{"id":"2ffe8588-c317-4fcd-afa5-1ee36588734f","arxiv_id":"2506.00916","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a family of two-time singularly perturbed PDEs, the analytic solution decomposes into a convergent part plus two parts with Gevrey asymptotic orders 1/k1 and 1/k2 in the perturbation parameter.","lead":"This paper proves that solutions of a family of singularly perturbed PDEs with two complex time variables split into a convergent piece and two Gevrey asymptotic pieces of different orders. It matters because the usual Borel-Laplace summability tool fails in this setting, and the authors replace it with an analytic continuation argument.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 7 applies the multilevel Ramis–Sibuya theorem with I1/I2 assigned backwards: Cases 1 and 3 give only exp(−C/|ε|^{k2}) decay but are put in I1, which requires exp(−M/|ε|^{k1}); Theorem 2's claimed Gevrey orders are therefore not established unless the index assignment is corrected.","rationale":"The paper contains a substantial construction: the auxiliary equation, three Banach spaces, analytic continuation result, and the J1/J2/J3 decomposition are detailed and mostly coherent. The core obstacle to the main theorem is localized in Proposition 7's invocation of the multilevel Ramis–Sibuya theorem. The mismatch between the decay orders in Proposition 6 and the index-set definition in Proposition 7 is not a matter of notation; it reverses which sector functions are assigned Gevrey order 1/k1 versus 1/k2. Swapping I1 and I2 would make the exponential hypotheses align, and one would still need to prove nonemptiness or explicitly allow degenerate one-level cases. Because the defect is confined to a key step but appears easily repairable, CONDITIONAL (rather than REJECT) is appropriate. The reader's nonemptiness objection is real but secondary; the printed index assignment is the more load-bearing issue. Minor typos (e.g., k2 in the exponent of Proposition 6 Case 2) do not affect this assessment.","tokens_in":44989,"tokens_out":11528,"duration_ms":115171,"concrete_test":"Perform the following formal re-derivation: write Δ_p=J_{1,p+1}-J_{1,p} and list the three cases of Proposition 6 with their decay exponents. Then substitute these bounds into Theorem 3 verbatim, keeping 0<k2<k1. If I1:= {p: Case 2 holds} and I2:= {p: Case 1 or Case 3 holds}, the exponential hypotheses are satisfied; if the printed assignment is retained, the I1 hypothesis fails for any p in Case 1 or 3. Also test nonemptiness: construct a good covering and a choice of (d_p,\\tilde d_p) with d_0=d_1≠d_2 and \\tilde d_0≠\\tilde d_1=\\tilde d_2, all satisfying the cosine conditions; both classes then occur, showing the nonemptiness issue is addressable, but it is not proven in the current text.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In Theorem 3 (§5.2), with 0<k2<k1, the hypothesis for p∈I1 is ||Δp(ε)|| ≤ Kp exp(−Mp/|ε|^{k1}) and for p∈I2 it is ≤ K̃p exp(−M̃p/|ε|^{k2}). Proposition 6 gives: Case 2 (only d_p changes): exp(−C_{p,4}/|ε|^{k1}); Case 1 (only \\tilde d_p changes): exp(−C_{p,2}/|ε|^{k2}); Case 3 (both change): exp(−C_{p,6}/|ε|^{k2}). Proposition 7 then defines I1 as 'Case 1 or Case 3' and I2 as Case 2. This is exactly reversed: the weak k2-decay indices are placed in I1 and the strong k1-decay indices in I2. Hence the hypotheses of Theorem 3 are not met as written. With the printed assignment, the component labelled J_{1,1,p} would only be produced from k2-decay differences and would admit Gevrey order 1/k2, not 1/k1; consequently Corollary 1 and Theorem 2, which attach J_{1,1,0} to order 1/k1, do not follow. A second, independent gap is that neither Proposition 6 nor Proposition 7 proves both index classes are nonempty; if all differences are of one type, the cited two-level theorem cannot be invoked (and the 'two-level' conclusion degenerates). The failure is in the proof of Proposition 7, not in the PDE construction itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a singularly perturbed nonlinear Cauchy problem in two complex time variables with a leading operator of the form Q(∂_z) - ε^{Δ_0}(t_1^{k_1+1}∂_{t_1})^{δ_1}(t_2^{k_2+1}∂_{t_2})^{δ_2} R(∂_z), with k_1 > k_2. The authors construct analytic solutions as double Laplace and inverse Fourier transforms of an auxiliary Borel-plane function, prove analytic continuation properties of that auxiliary function on products of sectors and discs, and then establish a three-term decomposition of the analytic solution: a convergent holomorphic part plus two parts admitting Gevrey asymptotic expansions of orders 1/k_1 and 1/k_2 in the perturbation parameter. The main technical engine is a two-level Ramis-Sibuya theorem applied to a family of truncated Laplace integrals J_{1,p} indexed over a good covering. The paper also gives Gevrey bounds for the remaining pieces J_2 and J_3.","tokens_in":45342,"tokens_out":5867,"duration_ms":59128,"significance":"If the main theorem is correct, the paper is a meaningful advance in parametric Gevrey asymptotics for PDEs with two complex time variables, since the auxiliary Borel function is only defined on a product of sectors and the proof must therefore avoid a direct double Laplace deformation. The fixed point arguments in Propositions 1, 3, and 4 are detailed, and the exponential difference estimates in Proposition 6 have the correct qualitative shape. The paper does not fit any data and relies on previously published theorems as tools. However, as written, the proof of the central two-level decomposition has a load-bearing gap in the application of the multilevel Ramis-Sibuya theorem, so the main conclusion is not yet established.","major_comments":[{"comment":"The assignment of the index sets I_1 and I_2 in Proposition 7 is inconsistent with the hypotheses of the multilevel Ramis-Sibuya theorem stated in Theorem 3. Proposition 6 shows that Case 1 and Case 3 give decay exp(-C/|ε|^{k_2}), while Case 2 gives decay exp(-C/|ε|^{k_1}). Proposition 7 places Case 1 and Case 3 in I_1 and Case 2 in I_2, but Theorem 3 requires p∈I_1 to satisfy exp(-M_p/|ε|^{k_1}) and p∈I_2 to satisfy exp(-M̃_p/|ε|^{k_2}). The roles are reversed. Consequently the stated proof does not establish that J_{1,1,p} admits a Gevrey expansion of order 1/k_1, and Corollary 1 and Theorem 2 inherit this gap.","section":"§4.1, Proposition 7 and §5.2, Theorem 3"},{"comment":"The paper never proves that both index classes I_1 and I_2 are nonempty for the chosen good covering and direction array (d_p, \\tilde d_p). The construction of the directions only requires the sector conditions involving cos(k_1(d_p - arg(εt_1))) and cos(k_2(\\tilde d_p - arg(εt_2))); it does not force the appearance of both Case 2 and Case 1/3 transitions. If all adjacent differences fell into a single decay class, Theorem 3 could not be invoked and the two-level conclusion of Proposition 7 would degenerate. A proof of existence of a good covering and directions realizing both classes, or a separate treatment of the degenerate case, is needed.","section":"§4.1, paragraph before Proposition 6"}],"minor_comments":[{"comment":"The word \"solucion\" should be \"solución\" or \"solution\".","section":"§3.1"},{"comment":"\"For very T∈C⋆\" should read \"For every T∈C⋆\".","section":"Lemma 1"},{"comment":"In the estimate preceding the conclusion of Case 2, the exponent contains (ρ_1/2 / r_{T_1})^{k_2}, but the derivation from the first variable yields an exponent k_1; this appears to be a typo.","section":"Proposition 6, Case 2 proof"},{"comment":"The phrase \"the solution (39) of (12)\" is imprecise: the function J_{1,p} is only the truncated piece J_1 of the full analytic solution, not itself a solution of the Cauchy problem (12).","section":"Proposition 7"},{"comment":"The definition of A_{k_1} contains an unmatched parenthesis and the notation r_{T_1}^{K_1} should presumably be r_{T_1}^{k_1}.","section":"Proposition 9"},{"comment":"\"Aknowledgements\" should be \"Acknowledgements\".","section":"Acknowledgements"}],"recommendation":"major_revision","confidential_remarks":"The central PDE construction and the fixed point estimates appear sound, and the reversed I_1/I_2 assignment in Proposition 7 looks repairable. The more serious issue is the missing proof that both exponential decay classes occur; this needs either a construction or a degenerate-case argument. I would be willing to review a revised version that addresses these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"X, here's my read. The paper proves existence of analytic solutions for a genuinely new family of two-time singularly perturbed PDEs, and the auxiliary Borel function lives only on a product of sectors, which blocks the standard Ramis-Sibuya route. That part is real: the fixed point arguments in the weighted Banach spaces are detailed, and the analytic continuation in Proposition 5 is a solid step. The Gevrey bounds for J2 and J3 also check out. So there is a genuine technical advance here.\n\nThe problem is Proposition 7. The multilevel Ramis-Sibuya theorem (Theorem 3) requires k1-decay for indices in I1 and k2-decay for I2, with k1 > k2. Proposition 6 gives exactly that: Case 2 has k1-decay, Cases 1 and 3 have k2-decay. Proposition 7 then defines I1 as 'Case 1 or Case 3' and I2 as Case 2. That is backwards. With the printed assignment, the component labelled J_{1,1} would be built from k2-decay differences, so it can only be expected to be Gevrey of order 1/k2, not 1/k1. Corollary 1 and Theorem 2 inherit this. This is not a typo in a constant; it is the load-bearing application of the summability theorem.\n\nThere is a second, smaller gap: even if the assignment were swapped, the paper never shows both I1 and I2 are nonempty. If all adjacent differences happen to have the same decay rate, the two-level decomposition degenerates and the distinct orders 1/k1 and 1/k2 are not obtained. That needs an explicit construction of the direction arrays or an alternative argument.\n\nMinor issues: several displayed formulas have typos (e.g., the exponent in the E4 estimate has a stray 'k2' in one place), and the proof sketch of Proposition 7 is only a paragraph. None of these matter compared with the index problem.\n\nBottom line: the analytic solution part is worth keeping, and the overall strategy may be repairable by swapping I1 and I2 and adding a nonemptiness argument. But as written, Theorem 2 is not established. I would send it to a referee, but the referee should be asked to focus on Section 4.1. I would not cite the decomposition theorem until the revision lands.","headline":"A technically rich paper whose central Gevrey decomposition is undermined by a reversed index assignment in the Ramis-Sibuya step.","tokens_in":45907,"tokens_out":2956,"would_cite":false,"duration_ms":26887,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35C10","35R10","35C15","35C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the analytic solution of a singularly perturbed nonlinear Cauchy problem in two complex time variables splits into a holomorphic part plus two parts carrying Gevrey asymptotic expansions of orders 1/k1 and 1/k2 in…","keywords":["Gevrey asymptotics","singularly perturbed PDE","two complex time variables","Borel-Laplace transform","multisummability","Ramis-Sibuya theorem","Cauchy problem","analytic continuation"],"falsifier":"Fix k1=2 and k2=1 and take a good covering in which each sector E_{p+1} is obtained from E_p only by rotating the first integration direction while leaving the second fixed, so every adjacent difference falls into Case 2 of Proposition 6. If the computed differences J_{1,p+1}-J_{1,p} all decay like exp(-C/|epsilon|^2), then only one decay class is present, the multilevel Ramis-Sibuya theorem cannot produce two distinct Gevrey orders, and Theorem 2's two-level splitting would not follow from the argument as written.","tokens_in":44746,"feed_emoji":"📐","tokens_out":6447,"duration_ms":60653,"temperature":0.7,"pith_summary":"The paper studies a singularly perturbed nonlinear Cauchy problem in two complex time variables, with epsilon as a small perturbation parameter. It tries to prove that the analytic solution, built as a double Laplace and inverse Fourier transform of an auxiliary Borel function, admits a three-way splitting: a part holomorphic in epsilon plus two parts with Gevrey asymptotic expansions of orders 1/k1 and 1/k2 in epsilon on a common sector. The usual route fails because the Borel function is only defined on a product of unbounded sectors, so the classical deformation of double integration paths is unavailable. The paper bypasses this by proving analytic continuation of the Borel solution into a neighborhood of the origin in each variable separately, then splitting the integration paths accordingly. If correct, this establishes a genuine two-level Gevrey structure for a class of PDEs that earlier treatments of symmetric, asymmetric, or truncated-Laplace configurations did not cover.","feed_headline":"Two Gevrey orders emerge in a two-time singular PDE","feed_subtitle":"Even though Borel data live only on a product of sectors, analytic continuation yields a two-level asymptotic expansion.","key_machinery":"The load-bearing construction is the double Laplace and inverse Fourier representation (16) together with the auxiliary Borel-Fourier convolution problem (21) for omega(tau1,tau2,m,epsilon). A Gevrey asymptotic expansion of order 1/k means that the N-th remainder of the expansion is bounded by C A^N Gamma(1+N/k) |epsilon|^N; the paper uses this standard notion repeatedly. $\\Omega$ is first solved in three Banach spaces: on a bidisc, on a product of unbounded sectors, and on their intersection. Proposition 5 shows that the sectorial solution continues analytically into the disc in each variable separately, not into a full bidisc, and this separate continuation allows the two integration paths to be deformed one at a time or through concatenated arcs. The exponential decay estimates of Proposition 6, feeding the multilevel Ramis-Sibuya theorem, are what convert those path deformations into the two distinct Gevrey orders 1/k1 and 1/k2.","core_discovery":"The paper's central claim is Theorem 2: under assumptions (3)-(11), the solution u_{d1,d2}(t1,t2,z,epsilon) of the Cauchy problem (12) decomposes as b(t1,t2,z,epsilon) + u_{d1,d2,1}(t1,t2,z,epsilon) + u_{d1,d2,2}(t1,t2,z,epsilon), where b is holomorphic in epsilon near the origin with values in the Banach space of bounded holomorphic functions on T1 x T2 x H_{$\\beta$'}, and each u_{d1,d2,j} admits a formal power series as its Gevrey asymptotic expansion of order 1/k_j with respect to epsilon on a sector E. The two orders are the reciprocals of the exponents k1>k2 appearing in the leading irregular time operator. The proof splits the double Laplace integral as J1+J2+J3: J2 is flat at order 1/k2, J3 is flat at order 1/k1, and J1 is completed to a family (J_{1,p}) on a good covering of sectors. Differences of consecutive members decay either like exp(-C/|epsilon|^{k1}) or exp(-C/|epsilon|^{k2}) depending on which integration directions move, and the multilevel Ramis-Sibuya theorem then converts these decay classes into the two distinct Gevrey levels.","pith_inferences":["The paper leaves open the explicit construction of direction arrays and good coverings that guarantee both decay classes occur; a natural test is to exhibit such a covering for the model example with k1=2 and k2=1, or to establish that generic coverings produce both classes.","One could expect the same two-level splitting to persist for higher-order multi-time operators or for q-analogs, with the exponents k_j replaced by the corresponding q-time exponents; this extension is not proved in the paper.","The separate-variable analytic continuation suggests a general principle: if a Borel solution extends into the origin along each time axis independently, a two-level Laplace summation may replace full summability even when the Borel domain is only S1 x S2."],"forward_implications":["The formal solution of (12) is not merely summable in a single Gevrey class: it carries a two-level structure whose orders are tied to the irregular time operators, even though ordinary two-variable Borel-Laplace summability fails in the sector-product geometry.","The result extends the earlier symmetric, asymmetric, and truncated-Laplace settings to a wider class of lower-order irregular operators satisfying (3)-(5), including terms of the form t^{ell1}1 partial_{t1}^{ell2} t^{ell3}2 partial_{t2}^{ell4}.","It provides a reusable mechanism: analytic continuation in each Borel variable separately can replace the unavailable deformation of a double integration path in other PDE problems whose Borel domain is only a product of sectors.","The decomposition b + u1 + u2 implies that the analytic solution and the formal power series are related by two distinct asymptotic levels, predicting two different accuracy regimes for the perturbation expansion in applications where epsilon is a small physical parameter."],"supporting_citations":[{"why":"Establishes the multilevel Gevrey phenomenon in the symmetric two-time setting that the present paper generalizes to the coupled leading operator.","marker":"[10]"},{"why":"Handles the asymmetric two-time case and proves the small-divisor obstacle that the present work bypasses.","marker":"[11]"},{"why":"Uses truncated Laplace transforms for a linearized version and supplies the integral representation templates reused here.","marker":"[6]"},{"why":"Provides the multilevel Ramis-Sibuya theorem (Theorem RS) used in Proposition 7.","marker":"[13]"},{"why":"Supplies Balser's decomposition viewpoint on multisummability that motivates the b + u1 + u2 splitting.","marker":"[18]"},{"why":"Furnishes the factorial bound Lemma 7 used to prove the Gevrey estimates for J2 and J3.","marker":"[19]"},{"why":"Provides the operator identity used to rewrite irregular time operators in the Borel equation.","marker":"[27]"},{"why":"Supplies the inverse Fourier transform and convolution properties underpinning the Borel-Fourier Banach spaces.","marker":"[12]"}],"fun_headline_variants":["Two Gevrey asymptotic levels from a two-time singular PDE","Formal and analytic solutions split into two Gevrey sectors","Two-time PDE yields double Gevrey asymptotics","Multilevel Ramis-Sibuya gives two asymptotic orders","Double Gevrey levels in a two-time singular problem"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument requires that, among the consecutive sectors of the good covering, both exponential decay classes actually occur: at least one adjacent pair producing the exp(-C/|epsilon|^{k2}) bound and at least one producing the exp(-C/|epsilon|^{k1}) bound, and the paper assumes these nonempty classes rather than proving that the chosen directions force them.","fun_headline_variants_meta":{"raw":{"variants":["Two Gevrey asymptotic levels from a two-time singular PDE","Formal and analytic solutions split into two Gevrey sectors","Two-time PDE yields double Gevrey asymptotics","Multilevel Ramis-Sibuya gives two asymptotic orders","Double Gevrey levels in a two-time singular problem"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000439,"raw_usage":{"total_tokens":2212,"prompt_tokens":913,"completion_tokens":1299,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":529,"completion_tokens_details":{"reasoning_tokens":1217}},"tokens_in":529,"tokens_out":1299,"duration_ms":9136,"temperature":1.0,"reasoning_tokens":1217,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:55:30.631441+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix k1=2 and k2=1 and take a good covering in which each sector E_{p+1} is obtained from E_p only by rotating the first integration direction while leaving the second fixed, so every adjacent difference falls into Case 2 of Proposition 6. If the computed differences J_{1,p+1}-J_{1,p} all decay like exp(-C/|epsilon|^2), then only one decay class is present, the multilevel Ramis-Sibuya theorem cannot produce two distinct Gevrey orders, and Theorem 2's two-level splitting would not follow from the argument as written.","supporting_citations":[{"cited_title":"Lastra, S","cited_arxiv_id":null,"evidence_quote":"Establishes the multilevel Gevrey phenomenon in the symmetric two-time setting that the present paper generalizes to the coupled leading operator."},{"cited_title":"Lastra, S","cited_arxiv_id":null,"evidence_quote":"Handles the asymmetric two-time case and proves the small-divisor obstacle that the present work bypasses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Uses truncated Laplace transforms for a linearized version and supplies the integral representation templates reused here."},{"cited_title":"Lastra, S","cited_arxiv_id":null,"evidence_quote":"Provides the multilevel Ramis-Sibuya theorem (Theorem RS) used in Proposition 7."},{"cited_title":"Loday-Richaud, Divergent series, summability and resurgence","cited_arxiv_id":null,"evidence_quote":"Supplies Balser's decomposition viewpoint on multisummability that motivates the b + u1 + u2 splitting."},{"cited_title":"Malek,On a partialq−analog of a singularly perturbed problem with fuchsian and irregu- lar time singularities, Abstract and Applied Analysis, vol","cited_arxiv_id":null,"evidence_quote":"Furnishes the factorial bound Lemma 7 used to prove the Gevrey estimates for J2 and J3."},{"cited_title":"Tahara, H","cited_arxiv_id":null,"evidence_quote":"Provides the operator identity used to rewrite irregular time operators in the Borel equation."},{"cited_title":"Lastra, S","cited_arxiv_id":null,"evidence_quote":"Supplies the inverse Fourier transform and convolution properties underpinning the Borel-Fourier Banach spaces."}],"review_version":1}