{"id":"64839b26-4ff4-4e44-82aa-b089d620aff6","arxiv_id":"2608.08902","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Vector-valued modular resurgent series are defined, and Eichler integrals of unary theta series are proven to be non-trivial examples whose Stokes constants are rotated by the modular S-matrix and whose median resummation reconstructs the series.","lead":"This paper introduces vector-valued modular resurgent series, a framework that tracks the divergent expansions and Stokes data of vectors of quantum modular forms. It proves the framework works for Eichler integrals of unary theta series, the false theta functions that appear throughout quantum topology and vertex algebra theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"General weight-1/2 and 3/2 Eichler-integral claim in §4.6 does not satisfy Definition 2.5: part 3 (Euler-product L-function decomposition) is explicitly unproved and the Borel tower is only one-sided; the abstract overstates the result as established 'in general'.","rationale":"The unary theta section (Propositions 4.7, 4.9, 4.10 and diagram (173)) appears well supported: explicit Borel transforms, Stokes constants proportional to Ω(S)ω^(ν), quantum modular cocycles, and median resummation are derived in detail. The q-Pochhammer section is honestly disclosed as a repackaging of scalar results. The load-bearing problem is the abstract and Conclusions claim that the same structure holds for arbitrary weight-1/2 or 3/2 vector-valued cusp forms. Section 4.6 Step 3 proves only parts 1–2 in a weakened one-sided sense and obtains meromorphic continuation of the Stokes–Dirichlet vector via the Hecke correspondence; Remark 4.5 explicitly says part 3 (linear combination of Euler-product L-functions) is not established. Since Definition 2.5 defines vector-valued MRS by parts 1–3, the general claim does not follow from the paper's own definitions. A concrete non-unary cusp form (e.g., the level-11 Shimura lift) can be checked to fail the required periodic/character decomposition, showing the overstatement is not merely a missing proof but a false statement as written. Thus we agree with the reader's conditional verdict: the paper needs to either restrict the 'in general' claim to the unary theta case (or to cusp forms with the extra Hecke-eigenform hypothesis) or prove the missing part 3. The internal admission in Remark 4.5 makes this a straightforward correction rather than a rejection of the entire contribution, which contains substantial correct results for the unary theta series.","tokens_in":47189,"tokens_out":10415,"duration_ms":102267,"concrete_test":"Take a concrete non-unary weight-3/2 vector-valued cusp form, e.g. the Shimura lift of the level-11 weight-2 newform f(q)=q∏_{n≥1}(1−q^n)^2(1−q^{11n})^2, embedded as a vector-valued form for Mp_2(Z) as in §4.6. Compute its Fourier coefficients a_n (equivalently the coefficients of LLL(s)). Part 3 of Definition 2.5 would force the Stokes–Dirichlet coefficient sequence A_m from Eq. (228) to be a finite linear combination of Dirichlet characters with possible rescaling by c_j, hence a periodic sequence modulo some M. Verify numerically whether A_m is periodic for the first several thousand m. For this g the a_n are known not to be periodic (e.g., a_2=−2, a_3=−1, a_5=1 with no repetition), so the periodicity check fails. That failure would disprove the general claim as stated and require restricting it to unary theta series or to cusp forms satisfying the extra Hecke-eigenform hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central new-object claim—that vector-valued modular resurgence applies to Eichler integrals of arbitrary weight-1/2 and 3/2 vector-valued cusp forms—is not supported by its own Definition 2.5. Part 1 of Definition 2.5 requires, for every component, a common two-sided equally spaced tower {ρ_m = A m}_{m∈Z∖{0}}. In §4.6 Step 3 the authors themselves note the tower is one-sided ({ξ_λ = 2πiλ} with λ ∈ (1/D)Z_{>0}), has missing levels for general g, and becomes two-sided only after the unary-theta folding ξ = ζ²; they call this a 'weak sense' of part 1. Part 3 requires the Stokes–Dirichlet vector to equal M diag(c_j^s) applied to a vector of genuine L-functions with Euler products. Remark 4.5 states verbatim that 'part 3 of Definition 2.5 is not established by Steps 1–7' and that an Euler product 'would need g to be related to Hecke eigenforms.' Eq. (228) only exhibits meromorphic continuation through the cusp-form Dirichlet vector LLL(s+κ−1). Thus the general statement in the abstract and Conclusions ('the same modular resurgent structure ... holds for the Eichler integral of any vector-valued cusp form of weight 1/2 or 3/2') fails the paper's own definition; the fully verified vector-valued MRS is the unary theta case. This is an internal inconsistency, not a disagreement with an external convention.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a vector-valued extension of the modular resurgence framework, defining vector-valued modular resurgent series (Definition 2.5), formulating a paradigm and two conjectures (median resummation and quantum modularity), and presenting two families of examples. For the q-Pochhammer vectors (Section 3), the vector-valued structure is explicitly identified as a repackaging of the scalar results of [2], with the Dirichlet series decomposed via Dirichlet characters. For Eichler integrals of unary theta series (Sections 4.1–4.5), the paper proves in detail that the folded vectors are vector-valued MRSs: it computes the Borel transform and Stokes constants (Proposition 4.7), shows the paradigm diagram (Eq. (173)), proves vector-valued quantum modularity (Proposition 4.9) and median resummation (Proposition 4.10), and discusses bimodular completions. The final subsection (Section 4.6) outlines an extension to arbitrary vector-valued cusp forms of weight 1/2 and 3/2, but with explicit caveats given in Remark 4.5. The central tension is that the abstract and conclusions assert the general weight-1/2 and 3/2 result without the qualification that part of Definition 2.5 is not verified in that generality.","tokens_in":47478,"tokens_out":3244,"duration_ms":36251,"significance":"The unary theta case is developed carefully and in detail, with explicit Borel transforms, Stokes constants rotated by the S-matrix, the closed paradigm diagram, and direct proofs of quantum modularity and median resummation. These computations are non-trivial and are a genuine contribution, as they provide a fully worked vector-valued resurgent structure for false theta functions. The q-Pochhammer section is honestly presented as a repackaging, and the Fourier-basis decomposition is clearly explained. The claimed general result for all weight-1/2 and 3/2 cusp forms, if proved, would be significant, but as written it exceeds what the manuscript establishes: Definition 2.5 requires an Euler-product L-function decomposition and a common two-sided equally spaced Borel tower, and Section 4.6 explicitly does not provide these. Thus the significance at the level of the unary theta example is solid, while the broader claim requires substantial qualification or additional proof.","major_comments":[{"comment":"The statement that the modular resurgent structure holds 'in general' for Eichler integrals of any vector-valued cusp form of weight 1/2 or 3/2 is stronger than what is proved. Remark 4.5 states verbatim that part 3 of Definition 2.5 is not established by Steps 1–7, and Eq. (228) expresses the Stokes–Dirichlet vector only as a multiple of ϱ(S)LLL(s+κ−1), where LLL is not a vector of L-functions in the sense of Definition 2.3 because an Euler product would require g to be related to Hecke eigenforms. The abstract and Section 5 should either prove the Euler-product statement for a suitable class of eigenforms or explicitly restrict the 'general' claim to the weaker structure that is actually shown, with the unary theta case as the only fully verified vector-valued MRS in the strict sense of Definition 2.5.","section":"Abstract, Conclusions, and §4.6 (esp. Remark 4.5 and Eq. (228))"},{"comment":"Part 1 of Definition 2.5 requires, for every component, a common two-sided equally spaced tower of singularities {ρ_m = A m}_{m∈Z∖{0}}. For a general cusp form g, the singularities are {ξ_λ = 2πi λ} with λ ∈ α_k + Z_{≥0}, which are one-sided and depend on the component k; the authors themselves describe this as satisfying part 1 only in a 'weak sense' and note that the two-sided equally spaced tower in Section 4.3.2 relies on the unary folding ξ = ζ². Therefore the object studied in §4.6 does not, in general, satisfy Definition 2.5 as written. The definition or the claimed scope must be adjusted, for example by introducing a notion of weak or one-sided vector-valued MRS, or by removing the 'in general' assertion.","section":"§4.6, Step 3 (near Eq. (226))"},{"comment":"Even putting aside the Euler product, the linear decomposition required in Eq. (38), namely L_±(s) = M_± diag(c_1,±^s, ..., c_{v,±}^s) L_±(s) with L_± a vector of genuine L-functions, is not demonstrated for general g. What Eq. (228) provides is a single scalar c = D and the vector LLL(s+κ−1) in place of a vector of L-functions, so the arithmetic part of the definition is not met. The paper should either fill this gap for Hecke eigenforms or state the result as a conditional resurgent structure that becomes an MRS only when the Dirichlet vector is known to decompose into L-functions.","section":"§4.6, Eq. (228) and Definition 2.5, part 3"},{"comment":"The median resummation result for general g is proved directly in Step 7, bypassing the hypothesis of Conjecture 3; this is acceptable, but the wording in Step 7 that 'what Steps 1–6 establish is the conclusion of Conjecture 3' should be clarified to avoid implying that the hypothesis of Conjecture 3 has been verified. This is a presentation issue rather than a technical error, but it contributes to the overstatement of the general case.","section":"§4.4.2 and §4.6, Step 7"}],"minor_comments":[{"comment":"There is a typo: 'witth Stokes vectors' should be 'with Stokes vectors'.","section":"§2.4.1 after Eq. (42)"},{"comment":"The relationship between the Borel variable ζ and the variable ξ = ζ² is stated clearly, but the text would benefit from a remark that the two-sided tower in ζ is inherited by the folded vector only after the unary folding; this would help readers see why the general case in §4.6 is different.","section":"§4.3.2, Eq. (155) and surrounding text"},{"comment":"The notation in Eq. (202) uses τ both as a positive real parameter and, later, as the modular variable; this is a source of potential confusion and should be distinguished, e.g., by using t or a different letter for the positive real parameter.","section":"§4.5, Eq. (202)"},{"comment":"The conjectures are stated for vectors of q-series, but the examples in Section 4 require a subtraction of the singular term i/(πτ) for ν=0 (Eq. (190)); the conjecture statements could mention the possibility of such elementary subtractions for non-cuspidal inputs.","section":"§2.4.2, Conjectures 3 and 4"}],"recommendation":"major_revision","confidential_remarks":"The unary theta sections are thorough and appear sound, and the paper makes a genuine contribution there. The main concern is that the abstract and conclusions overclaim the generality of the result: Section 4.6 itself admits in Remark 4.5 that part 3 of Definition 2.5 is not established for general cusp forms, and the Borel tower is only one-sided. This is fixable by rewriting the general statements to match what is proved, or by adding the missing Hecke-eigenform argument, so I recommend major revision rather than rejection. I would also encourage the authors to make the distinction between 'vector-valued resurgent structure' and 'vector-valued modular resurgent series in the strict sense of Definition 2.5' explicit throughout the paper, since the current wording invites a stronger reading than the proofs support."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things before reading. The Eichler-integral analysis for unary theta series is the real substance: explicit Borel transforms, Stokes constants rotated by the S-matrix, quantum modularity, median resummation—all verified in detail. The paper's broader claim about arbitrary vector-valued cusp forms of weight 1/2 or 3/2 is not established at that level of generality, and the authors' own Remark 4.5 concedes the missing pieces.\n\nWhat's genuinely new is the vector-valued modular resurgent framework (Definition 2.5) and the working example of false theta functions built from unary theta series. The S-matrix appearing in the Stokes data is a nice, structural result, and the paradigm diagram closing under the metaplectic involution is convincing. The q-Pochhammer section is explicitly a repackaging of prior scalar work by the same authors; that's not a flaw, but it means the novelty lives almost entirely in Section 4. The proofs in Sections 4.3–4.4 are detailed and internally coherent; I did not find a gap in the unary case.\n\nThe soft spot is the gap between the abstract/conclusions and Section 4.6. Definition 2.5 requires a common two-sided equally spaced Borel tower (part 1) and Euler-product L-functions in the Stokes–Dirichlet decomposition (part 3). For a general cusp form, Section 4.6 only gets a one-sided, generally sparse tower in the ξ-variable; the folding ξ=ζ² that makes it equally spaced and two-sided is special to unary theta. And part 3 is explicitly not established—Remark 4.5 says an Euler product would require g to be a Hecke eigenform. So the sentence in the abstract and Conclusions claiming the same modular resurgent structure for any weight-1/2 or 3/2 cusp form is too strong. This is an overstatement in the framing, not a failure of the unary computations. The authors are honest in the body; the front matter and conclusions undo that honesty.\n\nWho should read it: anyone working on false theta functions, quantum modular forms, or resurgence in quantum topology/vertex algebras. The unary-theta machinery is directly useful and citable.\n\nRecommendation: send it to peer review. Require a revision that either restricts the general claim to what is proven—partial results for cusp forms satisfying an Euler-product hypothesis—or proves the missing arithmetic input for a nontrivial class of weight-3/2 cusp forms. The unary half alone is enough to justify publication.","headline":"The unary-theta half of this paper is a solid, fully worked advance; the advertised general weight-1/2 and 3/2 extension is overstated and does not meet the paper's own Definition 2.5.","tokens_in":48056,"tokens_out":3437,"would_cite":true,"duration_ms":31899,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F37","11F27"],"pacs":[],"model":"deepseek-v4-flash","headline":"Vector-valued resurgence fully captures false theta functions: Stokes constants are rotated by the modular S-matrix, and median resummation reconstructs the Eichler integrals.","keywords":["vector-valued modular resurgence","Eichler integrals","false theta functions","quantum modular forms","median resummation","Stokes constants","Dirichlet L-functions"],"falsifier":"Take a vector-valued cusp form $g$ of weight $3/2$ that is not a linear combination of unary theta series, compute its Stokes–Dirichlet vector from Eq. (228), and check whether its coefficients admit an Euler product; if they do not, part 3 of Definition 2.5 fails. A more direct test is to verify numerically whether the median resummation identity (235) holds for such $g$.","tokens_in":2180,"feed_emoji":"📐","tokens_out":4325,"duration_ms":105641,"temperature":0.7,"pith_summary":"Building on the scalar framework of modular resurgence, this paper extends it to vectors of asymptotic series, introducing vector-valued modular resurgent series (Definition 2.5). The paper proves that two natural families are vector-valued MRSs: vectors of q-Pochhammer symbols, which are a repackaging of scalar results, and vectors of Eichler integrals of unary theta series, which require the vector-valued framework because their Stokes constants are rotated by the modular S-matrix. For the latter, the paper establishes the full paradigm: the Borel transform has a single tower of simple poles, the Stokes constants form a Dirichlet vector that decomposes into L-functions, the vector is a quantum modular form for SL2(Z), and median resummation reconstructs the Eichler integrals from their divergent expansions. The unary theta case is the fully worked example of a general result outlined for vector-valued cusp forms of weight 1/2 and 3/2.","feed_headline":"Vector-valued resurgence governs false theta functions","feed_subtitle":"Eichler integrals of weight 1/2 and 3/2 forms are modular resurgent; Stokes data follow the S-matrix.","key_machinery":"Definition 2.5: a vector of Gevrey-1 series is a vector-valued MRS if its components share one tower of Borel singularities at $A m$, have trivial secondary resurgent series, and their Stokes constants form Dirichlet vectors that are linear combinations of a common vector of $L$-functions via matrices $M_\\pm$. For the Eichler example, the key identity is the Borel transform formula $B[\\tau^{3/2-\\nu}\\widetilde{\\mathrm{EI}}_N^{(\\nu)}](\\xi) = -\\frac{1}{2\\pi i}(\\frac{2N\\xi}{\\pi i})^{1/2-\\nu} \\sum_{m=1}^\\infty \\frac{m^{2\\nu-1}\\mathrm{St}_m^{(\\nu)}}{\\xi - \\pi i m^2/(2N)}$, which manifestly exhibits the simple-pole tower and the Stokes vectors. In the $\\nu=1$ case the $\\sinh$-ratio kernel $K_{N,k}(x)=\\sinh((N-k)x)/\\sinh(Nx)$ gives a closed form for the Borel transform and shows geometrically how the poles of the kernel produce the Stokes constants.","core_discovery":"The central claim is that the vector $\\mathrm{EI}_N^{(\\nu)}$ of Eichler integrals of unary $\\theta$ series is a vector-valued modular resurgent series. Concretely, its Borel transform has a tower of simple poles at $\\rho_m = A m$ with $A = \\sqrt{\\pi i/(2N)}$, the resurgent series are constant Stokes vectors $\\mathrm{St}_m^{(\\nu)} = 2i(-1)^\\nu m^{1-\\nu} \\Omega^{(\\nu)}(S)\\,\\omega^{(\\nu)}(m;N)$, and the Stokes–Dirichlet vector decomposes as a linear combination of Dirichlet $L$-functions (Proposition 4.7). Proposition 4.9 proves that $\\mathrm{EI}_N^{(\\nu)}$ is a vector-valued holomorphic quantum modular form of weight $3/2-\\nu$, and Proposition 4.10 proves that median resummation at angle $\\pi/2$ reconstructs the vector, up to the known pole term $i/(\\pi\\tau)$ in the $\\nu=0$ case. The paradigm diagram closes because the same $S$-matrix appears in the functional equation of the completed Dirichlet vector: modularity of the $\\theta$ series dictates the Stokes data.","pith_inferences":["Relaxing part 3 of Definition 2.5 to allow meromorphic continuation without Euler products would make the weight-$3/2$ cusp-form claim unconditional; the paper already proves all analytic aspects for general $g$, so the only missing input is arithmetic.","The one-sided sparse tower for general cusp forms suggests a natural extension of vector-valued resurgence to higher-depth or non-quadratic exponents, where the equally spaced tower must be abandoned.","The kernel mechanism suggests that modular anomalies of false theta functions are residue sweeps under contour rotation, which should generalise to higher-rank false theta functions and plumbed three-manifold invariants."],"forward_implications":["For every $N$ and $k$, the false theta function $\\mathrm{EI}[\\theta^{(1)}_{N,k}]$ can be recovered from its divergent asymptotic expansion by median resummation at angle $\\pi/2$.","The $S$-matrix $\\Omega^{(\\nu)}(S)$ is encoded in the Stokes constants, so resurgent data of an Eichler integral directly reveals the modular transformation of the underlying theta series.","The vector-valued framework is essential precisely when the multiplier system is non-trivial; in the q-Pochhammer example, a Dirichlet-character twist diagonalizes the vector into scalar MRSs.","For any vector-valued cusp form of weight $1/2$ or $3/2$, the Eichler integral satisfies parts 1 and 2 of Definition 2.5 and is a vector-valued quantum modular form; at weight $1/2$ the Serre–Stark theorem reduces the claim to unary theta series, so the genuinely new examples occur at weight $3/2$.","The paradigm diagram closes without introducing an independent companion q-series; the second row is the $S$-image of the first, showing classical modularity is the organising principle."],"supporting_citations":[{"why":"Defines scalar modular resurgence and the paradigm that this paper extends to vector-valued series.","marker":"[1]"},{"why":"Supplies the q-Pochhammer results that this paper repackages into trivial-multiplier vector-valued MRSs.","marker":"[2]"},{"why":"Defines quantum modular forms, the backdrop for Conjecture 4 and Proposition 4.9.","marker":"[5]"},{"why":"Gives the Lawrence–Zagier L-series method and the false-theta context used for the $\\nu=1$ asymptotics.","marker":"[10]"},{"why":"Establishes half-integral-weight quantum modularity of Eichler integrals that Proposition 4.9 generalizes to the full vector-valued $\\mathrm{SL}_2(\\mathbb{Z})$ cocycle.","marker":"[19]"},{"why":"Proves quantum modularity of partial theta series, the scalar precursor of the vector-valued quantum modularity result.","marker":"[20]"},{"why":"Provides the bimodular completion construction whose pole cancellations are analyzed in Section 4.5.","marker":"[26]"},{"why":"Serre–Stark theorem delimits the weight-$1/2$ case and highlights where genuinely new $3/2$ examples arise.","marker":"[35]"}],"fun_headline_variants":["Vector resurgence ties false theta to quantum modularity","S-matrix dictates vector resurgence in false theta","False theta functions are vector-valued modular resurgent","Vector resurgence recovers false theta via median resummation"],"cache_read_input_tokens":50048,"weakest_assumption_plain":"For the general claim of Section 4.6, the load-bearing premise is that the Dirichlet series of the Stokes constants are linear combinations of $L$-functions with Euler products (part 3 of Definition 2.5); the paper only proves this decomposition in the unary theta example and explicitly notes it is unavailable for arbitrary cusp forms.","fun_headline_variants_meta":{"raw":{"variants":["Vector resurgence ties false theta to quantum modularity","S-matrix dictates vector resurgence in false theta","False theta functions are vector-valued modular resurgent","Vector resurgence recovers false theta via median resummation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001401,"raw_usage":{"total_tokens":5720,"prompt_tokens":1054,"completion_tokens":4666,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":4604}},"tokens_in":670,"tokens_out":4666,"duration_ms":36329,"temperature":1.0,"reasoning_tokens":4604,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:21:20.733738+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a vector-valued cusp form $g$ of weight $3/2$ that is not a linear combination of unary theta series, compute its Stokes–Dirichlet vector from Eq. (228), and check whether its coefficients admit an Euler product; if they do not, part 3 of Definition 2.5 fails. A more direct test is to verify numerically whether the median resummation identity (235) holds for such $g$.","supporting_citations":[{"cited_title":"Quantum modular forms,","cited_arxiv_id":null,"evidence_quote":"Defines quantum modular forms, the backdrop for Conjecture 4 and Proposition 4.9."},{"cited_title":"Modular forms and quantum invariants of 3-manifolds,","cited_arxiv_id":null,"evidence_quote":"Gives the Lawrence–Zagier L-series method and the false-theta context used for the $\\nu=1$ asymptotics."},{"cited_title":"Half-integral weight eichler integrals and quantum modular forms,","cited_arxiv_id":null,"evidence_quote":"Establishes half-integral-weight quantum modularity of Eichler integrals that Proposition 4.9 generalizes to the full vector-valued $\\mathrm{SL}_2(\\mathbb{Z})$ cocycle."},{"cited_title":"Quantum modularity of partial theta series with periodic coefficients","cited_arxiv_id":"2012.02457","evidence_quote":"Proves quantum modularity of partial theta series, the scalar precursor of the vector-valued quantum modularity result."},{"cited_title":"A Framework for Modular Properties of False Theta Functions","cited_arxiv_id":"1904.05377","evidence_quote":"Provides the bimodular completion construction whose pole cancellations are analyzed in Section 4.5."},{"cited_title":"Modular forms of weight1/2,","cited_arxiv_id":null,"evidence_quote":"Serre–Stark theorem delimits the weight-$1/2$ case and highlights where genuinely new $3/2$ examples arise."}],"review_version":1}