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Consider the following table regarding the functions of each wire in 1000BASE-T ethernet (source):

table with T568 pinout and their uses in gigabit ethernet

For every differential pair except pair C (the blue one, in the center), the white wire is the positive polarity one. Is there any reason for this exception? It seems to me that the standard went out of its way to avoid the "white = positive" convention here, which is also how the RJ25 standard does things.

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There is no obvious reason.

As it is now, all odd-numbered pins are the white-striped wires and all even-numbered pins are the solid colored wires. That comes from the wiring standard, not Ethernet. It makes sure the wire polarities alternate, even if the pairs are not organized so that two wires of a pair would be next to each other.

Also the Ethernet standard makes sense, with increasing pin numbering, always the positive wire of a pair comes first, no matter how the pairs are organized.

Finally, it does not matter much as all gigabit PHYs are supposed to automatically detect and correct channel polarity.

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