{"id":"104c85c2-f3a5-4997-bcfb-1a6e166b5e34","arxiv_id":"2608.08347","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A recurrence plus two tropical boundary limits reduce Shi and Wang's Conjecture 3.8 for Zagier's twelfth Nahm sum to two generalized-eta identities, which are proved by valence certificates.","lead":"This paper proves a conjecture of Shi and Wang about the modularity of a family of Nahm sums, objects from a famous problem in number theory and conformal field theory. It develops a general machine-guided method in which a computer searches for exact recurrences and boundary data, and every object found is then verified by traditional symbolic proof.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 8.3 rests entirely on the valence certificates, but the printed cusp-order formula (49) contains an undefined symbol 'n', so the certificates cannot be reproduced from the manuscript as written; this should be corrected and the certificates independently rechecked before acceptance.","rationale":"The reader's weakest-assumption analysis correctly identified the valence certificates as the main risk point: a single erroneous cusp order can change B* and break the reduction of the 2x2 system to the generalized-eta identities. My stress test sharpens this into a concrete defect: formula (49), the exact statement from which all cusp orders are supposedly computed, contains an undefined symbol 'n' and lacks the procedural details needed for independent reproduction. Since Proposition 8.3 has no other proof route, the written manuscript does not currently support an unconditional ACCEPT; the theorem may be true and the artifacts may be correct, but the displayed proof is not self-contained at its final computational step. I therefore recommend CONDITIONAL rather than REJECT: the central argument is otherwise well-structured, the recurrence certificate, tropical limits, and boundary-system logic are explicit and checkable, and the claimed defect is plausibly a typographical/specification issue. The condition is that (49) be stated correctly and unambiguously, and that both valence certificates be regenerated and independently verified against the corrected formula.","tokens_in":19496,"tokens_out":50003,"duration_ms":417631,"concrete_test":"Run an independent cusp-order computation for Gamma_1(300) and Gamma_1(100) using a separate implementation (e.g., Sage modular-symbol cusp classes together with the Frye-Garvan thetaids order formula), recompute B* in (50) for both identities, and compare the results with 1920, 216, and the shipped JSON tables. In the same pass, locate the cusp-order routine in the released code and identify exactly which formula it implements. If the code's formula differs from (49) as printed, or if (49) cannot be parsed because of the undefined 'n', the certificate must be regenerated with the corrected formula and the bound/vanishing checks repeated.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 1.1 is reduced to Proposition 8.3, and Proposition 8.3 is proved only by the two valence certificates in Section 9. The soundness of those certificates hinges on the cusp-order formula (49), the enumeration of all inequivalent cusps, and the bound B* in (50). As printed, (49) is not a well-defined formula: the numerator inside the fractional part is 'n ar / gcd(N,c)', and the symbol 'n' is never defined. It is also not stated how cusp representatives are chosen or reduced, nor how the 560 and 140 cusp-order rows were enumerated. Because B* is a sum over min(0, ORD) over all non-infinity cusps, any mis-specification or transcription error in (49) can change B* and invalidate the implication 'vanishing beyond B* implies the identity'. The SHA-256 hashes and self-reported regeneration show only that the code reproduces the shipped tables; they do not show that the shipped tables were computed from formula (49) as written, nor that (49) as written is mathematically correct. This is a concrete, load-bearing gap in the written proof, even though the underlying artifacts may well be correct.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a proof-carrying recurrence--boundary method for affine families of positive-definite Nahm sums. The general framework consists of a universal coordinate-contiguous relation for exact algebraic cells (Theorem 3.1), a tropical face-limit theorem for parameter rays satisfying a linear complementarity condition (Theorem 3.3), a recurrence--boundary determination principle (Theorem 3.6), and a finite modular certification criterion via generalized-eta valence bounds (Corollary 3.7). The main application proves Shi--Wang's Conjecture 3.8 for the Nahm sums dual to Zagier's twelfth rank-three example. An evolutionary search proposes a second-order recurrence q^{-t}F_t=q^{2t+1}F_{t+1}+F_{t+2}; a Q-learning agent finds a five-cell exact certificate; two tropical rays produce a binary and a unary theta boundary value; solving the resulting 2x2 system reduces the conjecture to two generalized-eta identities, which are certified by valence arguments on Gamma_1(300) and Gamma_1(100). The paper concludes that the dual Nahm sums are modular and that every member of the affine family lies in Z[q,q^{-1}]F_0+Z[q,q^{-1}]F_1.","tokens_in":19732,"tokens_out":10354,"duration_ms":101719,"significance":"If the proof is sound, this resolves a remaining open conjecture in Nahm-sum modularity and gives modularity of the full affine family, not only of the two named sums. The paper is also methodologically valuable: machine learning and reinforcement learning are used only for discovery, while every accepted statement is reduced to an exact finite certificate, a displayed algebraic identity, or a finite valence computation. The proof structure is unusually transparent, and the released artifacts include independently written checkers and regenerable tables. The general theorems, especially the tropical face-limit theorem and the recurrence--boundary determination principle, are likely to be reusable beyond this example. The paper's weakest point is the printed cusp-order formula (49), which currently cannot be applied as written; this affects the two valence certificates on which Proposition 8.3 rests.","major_comments":[{"comment":"The invariant-order formula in (49) is not a well-defined formula as printed: the symbol B(N,r;a,c) contains the expression n ar/gcd(N,c) inside a fractional part, and the symbol n has not been defined anywhere in the manuscript. Since Proposition 8.3 is proved solely by the two valence certificates, and every cusp-order row in those certificates is asserted to be computed from (49), the published proof cannot be reproduced from the manuscript as written. Please correct the formula (almost certainly to a r/gcd(N,c)), state explicitly how reduced cusp representatives a/c are chosen and enumerated for Gamma_1(N), and have the 560+140 cusp-order rows recomputed independently from the corrected formula. The SHA-256 hashes in Remark 9.1 show only that the regeneration code reproduces the shipped tables; they do not certify the mathematical correctness of (49) as printed or of the enumeration.","section":"§9.1, Eq. (49)"},{"comment":"The valence bound B* is asserted rather than proved. To justify the contrapositive statement that vanishing beyond B* forces f identically zero, one needs the explicit lower bound ord_a f >= min(0, ord_a f_2, ..., ord_a f_s) at every non-infinity cusp a, and hence the inequality sum_{a != infinity} ord_a f >= -B*. This is standard but should be stated and proved, especially because the normalized summands f_i may have zeros, poles, or cancellations at some cusps. The manuscript should also describe the algorithm by which the 560 and 140 inequivalent cusps were enumerated and how the representative a/c was reduced, so that the bound B* can be independently recomputed.","section":"§9.1, Eq. (50)"}],"minor_comments":[{"comment":"The notation f = 1 - sum_{i>=2} f_i is introduced informally; please define the normalized summands f_i explicitly before (50), including which term of each product identity is chosen as the base term.","section":"§9.1"},{"comment":"The certificate tables are said to contain all 560+140 cusp-order rows, but the manuscript itself shows only hashes and counts. A human-readable summary of the cusp enumeration, or a supplementary appendix listing the representative cusps and their orders, would substantially help an independent referee audit the certificates.","section":"§9.2--9.3"},{"comment":"The displayed product formulas (6)--(7) are dense and hard to check visually; a table with each quotient written in terms of J_{r,50} and J_{50} exponents would improve readability.","section":"§1.4, Theorem 1.1"},{"comment":"The two tropical-ray computations are concise but correct; in the revision, consider adding one sentence in each proof explaining which set I, J, K of Theorem 3.3 is being used, since the reader must reconstruct it from the displayed complementarity data.","section":"§6 and §7"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the central argument appears coherent; the main obstruction is the malformed cusp-order formula (49), which is load-bearing for Proposition 8.3. I would like to see the corrected formula, a precise statement of the cusp enumeration, and an independent regeneration of the 560+140 cusp-order rows before publication. The computational artifacts and the honest separation of machine search from exact proof are definite strengths."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is better than the title makes it sound. The general framework is real: a universal contiguous-cell relation, a tropical face-limit theorem that turns parameter rays into theta/hypergeometric boundary values, and a recurrence–boundary determination principle. These are stated cleanly and proved with enough detail to follow. The application to Zagier's twelfth dual is a genuine new result: the five-cell recurrence certificate, the two boundary limits, and the 2×2 system reducing everything to two generalized-eta identities all check out structurally. I also credit the authors for keeping the ML/RL machinery strictly in the discovery phase and replacing every learned output with an exact symbolic certificate. That is the right way to do machine-guided proof, and the artifact release is unusually complete.\n\nThe soft spot is the one the stress-test flagged, and it is real but not fatal. Formula (49), the cusp-order formula on which both valence certificates depend, contains an undefined symbol 'n' in the fractional part. As printed, the certificates cannot be reproduced from the manuscript alone, and the proof of Proposition 8.3 rests entirely on those certificates. The same paragraph also says nothing about how the 560 and 140 cusp representatives were chosen or enumerated. That is a genuine gap in the written proof. But it is a fixable gap: the artifact stores the full tables, the SHA-256 hashes are there, and independent checkers recompute the bounds. The most likely explanation is a typo—'n' should be 'N' or similar—and the code in the repo presumably has it right. Still, for a paper whose whole method is proof-carrying, the printed formula needs to be correct and the enumeration specified. That should be corrected before acceptance, not after.\n\nThe math itself appears sound. I traced the gauge transformation, the two tropical rays, the parity split in the backward limit, and the determinant argument for uniqueness. No circularity: the product side is verified independently against the same linear system. The computational nature of the valence certificates is the right kind of computational—exact rational arithmetic, finite check, regenerable.\n\nWho is this for? Anyone working on Nahm sums, q-series, or modularity of multisums, plus people interested in proof-carrying machine learning in mathematics. It deserves a serious referee. My advice: send it to review, but require the author to fix the undefined symbol in (49), describe the cusp enumeration, and ideally add a short proof or reference justifying that exact cusp-order formula. Those are revisions, not rejections.","headline":"A genuinely new proof machine for Nahm-sum modularity, applied to resolve Shi–Wang's Conjecture 3.8; the main written proof has a fixable but real reproducibility gap in the valence-certificate formula.","tokens_in":20236,"tokens_out":1917,"would_cite":true,"duration_ms":20562,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F03","11P84","33D15","68T05","90C33"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves Shi and Wang's Conjecture 3.8: the two Nahm sums dual to Zagier's twelfth rank-three example are explicit generalized-eta quotients, hence modular, and every member of the affine family is an explicit…","keywords":["Nahm sums","modularity","Rogers-Ramanujan functions","theta functions","generalized eta products","valence formula","tropical limits","computer-assisted proof"],"falsifier":"Independently recompute all cusp orders and the bounds $B^*$ from the stored generalized-eta exponent tables, then expand the raw integer forms of the two product-side boundary equations through $q^{2005}$ for the $\\Gamma_1(300)$ identity and through $q^{255}$ for the $\\Gamma_1(100)$ identity; any nonzero coefficient at a degree below the respective bound, or any discrepancy in a single cusp order, would show that the proof of Proposition 8.3 and hence of Theorem 1.1 fails.","tokens_in":19299,"feed_emoji":"🧮","tokens_out":6135,"duration_ms":57584,"temperature":0.7,"pith_summary":"The paper proves two exact product identities for a pair of rank-three Nahm sums—infinite $q$-series built from rational quadratic forms—that Shi and Wang had conjectured. These two sums are the objects dual, under Zagier's inversion map, to the twelfth entry in Zagier's list of rank-three Nahm sums with modular behavior. The proof places the two sums in a one-parameter family, finds and exactly proves a second-order recurrence among neighboring members, evaluates two tropical limits as $\\theta$ series, solves the resulting $2\\times2$ system, and certifies the remaining generalized-eta identities by valence arguments on $\\Gamma_1(300)$ and $\\Gamma_1(100)$. If correct, the dual of Zagier's twelfth example is modular and the entire affine family consists of explicit integer-Laurent combinations of the two product bases.","feed_headline":"Machine proof settles the twelfth Nahm-sum conjecture","feed_subtitle":"Two Nahm sums equal explicit generalized-eta products, making the entire dual family modular.","key_machinery":"The load-bearing object is the recurrence-boundary triple. First, the universal coordinate-contiguous relation (14) defines exact cells, identities among Nahm sums with shifted parameter vectors; any finite Laurent-polynomial combination of cells that cancels in the free module yields a genuine identity, so a certificate is sound by Proposition 3.2. Second, the tropical face-limit theorem (Theorem 3.3) describes what happens along a ray $\\beta + h\\delta$: a linear-complementarity solution $v$ partitions the summation variables into bilateral, unilateral, and face-forced coordinates, reducing a higher-rank Nahm sum to a lower-rank $\\theta$ or $\\theta$-hypergeometric boundary value. Third, the recurrence-boundary determination principle (Theorem 3.6) recovers the initial sums of a family from an order-$s$ recurrence together with $s$ independent boundary limits. In the application these appear as the five-cell certificate (21), the two complementarity rays of Sections 6 and 7, the $2\\times2$ boundary system (43), and the valence criterion with cusp-order formula (49) and bound (50).","core_discovery":"Theorem 1.1 asserts that the two Nahm sums $F_0$ and $F_1$ defined in (5) equal the three-term generalized-eta quotients displayed in (6) and (7), where each summand is, up to a power of $q$, a quotient of products $(q^a;q^m)_\\infty$. The argument introduces the affine family $F_t$ with parameter $t$, proves the recurrence $q^{-t}F_t = q^{2t+1}F_{t+1} + F_{t+2}$ through a five-cell exact certificate built from a universal coordinate-contiguous relation, and computes two independent boundary values along tropical rays: a binary $\\theta$ series $\\Theta_-$ and a unary $\\theta$ series $\\vartheta$. The boundary matrix has unit determinant, so the two initial sums are uniquely determined; substituting the conjectured product forms reduces the theorem to two generalized-eta identities, which are proved exactly by valence certificates. The paper thereby resolves Shi and Wang's Conjecture 3.8 and, as Corollary 1.2, establishes modularity of the dual and the explicit structure of the whole affine family.","pith_inferences":["I infer that the practical bottleneck for Nahm duality is not recognizing a candidate product but locating a low-order recurrence and enough independent tropical rays; if so, the same search protocol should apply to the remaining Shi-Wang targets and to any affine Nahm family with a small recurrence module.","The proof suggests a stronger structural statement than a single product identity: the whole affine family is governed by two boundary theta series, so modularity of the family is equivalent to invertibility of the boundary matrix rather than to any one closed form.","A testable extension is to run the same complementarity-guided search on the dual of Example 9, where only a fourth identity is missing; the expected object would be a finite recurrence module over several residue classes, to which the cell-certificate soundness applies verbatim."],"forward_implications":["The Nahm sum dual to Zagier's twelfth rank-three example is modular, not just conjecturally, because $F_0$ and $F_1$ are finite combinations of generalized-eta quotients on congruence subgroups.","Every member $F_t$ of the affine family lies in $\\mathbb{Z}[q,q^{-1}]F_0 + \\mathbb{Z}[q,q^{-1}]F_1$; for instance $F_2 = F_0 - qF_1$, so each $F_t$ is an explicit finite combination of the two base products.","The recurrence-boundary pipeline gives a six-step program for other affine Nahm families: find a recurrence, prove it by cells, find independent asymptotic rays, evaluate boundary limits, solve a finite linear system, and certify the remaining identities by valence bounds.","The remaining open Shi and Wang targets are natural next applications of the same machinery, though the paper does not claim to prove them and explicitly notes that the required certificates for those targets are not supplied."],"supporting_citations":[{"why":"States the conjecture being proved, Conjecture 3.8, and supplies the exact 5-dissections of the two proposed product identities.","marker":"[9]"},{"why":"Introduces Nahm's problem and the duality map $D$ whose twelfth dual is the application of this paper.","marker":"[15]"},{"why":"Supplies the valence formula for generalized Dedekind eta products used to certify both final product identities.","marker":"[8]"},{"why":"Automates the valence method; the paper's cusp-order and bound computations are a direct rational-arithmetic port of its approach.","marker":"[4]"},{"why":"Provides the Slater identities S.79 and S.96 that evaluate the interior sums becoming the boundary coefficients $D(Q)$ and $S(Q)$.","marker":"[10]"},{"why":"Supplies the Rogers-Ramanujan product forms and Schur polynomial identities used for $G(Q)$, $H(Q)$, and the forward fundamental solutions.","marker":"[1]"}],"fun_headline_variants":["Machine-guided proof settles twelfth Nahm-sum conjecture","Exact recurrences and certificates prove Shi-Wang conjecture 3.8","Two eta products identified: dual of Zagier's twelfth is modular","Machine search finds, exact proof confirms: Nahm sums solved","From tropical rays to valence: modularity for Rank-3 family"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's final step assumes that every cusp order in the two valence certificates—all 560 inequivalent cusps of $\\Gamma_1(300)$ and all 140 of $\\Gamma_1(100)$—is computed correctly by formula (49), and that the bound $B^*$ of (50) is applied to every non-infinity cusp; a single miscomputed cusp order could allow a nonvanishing $q$-expansion to be mistaken for zero.","fun_headline_variants_meta":{"raw":{"variants":["Machine-guided proof settles twelfth Nahm-sum conjecture","Exact recurrences and certificates prove Shi-Wang conjecture 3.8","Two eta products identified: dual of Zagier's twelfth is modular","Machine search finds, exact proof confirms: Nahm sums solved","From tropical rays to valence: modularity for Rank-3 family"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000882,"raw_usage":{"total_tokens":3876,"prompt_tokens":1073,"completion_tokens":2803,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":689,"completion_tokens_details":{"reasoning_tokens":2715}},"tokens_in":689,"tokens_out":2803,"duration_ms":20726,"temperature":1.0,"reasoning_tokens":2715,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T00:07:57.620831+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently recompute all cusp orders and the bounds $B^*$ from the stored generalized-eta exponent tables, then expand the raw integer forms of the two product-side boundary equations through $q^{2005}$ for the $\\Gamma_1(300)$ identity and through $q^{255}$ for the $\\Gamma_1(100)$ identity; any nonzero coefficient at a degree below the respective bound, or any discrepancy in a single cusp order, would show that the proof of Proposition 8.3 and hence of Theorem 1.1 fails.","supporting_citations":[{"cited_title":"Modularity of Nahm Sums Dual to Zagier's Rank-Three Examples","cited_arxiv_id":"2607.23257","evidence_quote":"States the conjecture being proved, Conjecture 3.8, and supplies the exact 5-dissections of the two proposed product identities."},{"cited_title":"Zagier,The dilogarithm function, in: Frontiers in Number Theory, Physics, and Geometry II, Springer, Berlin, 2007, pp","cited_arxiv_id":null,"evidence_quote":"Introduces Nahm's problem and the duality map $D$ whose twelfth dual is the application of this paper."},{"cited_title":"Robins,Generalized Dedekind η-products, in: The Rademacher Legacy to Mathematics, Contemp","cited_arxiv_id":null,"evidence_quote":"Supplies the valence formula for generalized Dedekind eta products used to certify both final product identities."},{"cited_title":"Automatic Proof of Theta-Function Identities","cited_arxiv_id":"1807.08051","evidence_quote":"Automates the valence method; the paper's cusp-order and bound computations are a direct rational-arithmetic port of its approach."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Slater identities S.79 and S.96 that evaluate the interior sums becoming the boundary coefficients $D(Q)$ and $S(Q)$."}],"review_version":1}