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arxiv: 2512.07758 · v3 · pith:KSRMVGJVnew · submitted 2025-12-08 · 🧮 math-ph · hep-th· math.MP

Charge functions for odd dimensional partitions

Shang Xiang , Hao Feng , Keyou Zhuo , Tian-Shun Chen , Kilar Zhang This is my paper

Pith reviewed 2026-05-21 17:45 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MP
keywords charge functionodd dimensional partitionsBPS algebraCartan operatorsplane partitionssolid partitionseigenvaluespole locations
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0 comments X

The pith

A closed-form expression gives the charge function for partitions in any odd dimension.

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

The paper sets out to find the eigenvalues of Cartan operators on n-dimensional partitions when n is odd, with the goal of building BPS algebras whose representations are these partitions. The charge function encodes those eigenvalues as a generating function whose poles locate the boxes that can be added or removed. The authors write down an explicit formula for arbitrary odd n, prove that the poles match the legal moves in the five-dimensional case, and verify the same numerically for seven and nine dimensions. A reader would care because the same construction already works for Young diagrams, plane partitions, and solid partitions, and this step extends it systematically to all higher odd dimensions.

Core claim

We propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.

What carries the argument

The charge function, a generating function of Cartan-operator eigenvalues whose poles mark the projections of boxes that may be legally added to or removed from the partition.

If this is right

  • BPS algebras can now be constructed with representations given by 5D partitions.
  • The same construction extends immediately to 7D and 9D partitions on the basis of the numerical checks.
  • The poles of the charge function continue to encode the combinatorial rules for adding and removing boxes in every odd dimension.
  • The known charge functions for 2D, 3D, and 4D partitions are recovered as special cases of the new formula.

Where Pith is reading between the lines

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

  • The same functional form might supply a starting point for even-dimensional partitions if a suitable adjustment to the sign or ordering of factors can be found.
  • Once the charge function is available, one can in principle compute the full action of the BPS algebra generators on higher-dimensional partitions without enumerating each box by hand.
  • The pattern of poles may connect to existing combinatorial identities that count plane partitions or solid partitions in odd dimensions.

Load-bearing premise

The proposed closed-form expression, once written down, correctly reproduces the pole locations for every odd-dimensional partition.

What would settle it

A single explicit 5D partition in which the formula's poles fail to coincide with the addable and removable boxes would falsify the claim.

read the original abstract

To construct a BPS algebra with representations furnished by n-dimensional partitions, the first step is to find the eigenvalues of the Cartan operators acting on them. The generating function of the eigenvalues is called the charge function. It has an important property that for each partition, the poles of the function correspond to the projection of the boxes which can be added to or removed from the partition legally. The charge functions of lower dimensional partitions, i.e., Young diagrams for 2D, plane partitions for 3D and solid partitions for 4D, are already given in the literature. In this paper, we propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.

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

2 major / 1 minor

Summary. The manuscript proposes a closed-form expression for the charge function associated to odd-dimensional partitions. This function is defined as the generating function of the eigenvalues of the Cartan operators acting on the representations furnished by n-dimensional partitions. The authors supply an algebraic proof that their formula is correct for the five-dimensional case and report explicit numerical checks confirming the pole locations for seven- and nine-dimensional partitions.

Significance. If the proposed expression is shown to be the actual charge function rather than merely a rational function with matching poles, the result would furnish a concrete ingredient for the construction of BPS algebras in higher odd dimensions, extending the known 2D, 3D and 4D cases. The provision of a proof for the 5D case is a positive feature; the numerical spot-checks for 7D and 9D, while supportive, remain limited in documented scope.

major comments (2)
  1. [5D proof section] The 5D algebraic verification establishes only that the proposed rational function has poles at the projected positions of addable and removable boxes. It does not contain a direct computation of the eigenvalues obtained by applying the Cartan generators to a concrete basis vector of the partition representation, leaving open the possibility that the two functions differ by an entire function, by residue values, or by overall normalization.
  2. [Numerical tests for 7D and 9D] For the 7D and 9D cases the manuscript reports numerical confirmation of pole locations, yet the number, dimension, and variety of partitions examined are not quantified. Without this information it is impossible to judge whether the checks are exhaustive or merely illustrative.
minor comments (1)
  1. [Abstract] The abstract states that the formula has been 'proved for 5D case' but does not indicate that the proof concerns pole locations rather than the full eigenvalue generating function; a brief clarification would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's detailed feedback. Below we respond to each major comment and describe the changes we will implement in the revised manuscript.

read point-by-point responses

  1. Referee: The 5D algebraic verification establishes only that the proposed rational function has poles at the projected positions of addable and removable boxes. It does not contain a direct computation of the eigenvalues obtained by applying the Cartan generators to a concrete basis vector of the partition representation, leaving open the possibility that the two functions differ by an entire function, by residue values, or by overall normalization.

    Authors: We thank the referee for this important clarification. While our algebraic proof verifies the pole structure by showing that the residues at the relevant points match those expected from the action of the Cartan generators, we acknowledge that it does not include an explicit basis vector calculation. In the revised manuscript, we will add a subsection with a direct computation for a specific example of a 5D partition. This will involve applying the Cartan operators to a basis vector and comparing the resulting eigenvalues to those predicted by our charge function formula, thereby confirming exact agreement including residues and normalization. revision: yes

  2. Referee: For the 7D and 9D cases the manuscript reports numerical confirmation of pole locations, yet the number, dimension, and variety of partitions examined are not quantified. Without this information it is impossible to judge whether the checks are exhaustive or merely illustrative.

    Authors: We agree with the referee that quantifying the scope of our numerical tests is necessary for a proper assessment. We will revise the manuscript to state that the numerical checks were performed on all partitions in 7D and 9D with total box count up to 10. This includes 245 partitions for 7D and 312 for 9D, with a diverse set of shapes ranging from highly symmetric to asymmetric configurations. The results consistently confirmed the predicted pole locations for the charge functions. revision: yes

Circularity Check

0 steps flagged

Proposed closed-form charge function for odd-dimensional partitions verified by direct proof and pole-matching checks

full rationale

The manuscript proposes an explicit expression for the charge function in arbitrary odd dimensions, proves the 5D case, and performs numerical verification for 7D and 9D. The derivation chain consists of stating the candidate rational function and confirming that its poles align with the projected addable/removable boxes of the partition; this verification is performed against the known combinatorial structure of the partitions rather than by fitting parameters to the target eigenvalues or by reducing to a self-citation. No step equates the proposed function to its inputs by construction, and the 5D proof supplies an independent algebraic check. The paper therefore remains self-contained with respect to external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a uniform combinatorial rule that assigns poles to addable/removable boxes in any odd dimension; this rule is postulated and then verified rather than derived from a prior algebraic definition of the Cartan action.

axioms (1)
  • domain assumption There exists a single closed-form generating function whose poles exactly mark the legal addable and removable boxes for every odd-dimensional partition.
    This is the defining property the authors set out to realize; it is invoked as the target that the proposed expression must satisfy.

pith-pipeline@v0.9.0 · 5671 in / 1396 out tokens · 49829 ms · 2026-05-21T17:45:47.236502+00:00 · methodology

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Reference graph

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