A recursive approach for the determination of the nice sections of width three which have a 4-crown stack as retract

Frank a Campo

arxiv: 2510.05640 · v4 · pith:HT6K5OI7new · submitted 2025-10-07 · 🧮 math.CO

A recursive approach for the determination of the nice sections of width three which have a 4-crown stack as retract

Frank a Campo This is my paper

Pith reviewed 2026-05-18 09:46 UTC · model grok-4.3

classification 🧮 math.CO

keywords posetsnice sectionswidth three4-crown stackretractrecursive approachisomorphism typesautomorphic posets

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The pith

A recursive method determines which nice sections of width three retract onto a 4-crown stack, and applies it to list all such posets in subclass N2 up to height six while proving an exponential count of isomorphism types.

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

The paper introduces a recursive approach to identify nice sections of width three that have a 4-crown stack as retract. This approach is applied to the subclass N2. It determines all posets in N2 of height at most six that satisfy the retract condition. It further shows that N2 contains exactly 2 to the power of n minus 2 distinct isomorphism types of posets of each height n at least 2. The work supports progress on the open problem of characterizing finite minimal automorphic posets of width three by handling a specific retract case.

Core claim

We develop a recursive approach for the determination of the nice sections of width three which have a 4-crown stack as retract. We apply the approach on a sub-class N2 of nice sections of width three and determine all posets in N2 with height up to six which have a 4-crown stack as retract. For each integer n >= 2, the class N2 contains 2^{n-2} different isomorphism types of posets of height n.

What carries the argument

The recursive approach that systematically generates candidate nice sections of width three and verifies the 4-crown stack retract property, restricted to subclass N2.

If this is right

All posets in N2 of height at most six with the 4-crown stack retract property receive an explicit determination.
The subclass N2 contains precisely 2^{n-2} isomorphism types at each height n for n at least 2.
The method supplies a concrete verification procedure for the retract property in this subclass up to the given height bound.
The results advance the reduction of the minimal automorphic posets problem by handling the 4-crown stack retract case for N2.

Where Pith is reading between the lines

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

If the recursion extends without loss of correctness, it could classify the retract property for all heights and thereby help close the open characterization problem.
The exponential growth rate in N2 suggests this subclass forms a substantial and systematically expanding collection of the posets relevant to width-three automorphy questions.
Analogous recursive constructions might be developed for other retract types such as longer crown stacks or different width classes.
The enumerated lists for small heights enable direct computational checks of additional poset properties such as automorphy on the classified objects.

Load-bearing premise

The recursive approach correctly generates or verifies every relevant nice section in N2 without omissions or duplicates when restricted to height at most six.

What would settle it

An explicit poset belonging to N2 of height six that possesses a 4-crown stack as retract but is absent from or duplicated in the list produced by the recursive method.

read the original abstract

The characterization of the finite minimal automorphic posets of width three is still an open problem. Niederle has shown that this task can be reduced to the characterization of the nice sections of width three which have a non-trivial tower of nice sections as retract. In our article, we develop a recursive approach for the determination of those nice sections of width three which have a 4-crown stack as retract. We apply the approach on a sub-class $\mathfrak{N}_2$ of nice sections of width three and determine all posets in $\mathfrak{N}_2$ with height up to six which have a 4-crown stack as retract. For each integer $n \geq 2$, the class $\mathfrak{N}_2$ contains $2^{n-2}$ different isomorphism types of posets of height $n$.

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

read the letter

The paper gives a recursive method to enumerate nice sections in the N2 subclass with 4-crown retracts, lists them explicitly up to height six, and states a closed-form count of 2^{n-2} types per height, though the general claim needs the recursion to hold without gaps for all n. It builds on Niederle's reduction of the open problem about minimal automorphic posets of width three and turns that reduction into something concrete for this one subclass. The lists for small heights are the most useful output because they let you inspect the actual posets and see how the retract condition plays out. The recursion itself looks like a practical tool that could be coded or extended to other subclasses. The soft spot is the step from the height-six check to the formula that holds for every n. The recursion is supposed to add exactly the right number of new isomorphism types at each level while keeping the nice-section and retract properties, but a finite list does not automatically confirm that no omissions, duplicates, or violations appear later. If the paper only verifies the pattern up to six without a clear inductive argument or normalization rule that scales, the exponential count rests on an assumption that still needs checking. This is narrow work aimed at people already working on the width-three automorphic posets question. A reader in that area can use the lists and the recursive construction to test ideas or generate more examples. I would send it for peer review because it supplies verifiable data on a reduced open problem and the details are concrete enough for referees to examine directly.

Referee Report

2 major / 2 minor

Summary. The paper develops a recursive approach for identifying nice sections of width three that have a 4-crown stack as retract. It applies the method to the subclass N2, explicitly determines all such posets of height at most six, and claims that N2 contains exactly 2^{n-2} distinct isomorphism types of height-n posets for every n ≥ 2.

Significance. If the recursion is shown to be exhaustive and free of accidental isomorphisms or omissions for arbitrary height, the result supplies a concrete enumeration and closed-form count in a subclass relevant to Niederle's reduction of the open problem on minimal automorphic posets of width three. The explicit list up to height six and the exponential formula would constitute a tangible advance if rigorously justified.

major comments (2)

[Abstract and main enumeration statement] The general claim that N2 contains precisely 2^{n-2} isomorphism types for every n ≥ 2 is central to the paper's contribution, yet the manuscript only determines the posets explicitly for heights ≤ 6. An inductive argument establishing that the recursion rule generates exactly one new isomorphism type per binary choice at each height step, while preserving the nice-section and 4-crown-stack-retract properties without omissions or duplicates, is required to support the formula beyond the finite check.
[Recursive approach and application to N2] The recursive construction must be shown to be complete for the subclass N2; any height-specific case analysis or normalization step that does not scale uniformly could invalidate the exponential count for n > 6.

minor comments (2)

[Introduction] Provide an explicit definition of the subclass N2 (or fraktur N_2) at the first point of use rather than deferring it.
[Enumeration for height ≤ 6] If tables or diagrams list the posets of height ≤ 6, include a brief justification (e.g., via height or width invariants) confirming that the listed objects are pairwise non-isomorphic.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below, indicating the revisions we will make to strengthen the presentation of the recursive construction and the general enumeration result.

read point-by-point responses

Referee: [Abstract and main enumeration statement] The general claim that N2 contains precisely 2^{n-2} isomorphism types for every n ≥ 2 is central to the paper's contribution, yet the manuscript only determines the posets explicitly for heights ≤ 6. An inductive argument establishing that the recursion rule generates exactly one new isomorphism type per binary choice at each height step, while preserving the nice-section and 4-crown-stack-retract properties without omissions or duplicates, is required to support the formula beyond the finite check.

Authors: The recursive construction is defined uniformly via binary choices at each height increment, with each choice producing a distinct isomorphism type by design. The preservation of the nice-section property and the 4-crown-stack retract follows directly from the extension rules, which are stated without height dependence beyond the base cases. We agree that an explicit inductive argument would make this clearer and will add a dedicated subsection proving by induction on n that the process generates precisely 2^{n-2} isomorphism types for all n ≥ 2, with no omissions or accidental isomorphisms. revision: yes
Referee: [Recursive approach and application to N2] The recursive construction must be shown to be complete for the subclass N2; any height-specific case analysis or normalization step that does not scale uniformly could invalidate the exponential count for n > 6.

Authors: The recursion is formulated as a uniform rule applicable at every height, with only the initial posets of heights 2 and 3 serving as base cases to start the process. Any normalization is embedded in the general definition and scales directly with height. We will revise the manuscript to include an explicit argument establishing completeness for N2 at arbitrary height, confirming that the construction enumerates all members of the subclass without height-dependent restrictions that would break uniformity. revision: yes

Circularity Check

0 steps flagged

No circularity: recursive construction and count are independent of inputs

full rationale

The paper explicitly builds on Niederle's external reduction result for minimal automorphic posets of width three, then introduces an independent recursive approach to identify nice sections with 4-crown stack retracts. It applies the method to the defined subclass N2, enumerates all members of height at most six, and states the general isomorphism count 2^{n-2}. No step reduces the central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain; the recursion and enumeration are presented as self-contained against the external Niederle benchmark and the explicit small-height verification. The general formula follows from the recursive structure rather than circular re-use of outputs as inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, ad-hoc axioms, or invented entities are stated. The work relies on standard background results from poset theory and the cited reduction by Niederle.

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discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We develop a recursive approach for the determination of those nice sections of width three which have a 4-crown stack as retract... For each integer n ≥ 2, the class N2 contains 2^{n-2} different isomorphism types of posets of height n.

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