Reimagined Efficiency (E):

E = (Desired Outcomes + Insights from Waste/By-products) / Total Resources Expended

This formula acknowledges that efficiency is not just about direct outcomes or minimizing waste, but also about leveraging all aspects of the process, including waste, to gain insights, improve, and innovate.

Adaptive Thresholding (T):

T(x, C) = Base Threshold + Adjustment Factor(C)

Where x is the variable under consideration, C represents the context or conditions, and Adjustment Factor(C) is a dynamic factor that adjusts the threshold based on the current context.

Innovation Detection through Weak Signal Amplification (I):

I(x) = { 1 if Amplified Signal(x) > Innovation Threshold, 0 otherwise }

Here, I(x) indicates the detection of innovation when an initially weak signal x, once amplified, crosses a predefined threshold of significance, suggesting a potential innovative insight or trend.

Collaborative Enhancement (CRAFT):

Collaborative Gain = Σ (Incremental Improvements from Collaboration_i) from i=1 to n

This equation represents the aggregated value generated through collaborative efforts, where each iteration i contributes incremental improvements to the solution or understanding of a problem.