The distinction between 56k and slower speeds was that 56k could not be transmitted over the digital part of the POTS network in its modem audio format. Telepho

The distinction between 56k and slower speeds was that 56k could not be transmitted over the digital part of the POTS network in its modem audio format. Telephone systems are generally analogue between the house and the exchange (or road-side cabinet), and then digital between that and the main telephone network. The digital encoding was 8000 samples per second, at eight bits per sample, with a logarithmic transfer function in order to reduce noise levels when the line is quiet. So, the digital part of a telephone line ran at 64kb/s. In order to support 56k speeds, the exchange (or road-side box) had to be upgraded. Normal exchanges used a simple 8-bit DAC and ADC, but when a 56k modem was attached it would have to switch to a different mode. Effectively, the remote modem was moved from the ISP to the exchange, with the digital part of the POTS network transmitting de-modem-ed data, rather than a digital representation of an analogue signal encoding digital data, which is what happens with slower speeds. So yes, the 56k signal was transmitted over an analogue line, but no the 56k signal was not transmitted over the main POTS network. Work beyond 56k was not pursued because the underlying digital telephone network could not actually transmit data at a higher rate. This is also where the 64kb/s speed of an ADSL line comes from by the way. That just extends the digital part of the POTS network to the house. ADSL allows a slightly higher data rate (64kb/s rather than 56kb/s) because it has a side channel that carries the control signals such as dialling and flow control, whereas 56k modems had to send flow control and error correction in-band.