{"id":"8dac6b80-2eb1-436e-9a96-b48aa705ca6a","arxiv_id":"2502.09359","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The flipping operator is claimed to send F_{1-k,D} and G_{-k,D} to their negatives, but the key step F(Q^{k-1})=-Q^{k-1} is false.","lead":"The paper proves a flipping operator negates two families of locally harmonic Maass forms built from binary quadratic forms. The proof contains a false algebraic identity, so the main theorem is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Per-summand identity (3.3) used to prove (3.2) is false; since (3.2) is the only step treating the local polynomial, Theorem 1.1 and Corollary 1.2 are not established.","rationale":"The reader's REJECT is justified as an evaluation of the proof. The central claim Theorem 1.1 depends on establishing F_{2-2k}(P_C) = -P_C. The paper proves the Eichler-integral part correctly, but the local-polynomial part is dispatched through (3.2), whose proof relies on the per-summand identity (3.3). That identity is numerically false, and the reason is structural: the paper mishandles the transformation of the nonholomorphic factor v^{2-2k} under the real matrix A. This is not a cosmetic slip; it is the only argument covering the local polynomial, and both Theorem 1.1 and Corollary 1.2 inherit the gap. I verified the specific counterexample for the smallest nontrivial case and found the same mismatch as the reader. At the same time, the theorem itself may still be true, since summing over the relevant forms at that same point gives the desired sign flip; the preprint simply lacks the required summation argument. Therefore the correct assessment is that the paper is not acceptable as a proof of the stated results, matching the reader's verdict.","tokens_in":9175,"tokens_out":19044,"duration_ms":192840,"concrete_test":"Recompute equation (3.3) for k=2, D=12, Q=[-1,2,2], τ=1/2+i: with f=Q(τ,1)=15/4+i and v=1, the formula R_{-2}^2 f = -8a + (-8iaτ-4ib)/v + 2f/v^2 gives R_{-2}^2 f = 15/2 - 2i, hence F_{-2}(f) = -v^2/2 R_{-2}^2 f = -15/4 + i, while -f = -15/4 - i. The two sides differ, so the claimed per-summand identity fails and the proof of (3.2) collapses. As a control, summing all Q in Q_D with a<0<Q_τ at this τ (exactly [-1,0,3] and [-1,2,2]) gives a real total, so the theorem is not disproved by this example, but the written reduction to (3.3) is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 splits F_{1-k,D} into the Eichler-integral part F and the local polynomial P_C. The F part is handled correctly via (2.8), (2.9), and (1.4). The entire argument for P_C is (3.2), and its proof reduces to the per-summand identity F_{2-2k}(Q(τ,1)^{k-1}) = -Q(τ,1)^{k-1}, equivalently (3.3). This identity is false. For k=2, D=12, Q=[-1,2,2], τ=1/2+i, one has Q(τ,1)=15/4+i and R_{-2}^2 Q = 15/2 - 2i, so F_{-2}(Q) = -15/4 + i, whereas -Q = -15/4 - i. The error enters when the paper transforms h(τ)=τ^{k-1}/v^{2k-2} under A: the nonholomorphic factor v does not transform by the simple formula claimed. For k=2, h|_2 A equals (aτ+b)(c\\bar τ+d)^2/((cτ+d)v^2), which is not proportional to Q(τ,1)/v^2. Thus the derivation of (3.2) is invalid, and the main theorems are not proved. A cancellation over the full Q_D-sum may well exist: for the same D=12,k=2,τ, the selected forms sum to the real constant 15/2, so p is real and F(p)=-p holds. But that summation argument is absent from the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the flipping operator F_{2-2k} acting on the locally harmonic Maass forms F_{1-k,D} and G_{-k,D} introduced in earlier work of Bringmann, Kane, and Kohnen and of Bringmann and Mono. Theorem 1.1 claims that F_{2-2k}(F_{1-k,D}(\\tau)) = -F_{1-k,D}(\\tau) for non-square D, k \\ge 2, and \\tau outside the exceptional set E_D, and Corollary 1.2 makes the analogous claim for G_{-k,D}. The proof in Section 3 splits F_{1-k,D} into an Eichler-integral part and a local polynomial P_C. The Eichler-integral part is handled through the known relations (2.8), (2.9), and (1.4), while the treatment of P_C is reduced to the identity (3.2), which is in turn derived from a per-summand identity F_{2-2k}(Q(\\tau,1)^{k-1}) = -Q(\\tau,1)^{k-1}.","tokens_in":9522,"tokens_out":20292,"duration_ms":180650,"significance":"If the main theorem were established, it would give a natural hyperbolic analogue of the known flipping behavior of parabolic Maass\\textendash Poincar\\'e series and could simplify the local-polynomial arguments used in [10] and [18]. The organization of the paper is transparent, and the splitting of F_{1-k,D} and G_{-k,D} into Eichler integrals plus elementary terms is a useful structural observation. However, the central claim currently rests on a false identity, so the paper's main result is not proved as written.","major_comments":[{"comment":"The identity F_{2-2k}(Q(\\tau,1)^{k-1}) = -Q(\\tau,1)^{k-1}, which is the per-summand statement used to prove (3.2), is false. For k=2, D=12, Q=[-1,2,2], and \\tau=1/2+i, one has Q(\\tau,1)=15/4+i and a direct computation gives R_{-2}^2(Q(\\tau,1)) = 15/2 - 2i, hence F_{-2}(Q(\\tau,1)) = -15/4 + i, whereas -Q(\\tau,1) = -15/4 - i. Since (3.2) is the only step that treats the local polynomial P_C, Theorem 1.1 is not established.","section":"Section 3, proof of Theorem 1.1, Eq. (3.3)"},{"comment":"The step labeled \"Rewriting yields\" after (3.5) is incorrect. For h(\\tau)=\\tau^{k-1}/v^{2k-2}, the Petersson slash with A=(\\alpha \\beta; \\gamma \\delta) gives h|_{2k-2}A(\\tau) = (\\alpha\\tau+\\beta)^{k-1}(\\gamma\\bar\\tau+\\delta)^{2k-2}/((\\gamma\\tau+\\delta)^{k-1} v^{2k-2}), which is not a constant multiple of Q(\\tau,1)^{k-1} v^{2-2k}. The nonholomorphic factor v does not transform by (c\\tau+d), so the displayed transformation formula used to derive (3.3) is invalid.","section":"Section 3, derivation of (3.3)"},{"comment":"There is also an exponent inconsistency in the slash relation: from \\tau|_{-2}A = -Q(\\tau,1)/\\sqrt{D}, the standard slash convention gives \\tau^{k-1}|_{2-2k}A = (-1)^{k-1}D^{-(k-1)/2}Q(\\tau,1)^{k-1}, not D^{(k-1)/2} as written in the paper. Even if this were only a typo, the false transformation of h(\\tau) noted above remains a separate obstruction to (3.3).","section":"Section 3, around Eq. (3.4)"}],"minor_comments":[{"comment":"The notation \"k\\in N\\ge 2\" should be written as \"k\\in\\mathbb{N}, k\\ge 2\" for clarity.","section":"Introduction"},{"comment":"There is a stray bracket in the citation \"[2, Theorem 6.11 iv)]\" after (1.1).","section":"Introduction"},{"comment":"The phrase \"Letting k\\mapsto k+1 in (3.1)\" is correct, but the dependence on k should be made explicit because the operators and the constant in (3.1) depend on k.","section":"Proof of Corollary 1.2"}],"recommendation":"major_revision","confidential_remarks":"The flaw is in the central step of the proof of Theorem 1.1, so the paper cannot be accepted in its current form. I recommend major revision rather than reject because the missing cancellation over b-symmetric forms, which would repair (3.2), is a plausible and well-scoped addition; however, if the authors cannot supply that argument, the paper should not be published."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: Theorem 1.1 is not proved. The proof of the local-polynomial step (3.2) rests on the per-summand identity F_{2-2k}(Q^{k-1}) = -Q^{k-1}, and that identity is false. For k=2, D=12, Q=[-1,2,2], τ=1/2+i, a direct computation gives F_{-2}(Q) = -15/4 + i, while -Q = -15/4 - i. The source of the error is (3.5), which says R^{2k-2}_{2-2k}(τ^{k-1}) = (2k-2)! (τ/v^2)^{k-1}. Taking k=2, (2.5) gives R^2_{-2}(τ) = 2\\bar{τ}/v^2, not 2τ/v^2. The binomial sum in the verification evaluates to (1 - 2iv/τ)^{k-1}, not 1. The paper drops the nonholomorphic factor's transformation; that's the load-bearing mistake.\n\nWhat the paper does well: the question is natural, the splitting from [3] and [5] is used cleanly, and the Eichler-integral part of the argument is standard and correct. The result is a plausible extension of (1.1), and if it can be fixed, it would be a useful addition to the locally harmonic Maass form story.\n\nOne correction to the reader's report: Corollary 1.2 does not inherit the error. Its proof uses only (3.1) and the splitting in Lemma 2.5(3), which has a constant term rather than the Q-polynomial. So the corollary likely goes through; the paper should have said so explicitly. The main theorem, however, is unproven as written. A cancellation over the full Q_D-sum might restore (3.2), but the paper does not supply that argument.\n\nCitation pattern is fine: the heavy reliance on [3] and [5] is legitimate because those are the sources of the decompositions. The paper is short and clearly written, but a central computation is wrong. I would not cite Theorem 1.1 in its current form, and I wouldn't bring the paper to reading group unless the authors post a corrected version.\n\nRecommendation: send it to a referee anyway. The error is specific and fixable in principle, and the result is worth having if the summation argument materializes. With that concern on the table, a referee can make the call. If the authors cannot fix (3.5), reject.","headline":"Theorem 1.1's proof relies on a false identity (3.5); the corollary may survive, but the main result is unproven as written.","tokens_in":10118,"tokens_out":11828,"would_cite":false,"duration_ms":101200,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F03","11F11","11F25","11F37"],"pacs":[],"model":"deepseek-v4-flash","headline":"A sign-reversal law for the flipping operator extends to hyperbolic Poincaré series.","keywords":["flipping operator","locally harmonic Maass forms","Eisenstein series","integral binary quadratic forms","modular forms","Poincaré series","Eichler integrals","Maass raising operator"],"falsifier":"Take $k=2$ and $D=12$, choose a $\\tau$ outside $E_D$ (for instance $\\tau=0.5+i$), and evaluate both sides of $F_{-2}(F_{-1,12}(\\tau)) = -F_{-1,12}(\\tau)$ using the series (1.3) and the explicit differential operator definition. Any mismatch would refute Theorem 1.1; the same numerical check can be run for $G$ and for other $k,D$. Because the proof's per-summand identity can fail while the summed identity holds, the check must use the full sum over $Q_D$, not a single quadratic form.","tokens_in":8941,"feed_emoji":"🔁","tokens_out":8776,"duration_ms":83210,"temperature":0.7,"pith_summary":"The paper aims to show that the flipping operator—the differential operator that exchanges the holomorphic and non-holomorphic halves of a harmonic Maass form—acts as sign reversal on two families of locally harmonic Maass forms associated to indefinite binary quadratic forms. These functions are hyperbolic Poincaré series attached to a non-square discriminant $D$, and they are connected to Shimura and Shintani lifts and to central $L$-values. The main theorem states that for $k\\ge 2$ and $\\tau$ outside a measure-zero exceptional set $E_D$, one has $F_{2-2k}(F_{1-k,D}(\\tau)) = -F_{1-k,D}(\\tau)$, with an analogous identity $F_{-2k}(G_{-k,D}(\\tau)) = -G_{-k,D}(\\tau)$. If true, this extends the sign-flipping law already known for parabolic Eisenstein series to the hyperbolic setting.","feed_headline":"Flipping operator negates two Maass form families","feed_subtitle":"The sign flip known for parabolic Eisenstein series now holds for quadratic-form Poincaré series of types F and G.","key_machinery":"The load-bearing machinery is the splitting identity from Lemma 2.4: on each connected component of $\\mathbb{H}\\setminus E_D$, the function $F_{1-k,D}$ equals a non-holomorphic Eichler integral minus a constant times a holomorphic Eichler integral, plus a local polynomial $P_C$. The proof feeds this splitting into the interchange identities (2.8)–(2.9), which say that the flipping operator swaps the roles of the Bol and shadow operators up to constants; this forces the two integral pieces to flip sign. The remaining local-polynomial piece is then shown to flip sign by a direct calculation with the iterated Maass raising operator. For $G_{-k,D}$, the analogous splitting from the companion paper carries the same argument.","core_discovery":"On the paper's own terms, the discovery is that the flipping operator reverses the sign of the locally harmonic Maass forms $F_{1-k,D}$ and $G_{-k,D}$ away from the exceptional set $E_D$. The proof decomposes $F_{1-k,D}$ into a holomorphic Eichler integral, a non-holomorphic Eichler integral, and a local polynomial $P_C$. The two integral pieces are handled by the known interchange between the flipping, Bol, and shadow operators; the local polynomial is handled by an explicit calculation with the iterated Maass raising operator. The same strategy, using the splitting for $G_{-k,D}$ from the companion paper, yields the corollary for $G_{-k,D}$.","pith_inferences":["If the summed identity survives while the termwise identity fails, the sign reversal must be produced by cancellations between the $b$ and $-b$ summands of $Q_D$; checking this pairing explicitly would give a purely combinatorial proof of the local-polynomial step.","The identity suggests the flipping operator acts like a global involution on the space spanned by the $F_{1-k,D}$ and $G_{-k,D}$, possibly with eigenvalues $\\pm 1$; extending the computation to other lifts or to vector-valued settings might reveal a family of such involutions.","One could probe whether the exceptional set $E_D$ can be removed by using the continuously removable singularities of $G_{-k,D}$, in which case the sign identity might hold on all of $\\mathbb{H}$ by continuity."],"forward_implications":["For every non-square discriminant and every $k\\ge 2$, the flipping operator negates $F_{1-k,D}$ on the complement of $E_D$, so the sign-reversal law previously known for parabolic Eisenstein series now holds for a family of hyperbolic Poincaré series.","The same sign reversal holds for the weight $-2k$ companion $G_{-k,D}$, with the same exceptional set excluded.","Because $F_{1-k,D}$ is a simultaneous preimage of $f_{k,D}$ under the Bol and shadow operators, the identity shows that flipping swaps those two roles while negating the function, exactly as in the parabolic case.","On each connected component of $\\mathbb{H}\\setminus E_D$, the local polynomial part $P_C$ is negated by the flipping operator, so the identity can be read off component by component.","The paper's remark suggests the identity may give a shortcut for detecting when the nonconstant part of the local polynomial vanishes, which is the criterion tied to twisted central $L$-values."],"supporting_citations":[{"why":"Defines the locally harmonic Maass form $F_{1-k,D}$ and supplies Lemma 2.4, the splitting into Eichler integrals plus a local polynomial that the proof acts on.","marker":"[3]"},{"why":"Supplies the flipping operator properties, the interchange identities (2.8)–(2.9) with the Bol and shadow operators, and the background on Maass–Poincaré series.","marker":"[2]"},{"why":"Gives the definition of the flipping operator used throughout the paper.","marker":"[4]"},{"why":"Introduces $G_{-k,D}$ and its splitting into Eichler integrals plus a constant, which is the basis for Corollary 1.2.","marker":"[5]"},{"why":"Provides the decay and local harmonic properties of $g_{k+1,D}$ that make the nonlocal part of $G$ vanish under flipping.","marker":"[19]"},{"why":"Defines the cusp form $f_{k,D}$ whose Eichler integrals appear in the splitting in Lemma 2.4.","marker":"[20]"}],"fun_headline_variants":["Flipping negates F and G Maass forms","Flipping flips two locally harmonic Maass forms","Flipping operator reverses sign of two Maass families","Flipping operator: sign change for F and G Maass forms","Flipping negates two locally harmonic Maass forms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs the flipping operator to reverse the sign of each term $Q(\\tau,1)^{k-1}$ in the local polynomial; if this termwise identity fails, the theorem would still be true only if a more delicate cancellation among the quadratic-form summands produces the sign reversal.","fun_headline_variants_meta":{"raw":{"variants":["Flipping negates F and G Maass forms","Flipping flips two locally harmonic Maass forms","Flipping operator reverses sign of two Maass families","Flipping operator: sign change for F and G Maass forms","Flipping negates two locally harmonic Maass forms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002544,"raw_usage":{"total_tokens":9701,"prompt_tokens":852,"completion_tokens":8849,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":8765}},"tokens_in":468,"tokens_out":8849,"duration_ms":54301,"temperature":1.0,"reasoning_tokens":8765,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T21:48:45.765798+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $k=2$ and $D=12$, choose a $\\tau$ outside $E_D$ (for instance $\\tau=0.5+i$), and evaluate both sides of $F_{-2}(F_{-1,12}(\\tau)) = -F_{-1,12}(\\tau)$ using the series (1.3) and the explicit differential operator definition. Any mismatch would refute Theorem 1.1; the same numerical check can be run for $G$ and for other $k,D$. Because the proof's per-summand identity can fail while the summed identity holds, the check must use the full sum over $Q_D$, not a single quadratic form.","supporting_citations":[{"cited_title":"Bringmann, B","cited_arxiv_id":null,"evidence_quote":"Defines the locally harmonic Maass form $F_{1-k,D}$ and supplies Lemma 2.4, the splitting into Eichler integrals plus a local polynomial that the proof acts on."},{"cited_title":"Bringmann, A","cited_arxiv_id":null,"evidence_quote":"Supplies the flipping operator properties, the interchange identities (2.8)–(2.9) with the Bol and shadow operators, and the background on Maass–Poincaré series."},{"cited_title":"Bringmann, B","cited_arxiv_id":null,"evidence_quote":"Gives the definition of the flipping operator used throughout the paper."},{"cited_title":"Mono,Locally harmonic Maass forms of positive even weight, Israel J","cited_arxiv_id":null,"evidence_quote":"Provides the decay and local harmonic properties of $g_{k+1,D}$ that make the nonlocal part of $G$ vanish under flipping."},{"cited_title":"Zagier,Modular forms associated to real quadratic fields, Invent","cited_arxiv_id":null,"evidence_quote":"Defines the cusp form $f_{k,D}$ whose Eichler integrals appear in the splitting in Lemma 2.4."}],"review_version":1}