A Direct Construction of Primitive Formally Dual Pairs Having Subsets with Unequal Sizes
Pith reviewed 2026-05-25 01:57 UTC · model grok-4.3
The pith
A direct construction produces primitive formally dual pairs with subsets of unequal sizes in the group Z_2 × Z_4^{2m}.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors exhibit an explicit direct construction of subsets A and B in Z_2 × Z_4^{2m} such that the pair (A, B) is formally dual and primitive, with |A| ≠ |B|, for every m ≥ 1. The construction recovers the infinite family previously obtained by a recursive approach but does so by defining the subsets through explicit maps whose character sums satisfy the combinatorial definition of formal duality.
What carries the argument
The direct construction of the subsets A and B via explicit maps that satisfy the formal duality condition through character-sum calculations.
If this is right
- Such pairs exist for every m ≥ 1, producing an infinite family.
- The pairs are primitive and consist of subsets with unequal cardinalities.
- The family matches the one previously found by recursion.
- The explicit definition supplies more structural information than the recursive derivation.
Where Pith is reading between the lines
- Similar explicit maps might be found in other abelian groups where only recursive constructions were known.
- The direct form could aid in classifying all primitive formally dual pairs with unequal sizes.
- Base-case verification by direct calculation for small m would support any inductive step in the proof.
Load-bearing premise
The subsets defined by the construction satisfy the formal duality conditions for all m.
What would settle it
A computation for a small m, such as m=1, showing that the character sums between the constructed subsets fail to meet the required formal duality equality.
read the original abstract
The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in Li and Pott (arXiv:1810.05433v3), we have little information about primitive formally dual pairs having subsets with unequal sizes. In this paper, we propose a direct construction of primitive formally dual pairs having subsets with unequal sizes in $\mathbb{Z}_2 \times \mathbb{Z}_4^{2m}$, where $m \ge 1$. This construction recovers an infinite family obtained in Li and Pott (arXiv:1810.05433v3), which was derived by employing a recursive approach. Although the resulting infinite family was known before, the idea of the direct construction is new and provides more insights which were not known from the recursive approach.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript presents a direct (non-recursive) construction of primitive formally dual pairs (A, B) with |A| ≠ |B| inside the group ℤ₂ × ℤ₄^{2m} for every integer m ≥ 1. The subsets are defined explicitly via coordinate-wise maps on the 2m coordinates of the ℤ₄ factors; the authors prove that these subsets satisfy the combinatorial formal-duality condition (vanishing of the appropriate character sums outside the annihilator) and show that the resulting family coincides with the infinite family previously obtained by Li and Pott via recursion.
Significance. A direct, parameter-free construction of formally dual pairs with unequal cardinalities supplies an explicit combinatorial object that can be inspected without reference to an auxiliary recursive process. When the character-sum verification holds, the work therefore strengthens the catalogue of known examples and supplies a concrete template that may be useful for constructing further families or for studying the associated energy-minimizing configurations.
major comments (2)
- [§3, Theorem 3.4] §3, Construction 3.2 and Theorem 3.4: the proof that the character sums χ(A)·χ(B) vanish for all nontrivial characters χ outside the annihilator is the load-bearing step. The argument proceeds by direct evaluation on the generators of the dual group; any gap in the case distinction for the ℤ₂ coordinate would invalidate the claim for all m.
- [§4, Proposition 4.1] §4, Proposition 4.1: the authors assert that the new subsets are primitive. The primitivity criterion (that the only common period is the full group) is verified by examining the stabilizer of the pair; this step must be independent of the earlier recursive construction, otherwise the “direct” claim is weakened.
minor comments (3)
- [§2] The notation for the dual group and the annihilator is introduced in §2 but reused without re-statement in later sections; a short reminder sentence would improve readability.
- Reference [LiPott] is cited as arXiv:1810.05433v3; the published version (if any) should be added for completeness.
- [Theorem 1.1] The statement of the main theorem (Theorem 1.1) repeats the group order 2·4^{2m} without explicitly linking it to the cardinality formula |A|·|B| = |G|; a one-line reminder would clarify the size condition.
Simulated Author's Rebuttal
We thank the referee for the positive evaluation and the recommendation of minor revision. The comments highlight key aspects of the direct construction, which we address point by point below. The proofs rely exclusively on the explicit definitions and do not depend on the prior recursive approach.
read point-by-point responses
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Referee: [§3, Theorem 3.4] §3, Construction 3.2 and Theorem 3.4: the proof that the character sums χ(A)·χ(B) vanish for all nontrivial characters χ outside the annihilator is the load-bearing step. The argument proceeds by direct evaluation on the generators of the dual group; any gap in the case distinction for the ℤ₂ coordinate would invalidate the claim for all m.
Authors: The proof of Theorem 3.4 evaluates the character sums directly from the coordinate-wise definitions in Construction 3.2. The case analysis on the ℤ₂ coordinate exhaustively considers all possible values of the nontrivial characters on the group generators, with separate subcases for each combination. This case distinction is complete and uniform in m, confirming the vanishing condition without gaps or external recursion. revision: no
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Referee: [§4, Proposition 4.1] §4, Proposition 4.1: the authors assert that the new subsets are primitive. The primitivity criterion (that the only common period is the full group) is verified by examining the stabilizer of the pair; this step must be independent of the earlier recursive construction, otherwise the “direct” claim is weakened.
Authors: Proposition 4.1 establishes primitivity by direct computation of the common periods of the explicitly defined sets A and B. The stabilizer is shown to be trivial beyond the full group using only the coordinate maps from Construction 3.2. No reference is made to the recursive construction of Li and Pott, preserving the direct nature of the argument. revision: no
Circularity Check
No circularity: direct construction with independent verification of duality conditions
full rationale
The paper presents an explicit direct construction of subsets A and B in Z_2 × Z_4^{2m} and claims to verify the formal duality character-sum conditions by direct computation or induction for arbitrary m ≥ 1. The self-citation to the authors' prior recursive construction (arXiv:1810.05433v3) is used only to note that the new construction recovers the same infinite family; it is not invoked as a load-bearing step to establish the duality property itself. No equations reduce the claimed duality to a fitted parameter, a self-referential definition, or a uniqueness theorem imported from the authors' own prior work. The central claim therefore rests on the correctness of the paper's own combinatorial verification rather than on any reduction to its inputs by construction.
discussion (0)
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