Dimensions of electric current

[Horace]

Gopi, user737

The goal is to prove that electric current has dimensions [S/T] to a normie who does not understand RST yet but understands dimensional analysis and math well.

In Larson's books I found only two thin mathematical arguments for the dimensions of current being S/T (speed):
The isomorphism of the E=½mv² and E=½Li² equations (kinetic energy vs energy stored in an inductance) and
The isomorphism of the E=mc² and E=i²Rt equations (mass-energy equivalence vs. joule heating).

However, this is not enough because the isomorphisms of products do not imply the isomorphisms of their factors !!! For example:
The general mathematical relation ab²=cd² does not necessarily mean that a=c and b=d. It could, of course, but there are infinite many other solutions to this equation where a≠c and b≠d and a,b,c,d are quantities of different dimensions.

So what is a solid mathematical argument that the dimensions of current are S/T (speed) if not these isomorphisms ?
...and I mean a "mathematical argument". Not an argument by analogy like the oscillations of the LC circuit vs. a spring-mass mechanical oscillator.

I heard that there is also a third mathematical argument involving the Faraday's constant but I could not find it and I cannot imagine how it would work since this constant merely relates the total charge of carriers to their number (the mole) and not to their mass. Notice that charge carriers can have different masses anyway, so the charge/mass ratio is not unambiguous.

Note that, the Heaviside-Lorentz unit system or ESU CGS system of units suggest different dimensions for current, which remain self-consistent despite being different from RST's.
Consider that in unit systems without a separate base unit for electric charge or current (such as the Gaussian or Heaviside-Lorentz unit system, commonly used in quantum physics contexts involving effects like the Josephson effect or kinetic inductance), electromagnetic quantities are reduced to mechanical dimensions of mass (M), space (S), and time (T).

The dimensional formula for the electric current in these systems is: M^½ S^³⁄₂ T^-2.
If we substitute the RST's dimension of mass (T³ S^-3) into that CGS dimensional formula, we obtain:
(T³ S^-3)^½ S^³⁄₂ T^-2
...and this simplifies to:
T^³⁄₂ S^-³⁄₂ S^³⁄₂ T^-2
T^-½ ...or ¹⁄√Time

The EMU is an example of yet another system of units that achieves self-consistency with the following dimensions of electric current:
Electric current: M^½ S^½ T^-1.
After substitution with the RST's dimension of mass (T³ S^-3):
(T³ S^-3)^½ S^½ T^-1
...and this simplifies to:
T^³⁄₂ S^-³⁄₂ S^½ T^-1
T^½ S^-1 ...or √Force

As you can see, the mere similarity between the equations E=½mv² and E=½Li² does not necessarily mean that i has the same dimensions as v and that m has the same dimensions as L.
Yeah I realize that the RST dimensions of mass might be incompatible with the dimensions of these Gaussian / CGS systems, but they seem to work out for the mechanical units like force, acceleration and kinetic energy.

OK, here it is again in the electrical world: P = IV (axiomatically) and V = IR (Ohm's law)
→ P = I²R
→ P = E/t = I²R
→ P = I²R
→ E = I²Rt
→ E =I²m (mass IS resistance for a period of time)
→ E = c²m (current IS motion at the speed of light, c)
→ E = mc²... looks familiar.

In furtherance of demonstration, note that when crossing the unit speed boundary speed (s/t) becomes 1/t²: substitute 1/t for s (as s/t = 1; this IS motion) and the result is 1/t² (equivalent speed).

In natural units that makes μεI² = 1 or μ₀ε₀ = 1/c²
Fully reduced (taking both 1 = c and I = c as current IS a speed of +1): μ = 1/ε

Doesn't get any more obvious than that.

That may be obvious to the disciples of the RST but it is not obvious to the normies who had not studied it and are stuck in the conventional aquarium of 3D space and 1D time and run screaming into the dark when encountering words like counterspace and unit speed boundary.

If I were to write to one of the normies that Rt = m or that i² and c² have the same dimensions just because the equations E = Rti² and E=mc² look similar then they would wipe the floor with me because of a logical fallacy stemming from the fact that isomorphism of the products does not necessarily mean the isomorphism of their corresponding factors !!! The same goes for the statement like "mass IS resistance for a period of time".
Mathematically, this happens because equations like E=ab² and E=cd² do not necessarily mean that a=b nor c=d ...nor that their dimensions are the same.
The symbolic relation ab² = cd² has many other solutions where a≠c and b≠d and a,b,c,d are quantities of different dimensions !

However, E = mi² and E = mc² collectively DO MEAN that i² and c² have the same dimensions because it is a relation of a different kind which can be symbolized by ab² = ac². Notice that this time, the same variable (m or a) appears on both sides of the equation.
Unfortunately, E = mi² cannot be constructed before proving that m=Rt.

[Horace]

The Ampere's Force Law F/L = 2k_A⋅i₁i₂/d can be algebraically transformed to i₁i₂ = Fd/(2k_A⋅L)
This is of no help either because it involves the dimensioned constant k_A = µ₀/4π.

Within its SI dimensions [MST^-2I^-2] , the µ₀ constant contains the Ampere^-2 which cancels out the currents i₁i₂ on the left side of the equation

Substituting the RST's mass dimensions (T³S^-3) into the SI definition of µ₀ yields the unhelpful [µ₀] = [TS^-2I^-2] ...or force per current² [F I^-2].

In order to prove that [i₁i₂] has the dimensions of [S²T^-2] ...or speed², the dimensions of the constants k_A and µ₀ would have to be [T³S^-4 ] ...or mass / length [MS^-1].

The proof that [F I^-2] = [MS^-1] eludes me, too.