How many signatures are required to start the formal decision making stage of a direct democracy mechanism?

Referendum Act (amended date: 2018/01/03)

Article 10. The number of proposers for the matters prescribed in Paragraph Two of Article 2 shall be not less than one-ten thousandth of the total electors in the most recent election of President and Vice President.

After receiving a proposal of referendum or a corrected proposal, the competent authority shall finish its examination within 30 days.

After the process of examination, the competent authority shall notify the leading proposer to make necessary corrections once within 30 days, specifying the reasons; if the necessary corrections are not made or still do not meet the relevant provisions, the proposal shall be rejected. […]

If the proposal referred to in the preceding Paragraph meets the relevant provisions, the leading proposer shall, within 10 days, receive a model list of joint signers or an authentication code of the electronic joint signing system from the competent authority to collect joint signatures. If the leading proposer does not receive the model list or an authentication code within the specified time limit, it will be regarded as abandonment of joint signatures.

Article 12. For the matters prescribed in Paragraph Two of Article 2, the number of proposers shall be not less than 1.5% of the total electors in the most recent election of President and Vice President.

The list of joint signers of a proposal of referendum shall be submitted by the leading proposer to the competent authority within 6 months commencing from the day after receiving the model list of joint signers or an authentication code of the electronic joint signing system; if the leading proposer does not submit it within the aforesaid time limit, it will be regarded as abandonment of joint signature.

The list of proposers referred to in the preceding Paragraph shall be completed column by column in the specified format with the signature or seal, the ID card number and the permanent address affixed and bound into books by municipality, county, and township (city/district). The original and a copy of the said list of proposers shall be submitted to the competent authority. […]

Civil Servants Election And Recall Act (amended date: 2018/05/09)

Article 81. The joint signers of a proposal of recall shall be the electors in the original electoral district where the person recalled has been elected, and the number of signers shall be not less than 10% of the total electors in the original electoral district. […]

The proposers may not be the joint signers of the same proposal of recall. The number of proposers and of joint signers shall be calculated separately.

Presidential and Vice Presidential Election and Recall Act (amended date: 2017/04/19)

Article 70. If a proposal of recall of the President or the Vice President is proposed by not less than 1/4 and agreed by the not less than 2/3 of the total members of the Legislative Yuan, the Legislative Yuan shall announce the proposal of recall is established. However, the President or the Vice President who has not been in the position for one year may not be recalled. […]

For the Citizens’ Initiative: in the first stage of submission of the proposal 0.01% of the total number of electors in the most recent presidential elections; in the second stage, after the proposal has been approved by the CEC, 1.5% of the total number of electors. For the General Recall Initiative: the number of signatures shall be not less than 10% of the total electors in the original electoral district. For the Recall of President or Vice-president: no signatures by electors are required to start the formal decision making stage of a recall. Recall process in this case is proposed and initiated by the Legislative Yuan.