SELLER will deregister the ownership from the bicycle registration for anti-theft purposes, and obtain 抹消登録書 (Proof of deregistration) or will fill out a letter of attorney for the deregistration of the bicycle ownership.
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BUYER needs 抹消登録 (Proof of deregistration) or a letter of attorney for the deregistration of the bicycle ownership, and 自転車譲渡証明書 (Transfer certificate for the bicycle ownership). And will register new ownership with the above documents at a bike shop. It costs 600 yen for a new registration. (From 2024, December, 1st, it costs 800 yen.)
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Also please be aware that it has been mandatory for bike riders to enroll in bike insurance in Osaka since July 1st, 2016.
You can buy it online. Here is a relatively reasonable insurance:
In the past, many JETs wanted to bring their bikes (ママチャリ) back home but they have abandoned them due to insane shipping costs.
The last possible option might be https://www.bikeflights.com/
You need to prepare a huge cardboard box disassemble your bike, and then pack them by yourself. Hopefully, this service is reasonable. BikeFlights.com is a US-born service so the website is all in English.
You need to go to Hankyu Toyonaka/Osaka-Umeda station to do this. Your refund claim form, your ID, and a handling fee of 220 yen are required at the counter. They are open 7:30-19:30 on weekdays and 9:00-17:00 on weekends.