimate and the Hopf-type argument used in Corollary 3.8 hold for the system.","section":"Section 3.1, Theorem 3.2"},{"comment":"The proof of the strong H^{s'} convergence in Theorem 6.7(i) is not written correctly. It bounds {u_k} in H^{s'}(Omega'') and then invokes a compact embedding H^{s''}(Omega'') subset of H^{s'}(Omega') for s'' < s'. The compact embedding goes in the opposite direction: H^{s'}(Omega'') embeds compactly into H^{s''}(Omega'') for s'' < s'. As written, the argument does not yield convergence in H^{s'}. The claim itself appears fixable: to prove convergence in H^{s'}, one should first obtain a uniform bound in H^t for some t in (s', min(2s,1/2)) and then apply compactness in H^{s'}; please rewrite the proof accordingly.","section":"Section 6.2, Theorem 6.7(i)"},{"comment":"The displayed formula for Per(T,D_2) in (7.37) is incorrect. For T = {|x'| < 1/2, |x_n| < epsilon}, which is compactly contained in D_1, the relative perimeter in D_2 is 2 omega_{n-1} 2^{1-n} + 2 epsilon (n-1) omega_{n-1} 2^{2-n}, not (omega_{n-1} + 2 epsilon omega_{n-2}) 2^{2-n}: the lateral boundary contributes the sphere measure in R^{n-1}, proportional to (n-1) omega_{n-1}, not omega_{n-2}. The contradiction argument appears repairable because it only uses positivity of the epsilon-linear term and the value omega_{n-1}/2^{n-2} for the flat part, so a suitably chosen epsilon still yields a contradiction, but the formula and the choice of epsilon must be corrected.","section":"Section 7.3, Lemma 7.13, Eq. (7.37)"}],"minor_comments":[{"comment":"The second proof after Lemma 7.13 is headed \"Proof of Theorem 1.5\" but it proves Theorem 1.6; please correct the heading.","section":"Section 7.3"},{"comment":"In the definition of the competitor F, the line \"E_{j0} := H^c \\ T\" should read \"F_{j0} := H^c \\ T\", since the sets of the competitor F are being defined.","section":"Section 7.3, Lemma 7.13"},{"comment":"In Corollary 6.6 and Theorem 6.7(ii), the notation W(v_k) should be W(u_k) for consistency with the Allen-Cahn equation and with Theorem 4.1.","section":"Sections 6.1-6.2"},{"comment":"The manuscript contains several typos and formatting glitches, for example \"Ginzburburg-Landau\", \"Corollay\", and \"V olume\" in the table of contents; a careful copyedit is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the omitted proof of Theorem 3.2: every compactness and stationarity statement in Theorem 1.1 depends on it, and the transfer from the scalar [51] is nontrivial. If the authors can supply the missing proof, the paper's central claims are likely sound; if not, Theorem 1.1 is not established. The perimeter-formula error in Lemma 7.13 is local and repairable, but it must be corrected because Theorem 1.6 relies on it. I recommend a major revision rather than rejection, given the quality and detail of the rest of the argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a serious, well-written paper. It generalizes the scalar fractional Allen-Cahn sharp-interface limit to vectorial multi-well systems and at the same time builds a regularity theory for nonlocal minimal partitions, including the surprising strict-inverse-triangle case. The main theorems are stated precisely, and most of the proofs are detailed. Before citing it, you should know one thing: the compactness and stationarity of the limit depend on Theorem 3.2, whose proof is omitted because it is claimed to follow line-by-line from [51, Theorem 3.3].\n\nWhat is genuinely new: the vectorial generalization is not cosmetic. The identification of the limit partition via the first variation, the clearing-out lemma, the quantitative stratification for stationary partitions, and the partial regularity results for minimizers — especially Theorem 1.6 for 3-partitions with a strict inverse triangle inequality — are real advances. The paper is also honest: it flags the omitted proof itself and does not hide the restriction in Theorem 1.6 that s must be close to 1/2.\n\nWhere I worry: Theorem 3.2 is load-bearing. It states C^infinity and Hölder regularity up to the boundary for the degenerate Allen-Cahn boundary-reaction system. From it the paper derives the monotonicity formula, clearing-out, compactness, and the stationarity of the limit partition. The transfer from the scalar result in [51] is not automatic: (3.9) is a system coupled through ∇W(v), and the paper only proves a bound on |u|, not componentwise. If any step in the scalar proof uses a comparison principle or a sign condition that does not extend to systems, then 'line to line' is not an argument. The stress-test note makes this point correctly. This is a genuine gap, not a cosmetic omission.\n\nThe rest of the architecture seems coherent. I found no internal contradictions, no fitted parameters, and no post-hoc data selection. The citation pattern is fine; citing [51] heavily is natural since this paper builds directly on it.\n\nWho gets value: researchers in nonlocal phase transitions, geometric measure theory, and fractional Allen-Cahn. A reader wanting a fully self-contained proof should wait for a revision that supplies Theorem 3.2.\n\nRecommendation: send it to peer review, with a referee explicitly tasked to verify the vectorial extension of [51, Theorem 3.3]. The paper deserves serious refereeing; the omitted proof should be filled before acceptance.","headline":"Strong vectorial sharp-interface limit and nonlocal partition regularity, with one load-bearing proof omitted — worth refereeing hard.","tokens_in":76653,"tokens_out":3111,"would_cite":true,"duration_ms":34721,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","49Q20","35B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Vectorial fractional Allen-Cahn limits are nonlocal minimal partitions, with regularity even if surface tensions violate the triangle inequality.","keywords":["fractional Allen-Cahn","nonlocal minimal partitions","multiple-well potentials","fractional perimeter","singular perturbations","phase transitions","partial regularity","sharp-interface limit"],"falsifier":"Run the full vectorial extension argument on a three-well potential $W(z)=|z-a_1|^2|z-a_2|^2|z-a_3|^2$ with $s=0.3$ and symmetric Dirichlet data, and check numerically whether the normalized energy in half-balls obeys the monotonicity formula and whether small-energy half-balls clear to a single well; a concrete counterexample would be a bounded weak solution for which the normalized energy decreases at some scale, or whose small-energy limit retains two wells in a half-ball. More directly for Theorem 1.6, compute the limiting 3-partition in the plane for $\\sigma_{12}>\\sigma_{13}+\\sigma_{23}$: the theorem predicts no triple junction and a smooth interface except possibly at isolated points, and a visible triple junction would refute it.","tokens_in":75652,"feed_emoji":"🧩","tokens_out":6587,"duration_ms":67865,"temperature":0.7,"pith_summary":"The paper establishes a sharp-interface limit for the vectorial fractional Allen-Cahn equation with an arbitrary number of wells. For exponents $s \\in (0,1/2)$, it shows that any sequence of uniformly bounded-energy weak solutions converges to a partition of the domain into regions occupied by single phases, and that the limiting partition is stationary for a nonlocal interfacial energy in which the cost of an interface between phases $i$ and $j$ is proportional to $|a_i-a_j|^2$. The same conclusion holds for minimizers, with the limit minimizing the nonlocal energy under the assigned exterior data. Because the limiting coefficients are squared distances, they need not satisfy the usual triangle inequality, which is why the paper also develops a partial regularity theory for minimizing 3-partitions, including the case where a triangle inequality fails strictly. The upshot is a variational model of multi-phase interfaces that is genuinely nonlocal and has no local counterpart.","feed_headline":"Fractional multi-phase limits are nonlocal minimal partitions","feed_subtitle":"For s<1/2, vectorial Allen-Cahn solutions concentrate on stationary nonlocal interfaces.","key_machinery":"Two mechanisms carry the argument. The first is the Caffarelli-Silvestre extension, which realizes $(-\\Delta)^s$ as the Dirichlet-to-Neumann operator of the degenerate elliptic operator $-\\mathrm{div}(z^{1-2s}\\nabla\\cdot)$ in the upper half-space; through it, solutions of the fractional Allen-Cahn equation become solutions of a boundary-reaction problem, and one obtains the monotonicity formula $\\Theta_{s,\\varepsilon}(v_\\varepsilon,x_0,r)=r^{-(n-2s)}E_{s,\\varepsilon}(v_\\varepsilon,B_r^+(x_0))$ non-decreasing in $r$. That formula yields the clearing-out lemma, compactness of energy, and convergence of the energy densities. The second is the non-infiltration property for minimizing partitions, which says that a chamber with small enough volume fraction in a ball must disappear from the half-radius ball. Together with the classification of tangent maps and a quantitative stratification of the singular set, non-infiltration reduces the 3-phase problem, including the strictly inverse-triangle case, to the known regularity theory for almost-minimizers of the fractional perimeter.","core_discovery":"On the paper's own terms, the central discovery is a compactness-and-identification theorem: under assumptions (H1)-(H3) on the multi-well potential, Theorem 1.1 proves that solutions $u_\\varepsilon$ of $(-\\Delta)^s u_\\varepsilon + \\varepsilon^{-2s}\\nabla W(u_\\varepsilon)=0$ in $\\Omega$, with uniformly bounded fractional Allen-Cahn energy and converging exterior data, subconverge to $u_* = \\sum_j \\chi_{E^*_j} a_j$, where $E^*$ is a partition of $\\Omega$ into open sets of finite $2s$-perimeter. The limit is stationary for the nonlocal partition energy $P^\\sigma_{2s}$, meaning $\\delta P^\\sigma_{2s}(E^*,\\Omega)[X]=0$ for every compactly supported field $X$; Theorem 1.2 adds that minimizers converge to minimizers of $P^\\sigma_{2s}$. For minimizers, the paper then proves partial regularity: in the nearly homogeneous case, and for 3-partitions either under a strict triangle inequality or, for $s$ close to $1/2$, under a strict inverse triangle inequality, the interface is $C^\\infty$ outside a relatively closed singular set of Hausdorff dimension at most $n-2$; in the inverse case the interface between the two dominant phases is contained in that singular set.","pith_inferences":["The paper leaves the threshold $s_*(\\sigma,n)$ in Theorem 1.6 implicit; a natural conjecture is that the inverse-triangle regularity holds for every $s \\in (0,1/2)$, with the present proof limited by the quantitative convergence of normalized fractional perimeter to classical perimeter as $s \\to 1/2$.","Since $\\sigma_{ij}=|a_i-a_j|^2$ depends only on the squared distances between wells, one can design the well geometry to realize any $\\ell^2$-embeddable matrix of coefficients, making the partition theory a laboratory for nonlocal clusters with exotic surface tensions.","In dimension two the authors conjecture that the interface between the two dominant phases is actually empty under a strict inverse triangle inequality; this is checkable by computing minimizers of $P^\\sigma_{2s}$ with symmetric three-phase exterior data.","The optimal rate $\\varepsilon^{\\min(4s,1)}$ for the potential energy, which the paper leaves open, could be tested by constructing a matched asymptotic expansion of the two-well transition profile in a half-space."],"forward_implications":["For $s \\in (0,1/2)$, the sharp-interface limit of critical points is a stationary nonlocal partition, giving the first vectorial multi-well analogue of the scalar stationary-interface theorem.","Minimizers of the fractional Allen-Cahn energy converge to minimizers of $P^\\sigma_{2s}$, so the nonlocal partition problem is a genuine variational limit of the phase-field model.","Transition layers have thickness of order $\\varepsilon$, and away from the interface the solution approaches its limit at rate $\\varepsilon^{2s}$, while the potential energy vanishes like $\\varepsilon^{\\min(4s,\\alpha)}$ for every $\\alpha \\in (0,1)$.","In the nearly homogeneous coefficient regime, minimizing partitions are $C^\\infty$ outside a singular set of Hausdorff dimension at most $n-2$, and locally finite when $n=2$.","For 3-partitions, even a strictly violated triangle inequality yields the same partial regularity for $s$ close enough to $1/2$, and the interface between the two phases with too-large a coefficient is itself contained in the singular set."],"supporting_citations":[{"why":"Supplies the scalar asymptotic strategy, including the extension, monotonicity formula, clearing-out lemma, and compactness that this paper generalizes to vector fields with multiple wells.","marker":"[51]"},{"why":"Establishes the $\\Gamma$-convergence and minimizer limit for scalar nonlocal phase transitions, providing the model for the limit identification in the vectorial case.","marker":"[57]"},{"why":"Introduces the fractional $2s$-perimeter and the regularity theory for nonlocal minimal surfaces, which is invoked after the partition is reduced to two phases.","marker":"[15]"},{"why":"Provides the homogeneous-case non-infiltration property and almost-everywhere regularity for fractional isoperimetric clusters, adapted in Theorem 1.4.","marker":"[24]"},{"why":"Gives the additive-case regularity and classification of triple junctions for nonlocal minimal clusters in the plane, which Theorem 1.5 recovers.","marker":"[19]"},{"why":"Supplies the bootstrap regularity for integro-differential operators used to upgrade $C^{1,\\alpha}$ interfaces to $C^\\infty$ once the nonlocal mean-curvature equation is known.","marker":"[9]"},{"why":"Provides the almost-minimizer regularity threshold for nonlocal sets, used as the final regularity input for the two-phase reduced problem.","marker":"[27]"},{"why":"Gives the convergence of normalized fractional perimeter to the classical perimeter as $s \\to 1/2$, used in the inverse-triangle exclusion argument of Lemma 7.13.","marker":"[25]"}],"fun_headline_variants":["Nonlocal Allen-Cahn limits are minimal partitions","Allen-Cahn nonlocal limits: minimal partitions","Fractional Allen-Cahn yields nonlocal minimal partitions","Minimal nonlocal partitions from Allen-Cahn","For s<1/2, Allen-Cahn yields nonlocal partitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Theorem 3.2, the vectorial regularity-and-stationarity theorem for the degenerate Allen-Cahn boundary reaction whose proof is omitted because it is said to follow line by line from [51, Theorem 3.3]; if that extension failed, the clearing-out lemma, the convergence of energy densities, and the stationarity of the limiting partition would not be available.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal Allen-Cahn limits are minimal partitions","Allen-Cahn nonlocal limits: minimal partitions","Fractional Allen-Cahn yields nonlocal minimal partitions","Minimal nonlocal partitions from Allen-Cahn","For s<1/2, Allen-Cahn yields nonlocal partitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001307,"raw_usage":{"total_tokens":5330,"prompt_tokens":948,"completion_tokens":4382,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":4305}},"tokens_in":564,"tokens_out":4382,"duration_ms":29758,"temperature":1.0,"reasoning_tokens":4305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:21:50.792412+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the full vectorial extension argument on a three-well potential $W(z)=|z-a_1|^2|z-a_2|^2|z-a_3|^2$ with $s=0.3$ and symmetric Dirichlet data, and check numerically whether the normalized energy in half-balls obeys the monotonicity formula and whether small-energy half-balls clear to a single well; a concrete counterexample would be a bounded weak solution for which the normalized energy decreases at some scale, or whose small-energy limit retains two wells in a half-ball. More directly for Theorem 1.6, compute the limiting 3-partition in the plane for $\\sigma_{12}>\\sigma_{13}+\\sigma_{23}$: the theorem predicts no triple junction and a smooth interface except possibly at isolated points, and a visible triple junction would refute it.","supporting_citations":[{"cited_title":"MILLOT, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the scalar asymptotic strategy, including the extension, monotonicity formula, clearing-out lemma, and compactness that this paper generalizes to vector fields with multiple wells."},{"cited_title":"SAVIN, E","cited_arxiv_id":null,"evidence_quote":"Establishes the $\\Gamma$-convergence and minimizer limit for scalar nonlocal phase transitions, providing the model for the limit identification in the vectorial case."},{"cited_title":"CAFFARELLI, J.M","cited_arxiv_id":null,"evidence_quote":"Introduces the fractional $2s$-perimeter and the regularity theory for nonlocal minimal surfaces, which is invoked after the partition is reduced to two phases."},{"cited_title":"COLOMBO, F","cited_arxiv_id":null,"evidence_quote":"Provides the homogeneous-case non-infiltration property and almost-everywhere regularity for fractional isoperimetric clusters, adapted in Theorem 1.4."},{"cited_title":"CESARONI, M","cited_arxiv_id":null,"evidence_quote":"Gives the additive-case regularity and classification of triple junctions for nonlocal minimal clusters in the plane, which Theorem 1.5 recovers."},{"cited_title":"BARRIOSBARRERA, A","cited_arxiv_id":null,"evidence_quote":"Supplies the bootstrap regularity for integro-differential operators used to upgrade $C^{1,\\alpha}$ interfaces to $C^\\infty$ once the nonlocal mean-curvature equation is known."},{"cited_title":"D ´AVILA: On an open question about functions of bounded variation,Calc","cited_arxiv_id":null,"evidence_quote":"Gives the convergence of normalized fractional perimeter to the classical perimeter as $s \\to 1/2$, used in the inverse-triangle exclusion argument of Lemma 7.13."}],"review_version":2}