{"id":"77d2c929-8e9c-40a6-bb15-14610cf5651a","arxiv_id":"2505.14799","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A sign error in the graph-reduction step invalidates the claimed Brouwer degree formula and the existence theory built on it.","lead":"This paper tries to prove when equations with sums of exponentials on finite graphs have solutions, by computing a topological invariant called the Brouwer degree. The claimed degree formula is flawed: a reduction step changes the sign of the degree when the number of vertices removed is odd, so the main theorems do not hold as stated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.3 omits the factor (-1)^{k-r}: with H = Δu + F, reducing k vertices to r flips the degree sign when k-r is odd, so Theorem 1.4's sign formula fails for odd |V| (a 3-vertex example gives d = -1 where the theorem says 1).","rationale":"I independently re-derived the reduction in Lemma 2.3. The determinant identity proved, det(L-D) = det R det(\\tilde L - \\tilde D), concerns the Jacobian of G(u) = L u - F(u), whereas the degree d in §2.2 is defined for H(u) = Δu + F(u) = -L u + F(u), whose Jacobian is -L + D. A direct block-reduction computation (or the identity det(-L+D) = (-1)^{k-r} det R det(-\\tilde L+\\tilde D)) shows deg(H on G) = (-1)^{k-r} deg(H on \\tilde G), with det R > 0. I verified this numerically on a two-vertex graph reducing to one vertex (n=1, f_1=(-1,0), c=1): the original degree is +1, the reduced degree is -1. The reader's weakest assumption is therefore exactly right, and I agree with the conclusion that the proof of Lemma 2.3 and the sign formula in Theorem 1.4 fail as stated. A concrete contradiction: on the 3-vertex path with n=1, f_1 ≡ -1, c=1, the strictly convex functional J has the unique critical point u=0; det DH(0) = det(-L-I) = -8, so d = -1, whereas Theorem 1.4 assigns d = 1. The error is repairable: if d is (re)defined as the degree of G = -Δu - F, then Lemma 2.3's identity is the correct Jacobian identity and the computed signs coincide with the theorems; existence statements, which depend only on zero/nonzero degree, are unchanged. Additional smaller gaps exist (case (b) in Theorem 4.1 assumes f_2(x_0)<0 although f_2 ≡ 0 is allowed; the reduced \\tilde f_0 in Lemma 2.3 is nonconstant and must be re-centered by the change of variables of §2.1, changing \\tilde f_i by positive factors). These do not alter the main verdict. Because the headline claim—an explicit formula for the Brouwer degree—is false as stated, the paper cannot be accepted without revision; the reader's REJECT verdict stands.","tokens_in":28791,"tokens_out":31633,"duration_ms":265049,"concrete_test":"Compute the Brouwer degree for the 3-vertex path graph (ω=1, m≡1) with n=1, f_1 ≡ -1, c=1, using the paper's definition H(u) = Δu - e^u + 1. The functional J(u) = ½Σω(u_i-u_j)^2 + Σ(e^{u_j} - u_j) is strictly convex, so u=0 is the unique zero; det DH(0) = det(-L-I) = -8, hence d = -1. Compare with Theorem 1.4, which gives d = 1 (q(F_1) = -∞ < +∞, c = 1 > 0). The mismatch confirms the sign error; equivalently, re-derive Lemma 2.3's determinant identity for -L+D and check the (-1)^{k-r} factor.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper defines d_{f_n,...,f_1,c} as the Brouwer degree of H(u) = Δu + Σ f_i e^{iu} + c (§2.2). With m≡1, Δ = -L, so DH(u) = -L + D, where D = diag(Σ i f_i(x)e^{iu(x)}). Lemma 2.3's proof establishes the Schur-complement identity det(L - D) = det R · det(\\tilde L - \\tilde D) and concludes the degrees are equal. But det(-L + D) = (-1)^{k-r} det R · det(-\\tilde L + \\tilde D), with det R > 0; equivalently, eliminating the k-r vertices replaces the eliminated block by -R, whose determinant contributes (-1)^{k-r}. Thus the reduced degree equals (-1)^{k-r} times the original degree, not the degree itself. Section 4 reduces every graph to two vertices, so all signs in Theorems 4.1, 4.2, 6.1, 6.2 and Theorem 1.4 are reversed whenever |V| is odd. The error is not cosmetic: the stated formula is false as a statement about the defined degree. Direct check on the 3-vertex path with n=1, f_1 ≡ -1, c=1: equation -Δu = -e^u + 1 has unique zero u=0, det DH(0) = det(-L-I) = -8, so d = -1, while Theorem 1.4 (q(F_1) < +∞, c > 0) assigns d = 1. The zero/nonzero dichotomy is unaffected, so the existence theorems probably survive a sign correction, but the headline degree computation, the paper's key novelty, is wrong as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies semilinear elliptic equations of the form -Δu = Σ_{i=1}^n f_i(x)e^{iu(x)} + c on finite connected weighted graphs, with fully general exponential nonlinearities. The authors prove a Brezis–Merle type alternative, derive a priori estimates under conditions on the leading coefficient and on c or the average of f_1, then compute the Brouwer degree of the associated map H(u) = Δu + Σ f_i e^{iu} + c by reducing the graph to two vertices via a Schur-complement argument. Nonzero degree yields existence, and the remaining cases are treated by sub- and supersolutions, including multiple-solution statements. The central claim is Theorem 1.4, giving degree 1, -1, or 0 depending on q(F_n), the sign of c, and the sign of the average of f_1 when c=0. The paper is clearly organized and contains substantial original material, but the main degree computation is invalid as written because Lemma 2.3 omits a dimension-dependent sign factor.","tokens_in":29148,"tokens_out":12700,"duration_ms":125606,"significance":"If the sign issue is corrected, the paper would be a useful contribution: it gives a unified topological-degree treatment of Kazdan–Warner, Chern–Simons, and more general exponential equations on finite graphs, and it combines degree theory with sub-/supersolution methods to cover degree-zero cases. The a priori estimates in Theorem 1.2 and the two-vertex reduction technique are genuine extensions of earlier work [SW22], [LSY24]. The reliance on those prior published results is explicit and not circular. However, the exact degree formula is the paper's headline novelty, and that formula is false as stated for odd vertex sets; the zero/nonzero dichotomy is unaffected, so the existence theorems are likely salvageable, but the revision must be substantive.","major_comments":[{"comment":"The reduction lemma is missing a sign factor. Since H(u)=Δu+Σ f_i e^{iu}+c = -Lu + F(u), the relevant Jacobian is -L + D, not L - D. The determinant identity at the end of the proof of Lemma 2.3, det(L-D)=det R · det(\\tilde L - \\tilde D), together with det(-L+D)=(-1)^k det(L-D) and det(-\\tilde L+\\tilde D)=(-1)^r det(\\tilde L-\\tilde D), yields det(-L+D)=(-1)^{k-r} det R · det(-\\tilde L+\\tilde D). Hence the Brouwer degree of the reduced map equals (-1)^{k-r} times the original degree, not the original degree. Since every reduction in Section 4 goes to r=2, all displayed degree values carry an undisplayed factor (-1)^{|V|-2}=(-1)^{|V|}. This is not a convention issue: d is defined as deg(H,...) with H=Δu+..., and the standard orientation of the Brouwer degree is being used throughout.","section":"§2.2, Lemma 2.3"},{"comment":"As a consequence of the missing sign, the displayed degree formulas are false as statements about the defined degree. On the three-vertex path with n=1, f_1≡-1, c=1, the equation -Δu = -e^u + 1 has unique zero u=0; the Jacobian of H at 0 is -L-I, whose determinant is -8, so d=-1. Theorem 1.4, on the other hand, classifies q(F_1)<+∞ with c>0 as giving d=1. The same parity factor reverses every nonzero value in Theorems 4.1, 4.2, 6.1, and 6.2 whenever |V| is odd. I stress that the zero/nonzero dichotomy is unaffected, so the existence results based on d≠0 or on d=0 may survive a corrected sign, but the headline degree computation is incorrect as written and must be fixed before the paper can be accepted.","section":"Theorems 1.4, 4.1, 4.2, 6.1, 6.2"}],"minor_comments":[{"comment":"In the displayed vector at the bottom of page 12, the Schur-complement term is written as Q^T R^{-1}(f_{0,r+1},...,f_{0,n})^T; this should be (f_{0,r+1},...,f_{0,k})^T, since the vertex set is indexed up to k while n denotes the number of exponential terms.","section":"Lemma 2.3 proof"},{"comment":"The statement defines \\tilde f_0 only by the condition Σ_V \\tilde f_0 = Σ_V f_0. The proof actually defines \\tilde f_0 through a Schur-complement formula involving f_0 on the removed vertices. Please state that formula explicitly in the lemma, since the reduced equation is not simply the restriction of f_0 to \\tilde V.","section":"Lemma 2.3 statement"},{"comment":"There is a small grammatical slip: 'Note that the t=0 is the unique solution if ϵ=0' should read 'Note that t=0 is the unique solution if ϵ=0.' This is stylistic and does not affect the mathematics.","section":"Example 3.4"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is real and lands exactly as stated: Lemma 2.3 omits the factor (-1)^{k-r}, so the degree formulas in Theorems 1.4, 4.1, 4.2, 6.1, and 6.2 are false for odd |V|. I nevertheless recommend major revision rather than rejection, because the error is local and systematic: inserting the parity factor (-1)^{|V|-2} into the degree formulas appears to repair the statements, and the existence theorems based on the zero/nonzero dichotomy do not depend on the exact sign. The revision must be substantive: the authors need to correct Lemma 2.3, all degree theorems, and any informal statements about the value of the degree, and they should re-verify the two-vertex examples with the parity factor taken into account. The paper is within the journal's scope and the reliance on [SW22] and [LSY24] is properly acknowledged; there is no circularity concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: the paper is a serious attempt at a general existence theory for polynomial exponential nonlinearities on finite graphs, and the a priori estimate half is worth reading. But the headline theorem is false as stated. Lemma 2.3's reduction forgets the sign (-1)^{k-r} between the Jacobian determinants of -L+D and -\\tilde L+\\tilde D. Since the degree is defined through H = Δ + F = -L + F, the missing sign flips the degree whenever the number of removed vertices is odd. A concrete check: on a three-vertex path with m≡1, ω=1, n=1, f_1≡-1, c=1, the equation -Δu = -e^u+1 has the unique solution u=0, and det DH(0) = det(-L-I) = -8, so the degree is -1. Theorem 1.4 predicts +1. That is not a cosmetic slip.\n\nWhat is genuinely new: the paper is the first to treat the full sum Σ f_i e^{iu} with arbitrary n on finite graphs. The a priori estimates in Section 3, based on classifying the leading coefficient at a single vertex, are a real contribution. The sub- and supersolution constructions for the zero-degree cases in Sections 5 and 6 are substantial and appear to survive independently of the degree sign. The references to SW22 and LSY24 are appropriate and not circular; those are external published results.\n\nThe main soft spot is exactly the sign error. It permeates every degree computation in Section 4, so Theorem 1.4 and Theorems 4.1/4.2, 6.1/6.2 as stated are wrong. The zero/nonzero dichotomy is unaffected, so the existence theorems probably survive a sign correction, but the paper's key novelty—the explicit degree formula—is not correct as written. A second, smaller issue: the reduction of nonconstant f_0 to a constant at the start of Section 2 is justified via a change of variables, but the details are skimpy and should be checked when the sign is fixed.\n\nWho is this for? Researchers working on PDEs on graphs and topological degree methods. The a priori estimate part is useful, and the overall program is worth pursuing. But the current version cannot be accepted; it needs a major revision to track signs, and the reduced two-vertex computations should be redone with the correct parity. I would send it to a serious referee—the topic is active and the error is precise enough to verify—but I would not advise acceptance until the degree formula is corrected and re-verified.","headline":"A serious paper with a false central degree formula: the reduction in Lemma 2.3 drops a sign, so the headline theorem contradicts a simple three-vertex example.","tokens_in":29730,"tokens_out":6508,"would_cite":false,"duration_ms":58494,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35G20","35J61","35J91","35R02"],"pacs":[],"model":"deepseek-v4-flash","headline":"A degree formula completely determines existence for general exponential equations on finite graphs.","keywords":["general exponential nonlinearity","finite weighted graph","Brouwer degree","a priori estimates","sub- and supersolutions","graph Laplacian","semilinear elliptic equation","multiplicity"],"falsifier":"On the three-vertex path with unit edge weights, $m\\equiv 1$, $c=0$, $f_1(x_1)=1$, and $f_1(x_2)=f_1(x_3)=0$, compute the Brouwer degree of $H(u)=\\Delta u+e^{u(x_1)}$ directly from the sign of $\\det(-L+D)$ at its zero, then apply Lemma 2.3 to remove $x_3$ and recompute the degree on the reduced two-vertex graph; if the two integers differ, the reduction lemma and hence Theorem 1.4 would be falsified.","tokens_in":28549,"feed_emoji":"📐","tokens_out":13960,"duration_ms":158702,"temperature":0.7,"pith_summary":"This paper proves a complete existence criterion for semilinear elliptic equations on finite connected weighted graphs when the right-hand side is any finite polynomial in $e^u$: $-\\Delta u=\\sum_{i=1}^n f_i(x)e^{iu(x)}+c$. The criterion is phrased through the Brouwer degree of $u\\mapsto \\Delta u+\\sum_{i=1}^n f_i e^{iu}+c$, which the authors show is $1$, $-1$, or $0$ depending only on $c$, the average of $f_1$ when $c=0$, and the asymptotic limit $q(F_n)=\\lim_{y\\to+\\infty}\\max_{x\\in V}F_n(x,y)$. Since a nonzero degree guarantees a solution, this settles existence in half of the parameter plane; for the zero-degree cases the paper constructs sub- and supersolutions and proves existence under explicit coefficient conditions. It also proves that when $|c|$ lies strictly below a coefficient-dependent threshold, at least two solutions exist. The argument is built on a graph-reduction lemma that deletes vertices where every $f_i$ vanishes and computes the degree on a two-vertex graph.","feed_headline":"Brouwer degree decides exponential graph equations","feed_subtitle":"A two-vertex reduction yields exact existence and multiplicity for -Δu = Σ f_i e^{iu} + c on finite networks.","key_machinery":"Lemma 2.3 is the load-bearing mechanism: it takes a finite connected weighted graph, removes every vertex at which all coefficients $f_1,\\ldots,f_n$ vanish, and constructs a new weighted graph on the remaining vertices by forming the Schur complement of the Laplacian. The paper shows the new edge weights are nonnegative, the reduced graph is connected, and the constant term is adjusted so that the Brouwer degree is claimed to be unchanged; this is what makes the two-vertex computation legitimate. Together with homotopy invariance, the reduction turns the degree calculation into a sequence of $2\\times 2$ determinants, each classified by the sign of $c$ and of $\\bar f_1$ and by the finiteness of $q(F_n)$, the leading-asymptotic limit of the nonlinearity. The sub- and supersolution machinery supplies existence in the zero-degree cases, and a homology-based critical-group argument supplies the second solution.","core_discovery":"The central claim is Theorem 1.4: with $f_1\\not\\equiv 0$ when $c=0$, the Brouwer degree $d_{f_n,\\ldots,f_1,c}$ equals $1$ if $q(F_n)<+\\infty$ and either $c>0$ or ($c=0$ and $\\bar f_1>0$); equals $-1$ if $q(F_n)=+\\infty$ and either $c<0$ or ($c=0$ and $\\bar f_1<0$); and equals $0$ otherwise. The proof path is: Theorem 1.1 gives a discrete blow-up alternative—any sequence of solutions either is bounded, tends uniformly to $-\\infty$, or tends uniformly to $+\\infty$; Theorem 1.2 rules out the unbounded alternatives under coefficient assumptions; homotopy invariance then lets the coefficients be reshaped; and Lemma 2.3 eliminates all vertices with zero coefficients, leaving a two-vertex graph whose degree is a $2\\times 2$ determinant. A nonzero degree forces a solution by the standard existence property of the degree, while the zero-degree regions are handled by subsolution–supersolution arguments: solutions appear for $c$ near $0$ when $f_1$ has the right sign and the other coefficients lie on the correct side of explicit threshold functions, and a second solution appears for $0<|c|0$ in case (B), solutions exist for all $c$ close enough to $0$, with a positive constant $c_n$ bounding the admissible interval.","For $0<|c|