How it becomes dimensionless matters.

Understanding Dimensionless ratios using Pi and Beta as examples 

--- Pi (pi) --- 

* Origin: Ratio of Circumference (Length) to Diameter (Length). 

* Calculation: pi = Circumference / Diameter 

* Dimensionlessness: Specific length units (like meters/meters or inches/inches) cancel out numerically. pi is approximately 3.14159... 

* Persistent Nature: Even though numerically dimensionless, pi inherently represents a comparison between two lengths. 

* Generic "Unit" / Character: It embodies a [Generic L/L Ratio], reflecting a fundamental geometric property independent of any specific measurement system. This just means that this L/L proportion is true independent of any definition of length.

--- Beta (beta) --- 

* Origin: Ratio of a specific velocity 'v' (Length/Time) to the fundamental Length/Time=1 proportion. 

* Calculation: beta = v / c * Dimensionlessness: Specific speed units (like m/s over m/s or PlanckLength/PlanckTime over PlanckLength/PlanckTime) cancel out numerically. Beta is a number between 0 and 1. Where c is just a unit conversion factor.

* Persistent Nature: Beta still fundamentally describes a rate of Length traversal per unit of Time. It doesn't become conceptually detached from L/T just because the unit symbols cancel. 

* Connection to L~T Equivalence: Beta represents a specific fraction of the universe's fundamental L~T equivalence rate, which is converted to a specific velocity in SI by the unit conversion factor 'c'. 

* The numerical value c_SI is purely an artifact of how the SI meter is defined relative to Natures L/T = 1 proportion by fixing a time unit and defining a length unit against that second. It is the conversion factor embedded in our length unit definition. It is defined against Nature's L/T=1 proportion, the actual "speed of light."

* Meaning of Beta's Value: A value 'beta = X' means that for any chosen time unit, the velocity 'v' covers X times the distance that light would cover in that same time unit. 

* Beta = 1: This represents the maximum possible rate, achieving the universe's fundamental L~T rate (1 natural unit of length per 1 natural unit of time. When we assume L/T=1 then that is the maximum rate of change in the universe. 

* Generic "Unit" / Character: Beta embodies a [Generic L/T Ratio relative to Nature's L/T Proportion]. It's a universal, unit-system-independent measure of motion relative to the inherent spacetime structure of space and time.

--- Summary --- 

* Pi is dimensionless because it compares Length to Length. It retains a [Generic L/L Ratio] character. 

* Beta is dimensionless because it compares a specific L/T rate (v) to the fundamental L/T=1 rate (c). It retains a [Generic L/T Ratio relative to L/T=1] character. 

* For Beta, its numerical value is independent of the unit system, but its meaning is always tied to the fraction of the fundamental L~T rate it represents.