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I ordered the battery replacement kit for the iPhone 5. The replacement battery has a larger connector than the original connector. The replacement battery connector has four (4) contacts. While the original has only three (3). What is the next step???

The battery you received is probably for the iPhone 5. That connector has 4 central contacts whereas the connector for the iPhone 5c/5S and SE have 3 contacts. Check your order form, you either incorrectly ordered an iPhone 5 battery or you received the wrong one. Contact your vendor to see how they can help you.