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arxiv: 2606.13590 · v1 · pith:MLDFBPIEnew · submitted 2026-06-11 · 🧮 math.NT · math.CO

Some new modular Nahm sums of ranks 3 and 4

Zhineng Cao , Liuquan Wang This is my paper

Pith reviewed 2026-06-27 05:25 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Nahm sumsmodular formsRogers-Ramanujan identitiesq-seriesrank threerank four
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The pith

Six new families of modular Nahm sums of ranks 3 and 4 are constructed and shown to equal modular infinite products via Rogers-Ramanujan identities.

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

The paper constructs six previously unknown families of Nahm sums that turn out to be modular. Two rank-3 families arise by altering two of Zagier's known examples. Three rank-4 families are obtained by applying a lift-dual operation to earlier tadpole sums, and one more rank-4 family is located by extracting constant terms. In each case the authors produce an explicit Rogers-Ramanujan-style identity that writes the sum as an infinite product known to be modular, thereby proving the sum itself is modular. A reader would care because these identities enlarge the stock of explicit modular objects that can be studied through q-series and partition theory.

Core claim

The authors exhibit six new families of Nahm sums in ranks 3 and 4 that admit Rogers-Ramanujan type product representations and are therefore modular forms.

What carries the argument

Rogers-Ramanujan type identities that equate each new Nahm sum to a modular infinite product.

If this is right

  • Each of the six families supplies an explicit modular form that can be written both as a multiple sum and as a product.
  • The lift-dual operation maps known modular rank-3 tadpole sums to new modular rank-4 sums.
  • The constant-term method yields at least one additional modular rank-4 family beyond those obtained by lifting.
  • The two modified rank-3 families enlarge the list of known modular examples originally found by Zagier.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modification and lift-dual techniques may generate further modular families in ranks greater than 4.
  • The product representations could be used to derive new congruences or partition identities for the coefficients of these sums.
  • If the constant-term method is applied to other known low-rank sums, it may locate additional modular examples not reachable by lifting.

Load-bearing premise

The chosen modifications to Zagier's sums, the lift-dual operation on tadpole sums, and the constant-term extraction each produce a Nahm sum that possesses a Rogers-Ramanujan product formula.

What would settle it

Compute the power series expansion of one new Nahm sum to sufficiently high order and compare it term-by-term with the proposed infinite-product formula; mismatch at any coefficient falsifies the identity for that family.

read the original abstract

We discover six new families of modular Nahm sums in ranks 3 and 4. Two of them are rank three sums obtained by modifying two of Zagier's rank three examples. Three rank four families are derived by applying the lift-dual operation to the rank three tadpole Nahm sums studied by Milas and Wang, while the other rank four family is found by the constant term method. To prove modularity, we establish Rogers-Ramanujan type identities that express these Nahm sums as infinite products which are modular.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to discover six new families of modular Nahm sums in ranks 3 and 4: two rank-3 families obtained by modifying Zagier's examples, three rank-4 families derived via the lift-dual operation applied to the rank-3 tadpole Nahm sums of Milas and Wang, and one rank-4 family found by the constant term method. Modularity is proved by establishing Rogers-Ramanujan-type identities that equate these Nahm sums to infinite products known to be modular.

Significance. If the claimed identities hold, the work enlarges the short list of explicitly modular Nahm sums in ranks 3 and 4, which remain scarce despite their importance for quantum modular forms and related q-series. The methods (modification of known examples, lift-dual, constant-term extraction) are standard in the literature, and the explicit product representations constitute a concrete, falsifiable contribution that could support further classification efforts.

minor comments (3)
  1. [Introduction / §2] The abstract states that the rank-3 families are obtained by 'modifying two of Zagier's rank three examples,' but the introduction does not specify the precise modifications (e.g., which parameters are altered or which of Zagier's five examples are used). Adding a short table or explicit parameter list in §2 would improve readability.
  2. [§§3–5] In the statements of the Rogers-Ramanujan-type identities (presumably in §§3–5), the convergence region or the precise range of the summation indices for the Nahm sums is not restated; a single sentence recalling the definition from the literature would prevent ambiguity.
  3. [§4] The lift-dual operation is invoked for three families without a self-contained one-paragraph recap of its definition and how it preserves modularity; a brief reminder citing Milas-Wang would help readers who are not specialists.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main results: the discovery of six new families of modular Nahm sums in ranks 3 and 4, obtained via modification of Zagier's examples, the lift-dual operation, and the constant-term method, together with the Rogers-Ramanujan-type identities used to prove modularity.

Circularity Check

0 steps flagged

Minor self-citation to co-author's prior work, not load-bearing

full rationale

The derivation proceeds by explicit modifications to Zagier's rank-3 examples, application of the lift-dual operation to the tadpole sums of Milas-Wang, use of the constant-term method for one family, and direct establishment of new Rogers-Ramanujan-type product identities to prove modularity. These steps are constructive and identity-based rather than fitted or self-referential. The single self-citation to Milas and Wang (co-author overlap) supplies base cases but does not justify the central claims; the new identities are independently verified. No equation reduces to a prior result by definition or renaming, and no parameter is fitted then relabeled as a prediction. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from abstract regarding free parameters, axioms or invented entities; the work builds on existing literature without introducing new ones explicitly.

pith-pipeline@v0.9.1-grok · 5606 in / 1222 out tokens · 32127 ms · 2026-06-27T05:25:43.195111+00:00 · methodology

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Reference graph

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21 extracted references · 2 canonical work pages

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