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If the 'bridge chip' is just a reasonably generic SATA port multiplier
A transparent bridging scenario such as this would not make much sense as an SSD can easily saturate the available bandwidth of a 6 Gb/s SATA port. A non-transparent bridging scheme which makes the device look more like a traditional spinning HD with a really big cache would make more sense as the SATA port driver would not have to route every sector write/fetch to specific device. Most likely the driver on the OS side only manages migration of data between the SSD and HD to balance read/write performance favoring keeping as much free space on the SSD as possible to enhance write performance.