{"id":"fe7fd8cc-de1b-4174-8908-31b977e3d3f9","arxiv_id":"2501.01408","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The classical periods of the intrinsic mirror algebra of a log Calabi-Yau Fano pair reproduce the regularized quantum periods of the Fano variety.","lead":"This paper proves that for many Fano varieties, the geometric counts encoded in quantum periods can be recovered from a canonical mirror algebra attached to the complement of an anticanonical divisor. It yields integrality of these periods and explicit Laurent polynomial mirrors for cluster-type Fano varieties.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central theorem is conditional on a non-automatic log Calabi-Yau compactification hypothesis; the proof is internally coherent but the abstract overstates the proved scope.","rationale":"The reader's verdict is conditional, and my reading supports that calibration. The mathematical core of the paper, Theorem 1.1 and its proof, appears internally coherent: the degeneration argument is intricate but the key steps are stated with references to the relevant log Gromov-Witten and tropical machinery, and the final comparison between the trace form of W_D and the regularized quantum periods is a genuine computation. No fitted parameters, circular definitions, or obvious sign/factorial errors surfaced in the main theorem. The strongest reason not to accept unconditionally is the restrictive existence hypothesis on U: Theorem 1.1 is an implication from the existence of a log Calabi-Yau compactification, not a proof that such a compactification always exists. The paper acknowledges this in Remark 1.2(1), and the abstract's 'in large generality' and 'to all Fano varieties' overstate the proven scope. I do not see a flaw that would force rejection of the main theorem within its stated hypotheses, and I agree with the reader that a corrected abstract and a clearer separation of conditional statements would resolve the main objections. The suggested computation in the P2 triangle case would be a useful check of the degeneration formula, but it is a verification step rather than a response to a demonstrated error.","tokens_in":31137,"tokens_out":29113,"duration_ms":331731,"concrete_test":"Verify Theorem 1.1 in the basic case X = P2 with D the union of three lines in general position, where the hypothesis is satisfied. Compute the relevant theta-function structure constants of the intrinsic mirror algebra R_(P2,D), form W_D = ϑ_{D1} + ϑ_{D2} + ϑ_{D3}, and compare its classical period series with the known P2 regularized quantum period 1 + 6t^3 + 90t^6 + ... up to the stated normalization. If the equality fails, the degeneration computation in Section 5 is unreliable; if it holds, the main mechanism is supported. Separately, verify for D three concurrent lines that no log Calabi-Yau compactification exists, confirming that the theorem's hypothesis is essential rather than automatic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise of Theorem 1.1 is the existence of a log Calabi-Yau compactification (X'', D'') of U = X \\ D in the sense of Definition 4.1. This is an external hypothesis, not a consequence of X being smooth Fano with D a reduced anticanonical divisor. Lemma 4.2 only transfers this property from one log Calabi-Yau compactification to any other snc compactification once existence is known; it does not establish existence. The paper itself flags the essentiality of the hypothesis in Remark 1.2(1): for X = P2 and D three concurrent lines, the resolution is not log Calabi-Yau and Theorem 1.1 does not apply. Since the construction of the intrinsic mirror algebra and the equality of periods in Equation 1.2 both pass through the log Calabi-Yau hypothesis, every corollary inherits this restriction. In particular, the abstract's phrases 'integrality of regularized quantum periods in large generality' and 'Laurent mirrors to all Fano varieties' are stronger than what the proved theorems deliver: they hold only for Fano pairs whose anticanonical divisor has a log Calabi-Yau compactification, i.e. for which the relevant log discrepancies are non-negative. This is a genuine scope limitation rather than an internal inconsistency of the proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a conditional mirror-symmetry theorem for a smooth Fano variety X equipped with a reduced anticanonical divisor D, assuming that the open Calabi-Yau U = X \\ D admits a log Calabi-Yau compactification in the sense of Definition 4.1. For any dominating simple normal crossings compactification (X', D') of U, the author constructs an element W_D in the intrinsic mirror algebra R_{(X',D')} as a sum of theta functions associated with the divisor components, and proves that its classical periods equal the regularized quantum periods of X. The main proof proceeds by log Gromov-Witten degeneration: it reduces the ordinary invariants on the general fiber to logarithmic invariants on a specially built central fiber, then identifies these with naive curve counts via the author's earlier comparison results. The paper derives corollaries on integrality of quantum periods, Laurent mirrors for cluster-type Fano compactifications, Newton-Okounkov bodies and toric degenerations, an explicit Grassmannian mirror recovering the Marsh-Rietsch Plücker superpotential, and a Frobenius-structure equivalence for smooth anticanonical divisors.","tokens_in":31309,"tokens_out":16763,"duration_ms":180497,"significance":"If Theorem 1.1 is correct, this is a substantial advance: it gives a canonical, parameter-free construction of Landau-Ginzburg mirrors for a large class of Fano pairs, using intrinsic mirror symmetry and logarithmic Gromov-Witten theory, and it connects the resulting mirrors to cluster theory, polytopes, and toric degenerations. The recovery of the Marsh-Rietsch Plücker mirror and the new enumerative interpretation of flow-polynomial coefficients are appealing concrete payoffs. The paper is honest about its main hypothesis, and the proof is internally coherent, but it relies heavily on external results, including the author's preprints [Joh] and [Joh24], and on a degeneration argument that is only sketched in places. The breadth promised in the abstract is somewhat wider than the proved theorems.","major_comments":[{"comment":"Theorem 1.1 is stated only under the hypothesis that U = X \\ D admits a log Calabi-Yau compactification (X'', D'') in the sense of Definition 4.1. This hypothesis is not automatic for X smooth Fano and D in |-K_X| reduced; Remark 1.2(1) itself notes that X = P^2 with D three concurrent lines fails it. The abstract, however, announces 'integrality of regularized quantum periods in large generality' and 'Laurent mirrors to all Fano varieties whose mirrors contain a dense torus' without this restriction. Since every corollary in Sections 5 and 6 inherits the log Calabi-Yau compactification hypothesis, and Corollary 6.1 also inherits the optimized-seed condition, the summary statements should be reworded to match the proved scope.","section":"§1, Theorem 1.1 and Remark 1.2(1); abstract"},{"comment":"Lemma 5.1 is the bridge that converts logarithmic Gromov-Witten invariants into the naive curve counts η, and hence into the period coefficients. Its proof says that 'the desired equality would then follow as in the final section of the proof of [Joh, Theorem 1.1]' after showing that no component maps into the boundary. But the hypotheses of [Joh, Theorem 1.1], as quoted in Theorem 4.5, require that D is the support of a nef divisor and contains a zero stratum; the setting of Lemma 5.1 does not assume a zero stratum, and the point constraint is a general point of U rather than a boundary point. The manuscript should either identify the precise statement in [Joh] that applies after the boundary-exclusion argument, or include the missing argument, because this identification is the only place where the equality between the two independently defined period series is actually established.","section":"§5, Lemma 5.1"},{"comment":"The degeneration formula is written with ψ_{x_out}^{d-2}, where d = -K_X · A. For curve classes with d = 0 or d = 1 this is a negative power of a ψ class, which is not defined in Equation (1.1), yet the inner sum in (1.1) ranges over all d ≥ 0. The paper does not state a convention (for example, that such terms vanish or are handled separately), so the displayed equality π_W = \\hat G_X is not literally meaningful for the low-degree terms. Since the proof of Theorem 1.1 computes the d-th coefficient via d! times a trace, the low-degree cases should be checked explicitly or a standard convention should be stated.","section":"§5, Eq. (5.3) and the degeneration analysis"},{"comment":"The proof claims that π_{W_D} is independent of the chosen dominating compactification because the contributing invariants are naive curve counts and are 'independent of the choice of compactification by definition'. This independence is not immediate: the maps entering η(p_1, ..., p_m, A) are maps to a chosen compactification with specified contact orders, and different compactifications have different boundary divisors. The proof of Lemma 5.1 prevents boundary components only after passing through the log Gromov-Witten identification; the independence of η itself should be spelled out rather than asserted by definition.","section":"§5, proof of Theorem 1.1, first paragraph"}],"minor_comments":[{"comment":"The numbering is inconsistent: the Frobenius structure statement is Theorem 1.5 in the introduction but is later called Corollary 1.5, and the statement labelled 'Proof of Theorem 8.4' is actually a Proposition 8.4. Please unify the numbering.","section":"Throughout"},{"comment":"The phrase 'Laurent mirrors to all Fano varieties whose mirrors contain a dense torus' is ambiguous; it should specify whether the condition is on X \\ D containing a dense torus, on the mirror family, or on the existence of an optimized seed as in Corollary 6.1.","section":"Abstract"},{"comment":"The proof relies on [AB23, Theorem D] but does not state the precise compatibility of the base changes between the canonical wall structure and the cluster scattering diagram, nor how the pullback of W_D to a formal torus preserves the trace form after setting z^A = 1. A precise statement would help the reader verify the formal Laurent polynomial claim.","section":"§6, proof of Corollary 6.1"},{"comment":"The section title contains the typo 'Grassmanian'; it should read 'Grassmannian'.","section":"Title of §7"},{"comment":"The notation 'D = \\sum_i D_i' is used both for the original anticanonical divisor on X and for the divisor on X' after pullback; the proof identifies a piecewise linear function also denoted D. Please clarify the notation, especially in the paragraph after Definition 5.2.","section":"§5, Definition 5.2"}],"recommendation":"major_revision","confidential_remarks":"The central theorem depends on two preprints by the author ([Joh] and [Joh24]) and on [AB23], [Gro23], [ACGS20a]. The editor may wish to verify that these references are in final or accessible form before acceptance. The abstract's breadth should be tempered to match the conditional Theorem 1.1 and the cluster/optimized-seed hypotheses of the corollaries."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main theorem is real and the paper deserves serious refereeing, but the abstract makes a broader claim than the hypotheses support. Johnston proves that for a smooth Fano X with reduced anticanonical divisor D such that X\\D admits a log Calabi-Yau compactification in the Gross-Siebert sense, any snc compactification carries an element W_D in the intrinsic mirror algebra whose classical periods equal the regularized quantum periods of X. That is a genuine, nontrivial mirror theorem: it extends Mandel to general dominating compactifications, matches Tonkonog's symplectic analogue, and gives a canonical Landau-Ginzburg mirror.\n\nWhat is new: Theorem 1.1 itself, plus the corollaries on Laurent mirrors for cluster compactifications, the Newton-Okounkov bodies from seed polytopes, the Grassmannian computation recovering the Marsh-Rietsch Pluecker superpotential, and the smooth-divisor Frobenius theorem. The paper is honest about overlap with You's work—Theorem 1.5 is essentially in You, stated differently, and Johnston derives it by degeneration instead of mirror-map arguments. That is a reasonable presentation, not a hidden gap.\n\nThe soft spots are proportional. The load-bearing hypothesis is the existence of a log Calabi-Yau compactification of U = X\\D. This is not automatic; Remark 1.2(1) gives P2 with three concurrent lines as a counterexample. Lemma 4.2 only transfers the property between compactifications once it exists. So the theorem applies to a genuine but circumscribed class of Fano pairs. The abstract's phrases \"in large generality\" and \"all Fano varieties\" overstate the proved scope—the corollaries inherit the log CY restriction. The proof is long and internally coherent, but it leans heavily on the author's prior results [Joh] and [Joh24] and on black-box degeneration and gluing formulas. That is normal in this area, but a referee needs to check the dependencies carefully. No fitted parameters, no circularity that I can see.\n\nThe paper is for people in algebraic mirror symmetry, log Gromov-Witten theory, and Fano degenerations. It deserves a serious referee; the central theorem is a major step even with the scope restriction. My recommendation: send it out, and ask the author to rewrite the abstract to match the hypotheses and clarify exactly which corollaries need the log CY compactification assumption.","headline":"A substantial and internally coherent mirror theorem, but the abstract sells a scope the hypotheses do not support.","tokens_in":31904,"tokens_out":1615,"would_cite":true,"duration_ms":16542,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J33","14N35","14N10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Fano varieties with a log Calabi-Yau anticanonical complement, the regularized quantum periods are classical periods of a single canonical theta-function sum in the intrinsic mirror algebra.","keywords":["Fano varieties","quantum periods","intrinsic mirror algebra","theta functions","log Gromov-Witten invariants","log Calabi-Yau pairs","cluster varieties","toric degenerations"],"falsifier":"Take $X=\\mathbb P^2$ and $D$ a smooth cubic, so the hypotheses of Theorem 1.1 hold. The theorem asserts $\\pi_{W_D}=\\widehat G_X$, with $W_D=\\vartheta_D$ the $\\theta$ function of the divisor in the intrinsic mirror algebra. The coefficient of $t^3$ on the right is a finite ordinary Gromov-Witten count of rational cubics through eight points in $\\mathbb P^2$, and the coefficient of $t^3$ on the left is computed by iterated $\\theta$ products in $R_{(X,D)}$; comparing these two finite enumerative numbers would settle the equality, and any disagreement would falsify the theorem.","tokens_in":30860,"feed_emoji":"🪞","tokens_out":14217,"duration_ms":128579,"temperature":0.7,"pith_summary":"This paper proves that the regularized quantum periods of a smooth Fano variety are not just an arbitrary collection of Gromov-Witten numbers: under a natural log Calabi-Yau hypothesis, they are the classical periods of a single canonical element $W_D$ in the intrinsic mirror algebra. Given a Fano variety $X$ and a reduced anticanonical divisor $D$ whose complement $U=X\\setminus D$ is log Calabi-Yau, the theorem constructs $W_D$ inside the intrinsic mirror algebra of any normal-crossings compactification of $U$ dominating $(X,D)$; $W_D$ is simply the sum of the $\\theta$ functions of the components of $D$. The equality $\\pi_{W_D}=\\widehat G_X$ is proved by degenerating $X$ to a normal-crossings special fiber and reducing each quantum period to naive counts of rational curves. A sympathetic reader should care because this gives Fano mirror symmetry a canonical, choice-free mirror potential in many cases where none was previously known, and the paper draws out concrete consequences: integrality of quantum periods, Laurent polynomial mirrors for cluster compactifications, and the known determinantal-coordinate superpotential for Grassmannians. In the smooth anticanonical case it also shows the quantum period sequence determines all $\\theta$-function structure constants of the mirror algebra.","feed_headline":"One theta-function sum computes Fano quantum periods","feed_subtitle":"The intrinsic mirror algebra supplies a canonical superpotential whose classical periods match all degree counts.","key_machinery":"The carrying object is the intrinsic mirror algebra $R_{(X',D')}$: a $k$-algebra over the monoid ring of effective curve classes, freely generated by a $\\theta$ basis $\\vartheta_p$ indexed by integral points of the essential skeleton of the log Calabi-Yau pair, with product structure constants $N^{r,A}_{p,q}$ given by logarithmic Gromov-Witten invariants. The superpotential $W_D$ is the finite sum of the $\\theta$ functions attached to the divisorial valuations of the components of the anticanonical divisor $D$. The argument that $\\pi_{W_D}=\\widehat G_X$ is carried by a degeneration of $X$ to a normal-crossings special fiber, together with a decomposition theorem for log Gromov-Witten invariants that expresses each degree-$d$ period as a sum over rigid decorated tropical types; the point constraint forces the relevant types to have a distinguished vertex on the main component with $d+1$ legs of contact order $1$, and all other vertices contribute fiber classes of projective bundles over components of $D$. A second mechanism, used in the smooth-divisor section, is the identity $\\vartheta_{p_1}\\cdots\\vartheta_{p_d}[\\vartheta_0]=\\eta(p_1,\\ldots,p_d,A)$ equating $\\theta$ products with naive curve counts.","core_discovery":"The central statement is Theorem 1.1: for $(X,D)$ with $X$ smooth Fano and $D\\in |-K_X|$ reduced, whenever $U=X\\setminus D$ admits some log Calabi-Yau compactification, every dominating snc compactification $(X',D')$ of $(X,D)$ carries an element $W_D=\\sum_i \\vartheta_{D_i}$ in the intrinsic mirror algebra $R_{(X',D')}$ whose classical periods $\\pi_{W_D}=\\sum_{d\\ge0} W_D^d[\\vartheta_0]$ equal the regularized quantum periods $\\widehat G_X$ after the natural base change to the curve-class monoid of $X$. The equality is independent of the chosen compactification, so the mirror potential is intrinsic to the pair $(X,D)$. The proof works by degenerating $X$ to a normal-crossings central fiber whose main component is $X'$ and whose extra components are projective bundles over the components of $D$, then applying the decomposition of degenerate log Gromov-Witten invariants; the only contributing tropical types have all quantum data concentrated in the main component, so the full period becomes a product of naive curve counts. From this the paper derives integrality of regularized quantum periods, Laurent mirrors for cluster-type compactifications, the Grassmannian superpotential, and, for smooth $D$, the equivalence of the quantum period sequence with all $\\theta$-function structure constants of $R_{(X,D)}$.","pith_inferences":["If the independence of the quantum period on the chosen snc compactification holds beyond the torus case (the paper proves it in the torus case and expects it generally), Theorem 1.1 would provide a canonical mirror object for every Fano variety with a log Calabi-Yau anticanonical complement, independent of charts or compactification choices.","The degeneration proof suggests that regularized quantum periods can be defined for singular Fano varieties or log pairs, and one could test whether the ordinary quantum periods of a Fano degeneration are recovered from generalized periods of a singular toric special fiber, a direction the author raises as future work.","The equality between flow-polynomial coefficients and non-archimedean cylinder counts indicates that the combinatorial counts of perfect matchings on plabic graphs admit a geometric bijection with non-archimedean disks; constructing such a bijection explicitly would be a natural test of the enumerative content.","The Frobenius-structure result for smooth anticanonical divisors suggests that for broader classes of log Calabi-Yau pairs the full mirror algebra might be determined by a single period series; checking this beyond the smooth case would be a direct extension of the paper's methods."],"forward_implications":["Every Fano pair $(X,D)$ satisfying the log Calabi-Yau hypothesis receives a canonical Landau-Ginzburg mirror potential $W_D$ inside the intrinsic mirror algebra, so mirror symmetry for such varieties no longer requires an ad hoc choice of Laurent polynomial.","The regularized quantum periods of such a Fano variety are non-negative integers, since each classical period of $W_D$ is a naive curve count of rational curves in an open Calabi-Yau.","For Fano compactifications of affine cluster varieties with optimized seeds for the divisorial valuations, the mirror potential restricts to a genuine Laurent polynomial on every seed torus, and the polar dual of its Newton polytope produces a Newton-Okounkov body and a toric degeneration of $X$.","For Grassmannians $Gr(n-k,n)$, the construction recovers the known determinantal-coordinate superpotential on the dual Grassmannian, whose coefficients are counts of non-archimedean cylinders.","When $D$ is a smooth anticanonical divisor, the regularized quantum periods determine the entire theta-function structure of $R_{(X,D)}$, equivalently all two-pointed logarithmic Gromov-Witten invariants of the pair."],"supporting_citations":[{"why":"Constructs the intrinsic mirror algebra $R_{(X,D)}$ with theta basis and structure constants; this is the algebra in which $W_D$ lives and whose trace form is used.","marker":"[GS21]"},{"why":"Establishes that theta-function structure constants equal naive curve counts under the affine log Calabi-Yau hypotheses, the key enumerative input for Lemma 5.1.","marker":"[Joh]"},{"why":"Provides the descendent log Gromov-Witten formula for theta structure constants used in Lemma 5.1 and in the Frobenius-structure section.","marker":"[Joh24]"},{"why":"Supplies the decomposition of degenerate Gromov-Witten invariants by tropical types used in the degeneration argument proving Theorem 1.1.","marker":"[ACGS20a]"},{"why":"Develops the Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus and the non-archimedean cylinder counts used for independence and for Grassmannian coefficients.","marker":"[KY23]"},{"why":"Provides canonical bases, optimized seeds, scattering diagrams and positive polytopes underlying the Laurent mirror and Newton-Okounkov body corollaries.","marker":"[GHKK18]"},{"why":"Gives the determinantal-coordinate Grassmannian superpotential that Section 7 recovers from the intrinsic mirror construction.","marker":"[MR20]"},{"why":"Supplies the cluster valuations and Newton-Okounkov body and toric degeneration framework for Grassmannians used to identify the mirror and its polytopes.","marker":"[R W19]"}],"fun_headline_variants":["One mirror superpotential reproduces all Fano quantum periods","Classical periods of intrinsic mirror potential equal quantum periods","Intrinsic mirror algebra gives universal superpotential for Fano","All Fano quantum periods arise from one classical superpotential","Mirror superpotential's periods encode all Fano degree counts"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire theorem rests on the assumption that the open variety $U=X\\setminus D$ admits at least one log Calabi-Yau compactification; if no such compactification exists, the intrinsic mirror algebra construction and the equality of periods are not established.","fun_headline_variants_meta":{"raw":{"variants":["One mirror superpotential reproduces all Fano quantum periods","Classical periods of intrinsic mirror potential equal quantum periods","Intrinsic mirror algebra gives universal superpotential for Fano","All Fano quantum periods arise from one classical superpotential","Mirror superpotential's periods encode all Fano degree counts"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00094,"raw_usage":{"total_tokens":4097,"prompt_tokens":1104,"completion_tokens":2993,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":720,"completion_tokens_details":{"reasoning_tokens":2912}},"tokens_in":720,"tokens_out":2993,"duration_ms":18128,"temperature":1.0,"reasoning_tokens":2912,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:28:37.914201+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $X=\\mathbb P^2$ and $D$ a smooth cubic, so the hypotheses of Theorem 1.1 hold. The theorem asserts $\\pi_{W_D}=\\widehat G_X$, with $W_D=\\vartheta_D$ the $\\theta$ function of the divisor in the intrinsic mirror algebra. The coefficient of $t^3$ on the right is a finite ordinary Gromov-Witten count of rational cubics through eight points in $\\mathbb P^2$, and the coefficient of $t^3$ on the left is computed by iterated $\\theta$ products in $R_{(X,D)}$; comparing these two finite enumerative numbers would settle the equality, and any disagreement would falsify the theorem.","supporting_citations":[],"review_version":1}