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arxiv: 2503.10829 · v2 · pith:5H5WKWW4new · submitted 2025-03-13 · 🧮 math.DS

Linear Relations of Finite Length Modules are Shift Equivalent to Maps

Bartosz Furmanek , Filip Oskar {\L}anecki , Mateusz Przybylski , Jim Wiseman This is my paper

Pith reviewed 2026-05-22 23:48 UTC · model grok-4.3

classification 🧮 math.DS
keywords linear relationsfinite length modulesshift equivalencebijective mappingsdynamical systemsmodule theory
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The pith

Linear relations defined on finite length modules are shift equivalent to bijective mappings.

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

The paper proves that any linear relation, viewed as a submodule of the direct sum of two modules, becomes shift equivalent to a bijective mapping when the modules have finite length. This equivalence lets objects that encode uncertainty in sampled dynamics be replaced by ordinary functions for the purpose of defining invariants. A reader would care because the result turns an algebraic relation into a deterministic map while preserving the dynamical information carried by the original object. The finite-length condition is essential to the alignment between relations and bijections under the shift equivalence relation.

Core claim

Linear relations defined as submodules of the direct sum of two modules carry dynamical information and enable subtle invariants. When the modules have finite length, every such relation is shift equivalent to a bijective mapping.

What carries the argument

Shift equivalence applied to linear relations (submodules of module direct sums) on finite-length modules, reducing them to bijective maps.

If this is right

  • Dynamical systems modeled by linear relations on finite length modules can be studied instead via bijective maps.
  • Invariants originally defined through relations become computable from the equivalent bijective mappings.
  • The uncertainty captured by relations is converted into deterministic behavior under shift equivalence.

Where Pith is reading between the lines

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

  • The equivalence may suggest similar reductions for other algebraic objects that model sampled dynamics.
  • Computational techniques developed for maps could now be applied directly to finite-length relations.
  • Testing the boundary of finite length could reveal where the equivalence breaks and new invariants appear.

Load-bearing premise

The modules under consideration have finite length.

What would settle it

Exhibit a linear relation on an infinite-length module that cannot be shown shift equivalent to any bijective mapping.

Figures

Figures reproduced from arXiv: 2503.10829 by Bartosz Furmanek, Filip Oskar {\L}anecki, Jim Wiseman, Mateusz Przybylski.

Figure 1
Figure 1. Figure 1: Linear relations over Z3 field with id0. Every box (color) shows one class of Szym-equivalent objects. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic depiction of the relation β considered in the Appendix of Proposition 3.2). In order to satisfy the formal constraints, we need to consider a linear relation. Let A be a free Z2-vector space with basis E and let α := ⟨β⟩ ⊆ A ⊕ A (A.1) be a subspace generated by the subset β. For k ∈ N let Nk be the set of natural numbers greater or equal than k. Lemma A.1. For α defined in (A.1) the following equ… view at source ↗
read the original abstract

Linear relations, defined as submodules of the direct sum of two modules, can be viewed as objects that carry dynamical information and reflect the inherent uncertainty of sampled dynamics. These objects also provide an algebraic structure that enables the definition of subtle invariants for dynamical systems. In this paper, we prove that linear relations defined on modules of finite length are shift equivalent to bijective mappings.

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 / 0 minor

Summary. The manuscript claims to prove that linear relations (defined as submodules of the direct sum of two modules) on modules of finite length are shift equivalent to bijective mappings. These relations are presented as carrying dynamical information and enabling subtle invariants for dynamical systems.

Significance. If correct, the result would establish an equivalence between linear relations on finite-length modules and bijective maps under shift equivalence, potentially simplifying the analysis of their dynamics and invariants. The finite-length hypothesis is explicitly required. However, with no derivations, definitions, or supporting arguments available for examination, the actual significance cannot be assessed.

major comments (1)
  1. [Abstract] Abstract (final sentence): the central theorem is asserted without any definitions of shift equivalence, linear relations beyond the one-sentence description, or a proof sketch. This prevents verification of whether the finite-length condition suffices for the claimed equivalence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the central theorem is asserted without any definitions of shift equivalence, linear relations beyond the one-sentence description, or a proof sketch. This prevents verification of whether the finite-length condition suffices for the claimed equivalence.

    Authors: The abstract is a concise summary of the result. The manuscript defines linear relations as submodules of the direct sum of two modules, introduces shift equivalence in the standard sense for relations, and provides a complete proof that any linear relation on a finite-length module is shift equivalent to a bijective map (using the finite-length hypothesis to guarantee the existence of suitable shifts and inverses). All supporting arguments appear in the body of the paper. revision: no

Circularity Check

0 steps flagged

No significant circularity; theorem proof is self-contained

full rationale

The paper states a theorem that linear relations on modules of finite length are shift equivalent to bijective mappings, with the finite length condition given explicitly as the setting. No equations, fitted parameters, self-citations, or ansatzes are present in the abstract or described claim that would reduce the result to its inputs by construction. As a pure existence/proof result in algebra, the derivation chain consists of standard mathematical arguments that do not invoke any of the enumerated circularity patterns. The result is presented as proved within the paper rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No full manuscript text is available, so free parameters, axioms, and invented entities cannot be extracted.

pith-pipeline@v0.9.0 · 5589 in / 880 out tokens · 24551 ms · 2026-05-22T23:48:53.200076+00:00 · methodology

discussion (0)

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Reference graph

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