the rates of future advancement of a technology if using the TRL step analogy, then that rate will become slower for technological concepts that are approaching their upper theoretical operating limits.
Indicators for Emerging Breakthroughs: History has repeatedly shown that the conditions are ripe for breakthrough technologies to emerge when the following two conditions have been met: 1) The prior technology is reaching the point of diminishing returns, and 2) Clues have emerged for how to seek those breakthroughs (that it's
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Millis 2018 Grant NNX17AE81G_for_CR.docx not just science fiction any more).

By "clues" it is meant that there are actionable and relevant research steps that can now be taken.
An arguable case in point is the emergence of the "Mach Effect Thruster" and other claims of propellantless propulsion. In this case the prior technology, rocketry, is at, or nearing, its point of diminishing returns. The degree of progress that can be gained for a given level of investment is far less now than in the early years of rocketry. The second criteria is the emergence of clues for alternatives. In this case, the correlation between the desired breakthrough of a propellantless thruster and specific unexplored topics of physics were identified and published spanning the mid-1990s through 2010 [13]. One of the relevant open questions in physics regards the origins and structure of inertial frames—clues for where to look [107].
Research on the open question about inertial frames led to the Mach Effect Thruster, now under test [85]. Consistent with historic patterns, the (unconfirmed) initial performance is not practical (~ 3 µN @ 30 W input), but this is only the first iteration. If the physics principles behind this device are found viable, then those new principles can be applied in other contexts leading to other advancements. And consistent with historical patterns if that verification is achieved, expect to hear common, but naive phrases like: "Yes, it might work, but it is not all that useful," and, "We don't need it for the mission." Mathematical Models: Mathematically, the "S-curve" is a "Logistic Function" following the form of Eq. 25. It should be noted that the x-axis is not time, but rather the level of resources devoted to the project.
⎞
y = L 1
⎠ ⎟ Eq. 26
⎝ ⎜
( ) ⎛
1+e−k x−x0
Where:
L = Upper asymptote, analogous to the physics limit of the concept k = Maximum rate of progress at the mid point of the concept's development
€ y = A measure of performance of the concept (Isp, thrust, w/kg, etc.) where the y=0 point is approximately the first demonstrated functionality of the device x-x0 = Level of resources expended In principle, the correlation of further advancements versus investment can be modeled by knowing the performance of the earliest embodiment of the method, along with the investment expended to reach that point, plus it's theoretical limit, and at least one other performance versus resources point.
For each key technology, it would be helpful to know the following: • Current performance value for TRL-9 technology, and the year that TRL threshold was reached • Next generation (TRL-6) performance value, and the year when that TRL threshold was reached • Predicted, new performance level currently in research, and its current TRL level. • Theoretical ultimate performance level