Thank you for the reply. I will ignore the insults and focus on the mathematics.

First, please note that you are responding to a machine translation of my English comments. If some Japanese phrasing sounds unnatural, that is a translation artifact and not the substance of my critique. My actual points are the mathematical ones below.

## 1) “Without loss of generality” and gcd assumptions

My point is not to claim (as a strawman) that you explicitly wrote “(gcd(x,z)=1)” as an assumption from the start. The point is narrower:

- In several places, arguments about “no common prime factors” and later divisibility claims only go through if certain gcd conditions hold (sometimes effectively pairwise).
- If your only reduction is the standard *primitive* one, (gcd(x,y,z)=1), then any step that uses stronger coprimality must be *derived* and the derivation must be stated.

So the concrete request is: **please cite the exact line(s) where you justify every coprimality claim you use** (e.g., “(x) and (z) are relatively prime”, or any equivalent statement used to conclude “there are no common prime factors of …”). If you believe you never use anything beyond (gcd(x,y,z)=1), then we can resolve this by pointing to the specific passage and checking what is actually required there.