{"id":"e92fd982-6fee-4c8d-b583-18f4d26257e8","arxiv_id":"2509.01598","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A double-copy (KLT-type) formula for genus-g hypergeometric integrals is derived from twisted homology intersection numbers and verified numerically at genus two.","lead":"The paper derives a 'double copy' formula that expresses certain complex, closed-string-like integrals on genus-g Riemann surfaces as bilinear sums of simpler open-string-like integrals, with the kernel built from twisted-homology intersection numbers. This generalizes known tree-level and one-loop double-copy identities toward a possible g-loop KLT relation, and ships numerical tools verified at genus two.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Basis of twisted homology rests on unproved determinant (4.35); an error there would make the KLT kernel H^{a,b}_{(2,2)} undefined and break the central double copy (5.24).","rationale":"The reader's weakest assumption identifies exactly the load-bearing point: the spanning set of twisted cycles is shown to be a basis only through the determinant formula (4.35), which is presented with a geometric sketch rather than a complete derivation. I agree with that assessment and do not find a more serious internal inconsistency. The central identity (5.24) is otherwise supported by an independent Stokes-theorem argument in appendix D, by the cited twisted Riemann bilinear relations [28, 30], and by genus-two numerical checks in section 7. Those supports are genuine: they would detect many errors in the intersection-number formulas, and they specifically validate the double-copy structure for g=2. They do not, however, establish the general-g,n basis property that is a prerequisite for the KLT kernel H^{a,b}_{(2,2)}. An error in the determinant or diagonal structure of H^{(2,n)} would not affect the direct integrals but would change or invalidate the bilinear kernel, so the conditional status of the paper is appropriate. I therefore recommend no change to the reader's CONDITIONAL verdict; the proposed test would settle whether the gap is only a missing proof or an actual error.","tokens_in":46425,"tokens_out":12839,"duration_ms":148593,"concrete_test":"Symbolically recompute the matrix H^{(2,n)}_{a,b}=[γ_{2a}|ˇγ_{nb}] for g=2, n=4 and g=3, n=4 directly from the regularized cycle expansions (4.14)-(4.15) and the dual cycles in figure 5, using the local intersection rule (4.19)-(4.20). Compare every entry and the determinant to (4.35). If any off-diagonal entry is nonzero, or the determinant differs in any factor or exponent, the basis statement of Theorem 4.2 fails in that case and the inverse matrix in (5.24) is not justified. If the cases match, repeat the same computation symbolically for general g,n to confirm the claimed diagonal form and product formula.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The double-copy identity (5.24)-(5.25) requires the inverse intersection matrix H^{a,b}_{(2,2)}. That inverse exists only if the chosen regularized cycles {γ_{2AJ}, γ_{2BJ}, γ_{2j}} form a basis of H_1(Σ*_g, L_s). The basis claim (Claim 4.1, Theorem 4.2) is justified solely by the asserted non-vanishing determinant (4.35) of the auxiliary matrix H^{(2,n)}_{a,b}=[γ_{2a}|ˇγ_{nb}], built from dual cycles based at z_n (figure 5). This determinant is stated as a product formula but not derived; the computations in appendix A concern intersection numbers among cycles based at z_2, not the entries of H^{(2,n)}. The geometric sketch (figures 9-11) indicates how the genus-one computations recycle, but it does not provide the full general-g,n computation of the off-diagonal entries or the determinant exponents. Because this determinant is the only demonstrated mechanism excluding additional linear relations, a sign, ordering, or counting error in (4.35) for some g or n would make H_{(2,2)} singular or would produce the wrong KLT kernel, invalidating (5.24) even though the integrals themselves are well-defined. The elementary proof in appendix D does not fully close this gap: it verifies the final double-copy expression explicitly for genus two (after eq. (D.11)) and asserts straightforward generalization, while the general-g formula again presupposes a valid homology basis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies one-fold hypergeometric integrals on a genus-g Riemann surface punctured at n-1 points, with a multivalued twist T(z1). It constructs regularized twisted cycles, computes homology intersection numbers among them, and uses twisted Riemann bilinear relations to derive a double-copy identity: the ``complex'' integral J^{φ_k,φ_l} = ∫ |T|² φ_l ∧ φ_k is expressed as a bilinear sum of ordinary integrals I^{φ}_{γ} weighted by the inverse of the homology intersection matrix H^{(2,2)}. The same structure is extended to Abelian Kronecker forms. Numerical checks are provided at genus two for both holomorphic differentials and a genus-two polylog kernel, using Schottky uniformization and Crowdy-Marshall methods.","tokens_in":46713,"tokens_out":6522,"duration_ms":80434,"significance":"If the central identity (5.24)/(6.20) holds for all g, the paper provides a concrete genus-g analogue of a KLT formula in which the inverse twisted-homology intersection matrix plays the role of the KLT kernel. The manuscript also contains useful technical contributions: explicit intersection numbers for a new family of twisted cycles, an analytic inverse matrix for genus two, a numerical implementation of the relevant integrals, and apparently the first publicly available evaluation of a genus-two polylogarithm kernel. The main claim is supported by three independent routes — a specialization of cited twisted Riemann bilinear relations, a Stokes-theorem argument in Appendix D, and numerics — but, as detailed below, two of these routes are incomplete for general g.","major_comments":[{"comment":"The proof of Claim 4.1/Theorem 4.2 is incomplete. The non-vanishing determinant of H^{(2,n)} is the only demonstrated mechanism excluding additional linear relations among the cycles {γ_{2AJ}, γ_{2BJ}, γ_{2j}}, and hence the only justification that H^{(2,2)} in (4.38) is invertible. However, the entries of H^{(2,n)} are not written down; the text simply asserts that the matrix is diagonal and gives a product formula. Appendix A computes [γ_{2a}|γ_{2b}] for cycles based at z2, not the dual cycles ˇγ_{nb} of figure 5 that enter (4.34). Thus the determinant is unverified for general g,n. Since H^{a,b}_{(2,2)} in (5.24) is the KLT kernel, this gap is load-bearing. Please supply the full computation or an alternative proof of the basis property.","section":"Section 4.3, Eq. (4.35)"},{"comment":"The elementary Stokes proof is performed explicitly only for genus two: the text after (D.11) states that the general formula is straightforward to generalize and that agreement with (7.7) was verified for g=2, n=3. The paper's central claim (5.24), however, is for all g. No explicit derivation of (D.11) for g>2 is shown, and the general formula is asserted rather than proved. This is a second incomplete justification of the higher-genus result. If the basis gap in Section 4.3 is fixed, this may be less critical, but as written the higher-genus double copy is not fully proved.","section":"Appendix D, Eq. (D.11)"},{"comment":"The determinant formula (4.35) contains factors (e^{2πis_{1j}}-1)^{-1} only for j=3,...,n-1, but the linear relation (4.16) has coefficients that can vanish for special values of s_{1AJ} and s_{1BJ}. The paper does not characterize the excluded locus where the chosen subset of cycles fails to be a basis, nor does it state whether the inverse kernel H^{a,b}_{(2,2)} has removable singularities at those points. For a claim about generic s-values this is acceptable, but it should be made precise.","section":"Section 4.3, around Eq. (4.35)-(4.37)"}],"minor_comments":[{"comment":"The notation is inconsistent: [AI|BJ]_{top} = δ_{IJ} = -[BI, AJ]_{top} mixes comma and bar notation. Please standardize.","section":"Eq. (2.2)"},{"comment":"The numerical comparison lacks a discussion of numerical accuracy or convergence criteria. Please state the estimated error of the direct-integration and double-copy results, and specify the number of sample points used.","section":"Section 7.1, Figure 8"},{"comment":"The text repeatedly refers to ancillary files for numerical checks. Please provide a permanent repository link or DOI to allow reproducibility.","section":"Ancillary files"},{"comment":"In the sentence introducing (7.14), “flowing vectors” appears to be a typo for “following vectors.”","section":"Section 7.2"},{"comment":"The examples in Section 7 use (g,n)=(2,3), which violates the stated condition n > max(2,2g-1). The text explains why this is harmless, but this should be remarked explicitly in the statement of the propositions to avoid confusion.","section":"Propositions 3.1 and 6.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is potentially important, but the higher-genus claim currently rests on an unproved determinant (4.35) and an asserted generalization (D.11). I would recommend major revision: the load-bearing issue is localized and fixable by a complete derivation of H^{(2,n)} or an alternative proof of the twisted-homology basis, together with a proof of the genus-general Stokes formula. The numerical evidence at genus two is a positive sign, but it does not replace the missing general-g argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper deserves a serious referee. The central result is a genus-g double-copy formula, eq. (5.24), for a family of hypergeometric integrals tied to chiral-splitting string amplitudes. This is a genuine extension of the genus-0 and genus-1 KLT-type results, and the authors are careful not to oversell it: they explicitly leave full g-loop KLT relations to future work and restrict to rank-1 Abelian local systems.\n\nWhat is actually new: the regularized twisted-homology basis (Theorem 4.2), the homology intersection numbers in (4.25)-(4.31) including the genuinely new I≠J cases, and the double-copy formula itself, with its extension to Abelian Kronecker forms. The paper is also honest about what it does not do.\n\nWhat it does well: the central identity is supported three independent ways. First, it is a specialization of known twisted Riemann bilinear relations. Second, appendix D gives an elementary Stokes-theorem proof, explicit for genus 2 and stated for general genus in (D.11). Third, the genus-2 numerical checks in figure 8 and eq. (7.17) agree, and the ancillary files provide reproducible code. None of this looks circular: J is defined by its own integral, the kernel is the inverse of an independently computed intersection matrix, and the equality is tested numerically.\n\nThe soft spots are real but not fatal. The main one is the stress-test concern: the claim that the chosen cycles form a basis of twisted homology rests on the determinant formula (4.35), which is stated with a geometric sketch rather than a complete derivation. If that determinant were wrong for some g or n, the inverse kernel would not be the correct KLT kernel. I think the formula is very likely correct -- it is a simple product of monodromy factors, and the genus-2 numerics are consistent -- but the proof is incomplete. Similarly, the general-g intersection numbers in appendix A.3 rely on a \"collapse\" argument that is sketched rather than fully written, and Proposition 6.1 is argued rather than proven. Numerical verification covers genus 2 only. These are presentation gaps, not evident errors.\n\nThe paper is honest about its scope and shows clear thinking. I would send it to peer review and ask the authors to either prove or supply a complete derivation of (4.35), and to expand the general-g Stokes proof. For a reader interested in string amplitudes, twisted cohomology, or the double copy, this is worth the time.","headline":"A serious genus-g double-copy paper with real new content and honest scope; the main gap is a homology basis claim that is plausible but not fully proved.","tokens_in":47363,"tokens_out":2569,"would_cite":true,"duration_ms":31495,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives a double-copy (KLT) formula at arbitrary genus: a single-valued 'closed-string' integral equals a bilinear of 'open-string' integrals, with the inverse homology-intersection matrix as kernel.","keywords":["double copy","twisted homology","twisted cohomology","KLT relations","genus-g hypergeometric integrals","homology intersection numbers","Abelian Kronecker forms","higher-genus polylogarithms"],"falsifier":"Compute the intersection matrix H_{(2,2)} for a genus-three surface from the cell decomposition (4.13)-(4.15) by direct implementation, and check that its determinant matches (4.35) and that the bilinear identity (5.25) holds numerically for several holomorphic differentials and parameter values. A sign error, a missed intersection, or a second linear relation among the cycles would surface as a mismatch or a singular matrix; probing the excluded locus s1j ∈ Z would show whether the double copy breaks there or is recovered by a different basis.","tokens_in":46204,"feed_emoji":"🧵","tokens_out":19416,"duration_ms":189681,"temperature":0.7,"pith_summary":"The paper establishes double-copy (KLT-type) quadratic relations for a family of one-fold hypergeometric integrals on a punctured Riemann surface of genus g - integrals that are the building blocks of g-loop open-string amplitudes in the chiral-splitting formalism. Its central claim is that the single-valued 'closed-string' analogue, J = ∫_{Σ*_g} |T(z1)|² φ_l ∧ φ_k, is exactly a bilinear combination of ordinary 'open-string' integrals I = ∫_γ T φ, with coefficients given by the inverse of the matrix of twisted-homology intersection numbers - so that inverse matrix is the genus-g double-copy kernel. To reach this, the paper constructs explicit bases of twisted homology from regularized integration cycles and computes all their intersection numbers, then verifies the identity numerically at genus two and extends it to quasiperiodic Abelian Kronecker forms, which generate integration kernels for higher-genus polylogarithms. If correct, this is the first double-copy formula valid at every genus and a concrete step toward g-loop KLT relations in string theory.","feed_headline":"Closed-string integrals become bilinears of open ones, at every genus","feed_subtitle":"The genus-g analogue of the KLT double-copy relations, with numerical checks at genus two.","key_machinery":"Four pieces carry the argument. (1) The twist T(z1) = Π_j [E(z1,zj)]^{s1j} exp(2πi Σ_I s1AI ν_I(z1)), a multivalued function on the punctured surface whose monodromies define the local system L_s and the twisted differential ∇_ω; it plays the role of the Koba-Nielsen factor. (2) Twisted homology H1(Σ*_g, L_s), spanned by regularized cycles γ_{2AJ}, γ_{2BJ}, γ_{2j} built from an explicit cell decomposition, subject to exactly one linear relation (4.16). (3) The homology intersection numbers (4.25)-(4.31), assembled into the matrix H_{(2,2)} whose inverse is the double-copy kernel. (4) Twisted Riemann bilinear relations, which convert the compatibility of integration and intersection pairings","core_discovery":"The paper's central result is the double-copy identity (5.24)-(5.25): for real monodromy parameters s1•, the single-valued ('closed-string') integral J^{φ_k,φ_l} = ∫_{Σ*_g} |T(z1)|² φ_l ∧ φ_k equals the bilinear sum Σ_{a,b∈K} H^{a,b}_{(2,2)} I^{φ_k}_{γ_b} I^{φ_l}_{γ_a} over twisted cycles, with H^{a,b}_{(2,2)} the inverse of the homology intersection matrix. For holomorphic differentials this reads ∫ |T|² ω_J ∧ ω_I = Σ_{a,b} H^{a,b} (∫_{γ_a} T ω_I)(∫_{γ_b} T ω_J). The structure extends to Abelian Kronecker forms (6.20)-(6.25), giving double-copy formulas for single-valued higher-genus polylogarithm kernels, and is verified numerically at genus two. The authors' reading: the inverse intersect","pith_inferences":["Because the kernel H^{a,b}_{(2,2)} depends only on the surface and the monodromies - not on which twisted 1-form is integrated - the same matrix should control double copies for every pair of cohomology classes on Σ*_g; extracting it once per surface would give all such quadratic identities at once.","A sharp next test would be a genus-three verification: at g = 3 the genuinely new I ≠ J intersection numbers (4.28)-(4.31) mix in ways absent at genus two, so a numerical check there would localize any error in the basis or determinant claim.","The paper restricts to Abelian (rank-one) quasiperiodic forms; extending the double copy to the full non-Abelian Kronecker generating function would presumably require higher-rank local systems and would deliver double-copy relations for individual higher-genus polylogarithm kernels rather than only symmetric combinations.","One implication the authors leave implicit: in degeneration limits of the surface, the inverse intersection matrix should factor, yielding inductive checks of the genus-g kernel against lower-genus (including tree-level) KLT kernels - a structure worth testing since it connects the all-genus formula to known factorization of string amplitudes."],"forward_implications":["The double copy holds at every genus g for the one-fold integrals: the inverse homology intersection matrix is a purely geometric KLT kernel, computable from the surface and the monodromy parameters alone.","At genus one the formula reproduces the known Riemann-Wirtinger double copy, matching the proposed one-loop KLT relation; the genus-g result is its natural continuation.","The extension to Abelian Kronecker forms puts the double copy inside reach of higher-genus polylogarithm kernels - concretely, the genus-two kernel f̃_2^1 satisfies a double-copy formula (7.17).","Under the reality conditions (5.31)/(6.22), |T|² becomes the exponential of the string Green's function G(z1,zj), so the double copy becomes a statement about g-loop chiral-splitting string integrands rather than purely mathematical integrals.","Numerically, the double copy provides the analytic continuation of the complex integral J beyond its naive convergence region - the genus-two example continues -1 < s12 < 1 to s12 > 1."],"supporting_citations":[{"why":"Supplies the twisted cohomology groups and their spanning basis on the punctured genus-g surface, the cone that the paper's new homology basis is paired with.","marker":"[29]"},{"why":"Supplies the twisted Riemann bilinear relations (their theorem 2) that turn intersection pairings into the bilinear identity behind the double copy.","marker":"[30]"},{"why":"Supplies the bilinear 'double copy' relations between a complex-conjugated cohomology and its dual that the paper's proposition 5.2 specializes.","marker":"[28]"},{"why":"Supplies the genus-one Riemann-Wirtinger integrals and their twisted (co)homology, the model the paper generalizes to genus g.","marker":"[25]"},{"why":"Supplies the genus-one intersection-number computations and the Stokes' theorem technique (their Proposition 4.22) reused for the elementary proof in appendix D.","marker":"[26]"},{"why":"Supplies the quasiperiodic intersection numbers and phase conventions that the α-substitution (6.11) extends to Abelian Kronecker forms.","marker":"[27]"},{"why":"The authors' earlier genus-one double-copy paper whose notation, intersection numbers, and regularization conventions are carried over to genus g.","marker":"[17]"},{"why":"Supplies the Schottky-uniformization numerical method (appendix C) used to evaluate the genus-two integrals and verify the double-copy identities.","marker":"[50]"},{"why":"Supplies the Schottky-Kronecker forms whose Abelian truncation provides the quasiperiodic integrands used in section 6.","marker":"[34]"},{"why":"Supplies the meromorphic integration kernels (e.g., g_2^1) entering the genus-two polylogarithm double-copy example.","marker":"[38]"}],"fun_headline_variants":["Twisted homology yields double copy for genus-g closed strings","Double copy from twisted (co)homology, checked at genus two","Genus-g closed-string integrals factor via homology intersections","Higher-genus KLT: closed strings as bilinears of open","At every genus, closed-string integrals obey double copy"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the explicitly listed regularized cycles span the twisted homology with exactly one linear relation, so the intersection matrix is invertible and the double-copy kernel is well defined; this is supported by a non-vanishing determinant (4.35) presented through geometric sketches rather than a complete derivation for general g.","fun_headline_variants_meta":{"raw":{"variants":["Twisted homology yields double copy for genus-g closed strings","Double copy from twisted (co)homology, checked at genus two","Genus-g closed-string integrals factor via homology intersections","Higher-genus KLT: closed strings as bilinears of open","At every genus, closed-string integrals obey double copy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1208,"prompt_tokens":724,"completion_tokens":484,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":468,"completion_tokens_details":{"reasoning_tokens":400}},"tokens_in":468,"tokens_out":484,"duration_ms":5291,"temperature":1.0,"reasoning_tokens":400,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:23:15.124094+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the intersection matrix H_{(2,2)} for a genus-three surface from the cell decomposition (4.13)-(4.15) by direct implementation, and check that its determinant matches (4.35) and that the bilinear identity (5.25) holds numerically for several holomorphic differentials and parameter values. A sign error, a missed intersection, or a second linear relation among the cycles would surface as a mismatch or a singular matrix; probing the excluded locus s1j ∈ Z would show whether the double copy breaks there or is recovered by a different basis.","supporting_citations":[{"cited_title":"Watanabe, Twisted cohomology of a punctured Riemann surface , Kumamoto J","cited_arxiv_id":null,"evidence_quote":"Supplies the twisted cohomology groups and their spanning basis on the punctured genus-g surface, the cone that the paper's new homology basis is paired with."},{"cited_title":"Cho and K","cited_arxiv_id":null,"evidence_quote":"Supplies the twisted Riemann bilinear relations (their theorem 2) that turn intersection pairings into the bilinear identity behind the double copy."},{"cited_title":"Hanamura and M","cited_arxiv_id":null,"evidence_quote":"Supplies the bilinear 'double copy' relations between a complex-conjugated cohomology and its dual that the paper's proposition 5.2 specializes."},{"cited_title":"Mano and H","cited_arxiv_id":null,"evidence_quote":"Supplies the genus-one Riemann-Wirtinger integrals and their twisted (co)homology, the model the paper generalizes to genus g."},{"cited_title":"Moduli spaces of flat tori and elliptic hypergeometric functions","cited_arxiv_id":"1605.02356","evidence_quote":"Supplies the genus-one intersection-number computations and the Stokes' theorem technique (their Proposition 4.22) reused for the elementary proof in appendix D."},{"cited_title":"Intersection numbers of twisted homology and cohomology groups associated to the Riemann-Wirtinger integral","cited_arxiv_id":"2206.03177","evidence_quote":"Supplies the quasiperiodic intersection numbers and phase conventions that the α-substitution (6.11) extends to Abelian Kronecker forms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Schottky-uniformization numerical method (appendix C) used to evaluate the genus-two integrals and verify the double-copy identities."}],"review_version":1}