Comment to Comment to Black Hole in Dehnen $\left(1,4,\frac{1}{2}\right)$ Dark Matter Halo: Exact Solution, Lensing, Light Ring, and Thermodynamics

TL;DR

This paper defends the validity of a previous exact solution related to black holes in a specific dark matter halo, clarifying misunderstandings and reaffirming the correctness of the original analysis.

Contribution

It provides a detailed clarification and correction of a critique regarding an exact black hole solution in a Dehnen dark matter halo.

Findings

The original solution remains valid despite claimed errors.

The critique claiming foundational errors is unfounded.

The paper clarifies the typographical nature of the alleged errors.

Abstract

The claim in \cite{Al-Badawi:2025ipr} that *"the errors in the foundational components (3) and (5) of Ref. [1] invalidate all subsequent analyses, numerical results, and physical interpretations that depend on them"* is **entirely unfounded**. This statement reflects a fundamental misunderstanding of the typographical nature of the error and appears to be a misplaced critique originating from a competing author, rather than a substantive assessment of the results, which remain fully valid.