'Fat' steel braided brake pipe

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We're going with steel braided PTFE lines for the ol' Triumph (the old lines would do a fair impersonation of balloon animals at this point) so I ordered an assortment of AN -3 fittings. Unsurprisingly they're meant for 3/16" ID and 1/4" OD pipe

I thought I'd be clever and get covered steel braided pipe as bare steel lines will nibble their way through paint and soft alloys that they rub up against. Unfortunately the stuff I got has an OD of ~7.5mm which is way too fat for standard AN -3 fittings (the ID is the correct 3/16" BTW.)

My bad for not reading the description and I've ordered correctly diameter pipe but what is this 'fat' pipe used for? Are there such things as AN -3 fittings that accept 5/16" OD pipes?
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Royal Enfield Continental GT 535, Husqvarna Vitpilen 401, Yamaha XSR700, Honda Rebel, Yamaha DT175, Suzuki SV650 (loan) Fazer 600, Keeway Superlight 125, 50cc turd scooter