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arxiv: 2604.20708 · v2 · submitted 2026-04-22 · 🧮 math.CO · math.GR

Lifting Cubic Realizations of Weak Orders in Types A and B

Daria Poliakova This is my paper

Pith reviewed 2026-05-09 23:58 UTC · model grok-4.3

classification 🧮 math.CO math.GR
keywords cubic realizationsweak orderstypes A and Bpre-Reeb graphsposet projectionsorder embeddingscombinatorial uniquenessdeletion towers
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The pith

Weak orders in types A and B admit combinatorially unique order-embedding cubic coordinates under cylindrical projections.

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

The paper constructs pre-Reeb graphs and augmented pre-Reeb graphs to track how cubic realizations of a poset can be lifted while respecting a cylindrical projection, which acts by deleting the last coordinate. When these graphs are built on the deletion towers of weak orders in types A and B, the pre-Reeb graphs recover the 1-skeleta of cubes and certain zonotopes. Their augmented versions have reachability posets that form total orders, which forces any two compatible order-embedding cubic lifts to coincide. A reader cares because this supplies an explicit combinatorial criterion for uniqueness instead of solving systems of inequalities.

Core claim

For cylindrical projections on posets, the pre-Reeb graph encodes the existence of compatible cubic lifts while the augmented pre-Reeb graph encodes the existence of compatible order-embedding cubic lifts. Applied to the deletion towers of the weak orders of types A and B, the pre-Reeb graphs are the 1-skeleta of cubes (type A) and of certain zonotopes (type B). In both cases the reachability poset of the augmented pre-Reeb graph is a total order, which immediately implies that the order-embedding cubic coordinates are combinatorially unique.

What carries the argument

The augmented pre-Reeb graph, whose reachability poset determines whether order-embedding cubic lifts are unique.

Load-bearing premise

That a total-order reachability poset on the augmented pre-Reeb graph is sufficient to guarantee combinatorial uniqueness of the order-embedding cubic lift.

What would settle it

An explicit weak order in type A or B for which two distinct order-embedding cubic coordinate assignments both respect the cylindrical projections.

Figures

Figures reproduced from arXiv: 2604.20708 by Daria Poliakova.

Figure 1
Figure 1. Figure 1: A cubic construction which does not admit an order-embedding deformation since its [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The directed flip graph Γ(F3) ∼= R(π : W3 → W2). • if i /∈ |A|, orient T → i → H; • if +i ∈ A, orient T → i and H → i; • if −i ∈ A, orient i → T and i → H. This produces an acyclic orientation by Step 1. Conversely, given O ∈ AO(Fn), recover ε from the direction of LR, and recover A by T → i → H ⇒ ±i /∈ A, (T → i and H → i) ⇒ +i ∈ A, (i → T and i → H) ⇒ −i ∈ A. This is inverse to Φ, so Φ is a bijection. St… view at source ↗
read the original abstract

We study cubic realizations of posets compatible with projection maps, meaning that the projection is represented by deletion of the last coordinate. For cylindrical projections, we introduce the pre-Reeb graph and the augmented pre-Reeb graph, which control compatible cubic lifts and compatible order-embedding cubic lifts, respectively. We apply this construction to the deletion towers in weak order of types A and B. The pre-Reeb graphs are the 1-skeleta of, respectively, cubes and certain zonotopes. In both cases, the augmented pre-Reeb graphs have reachability posets that are total orders, yielding combinatorial uniqueness of the compatible order-embedding cubic coordinates.

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

1 major / 1 minor

Summary. The paper introduces pre-Reeb graphs and augmented pre-Reeb graphs to study cubic realizations of posets that are compatible with cylindrical projections (where the projection deletes the last coordinate). It applies the construction to deletion towers of weak orders in types A and B, showing that the pre-Reeb graphs coincide with the 1-skeleta of cubes (type A) and certain zonotopes (type B). The central result is that the augmented pre-Reeb graphs have reachability posets that are total orders, from which the authors conclude combinatorial uniqueness of the compatible order-embedding cubic coordinates.

Significance. If the constructions and the implication from total reachability to uniqueness are rigorously verified, the work supplies a concrete combinatorial tool for controlling lifts of realizations under projection in Coxeter combinatorics. The explicit identification of the pre-Reeb graphs with well-known geometric 1-skeleta is a positive feature that may allow future geometric or representation-theoretic interpretations.

major comments (1)
  1. [Abstract and main theorem statement] The inference that a total order on the reachability poset of the augmented pre-Reeb graph forces uniqueness of the order-embedding cubic lift is load-bearing for the main claim, yet the abstract and the sketched argument do not supply an explicit bijection or inductive forcing step that uses totality to determine every coordinate. Without this step, it remains possible that the definition of order-embedding cubic coordinate (via cylindrical projection and the pre-Reeb construction) still permits multiple choices even when reachability is total.
minor comments (1)
  1. [Introduction / Definitions] The notation for the augmented pre-Reeb graph and its reachability relation should be introduced with a self-contained definition before the application to types A and B, rather than being presupposed in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback. We address the major comment below and will revise the paper to strengthen the exposition of the central implication.

read point-by-point responses

  1. Referee: [Abstract and main theorem statement] The inference that a total order on the reachability poset of the augmented pre-Reeb graph forces uniqueness of the order-embedding cubic lift is load-bearing for the main claim, yet the abstract and the sketched argument do not supply an explicit bijection or inductive forcing step that uses totality to determine every coordinate. Without this step, it remains possible that the definition of order-embedding cubic coordinate (via cylindrical projection and the pre-Reeb construction) still permits multiple choices even when reachability is total.

    Authors: We agree that the abstract and the current sketch of the argument do not make the forcing mechanism fully explicit. The manuscript establishes that the augmented pre-Reeb graphs have total-order reachability posets and concludes combinatorial uniqueness from this fact, but a detailed inductive step or explicit coordinate-by-coordinate determination is only sketched. In the revised version we will insert a dedicated lemma (placed immediately before the main theorem) that uses the totality of the reachability poset, the order-embedding condition, and the recursive definition of the augmented pre-Reeb graph under cylindrical projection to show that each successive coordinate is uniquely forced. This will supply the missing bijection between the total order and the unique lift. revision: yes

Circularity Check

0 steps flagged

No circularity: uniqueness follows from explicit total-order property of newly defined augmented pre-Reeb reachability posets

full rationale

The paper defines pre-Reeb graphs and augmented pre-Reeb graphs to control compatible cubic lifts for cylindrical projections on deletion towers. It then exhibits that, for weak orders of types A and B, the reachability posets of the augmented graphs are total orders and directly concludes combinatorial uniqueness of the order-embedding cubic coordinates from this property. No step reduces a prediction to a fitted parameter, renames a known result, or relies on a load-bearing self-citation whose content is itself unverified. The derivation is self-contained: the total-order claim is a combinatorial fact about the constructed graphs, and the uniqueness inference is presented as a consequence of that fact rather than an input smuggled into the definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper relies on standard definitions of weak orders, posets, cubic realizations, and projection maps from the literature on Coxeter groups and order theory. It introduces two new graph objects whose properties are asserted to yield uniqueness, but without the full text the precise axioms invoked for reachability and order-embedding are unknown.

axioms (2)
  • domain assumption Weak orders in types A and B admit deletion towers compatible with cylindrical projections.
    Invoked when applying the pre-Reeb construction to these specific posets.
  • ad hoc to paper The reachability relation on the augmented pre-Reeb graph forms a total order.
    This is the key property asserted to produce combinatorial uniqueness; its verification is not shown in the abstract.
invented entities (2)
  • pre-Reeb graph no independent evidence
    purpose: Controls which cubic realizations are compatible with a given cylindrical projection.
    Newly defined object whose 1-skeleton is claimed to be a cube or zonotope for the weak-order towers.
  • augmented pre-Reeb graph no independent evidence
    purpose: Controls order-embedding cubic lifts and encodes reachability that forces uniqueness.
    Stronger version of the pre-Reeb graph introduced to obtain the total-order property.

pith-pipeline@v0.9.0 · 5396 in / 1640 out tokens · 27643 ms · 2026-05-09T23:58:51.214353+00:00 · methodology

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Reference graph

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