{"id":"f5e6d8fa-29ca-48a2-add0-f7395ecf72d5","arxiv_id":"2502.06187","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A reconstruction theorem expresses genus-one permutation-equivariant quantum K-invariants of any compact Kähler manifold in terms of genus-zero data and residues.","lead":"Quantum K-theory counts curves on a space, and this paper proves a formula for the genus-one (one-hole) case, rewriting complicated invariants as simpler pieces. The result completes the reconstruction program for genus-one permutation-equivariant invariants of any compact Kähler manifold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Fix the sign inconsistency in the definitions of \\bar{y}_r and \\bar{y}_L^2 between Section 0.3 and Lemma 2.4; the residue formula in Theorem 1 depends on it.","rationale":"The reader's CONDITIONAL verdict is supported by multiple issues, including the classification completeness and this sign inconsistency. I agree that the paper should be accepted only after careful revision. The sign inconsistency is the most load-bearing because it is an internal contradiction between the definitions in the theorem statement and the proof of the key lemma. If the sign is wrong, the main formula is incorrect; if it is a typo, the proof is still flawed as written. This is more concrete and more directly tied to the central claim than the imported classification, although the classification completeness is also a serious concern. The recommended verdict remains CONDITIONAL, requiring the author to resolve the sign and provide the omitted computational details.","tokens_in":13523,"tokens_out":5886,"duration_ms":44613,"concrete_test":"Independently compute the q → -1 expansion of \\bar{y}_2(q) and \\bar{y}_L^2(q) directly from Proposition 1.1, the definition of \\bar{t}^{new}_2(q), and the algorithm for τ in Section 1.7, without using Lemma 2.4. Verify whether \\bar{y}_2(q) expands as \\bar{y}_{1,-1} + '\\bar{x}^{2a}_{1,-1}(-q-1) + O((-q-1)^2) and whether \\bar{y}_L^2(q) expands as \\bar{x}^{2c}_{1,-1} + (\\bar{x}^{2a}_{1,-1} + \\bar{x}^{2b}_{1,-1})(-q-1) + O((-q-1)^2). If the signs do not match the expansions claimed in Lemma 2.4, the residue identity fails, and Theorem 1's formula must be corrected.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 0.3, \\bar{y}_r is defined with a plus sign before the correlation term, and \\bar{y}_L^2 is defined with a plus sign before its correlation term. In Lemma 2.4, however, both are written with a minus sign: \\bar{y}_2(q) = \\bar{x}_1(q) - \\sum ... and \\bar{y}_L^2(q) = \\bar{x}_1(q) - \\sum ... . Theorem 1 uses the Section 0.3 definitions. If the minus sign in Lemma 2.4 is the correct one, then the F^{perm}_{1,2} term in Theorem 1 has the wrong sign. If the plus sign is correct, Lemma 2.4's expansion of \\bar{y}_2 near q=-1 is wrong, and the residue computation that produces the Case 2 contribution does not establish the claimed identity. Either way, the central formula is not reliably derived from the proof as written. This is an internal inconsistency, not a dispute with external conventions, and it directly enters the final reconstruction formula. The proof also uses dimension arguments to discard higher-order terms, but the sign issue is more concrete and immediately checkable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a reconstruction theorem for genus-one permutation-equivariant quantum K-invariants of a compact Kähler manifold. The main result, Theorem 1, expresses the full genus-one descendant generating function F_1(t) as the primitive no-descendant potential F_1(τ), a logarithmic Jacobian term (1/24) log(∂τ_1/∂t^1_{1,0}), and a sum over M=2,3,4,6 of residues at 0 and ∞ of explicit rational functions F^{perm}_{1,M}(x) built from genus-zero correlators and genus-one correlators with restricted descendant inputs. The proof uses the permutation-equivariant ancestor-descendant correspondence, the splitting axiom, a vanishing theorem and classification of non-vanishing contributions from the author's earlier work [10], and dilaton equations. The paper is computational and does not introduce new abstract machinery.","tokens_in":13707,"tokens_out":8668,"duration_ms":66840,"significance":"If correct, this result is a substantial generalization of the author's earlier reconstruction theorems for non-permutative quantum K-theory and for the point target space, and it provides an analog of Dijkgraaf-Witten's theorem in permutation-equivariant quantum K-theory. The formula is explicit and potentially useful for computations. The paper does not fit parameters or assume its conclusion; it derives the formula from previously established lemmas. A strength is that the proof is organized into a clear case analysis and the final assertion is concrete and checkable. However, the paper's reliance on prior results and a few opaque computational steps makes independent verification difficult.","major_comments":[{"comment":"In Section 0.3 the quantities \\bar{y}_r and \\bar{y}^L_2 are defined with a plus sign before the correlation sum, but in the proof of Lemma 2.4 (Section 2.6) the same quantities are written with a minus sign. Since Theorem 1 uses the Section 0.3 definitions to construct F^{perm}_{1,2}(x), and Lemma 2.4 identifies the Case 2 contribution with the residue of F^{perm}_{1,2}(x), the two signs cannot both be correct. The proof as written leaves the sign of the contribution undetermined; the author must reconcile the definitions and rerun the residue computation, or the final formula may be incorrect.","section":"Section 0.3 and Section 2.6 (Lemma 2.4)"},{"comment":"The sentence 'Directly computing the right-hand side's contribution yields the right-hand side of our claim' is the central combinatorial step that converts the sum over curves with ∼-chains into the correlator \\langle\\langle \\bar{x}^{2a}_{1,-1}(-\\bar{L}-1), \\bar{y}_{1,-1}, \\bar{y}_{1,-1}, \\bar{y}_{1,-1}\\rangle\\rangle_{1,4_1}. This equality is not demonstrated, and it involves signs and combinatorial factors that are essential for the final formula. Without a detailed derivation, the proof of Lemma 2.2 is not verifiable; the author should spell out the counting and the manipulation of the correlator.","section":"Section 2.3, Lemma 2.2, Step 2"},{"comment":"The proof of Theorem 1 relies on the completeness of the classification of non-vanishing super-traces listed in the table. This classification is cited from the author's prior paper [10] and is not proved in the present text. Since the sum over M=2,3,4,6 in Theorem 1 is exhaustive only if every non-vanishing stratum is captured by the table, the author should either state the classification as a precise theorem (or cite the specific theorem in [10] with a restatement) and indicate how it applies to the present setting, or provide a proof in an appendix. As written, a missing row in the table would silently omit contributions.","section":"Section 1.4 (vanishing theorem and non-vanishing table)"}],"minor_comments":[{"comment":"In the definition of \\bar{F}_1(\\bar{t}), the notation \\bar{\\ell} is used without being explicitly introduced; presumably it denotes the vector of cycle counts for the \\tau inputs. Please define it.","section":"Section 0.2"},{"comment":"Reference [2] (Givental, 'Gromov-Witten invariants and quantization of quadratic Hamiltonians') appears in the bibliography but is not cited in the text. Either cite it where relevant or remove it.","section":"References"},{"comment":"The final equality in Section 2.6, which converts a residue at q=-1 to residues at x=0 and x=\\infty, is written without explanation; a short comment on the change of variables would improve readability.","section":"Section 2.6"},{"comment":"The table in Section 1.4 lists the non-vanishing cases but does not explain the meaning of the superscripts and bars in the last column (e.g., \\bar{M}^{\\pm i}_{1,2_1+1_2+...}); a parenthetical explanation in the text would help.","section":"Section 1.4 table"},{"comment":"The statement 'the contributions of the following curves agree' is accompanied by figures that are not rendered in the arXiv version; please ensure all figures are embedded and referenced.","section":"Section 2.5"}],"recommendation":"major_revision","confidential_remarks":"The paper is a direct sequel to the author's own arXiv:2411.06053 and arXiv:2411.07487, and it relies heavily on those results. The editor may wish to verify that these manuscripts are in a publicly accessible, citable form, and that the present paper includes a precise statement of the vanishing theorem used in Section 1.4. The sign inconsistency and the missing computation in Lemma 2.2 should be resolved before the paper is accepted; these are not purely cosmetic issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper extends the author's genus-one reconstruction theorem for permutation-equivariant quantum K-invariants from the non-permutative case and the point target to arbitrary compact Kähler targets. If the formula holds, it completes a natural program: all genus-one descendant invariants are expressed in terms of genus-zero data, the primitive genus-one potential, and explicit residues. The framework comes from the author's previous work and from Givental's papers, and the new contribution is a long case-by-case residue computation.\n\nThe paper does several things well. It states the theorem cleanly, uses the ancestor-descendant correspondence and splitting axiom in a straightforward way, and the reliance on the vanishing theorem and non-vanishing classification from [10] is explicit rather than hidden. There is no parameter fitting or circular reasoning; the burden on previously established results is clear.\n\nThe immediate problem is a sign inconsistency. Section 0.3 defines \\bar{y}_r and \\bar{y}_L^2 with a plus sign before the correlation terms. Lemma 2.4, which assembles the order-2 contribution, writes both with a minus sign. Since Theorem 1 uses the Section 0.3 definitions, the residue formula for F^perm_{1,2} is not derived from the proof as written. Either the minus signs are typos and the expansions in Lemma 2.4 are wrong, or they are correct and the sign in the final formula is off. This is load-bearing, not cosmetic.\n\nBeyond that, several computational steps are asserted rather than shown: 'Directly computing the right-hand side's contribution yields' in Lemma 2.2, 'One check' in Lemma 2.4, and the use of 'dimension arguments' to discard higher-order terms in multiple places. For a theorem of this precision, a referee will want those steps spelled out or explicitly tracked to equations in [9] or [10].\n\nThe intended audience is people working in quantum K-theory, particularly permutation-equivariant invariants. The paper deserves a serious referee. My recommendation is to send it to peer review, with the expectation that the author fixes the sign issue and supplies the omitted details. I would not cite the formula as proven until that happens.","headline":"Solid extension of the genus-one reconstruction program, but a load-bearing sign inconsistency between Section 0.3 and Lemma 2.4 means the proof as written does not establish the main formula.","tokens_in":14248,"tokens_out":8009,"would_cite":false,"duration_ms":64516,"reading_group":"yes","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that all genus-one permutation-equivariant quantum K-invariants of a compact Kähler target are determined by genus-zero invariants, the primitive genus-one potential, and explicit residues of rational correlators.","keywords":["quantum K-theory","permutation-equivariant invariants","genus one","Gromov-Witten invariants","reconstruction theorem","ancestor-descendant correspondence","residues","compact Kähler target"],"falsifier":"Directly evaluate both sides of Theorem 1 for a compact toric target such as the complex projective line, truncated at a fixed low degree in the Novikov variables and with a small number of marked points carrying descendant classes, using the definition of the super-traces; a mismatch in the constant term or the leading q-expansion would show that the non-vanishing classification is incomplete or that a dimension argument discarding terms of order $(L-1)^2$ has failed.","tokens_in":13279,"feed_emoji":"🧮","tokens_out":8214,"duration_ms":73001,"temperature":0.7,"pith_summary":"This paper establishes a reconstruction theorem for the genus-one generating function in permutation-equivariant quantum K-theory: for any compact Kähler target, all descendant invariants (those with insertions of the universal cotangent line bundle) are determined by genus-zero invariants, the primitive genus-one potential with no descendant insertions, and explicit residue terms. The theorem is the genus-one analogue of the classical reconstruction theorem, and it extends the author's earlier results from the point target to a general target. A sympathetic reader should care because it reduces a complicated infinite set of invariants to a finite, explicit formula built from more accessible data, so in principle every genus-one invariant becomes computable once the genus-zero theory and the primitive potential are known.","feed_headline":"Formula rebuilds genus-one quantum K-invariants from genus-zero data","feed_subtitle":"All descendant invariants are fixed by the primitive potential plus explicit residues of rational correlators.","key_machinery":"The load-bearing objects are the permutation-equivariant ancestor-descendant correspondence and the classification of cyclic symmetry strata. The correspondence splits the descendant potential into $F_1(t)=F_1(\\tau)+\\bar F_1(\\bar t)$, and a recursive contraction map chooses $\\tau$ so that $\\bar t(1)=0$. On the ancestor side $\\bar F_1(\\bar t)$, a vanishing theorem from the author's preceding work lists all possible non-zero contributions to the super-trace by the order of the permutation acting on marked points; only the trivial order and cyclic orders $2,3,4,6$ survive. Each surviving contribution is evaluated by separating the base curve in the fixed locus from the map to the target, then using a Riemann-Roch type super-trace formula and dilaton equations for permutable inputs to convert chains of rational components into explicit genus-one correlators with inputs $\\bar y_{1,\\zeta}$, $\\bar y_2$, $\\bar y_3$, $\\bar y_4$, $\\bar y_6$ and $\\bar x_i$. The residues at $0$ and $\\infty$ replace powers of $(q^{-1}-1)$ by the descendant class $\\bar L$, which is what produces the final compact formula.","core_discovery":"The central claim is Theorem 1: the genus-one descendant potential $F_1(t)$ equals $F_1(\\tau)$, the no-descendant genus-one potential at a recursively reconstructed input $\\tau$, plus $\\frac{1}{24}\\log\\det(\\partial\\tau_1/\\partial t_{1,0})$, plus a sum over $M=2,3,4,6$ and $a=0,\\infty$ of residues $\\operatorname{Res}_a F^{\\mathrm{perm}}_{1,M}(x)\\,dx/x$ of explicit correlators. Here $\\tau$ is chosen so that $\\bar t(1)=0$, and each $F^{\\mathrm{perm}}_{1,M}(x)$ is built from genus-zero two-point and three-point correlators together with genus-one correlators that carry at most one descendant insertion of the universal cotangent line $\\bar L$. In words, the theorem claims that no higher-descendant genus-one information is needed: the entire genus-one descendant theory is controlled by genus-zero data, one primitive genus-one potential, and residues of rational functions whose poles and zeros sit at $0$ and $\\infty$ after the change of variable $x=q^{-1}$.","pith_inferences":["The author leaves implicit that the same structural decomposition may extend to higher genus: the fixed loci of cyclic permutations that control genus one suggest that a genus-$g$ reconstruction would be organized by cyclic automorphisms of genus-$g$ curves, a testable direction but not a claim made here.","The appearance of orders $2,3,4,6$ matches the torsion orders of points on an elliptic curve, so the formula may admit a reorganization as a single expression over the moduli of elliptic curves with level structure; this is an interpretive hypothesis, not a statement in the paper.","For practical computation, the theorem implies that on a toric target both sides can be expanded in Novikov variables to any fixed degree, so a direct low-degree numerical check on a simple target would either confirm the residue structure or expose a missing vanishing-stratum."],"forward_implications":["Every genus-one invariant carrying one or more descendants can be expressed explicitly in terms of genus-zero correlators and the primitive genus-one potential.","The only cyclic symmetries of marked points that can contribute are orders $1,2,3,4,6$, so the residue sum is finite and explicitly listed.","The recursive algorithm for $\\tau$ makes the reconstruction effective rather than merely existential.","Specializing the theorem to a point target recovers the author's earlier reconstruction theorems, and dropping the permutation data recovers the non-permutative genus-one reconstruction."],"supporting_citations":[{"why":"Establishes the genus-one reconstruction in non-permutative quantum K-theory, which supplies the order-one case treated identically here.","marker":"[9]"},{"why":"Supplies the vanishing theorem and non-vanishing case list, the dilaton equations for permutable inputs, and the recursive algorithm for computing tau.","marker":"[10]"},{"why":"Provides the permutation-equivariant ancestor-descendant correspondence that splits F_1(t) into F_1(tau) and the forced-limit term.","marker":"[3]"},{"why":"Supplies the super-trace formula of Riemann-Roch type used to evaluate every non-vanishing contribution.","marker":"[4]"},{"why":"Supplies the virtual structure sheaf and the string, dilaton, and associativity foundations for quantum K-invariants.","marker":"[7]"}],"fun_headline_variants":["Genus-one K-invariants rebuilt from genus-zero data","Permutation-equivariant K-invariants: genus-one from genus-zero","No extra descendants needed for genus-one K-invariants","Explicit residue formula fixes genus-one quantum K-invariants","Genus-one descendant potential from primitive genus-zero data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on the completeness of the classification, taken from the author's previous paper, of which symmetry strata and input patterns can give nonzero contributions to the ancestor-side generating function; if that list misses even one stratum, the final residue sum would omit those contributions.","fun_headline_variants_meta":{"raw":{"variants":["Genus-one K-invariants rebuilt from genus-zero data","Permutation-equivariant K-invariants: genus-one from genus-zero","No extra descendants needed for genus-one K-invariants","Explicit residue formula fixes genus-one quantum K-invariants","Genus-one descendant potential from primitive genus-zero data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000717,"raw_usage":{"total_tokens":3159,"prompt_tokens":818,"completion_tokens":2341,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":434,"completion_tokens_details":{"reasoning_tokens":2258}},"tokens_in":434,"tokens_out":2341,"duration_ms":14533,"temperature":1.0,"reasoning_tokens":2258,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T16:27:46.735836+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly evaluate both sides of Theorem 1 for a compact toric target such as the complex projective line, truncated at a fixed low degree in the Novikov variables and with a small number of marked points carrying descendant classes, using the definition of the super-traces; a mismatch in the constant term or the leading q-expansion would show that the non-vanishing classification is incomplete or that a dimension argument discarding terms of order $(L-1)^2$ has failed.","supporting_citations":[{"cited_title":"A formula on $g=1$ quantum K-invariants","cited_arxiv_id":"2411.06053","evidence_quote":"Establishes the genus-one reconstruction in non-permutative quantum K-theory, which supplies the order-one case treated identically here."},{"cited_title":"Two reconstruction theorems in permutation equivariant quantum K-theory","cited_arxiv_id":"2411.07487","evidence_quote":"Supplies the vanishing theorem and non-vanishing case list, the dilaton equations for permutable inputs, and the recursive algorithm for computing tau."},{"cited_title":"Permutation-equivariant quantum K-theory VII. General theory","cited_arxiv_id":"1510.03076","evidence_quote":"Provides the permutation-equivariant ancestor-descendant correspondence that splits F_1(t) into F_1(tau) and the forced-limit term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the virtual structure sheaf and the string, dilaton, and associativity foundations for quantum K-invariants."}],"review_version":1}