Corrigendum for Han's Corrigendum

Imagine a digital world made of tiny, blocky pixels, like a giant grid of Legos. Mathematicians who study this world (called "digital topology") need special rules to describe how these blocks connect and how you can travel from one to another. One of the most important tools they use is something called a "covering map."

Think of a covering map like this. Imagine a building with many floors that all look identical. The ground floor is the 'base'. A covering map takes a copy of each floor and lays it onto the base. Because every floor is identical to the base, this map makes no change in what the base looks like.

Uninformed readers might wonder why this matters, but topologists know covering maps are useful — for instance, they help compute the FUNDAMENTAL GROUP of an object, which is an important characteristic of its form (think of it as a fingerprint of the object's shape).

The Problem: A "Fix" That Broke Things

A researcher named S.-E. Han published a paper claiming to fix errors in his earlier work about these "covering maps." He called his new paper a "Corrigendum" (which just means "a correction").

However, Laurence Boxer, the author of the paper you provided, argues that Han's "correction" is actually full of new mistakes, copied ideas, and bad citations. It's like someone trying to fix a leaky roof by nailing a new, broken piece of wood on top of it, while also claiming they invented the wood.

Here is a breakdown of Boxer's main complaints, using simple analogies:

1. The "Fake" New Invention

Han tried to invent a new type of map called a "pseudo-covering." Han thought he had a new kind of map to make the floors and the base "almost" alike.

Boxer's Finding: Han's "new" system wasn't new at all. It was actually just the original, standard floor mapping he had already described years ago.
The Analogy: It's like someone inventing a "new" type of bicycle that turns out to be exactly the same as a regular bicycle, but they give it a fancy new name and claim it's a breakthrough.
The Twist: Han later realized his first definition was useless. So, in his "correction" paper, he quietly adopted a different definition that had already been invented by another researcher named Pakdaman. Han didn't give Pakdaman credit, acting as if he came up with it himself.

2. The Broken Proof

In his "correction" paper, Han tried to prove a specific mathematical fact about how these maps work.

Boxer's Finding: Han's logic was wrong. He tried to prove that a certain pattern doesn't exist, but Boxer showed that the pattern does exist and that Han's proof was flawed.
The Analogy: Imagine Han claiming, "I proved that you cannot walk in a circle in this park." Boxer walks in a circle, points at Han, and says, "I just walked in a circle. Your proof that you can't is wrong."
The Silver Lining: Boxer admits that Han's conclusion (the final answer) was actually correct, even though the proof (the steps to get there) was garbage. Boxer then provides a clean, correct proof to fix the mess.

3. The Citation Scandal

Boxer points out that Han is very bad at giving credit where it's due.

The Issue: Han often cites his own papers incorrectly or fails to mention that he is using ideas from other people (or even his own earlier work) without saying so.
The Analogy: It's like a chef writing a cookbook, claiming they invented a famous soup, but actually copying the recipe from a neighbor's book and forgetting to write the neighbor's name on the page.

4. The "Local Publication" Issue

Boxer highlights a particularly shady paper by Han published in a journal that isn't really a real international journal, but rather a local newsletter from Han's own university.

The Issue: This paper was largely plagiarized (copied) from other famous mathematicians. Han later promised not to cite his plagiarizing paper again, but has done so often since giving this pledge.
The Analogy: It's like a student promising not to cheat on a test, then copying the answers from the teacher's answer key and submitting it as their own work, all while hiding the fact that they did it in a secret, unmonitored classroom.

The Bottom Line

Boxer isn't saying that Han's ideas are useless. In fact, Boxer admits Han has contributed some great ideas to the field (like the concept of covering maps and measuring shortest paths).

However, Boxer argues that Han's recent work is mathematically sloppy and ethically questionable.

The Message: When Han submits a paper, it shouldn't be rejected immediately just because it's by him. But it needs to be checked extra carefully by experts who know the history of the field, because Han has a habit of making mistakes, copying others, and failing to fix his own errors properly.

In short: Han tried to fix his mistakes, but he made more mistakes, stole credit, and wrote bad proofs. Boxer is here to clean up the mess and set the record straight.

Technical Summary: Corrigendum for Han's Corrigendum

Problem Statement
This paper addresses significant flaws in S.-E. Han's 2026 paper, "Remarks on Pseudocovering Spaces in a Digital Topological Setting: A Corrigendum" (referenced as [19]). While Han's work was intended to correct errors in his previous publications ([16, 18]) regarding variants of covering spaces in digital topology, the current author, Laurence Boxer, demonstrates that Han's "Corrigendum" itself contains mathematical errors, presents unoriginal content without proper attribution, and includes improper citations. Specifically, the paper investigates the validity of Han's definitions of "pseudo-covering maps" and "local isomorphisms," arguing that Han's corrections fail to resolve foundational issues and, in some cases, introduce new inaccuracies.

Methodology
The paper employs a rigorous analytical approach grounded in digital topology, utilizing definitions and theorems from established literature (specifically citing works such as [6], [3], [14], and [21]). The methodology involves:

Critical Review of Definitions: Comparing Han's definitions of digital covering maps, pseudo-covering maps, and local isomorphisms against previously established, equivalent definitions found in the literature.
Counter-Example Construction: Constructing specific digital images and mappings to test the validity of assertions made in Han's [19]. The author utilizes a mapping $p: \mathbb{N}^* \to C_n$ (where $C_n$ is a digital simple closed curve) to disprove specific claims regarding neighborhood pre-images.
Logical Deduction: Applying known theorems (e.g., Theorem 3.4 from [6]) to demonstrate equivalences between different types of digital maps, thereby showing that Han's proposed "new" objects often coincide with existing, well-understood concepts.
Citation Audit: Examining the bibliographic references in Han's work to identify instances where original sources were omitted or misattributed.

Key Contributions and Results
The paper identifies and rectifies three primary categories of flaws in Han's [19]:

Mathematical Correction: The author disproves "Assertion 4.1" from Han's [19], which claimed that for a specific mapping, the union of neighborhoods of pre-images does not equal the pre-image of the neighborhood. Through a detailed case analysis (Proposition 4.2), the author proves that the assertion is false and that the equality holds. Consequently, a corollary derived by Han from this false assertion is shown to be unproven; the author provides a correct proof for the corollary using established theorems.
Clarification of Equivalences: The paper reinforces the result (originally shown by Pakdaman in [21]) that Han's original definition of a "digital pseudo-covering map" (Definition 3.1) is mathematically equivalent to a standard digital covering map. The paper further clarifies that the only definition of a pseudo-covering that yields a distinct object is the one proposed by Pakdaman (Definition 3.5), which lacks the unique path lifting property.
Attribution and Ethics: The paper highlights that Han's [19] presents Pakdaman's alternative definition of a pseudo-covering map (Definition 3.5) as a new contribution without attribution, despite acknowledging Pakdaman's existence in the text. Furthermore, the paper notes that Han's definition of digital covering maps (Definition 2.3) is attributed to Han's own papers in [19], ignoring its actual origin in works by Boxer and others ([3, 14]).

Significance and Claims
The paper claims that Han's "Corrigendum" is fundamentally flawed and requires its own correction. The author asserts that Han's work in this area is characterized by:

Mathematical Errors: Incorrect proofs and assertions.
Lack of Originality: The introduction of variants that are either trivial or identical to existing concepts, presented without credit to the original authors.
Citation Improprieties: Failure to cite the true origins of definitions and, in broader context, a pattern of unethical citation practices across multiple papers (including [10], [11], [13], [15], [20]).

The paper concludes that while Han has contributed useful ideas to digital topology (such as covering maps and shortest-path metrics), his submissions require the most rigorous review by experts familiar with the relevant literature to prevent the publication of "mathematical trash" and to ensure ethical standards in citation and originality are maintained. The work serves as a necessary correction to the literature, ensuring that the definitions and theorems regarding digital covering spaces are accurately attributed and mathematically sound.