Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities

Changsong Shi; Liuquan Wang

REVIEW 2 major objections 2 minor 1 cited by

Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities

T0 review · 2 major / 2 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read The Nahm sums dual to Zagier's rank-three modular examples are themselves modular in all but two cases, with Example 9 modular conditional on a single unproved identity and Example 12 supported by a conjectural one.

desk verdict Real progress on duals of Zagier's rank-three Nahm sums, but the proof of Theorem 1.4 has a substitution mismatch that also undermines Theorem 3.3, leaving Example 9 conditionality doubled. read the letter →

arxiv 2607.23257 v2 pith:XJQ26R37 submitted 2026-07-25 math.NT math.CO

Changsong Shi , Liuquan Wang This is my paper · ORCID

classification math.NTmath.CO MSC 11P8433D1533D4511F03

keywords Nahmsumsmodularq-seriesRogers–RamanujantypeidentitiesZagierrank-threeexamplesdualityofBaileypairsconstant-termmethodinfiniteproducts

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper takes the twelve rank-three Nahm sums that Zagier listed as modular and asks whether their duals—obtained by inverting the quadratic-form matrix and shifting the constant term—are modular too. It proves the duals of Examples 7, 8, 10, and 11 are modular by expressing each as a finite combination of infinite products via Rogers–Ramanujan-type identities, completing earlier work so that every dual except those for Examples 9 and 12 is settled. For Example 9 it proves three of the four identities needed and obtains a conditional modularity formula; for Example 12 it states conjectural product identities that would imply modularity. Along the way the paper proves four previously conjectured rank-four tadpole Nahm sum identities and finds several new rank-three modular Nahm sums. If the two conjectures hold, all non-singular duals from Zagier's rank-three list are modular.

What carries the argument

The engine is the Rogers–Ramanujan-type identity: an expression of a Nahm sum as a finite sum of quotients of infinite products J_m=(q^m;q^m)_∞ and J_{a,m}=(q^a,q^{m-a},q^m;q^m)_∞. Such identities make modularity visible, since with q=e^{2πiτ} each J-product is a modular form of weight 1/2 on a suitable congruence subgroup. The proofs combine the constant-term method (extracting the constant coefficient of a Laurent series), Bailey pairs (a systematic way to evaluate q-hypergeometric sums), classical q-series transformations, and the Jacobi triple product.

What would settle it

Compute enough q-series coefficients of b·D^(h)(q) − a·A^(h)(q) − R^(h)_4(q) for h=1,2,3,4; any nonzero coefficient disproves Conjecture 3.4. Similarly, expand the difference between each side of (3.157) and (3.158) to high order; a nonzero coefficient disproves Conjecture 3.8.

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Extended reading notes

Core claim

The central discovery is that the duality operation preserves modularity for most of Zagier's rank-three triples, and that the mechanism is explicit: each dual Nahm sum, after an appropriate rescaling of q, equals a small linear combination of infinite products built from (q^m;q^m)_∞ and (q^a,q^{m-a},q^m;q^m)_∞ blocks. For Example 8, seven product identities are proved; for Example 11, the last remaining identity from earlier work is proved; for Example 9, the proof reduces to a four-by-four linear system in which three relations are proved and the fourth is left as Conjecture 3.4; for Example 12, two conjectural product identities are given. Together, these identities make the modularity of

Load-bearing premise

The load-bearing premise is the unproved identity Conjecture 3.4 for Example 9 (and, for Example 12, Conjecture 3.8), since the modularity formulas for those duals are derived from a linear system that is underdetermined without it.

Editorial extensions

If this is right

The duals of Zagier's Examples 7, 8, 10, and 11 are modular, closing the last gap for Example 11 through identity (1.19).
The four rank-four tadpole Nahm sum identities conjectured in earlier work are now theorems.
Several new rank-three Nahm sums are modular; their duals are conjecturally modular and represented by explicit three-term product formulas.
If Conjecture 3.4 holds, the dual of Example 9 is modular, with a closed product formula (3.116); if Conjecture 3.8 holds, the dual of Example 12 is modular.
The conditional method reduces a Nahm-sum modularity proof to a finite linear system of theta identities—three proven, one conjectured—so numerical verification of the remaining relation would complete the proof.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same four-by-four linear-system organization used for Example 9 could be applied to Example 12; the missing piece is likely a parallel fourth theta identity rather than a new idea.
The proved Example 8 identities directly imply the tadpole identities, suggesting a general transfer: modularity of a dual rank-three sum can certify rank-four sums attached to tadpole diagrams.
The conjectured formulas for Example 12 were found by computing 5-dissections; checking the first several dozen coefficients of the stated products would provide a computational test short of a full proof.
If both conjectures hold, the duality principle holds for all non-singular Zagier rank-three triples in its strong form—the dual is itself a modular Nahm sum—leaving the known higher-rank counterexamples outside this original list.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does genuine work. It proves new modularity identities for the duals of Zagier's Examples 7, 8, 10, and 11, settles the remaining case (1.19) for Example 11, and proves four tadpole Nahm sum conjectures (1.10), (1.12)–(1.14). The techniques—Bailey pairs, constant-term extraction, dissections—are standard but applied with real skill. The authors also honestly flag the conditional parts: Example 9 rests on Conjecture 3.4 and Example 12 on Conjecture 3.8. That transparency is good.

But the stress-test concern lands. In the proof of Theorem 1.4, equation (4.28) with q replaced by q^4 does not match the inner sum in (4.27): the exponents in the outer and inner sums are off by factors of 4, the factor (-q^2;q^4)_j is missing, and the denominator (q^4;q^4)_{2j} differs from (q^2;q^4)_j(q^8;q^8)_j. Moreover, deriving (4.28) from the Bailey pair (2.69) via (2.65) yields an extra (-1)^j in the summand, which is absent from the displayed identity. So the deduction of (4.29) is not valid as written. Since Theorem 3.3's proof leans on Theorem 1.4 through (3.111)–(3.114), the proof of relations (3.93)–(3.95) for Example 9 is also unsupported by the written argument. The identities might still be true, and the unconditional results for Examples 7, 8, 10, and 11 do not depend on this gap, but it is a real hole.

A second soft spot is the repeated reliance on "the Maple approach in [16]" with no code or certificate. That makes key final simplifications unattestable, which matters in a field where known identities are sometimes misquoted.

Verdict: this deserves a serious referee, but the referee should be told to look hard at Theorem 1.4. If the proof cannot be repaired, Theorem 1.4 and the unconditional status of Example 9's relations should be weakened. The paper's other contributions look solid enough to warrant revision rather than rejection.

Referee Report

2 major / 2 minor

Summary. The paper investigates Nahm sums dual to Zagier's rank-three modular examples. It proves modularity for the duals corresponding to Examples 7, 8, 10, and 11, gives a conditional modularity result for Example 9 subject to Conjecture 3.4, and formulates conjectural product identities for Example 12. It also proves four previously conjectured rank-four tadpole Nahm sum identities (Theorem 1.3) and exhibits new rank-three modular Nahm sums (Theorem 1.4). The proofs rely on Bailey pairs, constant-term extraction, and Rogers-Ramanujan-type identities.

Significance. If the proofs are correct, the paper significantly advances Zagier's duality principle for rank-three Nahm sums: it establishes modularity for four of the six nontrivial duals and reduces Example 9 to a single explicit conjecture. The paper is honest in labeling conditional statements as conjectures. However, the proof of Theorem 1.4 has a serious gap in the Bailey-pair substitution, and this theorem is load-bearing for Theorem 3.3. The frequent reliance on unverified Maple simplifications also limits verifiability.

major comments (2)

[§4.2, Eqs. (4.27)–(4.29)] The substitution of (4.28) with q replaced by q^4 into (4.27) is invalid as written. With q→q^4, (4.28) gives an inner sum Σ_{j=0}^n q^{6j^2}/((q^4;q^4)_{n-j}(q^2;q^4)_j(q^8;q^8)_j), whereas (4.27) has Σ_{j=0}^n q^{6j^2}(-q^2;q^4)_j/((q^4;q^4)_{n-j}(q^4;q^4)_{2j}). The numerator factor (-q^2;q^4)_j is missing and the denominator differs. Moreover, applying (2.65) to the Bailey pair (2.69) with q^{1/2}→−q^{1/2} yields an extra (−1)^j. Thus (4.29) is not derived. Since Theorem 3.3 uses Theorem 1.4 via (3.111)–(3.114), the identities (3.93)–(3.95) are unsupported.
[Multiple sections (Lemma 2.1, (2.79), (3.2), (3.5), (3.116), (4.28)–(4.33))] Several final simplifications are delegated to 'the Maple approach in [16]' without code or certificates. These steps are load-bearing for the product identities and modularity conclusions. Please provide the Maple code, explicit certificates, or enough detail to make each step independently verifiable.

minor comments (2)

[§3.2] Section header says 'Dual of Example 2' but the first sentence says 'We list Zagier's Example 1 and its dual'; likely a typo.
[§3.6, Theorem 3.6] The conditional status of Example 9 should be stated more carefully in the abstract given the dependence on Conjecture 3.4 and the proof gap in Theorem 1.4.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the proof chain is self-contained and the conditional parts are explicitly labeled conjectures.

full rationale

The paper's central derivations are self-contained q-series computations. Theorem 1.1 (Example 8 dual identities) is proved directly from constant-term manipulations, Bailey pairs, and classical identities; Theorem 1.2 proves the previously open identity (1.19) using a rank-reduction formula from [36] and Theorem 1.1, none of which assumes (1.19); Theorem 1.3 proves the tadpole conjectures (1.10), (1.12)-(1.14) from identities already established in the paper. Although [36, Theorem 1.5] had proved (1.10) assuming (1.19), the paper supplies a direct proof, so there is no circular dependence. The only unproved load-bearing items are openly flagged: Conjecture 3.4 supplies the missing fourth relation (3.96) for Example 9, and Theorem 3.6 is stated conditionally on it; Conjecture 3.8 is explicitly conjectural for Example 12. These are completeness gaps, not hidden inputs. Self-citations (e.g., [32], [40]) point to prior work with independent proof contexts and are not used to assume the target results. The possible algebraic substitution issue in §4.2 around Eq. (4.28) into (4.27) is a proof-repair concern, not a circularity: no fitted constant is renamed as a prediction and no definition forces the conclusion. Overall, the derivation chain does not reduce to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted numerical constants or invented physical/mathematical entities appear; all parameters are symbolic. The main external assumptions are cited classical identities, computer-algebra verifications, and two unproved conjectures that the paper itself labels as conjectural.

assumptions (5)

standard math Standard q-binomial theorem, Jacobi triple product, Heine's transformation, and Bailey's lemma are valid formal power series identities.
Section 2 invokes these background results without proof; they are classical and accepted in the field.
domain assumption All cited Rogers–Ramanujan-type identities from Slater's list and other references (e.g., [5, 7, 24, 31]) are correct as stated.
The paper substitutes numerous single-sum identities from the literature; any error in those would propagate to the theorems.
domain assumption The 'Maple approach in [16]' correctly proves the theta-function identities used in the paper.
The paper repeatedly delegates final theta-identity verification to a computer-algebra method without shipping code or certificates, e.g., Lemma 2.1 and the last equalities in Theorems 1.2 and 1.4.
ad hoc to paper Conjecture 3.4 holds for h=1,2,3,4.
Theorem 3.6 derives modularity of the dual of Example 9 by solving (3.93)–(3.96); (3.96) is exactly Conjecture 3.4 and is unproved.
ad hoc to paper Conjecture 3.8 holds.
The modularity of the dual of Example 12 is asserted only conditional on these conjectural product identities.

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Pith. "Pith review of Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities." pith.science (2026). https://pith.science/paper/XJQ26R37

@misc{pith2026260723257,
  author       = {Pith},
  title        = {Pith review of: Nahm Sums Dual to Zagier's Rank-Three Examples and Related Identities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJQ26R37}},
  note         = {Machine review of arXiv:2607.23257}
}

read the original abstract

In 2007, Zagier identified twelve families of candidates for rank-three modular Nahm sums and established the modularity of three of them. The remaining cases were subsequently confirmed by Wang. Zagier's duality observation indicates that the Nahm sums associated with the duals of these examples might still be modular. We confirm that this is indeed true. The modularity of the dual of the sixth example was previously established by Milas and Wang, while that of the dual of the eleventh example was partially established by the present authors. We settle all remaining cases, thereby establishing the modularity of every defined dual in Zagier's rank-three list. We achieve this by proving new Rogers--Ramanujan type identities that express the relevant Nahm sums as finite sums of infinite products. Along the way we discover two new families of rank-three Nahm sums. As applications, we prove some Nahm sum identities conjectured by Cao--Wang, Li and the present authors.

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Machine-Guided Recurrence Boundary Theory for Nahm Sums
math.NT 2026-08 accept novelty 7.0 of 10

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.

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Reviewed July 31, 2026 · model on record in the stance chip above.