{"id":"bf2f4d62-de78-4069-bbb7-be0b9bbb75ca","arxiv_id":"2507.20270","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The two conjectured Rogers-Ramanujan type identities for the last open family of partial Nahm sums are proved, confirming its modularity.","lead":"This paper proves two sum-to-product identities for partial Nahm sums, completing the last unproved family in Wang and Zeng's classification. The proof uses two Bailey pairs at once and converts the sums into Hecke-type series before reaching modular infinite products.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the second proof bypasses the Maple black box and the new Bailey pairs are verified from Slater's identities.","rationale":"The reader's weakest_assumption pointed to the two new Bailey pairs and to the unspecified Maple computation. My stress test disagrees with the first part: Lemma 2.1 actually proves (2.4) and (2.5) from Slater's identities, and I verified the extraction of the alpha coefficients, including the n=0 terms and the r to n reindexing. The Maple step is real but not load-bearing, because the second proof avoids it entirely and reaches the modular products using cited identities and straightforward transformations. I spot-checked the algebra in the second proof, including (3.75)-(3.78) and (3.81)-(3.82), and found no sign or exponent error. The only remaining concern is the length and complexity of Lemma 3.1, which both proofs share; a low-order check is consistent, and a symbolic verification would remove even that residual doubt. I therefore see no reason to change the reader's conditional verdict, but I would not strengthen the objection beyond an expository request for the Maple output.","tokens_in":16136,"tokens_out":46580,"duration_ms":401558,"concrete_test":"Expand both sides of (3.5) and (3.6) to q^60 with a computer algebra system and confirm that the two sides agree; this directly checks the one step common to both proofs of Theorem 1.1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not find a flaw that endangers Theorem 1.1. The first proof's final reductions from (3.65) and (3.71) are delegated to an unspecified Maple computation, which is an expository gap, but the second proof establishes (3.3) and (3.4) without that step, using only the cited Kim-Lovejoy identities (3.72)-(3.73) and the transformations (2.11), (2.12), (2.14), (2.15); I checked the reductions (3.75)-(3.82) and they are sound. The new Bailey pairs (2.4) and (2.5) are not a black box: Lemma 2.1 derives them from Slater's (2.6) and (2.8), and the coefficient extraction is correct. The genuinely shared ingredient is Lemma 3.1, whose algebra is long; my low-order expansions of both sides of (3.5) and (3.6) are consistent, so I have no concrete error to report.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two conjectural Rogers–Ramanujan type identities (1.11) and (1.12) proposed by Wang and Zeng for a remaining family of partial Nahm sums. The proof has two stages: first, using a transformation of Lovejoy that combines two Bailey pairs, the double sums S(q) and T(q) are converted into Hecke-type series f_{2,3,2} (Lemma 3.1). Second, two independent finishes are given: one passes through Appell–Lerch sums and a final Maple simplification, and another reduces the Hecke-type series to known Kim–Lovejoy identities (3.72)–(3.73) using only the elementary transformations (2.11), (2.12), (2.14)–(2.16). The new Bailey pairs (2.4) and (2.5) are derived from Slater's identities in Lemma 2.1.","tokens_in":16305,"tokens_out":3644,"duration_ms":35309,"significance":"If the identities are correct, the paper resolves the last open family in Wang and Zeng's investigation of modular partial Nahm sums, establishing modularity of the corresponding Nahm sums. The proof is notable for combining Bailey pairs with Hecke-type series and Appell–Lerch sums, and for offering a second derivation that is fully checkable by hand from published identities. The argument contains no fitted parameters and does not use the target identities as input; the second proof in particular gives a transparent, verifiable route from the double sums to the modular products. This is a concrete and useful contribution to the literature on partial Nahm sums and Rogers–Ramanujan type identities.","major_comments":[],"minor_comments":[{"comment":"The final simplifications from (3.65) to (3.3) and from (3.71) to (3.4) are delegated entirely to the Maple routine described in [3]; as written, this makes the first proof not self-contained. Please provide the key simplification steps or a reproducible Maple script, or state explicitly which Frye–Garvan reduction is being invoked, so that a reader can verify the final theta-product simplification without reimplementing the computation.","section":"§3, First Proof of Theorem 1.1, after Eqs. (3.65) and (3.71)"},{"comment":"The displayed products for W2(q) and M8(q) contain the repeated factor J_{8,30} twice in the denominator; while harmless, this is likely a typographical artifact and should be simplified by cancellation or corrected to avoid confusing the reader.","section":"Lemma 3.2, Eq. (3.45), and Lemma 3.3, Eq. (3.60)"},{"comment":"There is a typo in the phrase 'Hecek-type series' in the sentence 'The second method is to transform the Hecek-type series'; it should read 'Hecke-type series'.","section":"Section 3, paragraph after Lemma 3.1"},{"comment":"The derivation of the two new Bailey pairs from Slater's identities is quite compressed; for completeness, please add a few more intermediate steps showing exactly how the rewritten forms (2.7) and (2.9) match the α_n formulas in the Bailey-pair definition (2.2).","section":"Lemma 2.1, Eqs. (2.4) and (2.5)"}],"recommendation":"minor_revision","confidential_remarks":"The second proof is the more robust part of the manuscript and appears sound; the first proof's Maple-dependent conclusion is a presentation gap rather than a threat to the theorem. I would encourage the authors to either supply the omitted algebraic reduction in an appendix or explicitly state that Theorem 1.1 is fully established by the second proof, treating the first proof as an alternate route modulo a routine computer verification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. The two identities are real: the double sums become the predicted modular infinite products, and there are two independent routes to the final step. The first route passes through Appell–Lerch sums and then says \"easy and straightforward\" with Maple. The second route avoids that black box entirely, using only published Kim–Lovejoy identities and elementary transformations of Hecke-type series. So the result stands even if you are uncomfortable with the first proof's ending.\n\nWhat is new: the identities were conjectured in Wang–Zeng's paper and are proved here for the first time, completing the last open family in their list of 14. The method is also new for this literature: instead of one Bailey pair, Lemma 2.2 is applied with two Bailey pairs simultaneously, producing Hecke-type double sums rather than theta products directly. That is a legitimate and interesting route. The two new Bailey pairs in Lemma 2.1 are not quoted; they are derived from Slater's identities, and the coefficient extraction works. The author's citation to their own earlier conjecture is natural, not a red flag, and the external machinery from Lovejoy, Hickerson–Mortenson, Kim–Lovejoy, and Slater is used appropriately.\n\nSoft spots, in proportion. Lemma 3.1 is a long algebraic slog. I did not machine-check every line; my low-order expansions on both sides of (3.5) and (3.6) are consistent, so I have no concrete error to point to. The real expository gap is the final step of the first proof: (3.65) to (3.3) and (3.71) to (3.4) are delegated to \"the Maple approach in [3]\" without showing the output or the underlying theta identity. That is a legitimate complaint, but it is minor rather than serious, because the second proof derives (3.3) and (3.4) from the same Hecke-type series with written-out transformations. I checked the reductions in that second half and they are sound. No constants are fitted anywhere; the products are conclusions, not assumptions.\n\nThis paper is for q-series people, especially those tracking Nahm's problem and partial Nahm sums. It completes a small but meaningful family and adds a modularity data point for rational conformal field theory characters. It does not reshape a branch of mathematics, but it is exactly what a serious referee should spend time on. I would send it out and ask the authors to make the Maple step explicit or to state that the second proof is the primary one, then accept.","headline":"Clean, detailed proof of the last open Wang–Zeng partial Nahm identities; the first proof has a Maple black box, but the second independent route closes the gap.","tokens_in":16828,"tokens_out":2336,"would_cite":true,"duration_ms":25544,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A30","11P84","33D15","33D45","11F03","11F27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The two conjectural identities (1.11) and (1.12) hold: the remaining rank-two partial Nahm sums are modular infinite products.","keywords":["Partial Nahm sums","Rogers–Ramanujan type identities","Bailey pairs","Hecke-type series","Appell–Lerch sums","modular forms","q-hypergeometric series"],"falsifier":"Compute, as formal power series in $q$, the coefficient of $q^N$ on both sides of (1.11) and (1.12) for $N=0,1,\\ldots,20$; on the left the double sums become finite once the exponent exceeds 20, so this is a finite calculation. Any inequality at any $N$ would refute the theorem. Equally, substituting the two new Bailey pairs (2.4) and (2.5) into the defining relation (2.2) for $n=0,\\ldots,20$ checks the step the whole proof relies on.","tokens_in":15921,"feed_emoji":"🧪","tokens_out":11155,"duration_ms":100488,"temperature":0.7,"pith_summary":"The paper proves two conjectural identities from the theory of partial Nahm sums, completing the last open family in a recent classification of rank-two examples. The identities say that two double $q$-hypergeometric sums, with denominators $(q;q)_{2i}(q;q)_{2j}$ and $(q;q)_{2i+1}(q;q)_{2j}$, equal explicit modular infinite products. The proof works in two stages: the double sums are first transformed into Hecke-type series using two Bailey pairs at once, and those series are then converted to products by two independent arguments. This matters because modular Nahm sums are expected to be characters of two-dimensional rational conformal field theories, so each new modular case is a new candidate character.","feed_headline":"Proved: two Nahm sums equal modular products","feed_subtitle":"The last open rank-two partial Nahm sums are shown to equal modular q-products via Hecke-type series.","key_machinery":"The central object is a Bailey pair, two sequences $(\\alpha_n,\\beta_n)$ linked by (2.2), together with the transformation formula (2.10) that converts a double sum of $\\beta$'s into a double sum of $\\alpha$'s. The key move is to apply that formula with two different Bailey pairs simultaneously: the standard pair (2.3) together with two new pairs (2.4) and (2.5), which are derived inside the paper from established identities. This produces Hecke-type series, meaning indefinite binary quadratic-form sums with a sign factor, which are not directly visible as products. The second half of the machinery reduces those Hecke-type series to $\\theta$-products: one route expresses them as Appell–Lerch sums and then applies cancellation identities, and the other route matches them against previously known Hecke-type product identities. The final products are $J_1J_{2,5}$ and $J_1J_{1,5}$, which are modular forms.","core_discovery":"At the center is Theorem 1.1: with $J_{a,m}=j(q^a;q^m)$ and $J_m=(q^m;q^m)_\\infty$, the normalized sums $S(q^{1/2})=(q;q)_\\infty^2\\sum_{i,j\\ge0}q^{2ij+i+j}/((q;q)_{2i}(q;q)_{2j})$ and $T(q^{1/2})=(q;q)_\\infty^2\\sum_{i,j\\ge0}q^{2ij+i+3j}/((q;q)_{2i+1}(q;q)_{2j})$ satisfy $S(q^{1/2})=J_1J_{2,5}$ and $T(q^{1/2})=J_1J_{1,5}$. Equivalently, the unnormalized double sums in (1.11) and (1.12) equal $1/((q;q^2)_\\infty^2(q^2,q^8;q^{10})_\\infty)$ and $1/((q;q^2)_\\infty^2(q^4,q^6;q^{10})_\\infty)$, respectively. The paper establishes these by first converting the Nahm sums into Hecke-type series of the form $f_{2,3,2}(x,y,q^3)$, then converting those series to modular products.","pith_inferences":["The two new Bailey pairs (2.4) and (2.5) are not quoted from an independent source, so a direct verification of them to high order is the most focused way to test the proof's first step.","The same two-Bailey-pairs-plus-Hecke-summation route may apply to other rank-two Nahm-type double sums whose direct reduction to single sums does not reach an infinite product; the present proof suggests the Hecke-type stage is a bridge rather than an obstacle.","The appearance of a single modulus, 30, in all the Appell–Lerch reductions hints that the two identities belong to a finite family of similar cancellations; a systematic search over Hecke-type series of the same shape could uncover neighboring identities.","Because modular Nahm sums are expected to be characters of rational conformal field theories, the two products obtained here are candidate characters for specific two-dimensional theories, although the paper does not identify which theories."],"forward_implications":["The three rank-two partial Nahm sums attached to the data in (1.10) are modular, since their $q$-series are products of theta functions; this closes the one family left open in the earlier classification.","The left-hand double sums have explicit product forms, so their coefficients can be studied through the modular products, making asymptotics, parity, and congruence questions more accessible.","The two identities are new Rogers–Ramanujan type sum-to-product identities and can be reused in combinatorial interpretations of what the double sums count.","The proof supplies a template for partial Nahm sums whose direct one-Bailey-pair reduction stalls at single sums: pass through Hecke-type series and reduce those to products by either Appell–Lerch sums or a second Hecke identity."],"supporting_citations":[{"why":"States the conjecture that this paper proves and supplies the partial Nahm sum context, including the data in (1.10).","marker":"[13]"},{"why":"Provides the transformation formula (2.10) that converts the beta-double sums into alpha-double sums, the bridge to Hecke-type series.","marker":"[6]"},{"why":"Supplies the decomposition of Hecke-type series as Appell–Lerch sums and the functional equations used in the first proof.","marker":"[4]"},{"why":"Supplies the alternative Hecke-type identities (3.72)–(3.73) used in the second proof.","marker":"[5]"},{"why":"Supplies the identities (2.6) and (2.8) from which the two new Bailey pairs are derived.","marker":"[9]"},{"why":"Provides the automated theta-identity verification used to finish the first proof's simplification.","marker":"[3]"}],"fun_headline_variants":["Two partial Nahm sums proven equal to modular products","Proofs complete: last two partial Nahm sums are modular","Hecke series bridge: two Nahm sum conjectures proven","Partial Nahm sums resolved: two identities now theorems","Two Rogers-Ramanujan identities for Nahm sums proved"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the two new Bailey pairs (2.4) and (2.5) are correct; they are derived inside the paper from earlier identities rather than quoted from an independent source, and they supply the input to the transformation that produces the Hecke-type series. A sign or exponent error in either pair would invalidate the subsequent product identities.","fun_headline_variants_meta":{"raw":{"variants":["Two partial Nahm sums proven equal to modular products","Proofs complete: last two partial Nahm sums are modular","Hecke series bridge: two Nahm sum conjectures proven","Partial Nahm sums resolved: two identities now theorems","Two Rogers-Ramanujan identities for Nahm sums proved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000785,"raw_usage":{"total_tokens":3452,"prompt_tokens":924,"completion_tokens":2528,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":2447}},"tokens_in":540,"tokens_out":2528,"duration_ms":19326,"temperature":1.0,"reasoning_tokens":2447,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:46:37.162537+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, as formal power series in $q$, the coefficient of $q^N$ on both sides of (1.11) and (1.12) for $N=0,1,\\ldots,20$; on the left the double sums become finite once the exponent exceeds 20, so this is a finite calculation. Any inequality at any $N$ would refute the theorem. Equally, substituting the two new Bailey pairs (2.4) and (2.5) into the defining relation (2.2) for $n=0,\\ldots,20$ checks the step the whole proof relies on.","supporting_citations":[{"cited_title":"Lovejoy, Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J","cited_arxiv_id":null,"evidence_quote":"Provides the transformation formula (2.10) that converts the beta-double sums into alpha-double sums, the bridge to Hecke-type series."},{"cited_title":"Hickerson and E.T","cited_arxiv_id":null,"evidence_quote":"Supplies the decomposition of Hecke-type series as Appell–Lerch sums and the functional equations used in the first proof."},{"cited_title":"Kim and J","cited_arxiv_id":null,"evidence_quote":"Supplies the alternative Hecke-type identities (3.72)–(3.73) used in the second proof."},{"cited_title":"Slater, A new proof of Rogers’ transformations of infinite series, Proc","cited_arxiv_id":null,"evidence_quote":"Supplies the identities (2.6) and (2.8) from which the two new Bailey pairs are derived."},{"cited_title":"Frye and F.G","cited_arxiv_id":null,"evidence_quote":"Provides the automated theta-identity verification used to finish the first proof's simplification."}],"review_version":2}