{"id":"6c38d7bf-e134-442e-83f6-c638c0d9fc34","arxiv_id":"2411.16048","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For stationary solutions of Δu=u^{-p}+f, the rupture set {u=0} is (n-2)-rectifiable with a sharp Minkowski-type estimate, and derivatives satisfy optimal Lorentz integrability.","lead":"This paper proves that the set where a solution of a singular elliptic equation touches zero has a precise geometric structure: it can be covered by smooth pieces and its size can be measured sharply. The result matters because it upgrades previous rough dimension bounds to exact rectifiability and optimal regularity estimates, mirroring what is known for harmonic maps.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The a priori C^{0,alpha} hypothesis gates all of Theorem 1.7; Proposition 3.22 yields uniformity only inside the already-Holder class, so the fine-structure conclusions do not cover arbitrary stationary solutions.","rationale":"I read the paper as a conditional but substantial upgrade of [12]: within the class of stationary solutions that are already C^{0,alpha} (the same standing assumption as Theorem 1.2 of [12]), the authors prove a sharp Minkowski-type sublevel estimate, (n-2)-rectifiability of the rupture set, and optimal Lorentz-space regularity for derivatives. I examined the internal scaffolding at the points where an argument most plausibly fails. The monotonicity formula for the mollified density (Proposition 2.16) and the rupture-set characterization (Proposition 2.11) are consistent; the alternative lemma (Lemma 6.13) together with Lemma 6.20 convincingly splits the analysis into the 'u small' and 'u positive' regimes; the covering lemmas of Section 9 apply the Reifenberg-type theorem with the correct scale normalizations (for example, Lemma 9.7 first reduces to radii at most 10s using the disjointness of {B_{r_z/10}(z)}); and the reductions in Section 10.2 (L^q embedded into the needed Morrey spaces via Lemma A.2, and the j = 1 case of (1.10)) are arithmetically consistent with the stated exponents. I also checked a potential scaling hazard in Proposition 3.22: the written inference from L_i -> infinity and r_i -> 0 to dist(0, partial Omega_i) -> infinity is not justified by L_i r_i^{alpha-1} -> infinity, but the conclusion is nevertheless true because the blow-up centers lie in B2 while the domain is B5, so the rescaled domains contain balls of radius about 1/r_i and converge to R^n. Thus I found no fatal flaw in the conditional argument. The weakest point is the one the reader identified: the a priori C^{0,alpha} hypothesis. Proposition 3.22 is a conditional uniformity statement, bounding the seminorm by the L^1 and L^q data only for solutions already assumed alpha-Holder, and every quantitative estimate downstream (Theorem 5.8 with (5.6), then Theorem 1.7) inherits this. Remark 1.8 is explicit that deriving this regularity is open. This is a scope limitation rather than an internal inconsistency, but it is load-bearing because the title and abstract present the fine structure of the rupture set for stationary solutions of (1.1), while the results apply to the C^{0,alpha} subclass; if non-Holder stationary solutions exist (for example, rupture profiles with a different homogeneity), the conclusions simply do not follow for them. The secondary issues raised by the reader also hold: (1.10) is proven only for j = 1 and f = 0, and the tangent-symmetry half of Theorem 5.8(3), needed for the common-tangent input in the rectifiability argument, is consigned to 'similar arguments' in [47] and the authors' unpublished preprint [24]. Neither issue overturns the main geometric conclusions, but both support keeping the verdict at CONDITIONAL rather than ACCEPT: the general case of (1.10) should be written out and the deferred argument made self-contained.","tokens_in":79184,"tokens_out":46465,"duration_ms":383472,"concrete_test":"Attempt to prove Proposition 3.22 for the wider class u in H^1_{loc} cap L^{-p}_{loc} cap L^1(B2), f in L^q, without assuming any Holder continuity: run the compactness contradiction of Section 3.3.2 using only the weak compactness available from the L^1 and L^q bounds. If the normalized blow-ups v_i = L_i^{-1}T_{x_i,r_i}(u_i - u_i(x_i)) admit a harmonic limit (Case 1) or a stationary solution of wDelta w = 0 (Case 2) with unit C^{0,alpha} seminorm, then the Liouville steps (Corollary A.7, Lemma A.6) give the contradiction, so the uniform bound is a priori and Theorem 1.7 becomes unconditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing concern is the standing hypothesis u in C^{0,alpha}_{loc}(B4), alpha = 2/(p+1), in Theorem 1.7 (and its parabolic corollary, Theorem 1.12). The entire quantitative program, the mollified density and monotonicity formula (Proposition 2.16), the a priori interior estimates (Lemmas 2.1 and 2.3, whose conclusions scale as r^{alpha+n-2} and r^{2alpha+n-2}), the compactness and blow-up analysis of Section 3, the quantitative stratification estimate of Theorem 5.8, and the covering and Reifenberg lemmas of Sections 8-9, requires the uniform bound (5.6), [u]_{C^{0,alpha}(B_{2R0})} + [f]_{M^{2alpha+n-4+gamma,2}(B_{2R0})} <= Lambda, as input. The transition from the hypotheses of Theorem 1.7 to (5.6) is made by Proposition 3.22, but that proposition assumes C^{0,alpha}_{loc} in its statement; its compactness contradiction yields a uniform seminorm bound only for solutions already known to be alpha-Holder. It is not a regularity theorem. Consequently, the Minkowski-content estimate (1.9), the Lorentz-space integrability (1.10), and the (n-2)-rectifiability of {u=0} are all conditional on an open regularity problem, as the authors concede in Remark 1.8. For a stationary solution that is not C^{0,alpha} at a rupture point, the alpha-scaling used to define blow-ups and the sublevel set {u < epsilon r^alpha} has no justification, so the paper establishes nothing about the rupture set of such a solution; the framing 'fine structure of the rupture set for stationary solutions of (1.1)' therefore overshoots what is proved. Two secondary gaps compound this: the proof of (1.10) in Section 10.2 treats only j = 1 and f = 0, declaring the general case 'almost the same argument' despite the need for a Schauder bootstrap; and the tangent-symmetry conclusion of Theorem 5.8(3), used as the common-tangent input for rectifiability, is delegated to 'methods in [47]' plus the authors' unpublished preprint [24], so the rectifiability claim is not fully self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies stationary solutions of the singular semilinear elliptic equation Δu = u^{-p} + f, u ≥ 0 in Ω ⊂ R^n, under the standing hypothesis u ∈ C^{0,α}_{loc}, α = 2/(p+1). The main theorem, Theorem 1.7, asserts three conclusions: (1) the Minkowski-type volume estimate L^n(B_r({u < ε r^α} ∩ B_1)) ≤ C r^2, which implies that the Minkowski dimension of {u = 0} is at most n − 2; (2) optimal Lorentz-space integrability D^j u ∈ L^{2/(j−α),∞} when f ∈ W^{j−1,∞}; and (3) (n−2)-rectifiability of the rupture set, which is discrete when n = 2. The proof develops a mollified density ϑ_f, a monotonicity formula, compactness and blow-up analysis, classical stratification à la White, quantitative stratification in the style of Naber–Valtorta, Reifenberg-type theorems, L^2 best-approximation estimates, and covering lemmas, and culminates in Theorem 5.8 and Theorem 1.7. A parabolic application, Theorem 1.12, gives 1-rectifiability of the rupture set for almost every time slice in n = 3, p > 3, for suitable weak solutions that are a priori Hölder.","tokens_in":79568,"tokens_out":11966,"duration_ms":117179,"significance":"If the main theorem is correct, it constitutes a substantial improvement over the earlier Hausdorff-dimension bounds for rupture sets obtained in [12, 18, 29, 13]: it gives quantitative Minkowski-content control and rectifiability of the rupture set, and it upgrades the integrability of derivatives to the expected Lorentz spaces. The technical apparatus contains genuinely useful ideas: a mollified density adapted to a density that can take negative values, a blow-up based on u − u(x) rather than u itself, and a quantitative stratification framework that avoids the nonnegativity assumptions used for harmonic maps. The paper is careful and detailed in its main estimates, and I did not find a definite internal contradiction in §§2–9. However, the correctness risk is concentrated in three places: the proof of Theorem 1.7(2) is only carried out for j = 1 and f ≡ 0; the H^k-a.e. tangent-symmetry conclusion in Theorem 5.8(3) relies on imported arguments from the authors' unpublished preprint [24]; and the whole theorem is conditional on the a priori Hölder regularity assumption, as the authors themselves state in Remark 1.8.","major_comments":[{"comment":"The proof of (1.10) is presented only for the case j = 1 and f ≡ 0; the text says that 'the general case follows from almost the same argument'. This is not a proof for a stated main theorem. The Lorentz-space integrability for arbitrary j and for nonvanishing f ∈ W^{j−1,∞} is one of the three central claims of Theorem 1.7, so the missing argument must either be supplied in full or the theorem must be restricted to the proved case.","section":"§10.2, proof of Theorem 1.7(2)"},{"comment":"The H^k-a.e. statement that tangent functions are k-symmetric with respect to a fixed subspace is obtained by 'almost the same methods in the proof of Theorem 1.4 and 1.5 of [47]' and by 'similar arguments in [24]', where [24] is an unpublished preprint by the same authors. This tangent-symmetry property is load-bearing for the rectifiability of the strata and hence of {u = 0}. The manuscript should contain a complete, self-contained proof, or at minimum should delineate precisely which steps are imported from [24] and make the preprint available and verifiable.","section":"§10.1, proof of Theorem 5.8(3)"},{"comment":"Proposition 3.22 is used in §10.2 to pass from the hypotheses of Theorem 1.7 to the uniform bound (5.6), but its statement already assumes u ∈ C^{0,α}_{loc}. It therefore proves a uniform Hölder estimate only within the class of solutions that are already known to be α-Hölder; it does not prove Hölder regularity for arbitrary stationary solutions. Since Remark 1.8 explicitly leaves that regularity as an open problem, all conclusions of Theorems 1.7 and 1.12 are conditional on an a priori Hölder assumption. The abstract and introduction should state this conditionality prominently and should not imply that the fine-structure theorems apply to all stationary solutions of (1.1).","section":"§3.3.2, Proposition 3.22, and Remark 1.8"},{"comment":"The claim that the estimates are sharp is justified only by the two-dimensional radial example (1.5). For n > 2 this example has a point rupture set and does not by itself demonstrate that the exponent 2 in (1.9) is optimal. Sharpness for n > 2 presumably follows from the trivial extension construction in Lemma 4.10, but this is not explicitly stated; the sharpness discussion should be made precise.","section":"§10.2, Remark 1.9 and the 'sharpness' paragraph"}],"minor_comments":[{"comment":"The abstract says the paper studies 'stationary solutions' without mentioning the standing a priori assumption u ∈ C^{0,α}_{loc}; this should be stated in the abstract to avoid overstating the scope.","section":"Abstract"},{"comment":"The sentence 'By Proposition 3.22 and Lemma A.2, without loss of generality, we can assume...' should explain the scaling/restriction that turns the L^1(B_2) + L^q(B_2) bound in Theorem 1.7 into the stronger local Morrey-plus-Hölder bound used later.","section":"§10.2, beginning"},{"comment":"The definition of S^k_{ε,r}(u) restricts to x ∈ B_{R_0} with r ≤ s < 1, but several later statements use balls B_s(x) that leave B_{4R_0}; the domain conventions for all radii should be listed once to avoid ambiguity.","section":"§5.1, Definition 5.5"},{"comment":"The proof of Lemma 6.4 uses compactness and Proposition 3.1, but the statement allows s = 1 with B_{20}(x) and requires an L^2-type bound; the role of Lemma 2.1 in providing that bound should be cited explicitly at the point where sup_i ‖u_i‖_{L^2} is asserted.","section":"§6.2, Lemma 6.4"},{"comment":"The phrase 'repeating this procedure to S for countably times' should be expanded: the argument produces, for each σ, a subset F of S of large H^k-measure that is rectifiable; the countable iteration needed to cover H^k-almost all of S should be written out.","section":"§10.1, rectifiability argument"}],"recommendation":"major_revision","confidential_remarks":"The main reason for major revision rather than rejection is that the central quantitative framework appears coherent and the explicit gaps are fillable: the full proof of Theorem 1.7(2) for all j, and a self-contained treatment of the tangent-symmetry step currently imported from [24]. I would also ask the editor to weigh whether the paper's conditional nature, already conceded in Remark 1.8, is sufficiently visible in the abstract and introduction. The reliance on the authors' own unpublished preprint [24] for a load-bearing step is a publication-policy concern independent of the mathematical validity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper makes a real advance—quantitative stratification applied to a density that can be negative, yielding a sharp Minkowski estimate and rectifiability of the rupture set—but all of Theorem 1.7 is conditional on a C^{0,α} a priori bound that is still an open problem. That is not a dishonest paper; Remark 1.8 says it plainly. But it means the title's “stationary solutions” overshoots what is proved.\n\nWhat is genuinely new: the modification of the blow-up to u−u(x), the alternative Lemma 6.13 dealing with points where u(x)>0, the mollified density and monotonicity formula, and the first optimal Lorentz-space integrability for D^j u. The argument is long but structured. The classical stratification part is careful, and the quantitative part follows Naber–Valtorta with nontrivial changes. The parabolic slicing corollary is a nice byproduct, though it inherits the same a priori regularity assumption.\n\nSoft spots, in proportion:\n- The main theorem does not apply to all stationary solutions. Proposition 3.22 does not prove Hölder regularity from weaker assumptions; it takes C^{0,α} as input. The stress-test gets this right. If the a priori Hölder problem is solved later, the theorem becomes unconditional; until then, treat Theorem 1.7 as “for stationary solutions in C^{0,α}.”\n- Theorem 1.7(2) is not proved in the stated generality. Section 10.2 treats only j=1, f=0, and the general case is deferred with “almost the same argument.” For a paper of this length, that is an unnecessary gap.\n- The tangent-symmetry conclusion used for rectifiability (Theorem 5.8(3)) leans on “methods in [47]” plus the authors’ own unpublished preprint [24]. This is load-bearing delegation. It may be repairable, but the referee should not have to chase an unpublished source.\n- The dependence on [24] is understandable—self-citation is not by itself a flaw—but here the cited result is not externally reproduced.\n\nFor whom: specialists in quantitative stratification and singular sets of elliptic free-boundary-type problems. They will get value from the techniques even if the theorem is conditional.\n\nRecommendation: send it to a serious referee. I would not accept as is. Ask the authors to either prove the general j case and import the [24] argument into the paper, or reword the theorem so the a priori assumption is explicit in the statement rather than only in remarks. The core mathematics looks sound and important enough to keep in the pipeline.","headline":"Genuinely nontrivial quantitative stratification upgrade for rupture sets, but every headline conclusion is conditional on a C^{0,α} a priori bound that remains open; worth a serious referee, not an accept as is.","tokens_in":80257,"tokens_out":3155,"would_cite":true,"duration_ms":33800,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J61","35J75","35B65","28A78"],"pacs":[],"model":"deepseek-v4-flash","headline":"For singular elliptic equations, the rupture set is (n−2)-rectifiable.","keywords":["semilinear elliptic equation","singular nonlinearity","rupture set","quantitative stratification","rectifiability","Minkowski dimension","stationary solutions","Lorentz spaces"],"falsifier":"Exhibit an admissible stationary solution (satisfying the stated $C^{{0,α}}$, $L^{1}$, and L^q hypotheses) whose rupture set {u=0}∩B1 has upper Minkowski dimension strictly larger than n−2, or fails to be (n−2)-rectifiable; either would refute Theorem 1.7. A complementary check is to test the sharp Lorentz exponent: if some admissible solution had D^j u outside $L^{{2/(j−α),∞}}$, the claimed integrability bound would fail.","tokens_in":78918,"feed_emoji":"📐","tokens_out":9554,"duration_ms":82956,"temperature":0.7,"pith_summary":"This paper studies stationary solutions of the semilinear elliptic equation Δu = $u^{{-p}}$+f, whose nonlinearity blows up as u approaches zero, and asks how large and how regular the rupture set {u=0} can be. It claims that, under a priori Hölder regularity and mild integrability assumptions on f, the sublevel set {u<εr^α} inside B1 has Lebesgue measure at most $Cr^{2}$, which forces the Minkowski dimension of the rupture set to be at most n−2; that the derivatives D^j u belong to an optimal Lorentz space when f is bounded in $W^{{j−1,∞}}$; and that {u=0} is (n−2)-rectifiable, with a discrete rupture set in two dimensions. The step beyond earlier work is that the rupture set is not just small in Hausdorff dimension but has a quantitative, stratified structure: each k-stratum of points classified by the symmetry of tangent functions is k-rectifiable. This matters because a rectifiable, quantitatively controlled singular set is what one needs to study tangent behaviour, sharp integrability, and time-sliced singular sets in the associated parabolic problem.","feed_headline":"Rupture sets of singular elliptic equations are (n−2)-rectifiable","feed_subtitle":"Quantitative stratification bounds the zero set's Minkowski dimension by n−2 and shows each stratum is rectifiable.","key_machinery":"The carrying object is the quantitative stratification S^k_{ε,r}(u), which collects the points where u fails to be approximately (k+1)-symmetric at every scale between r and 1. The proof pairs this with a modified blow-up sequence $r^{{-α}}$(u(x+ry)-u(x)), chosen so that points with u(x)>0 also have well-behaved limits, and with a mollified density ϑ_f(u;x,r) that admits a monotonicity formula even though it can take negative values. Because the density is not positive, the argument restricts quantitative stratification to the rupture set, where the density is bounded, and splits points into small-u and large-u cases that are handled by separate compactness and regularity arguments. This machinery produces the covering and Reifenberg-type estimates that yield both the Minkowski-content bound and the rectifiability of each stratum.","core_discovery":"The central discovery, stated as Theorem 1.7, is that a stationary solution u ∈ $C^{{0,α}}$_{loc} ∩ $H^{1}$_{loc} ∩ $L^{{-p}}$_{loc} of Δu = $u^{{-p}}$+f in B4, with f ∈ L^q and uniform $L^{1}$ and L^q bounds, has a rupture set with sharp quantitative geometry. Part (1) gives L^n(B_r({u<εr^α}∩B1)) ≤ C $r^{2}$ for some ε>0, implying that the Minkowski dimension of {u=0}∩B1 is at most n−2. Part (2) states that if f ∈ $W^{{j−1,∞}}$_{loc}, then D^j u belongs to the Lorentz space $L^{{2/(j−α),∞}}$, the sharp integrability level measured against the explicit solution u(x)=$α^{{-α}}$|x|^α. Part (3) states that {u=0} is (n−2)-rectifiable and, for n=2, is discrete. The proof achieves this by stratifying the rupture set according to the symmetry of tangent functions, proving each k-stratum is k-rectifiable, and by a quantitative stratification argument that works with a modified density which may be negative.","pith_inferences":["If the open regularity question flagged in Remark 1.8 — whether stationary solutions are automatically C^{0,α} — is resolved positively, then Theorem 1.7 would apply to all stationary solutions in the stated class without an a priori Hölder assumption.","The sublevel estimate (1.9) says more than controlling the zero set: it controls the region where u is merely smaller than εr^α, which is a quantitative nondegeneracy statement that could bear on free-boundary and obstacle-type questions.","The framework appears transferable to other singular nonlinearities and multi-phase models whose energy densities are signed; the structural requirement is boundedness of the density on the singular set rather than positivity.","A concrete testable extension suggested by the parabolic application: for the evolution problem with p≤3, determine whether the almost-every-time slice rectifiability persists or fails; the paper leaves this parameter range open."],"forward_implications":["The lower-dimensional bound is upgraded from Hausdorff dimension to Minkowski dimension: the rupture set has upper Minkowski dimension at most n−2, so its r-neighbourhood volume in B1 decays like r^2.","The rupture set is (n−2)-rectifiable: up to a null set it is covered by countably many Lipschitz images of R^{n−2}, and in dimension two it is a discrete set.","The derivative estimate D^j u ∈ L^{2/(j−α),∞} is sharp, with the model solution u(x)=α^{-α}|x|^α showing that the exponent cannot be improved.","For the three-dimensional parabolic equation ∂_t u = Δu − u^{-p} with p>3, the zero set {u(·,t)=0} is 1-rectifiable for almost every time t."],"supporting_citations":[{"why":"supplies the monotonicity formula and the prior Hausdorff-dimension bound n−2 that this paper upgrades to Minkowski-content and rectifiability results.","marker":"[12]"},{"why":"gives the improved Hausdorff-dimension estimate n−2+2/(p+1) for weak solutions, the baseline the new quantitative bounds sharpen.","marker":"[18]"},{"why":"introduces stationary solutions for this equation and the density whose monotonicity the present proof adapts.","marker":"[27]"},{"why":"defines weak solutions of the singular equation and supplies the first Hausdorff-dimension estimate for the rupture set.","marker":"[32]"},{"why":"provides the Reifenberg-type theorem and quantitative stratification framework used to obtain the Ahlfors-regular and covering estimates.","marker":"[47]"},{"why":"gives the simplified quantitative stratification and approximation machinery adapted here to a signed, possibly negative density.","marker":"[48]"},{"why":"supplies the classical stratification theory that identifies strata and their Hausdorff dimensions in Part I.","marker":"[67]"},{"why":"supplies the characterization of rectifiability via Jones-type square functions used in proving the k-rectifiability of the strata.","marker":"[1]"},{"why":"cited for concluding 'similar arguments' in the proof of the tangent-function symmetry statement; it is the authors' own unpublished preprint.","marker":"[24]"}],"fun_headline_variants":["Rupture sets: sharp Minkowski estimate and rectifiability","Singular elliptic equation: rupture set is (n−2)-rectifiable","Optimal integrability for gradients of singular elliptic solutions","3D singular evolutions: rupture sets are 1-rectifiable a.e.","Zero sets of singular elliptic equations: rectifiable and sharp"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All of the conclusions are conditional on the solution being a priori $C^{{0,α}}$-Hölder continuous with α=2/(p+1); the paper does not prove that stationary solutions enjoy this regularity, and Remark 1.8 states that it remains an open problem whether they do.","fun_headline_variants_meta":{"raw":{"variants":["Rupture sets: sharp Minkowski estimate and rectifiability","Singular elliptic equation: rupture set is (n−2)-rectifiable","Optimal integrability for gradients of singular elliptic solutions","3D singular evolutions: rupture sets are 1-rectifiable a.e.","Zero sets of singular elliptic equations: rectifiable and sharp"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001565,"raw_usage":{"total_tokens":6352,"prompt_tokens":1148,"completion_tokens":5204,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":764,"completion_tokens_details":{"reasoning_tokens":5112}},"tokens_in":764,"tokens_out":5204,"duration_ms":35622,"temperature":1.0,"reasoning_tokens":5112,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:36:42.045255+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit an admissible stationary solution (satisfying the stated $C^{{0,α}}$, $L^{1}$, and L^q hypotheses) whose rupture set {u=0}∩B1 has upper Minkowski dimension strictly larger than n−2, or fails to be (n−2)-rectifiable; either would refute Theorem 1.7. A complementary check is to test the sharp Lorentz exponent: if some admissible solution had D^j u outside $L^{{2/(j−α),∞}}$, the claimed integrability bound would fail.","supporting_citations":[{"cited_title":"D´ avila, K","cited_arxiv_id":null,"evidence_quote":"supplies the monotonicity formula and the prior Hausdorff-dimension bound n−2 that this paper upgrades to Minkowski-content and rectifiability results."},{"cited_title":"Dupaigne, A","cited_arxiv_id":null,"evidence_quote":"gives the improved Hausdorff-dimension estimate n−2+2/(p+1) for weak solutions, the baseline the new quantitative bounds sharpen."},{"cited_title":"Guo and J","cited_arxiv_id":null,"evidence_quote":"introduces stationary solutions for this equation and the density whose monotonicity the present proof adapts."},{"cited_title":"Jiang and F","cited_arxiv_id":null,"evidence_quote":"defines weak solutions of the singular equation and supplies the first Hausdorff-dimension estimate for the rupture set."},{"cited_title":"Naber and D","cited_arxiv_id":null,"evidence_quote":"provides the Reifenberg-type theorem and quantitative stratification framework used to obtain the Ahlfors-regular and covering estimates."},{"cited_title":"Naber and D","cited_arxiv_id":null,"evidence_quote":"gives the simplified quantitative stratification and approximation machinery adapted here to a signed, possibly negative density."},{"cited_title":"White, Stratification of minimal surfaces, mean curv ature flows, and harmonic maps, Journal f¨ ur die Reine und Angewandte Mathematik , 488 (1997), 1-35","cited_arxiv_id":null,"evidence_quote":"supplies the classical stratification theory that identifies strata and their Hausdorff dimensions in Part I."},{"cited_title":"Azzam and X","cited_arxiv_id":null,"evidence_quote":"supplies the characterization of rectifiability via Jones-type square functions used in proving the k-rectifiability of the strata."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"cited for concluding 'similar arguments' in the proof of the tangent-function symmetry statement; it is the authors' own unpublished preprint."}],"review_version":1}