What I’ve called the characteristic rate (modal rate) of travel or motion may be any rate independent of the travel mode, such as the minimum or maximum rate. The best-known example is the speed of light in a vacuum, c, which is generally considered the maximum speed for physics.
A characteristic rate that is a minimum speed may be appropriate in some contexts. A simple example is on expressways or freeways in which a minimum speed is posted. Another example is racing, such as automobile racing, in which a car going too slowly is black-flagged or pulled from the race.
Some have suggested that there is a minimum speed for physics (for example, here). I take the same approach with a minimum speed as with superluminal particles: the concept makes sense in a larger context, whether or not it is confirmed in physical science. There may also be reasons why the concept is valid for physics.
If there is a minimum speed, it transforms like the superluminal transformation, which has been shown here before:
r′ = γ (r − c² t/v), t′ = γ (t − r/v) with γ = (1 − c²/v²)–1/2 for |v| > |c|.
The maximum speed, c, is a ratio of travel distance and travel time: c = Δr/Δt. This may also be expressed as a pace Δt/Δr = 1/c. In the case of c, it is the ratio of a large travel distance and a small travel time, say rhigh and tlow. Convert these into rlow and thigh with the conversion factor, c. That is, rlow = ctlow and thigh = rhigh/c. Then the ratio of rlow and thigh must also be a constant independent of reference frame. That is:
rlow / thigh = ctlow / (rhigh/c) = c (tlow / rhigh)/c = c(1/c)/c = 1/c,
which equals the minimum speed independent of reference frame.
But what about a speed of zero, the state of rest? That would be smaller than 1/c. It may well be that for most purposes, 1/c ≈ 0. But physical phenomena at nanoscopic scales may show a minimum speed of 1/c. It could be called the speed of matter. Its existence is an empirical question.
Addendum: It would be arbitrary to derive the Lorentz transformation with time as the independent variable and not accept the equally valid dual Lorentz transformation derived with space as the independent variable. So, if there is a maximum speed, there is a maximum pace as well. It would not make sense if they were the same value.