Complex Circles of Partition and the Expansion Principles

Berndt Gensel; Theophilus Agama

arxiv: 2304.13371 · v6 · submitted 2023-04-26 · 🧮 math.GM

Complex Circles of Partition and the Expansion Principles

Berndt Gensel , Theophilus Agama This is my paper

Pith reviewed 2026-05-24 09:20 UTC · model grok-4.3

classification 🧮 math.GM

keywords complex circles of partitioncircles of partitionsqueeze principlepartition theorycomplex planegeneralizationexpansion principles

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The pith

Circles of partition extend from natural numbers to subsets of the complex plane, with the squeeze principle as the investigative tool.

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

The paper introduces complex circles of partition to generalize the existing theory of circles of partition. The complex plane becomes both the base set and the bearing set for partitions, replacing subsets of the natural numbers. The squeeze principle is applied to examine whether numbers can be partitioned under these new conditions. A sympathetic reader would care because the change opens partition questions to an entirely different number system.

Core claim

The central claim is that complex circles of partition generalize the classical framework by allowing partitions whose base set is any chosen subset of the complex plane. The squeeze principle supplies the means to investigate these partitions rigorously, and expansion principles are developed to support the new setting.

What carries the argument

Complex circles of partition, the structure that carries the generalization by treating the complex plane as both base and bearing set.

If this is right

Partitions can now be defined and studied when the base set is taken from the complex plane.
The squeeze principle becomes the central device for proving results about these complex partitions.
Expansion principles accompany the new circles and support further development of the theory.
The classical restriction to natural-number base sets is removed.

Where Pith is reading between the lines

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

The move may allow partition statements to incorporate magnitude or argument information from complex numbers.
Specific subsets such as the Gaussian integers could be tested to see whether the new circles produce previously unseen partition identities.
The framework might later connect to questions in complex analysis that involve counting or grouping points in the plane.

Load-bearing premise

The squeeze principle can be applied directly to partitions whose base set lies inside the complex plane.

What would settle it

An explicit subset of the complex plane for which the squeeze principle yields no definite classification of the partitions.

Figures

Figures reproduced from arXiv: 2304.13371 by Berndt Gensel, Theophilus Agama.

**Figure 2.** Figure 2: The ”Big Bang” Proof. Let [x] be a common point of C(m, M) and C(n, M). Then m−x and n−x are members of M, and m−x−iym and n−x−iyn are members of CM. Consequently, their axis partners x + iym and x + iyn are also members of CM. Therefore, with zm = x + iym and zn = x + iyn, we have [zm] ∈ Co (m, CM) and [zn] ∈ Co (n, CM) with x = ℜ(zm) = ℜ(zn). This reasoning can be reversed, and thus, from x = ℜ(zm) = ℜ(z… view at source ↗

**Figure 3.** Figure 3: Forecasting of C o (34, CP) by C o (30) and C o (32) 5. The Expansion Principles In this section, we do not distinguish between the axes L[z],[n−z] and L[n−z],[z] , as axes are considered equivalent under rearrangement of resident points. Subsequently we will consider only axes L[z],[n−z] with ℜ(z) < ℜ(n − z). Lemma 5.1 (Axial Points Ordering Principle). Let M ⊆ N and C o (n, CM) and C o (n + t, CM) be non… view at source ↗

**Figure 4.** Figure 4: Squeezing of C o (34, CP) by C o (30) and C o (38) Squeeze Principle. If 0 < s < t, then we have determined from two ”known” non–empty cCoPs C o (n, CM) and C o (n+t, CM) between them a ”new” non–empty cCoP C o (n + s, CB). By virtue of (5.3) holds ℜ(z) < ℜ(w) and ℜ(n − z) < ℜ(n + t − w) and by virtue of Lemma 5.1 ℜ(z) < ℜ(w) < ℜ(z) + t. If we set N instead of M and P instead of B, we get the following cor… view at source ↗

read the original abstract

In this paper, we further develop the theory of circles of partition by introducing the notion of complex circles of partition. This work generalizes the classical framework, extending from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set. We employ the squeeze principle as a central tool for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.

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.

Desk Editor's Note private letter to a colleague

The paper announces complex circles of partition via the squeeze principle but gives no definition or derivation for how that principle applies in C.

read the letter

This paper is basically announcing that they are extending circles of partition into the complex plane by using the squeeze principle there. That's the headline. The abstract frames it as a generalization where the complex plane becomes both the base and the bearing set for the partitions. What is new is the introduction of complex circles of partition as a concept. The authors say they employ the squeeze principle to investigate partitions with base sets that are subsets of the complex numbers. If this were fleshed out with actual derivations, it might be a reasonable next step from their earlier work. The paper does well in identifying that direction for the theory. Extending from natural numbers to complex is a clear move, and the idea itself is straightforward to state. The soft spot is that nothing is shown about how the squeeze principle is made to work in the complex setting. The stress test is on point here. Squeeze relies on some form of ordering or limits, and without specifying if it's via modulus or real and imaginary parts separately or something else, the claim that this rigorously investigates the partitions doesn't have support. The abstract just asserts it without examples or definitions. That makes the whole thing hard to evaluate. This is for people who are already following the circles of partition literature. A reader who knows the prior papers might see this as continuing that line, but without the details it's not useful to anyone else. I wouldn't bring this to the reading group. It doesn't look ready for peer review because the key step is missing. The recommendation is to desk reject or request a major revision that includes the definition of the squeeze in C and some concrete checks.

Referee Report

1 major / 0 minor

Summary. The paper introduces the notion of complex circles of partition as a generalization of the classical theory of circles of partition. It extends the framework from subsets of the natural numbers as base sets to partitions defined within the complex plane, which now serves as both the base and bearing set, and employs the squeeze principle as the central tool for investigating such partitions.

Significance. If the claimed generalization holds with a well-defined application of the squeeze principle to the complex plane, the work would provide a new analytic framework for partitions over C. However, the manuscript as presented offers no derivations, explicit constructions, examples, or verification that the partition relation remains well-defined under the proposed tool, limiting any assessment of novelty or impact.

major comments (1)

[Abstract] Abstract: The central claim that the squeeze principle can be applied directly to rigorously investigate partitions when the base set is a subset of the complex plane is asserted without any definition of the ordering, metric, or limiting process to be used on C. The squeeze principle classically relies on the total order of R; without specifying whether squeezing occurs via modulus, real/imaginary parts, a chosen total order on C, or componentwise application, it is impossible to verify that the resulting partition relation is well-defined or that the claimed generalization to C as both base and bearing set has been demonstrated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit definitions when extending the squeeze principle to the complex plane. We address the single major comment below and will make the corresponding revisions.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the squeeze principle can be applied directly to rigorously investigate partitions when the base set is a subset of the complex plane is asserted without any definition of the ordering, metric, or limiting process to be used on C. The squeeze principle classically relies on the total order of R; without specifying whether squeezing occurs via modulus, real/imaginary parts, a chosen total order on C, or componentwise application, it is impossible to verify that the resulting partition relation is well-defined or that the claimed generalization to C as both base and bearing set has been demonstrated.

Authors: We agree that the current manuscript does not supply an explicit definition of the ordering, metric, or limiting process on C, nor does it verify well-definedness of the partition relation under the squeeze principle. This omission prevents independent verification of the claimed generalization. In the revised manuscript we will add a preliminary section that (i) identifies C with R^2 equipped with the product order and the Euclidean metric, (ii) states that the squeeze principle is applied componentwise to the real and imaginary parts, and (iii) supplies the required limit definitions together with at least one explicit construction and numerical example confirming that the resulting partition relation is well-defined. These additions will be placed before the main results so that the abstract claim can be rigorously justified. revision: yes

Circularity Check

0 steps flagged

No circularity; central claim applies external squeeze principle without self-referential reduction or fitted inputs

full rationale

The provided abstract and context contain no equations, no fitted parameters presented as predictions, and no self-citations that bear the load of the generalization. The squeeze principle is invoked as an external tool from real analysis, and the extension to complex base sets is asserted without any visible definitional loop (e.g., no partition relation defined in terms of itself or renamed empirical pattern). The derivation chain therefore remains open to external verification and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5586 in / 941 out tokens · 20823 ms · 2026-05-24T09:20:05.888805+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We employ the squeeze principle as a central tool for rigorously investigating the possibility to partition numbers with base set as a certain subset of the complex plane.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.6. ... circle with its center on the real axis at n/2 and a diameter n.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 2 internal anchors

[1]

2:1, Wiley Online Library, 1938, pp

Estermann, Theodor On Goldbach’s problem: Proof that almost all even positive integers are sums of two primes , Proceedings of the London Mathematical Society, vol. 2:1, Wiley Online Library, 1938, pp. 307–314

work page 1938
[2]

Agama, Theophilus and Gensel, Berndt Studies in Additive Number Theory by Circles of Partition, arXiv:2012.01329, 2020

work page internal anchor Pith review Pith/arXiv arXiv 2012
[3]

Helfgott, Harald A The ternary Goldbach conjecture is true , arXiv preprint arXiv:1312.7748, 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013
[4]

Green, Ben and Tao, Terence The primes contain arbitrarily long arithmetic progressions , Annals of Mathematics, JSTOR, 2008, pp. 481–547

work page 2008
[5]

Chen, Jing-run On the representation of a larger even integer as the sum of a prime and the product of at most two primes, The Goldbach Conjecture, World Scientific, 2002, pp. 275–294

work page 2002
[6]

Math, vol: 6(3), 2002, 535–566

Heath-Brown, D Roger and Puchta, J-C Integers represented as a sum of primes and powers of two, Asian J. Math, vol: 6(3), 2002, 535–566

work page 2002
[7]

2:6, Russian Academy of Sciences, Steklov Mathematical Institute of Rus- sian

Shnirel’man, Lev Genrikhovich, On the additive properties of numbers , Uspekhi Matematich- eskikh Nauk, vol. 2:6, Russian Academy of Sciences, Steklov Mathematical Institute of Rus- sian . . . , 1939, pp. 9–25

work page 1939
[8]

4, Russian Academy of Sciences, Steklov Mathematical Institute of Russian

Chudakov, Nikolai Grigor’evich, The Goldbach’s problem , Uspekhi Matematicheskikh Nauk, vol. 4, Russian Academy of Sciences, Steklov Mathematical Institute of Russian . . . , 1938, 14–33

work page 1938
[9]

Partitio Numerorum

Hardy, Godfrey H and Littlewood, John E Some problems of “Partitio Numerorum”(V): A further contribution to the study of Goldbach’s problem , Proceedings of the London Mathe- matical Society, vol. 2(1), Wiley Online Library, 1924, pp. 46–56. Carinthia University of Applied Sciences, Spittal on Drau, Austria Email address: dr.berndt@gensel.at Department of...

work page 1924

[1] [1]

2:1, Wiley Online Library, 1938, pp

Estermann, Theodor On Goldbach’s problem: Proof that almost all even positive integers are sums of two primes , Proceedings of the London Mathematical Society, vol. 2:1, Wiley Online Library, 1938, pp. 307–314

work page 1938

[2] [2]

Agama, Theophilus and Gensel, Berndt Studies in Additive Number Theory by Circles of Partition, arXiv:2012.01329, 2020

work page internal anchor Pith review Pith/arXiv arXiv 2012

[3] [3]

Helfgott, Harald A The ternary Goldbach conjecture is true , arXiv preprint arXiv:1312.7748, 2013

work page internal anchor Pith review Pith/arXiv arXiv 2013

[4] [4]

Green, Ben and Tao, Terence The primes contain arbitrarily long arithmetic progressions , Annals of Mathematics, JSTOR, 2008, pp. 481–547

work page 2008

[5] [5]

Chen, Jing-run On the representation of a larger even integer as the sum of a prime and the product of at most two primes, The Goldbach Conjecture, World Scientific, 2002, pp. 275–294

work page 2002

[6] [6]

Math, vol: 6(3), 2002, 535–566

Heath-Brown, D Roger and Puchta, J-C Integers represented as a sum of primes and powers of two, Asian J. Math, vol: 6(3), 2002, 535–566

work page 2002

[7] [7]

2:6, Russian Academy of Sciences, Steklov Mathematical Institute of Rus- sian

Shnirel’man, Lev Genrikhovich, On the additive properties of numbers , Uspekhi Matematich- eskikh Nauk, vol. 2:6, Russian Academy of Sciences, Steklov Mathematical Institute of Rus- sian . . . , 1939, pp. 9–25

work page 1939

[8] [8]

4, Russian Academy of Sciences, Steklov Mathematical Institute of Russian

Chudakov, Nikolai Grigor’evich, The Goldbach’s problem , Uspekhi Matematicheskikh Nauk, vol. 4, Russian Academy of Sciences, Steklov Mathematical Institute of Russian . . . , 1938, 14–33

work page 1938

[9] [9]

Partitio Numerorum

Hardy, Godfrey H and Littlewood, John E Some problems of “Partitio Numerorum”(V): A further contribution to the study of Goldbach’s problem , Proceedings of the London Mathe- matical Society, vol. 2(1), Wiley Online Library, 1924, pp. 46–56. Carinthia University of Applied Sciences, Spittal on Drau, Austria Email address: dr.berndt@gensel.at Department of...

work page 1924