How does gross weight affect the Minimum Controllable Airspeed (Vmc)? I am about to get my MEL added to my CFI, and I cannot get my head around the multiple explanations for how gross weight affects Vmc.
The accepted answer is that The higher the gross weight, the lower is the Vmc.
I have been proffered two contradictory explanations, from both internet articles, manufacturer documents, and other CFIs, both of which seem to me (I have an MSME masters degree in Aero) to be flawed.

That because the aircraft is slightly banked into the direction of the good engine, the horizontal component of Lift (HCL) opposes the yaw from the rudder, thus allowing the aircraft to slow to a lower airspeed before full rudder authority is required to counteract the yaw from the operating engine.
This is wrong, because if you're in a bank, the HCL (by definition, it is HORIZONTAL!), is no longer lined up with the lateral axis of the aircraft. It is misaligned by exactly the bank angle, so, although there is a component of the HCL opposing the yaw force from the rudder, there is an exactly equal and compensating component of the Vertical Component of Lift (VCL) that will be augmenting the Yaw force.  This argument is bogus.


I mean if you think about it simpler, the Lift vector is always, ALWAYS, perpendicular to the wings, so that any effect of Lift along the lateral axis (parallel with yaw forces), must necessarily be zero.
The fact that is not aligned with the earth horizon is irrelevant.

The other rationale I have been presented with is that as the gross weight increases, the resistance to motion (Engineers would call this rotational inertia) increases, and makes the aircraft more stable. This is true, but it is an argument about dynamic stability, not about static stability. In other words, this affects the aircrafts resistance to changes in yaw/sideslip angle. Vmc is about static stability, i.e., at what airspeed does the aircraft, in a static, (unchanging) zero sideslip angle, require full rudder defection to hold the aircraft (STATICALLY) at that zero sideslip angle against all the yaw forces being produced by all the factors resulting from Asymmetric thrust?
So this argument or rationale also seems to be incorrect to me.


Where is the flaw in this reasoning?
By the way, the only effect of Gross weight on Vmc that I can think of is the obvious one, The higher the gross weight, the greater the Angle of Attack required to hold 1 G (Level flight), and obviously, both P-Factor and adverse yaw increases with AOA. So the higher the Gross weight, the greater the P-Factor, and the greater the yaw-induced aerodynamic effects of the asymmetric thrust. So, if this logic is correct, Higher Gross weight means Higher Vmc, not Lower.
I think your objection to #2 is exactly correct, and your #1 is where the misunderstanding may lie.

The point of VMCA is essentially to maintain a constant heading, and when you have no more rudder authority to make that happen, you're there -- and any slower, with slightly less rudder authority, your heading starts to change in the direction of the dead engine.  With no bank, it's all rudder; the lift from the wings doesn't tend to move the nose in any direction.  With bank, there is now a component of the lift from the wings that does tend to affect heading (which is the dynamic of banking when you turn -- the heading change comes from the lift created by the wings, not from the lift generated by the rudder).  

If you are banking into the good engine, that component of lift tends to change your heading toward the good engine and away from the failed engine.  Since the asymmetric thrust is working to change your heading toward the failed engine while you hold rudder to oppose that change, you now have the component of lift from the wings that is assisting your rudder.

If you consider a case of symmetric thrust, you can hold a heading with right wing down and left rudder or vice versa; the lift from the wings is working to change your heading to the right, and your rudder is counteracting that drift. At some point, with enough bank, you run out of rudder authority &amp; the airplane will turn, although pretty badly uncoordinated.  With an engine out, you CAN fly straight ahead holding bank into the dead engine, you just need a lot of rudder authority (i.e. lots of speed above VMCA) to hold your heading -- the rudder is fighting both the asymmetric thrust and the component of the wings' lift. And doing that, you'll run out of rudder authority a lot sooner (i.e. at a higher airspeed).  If you switch to bank into the good engine, the component of lift from the wings is now working with you, and that's the usual case for computing and demonstrating VMCA.

I suspect that the confusion on your point #1 comes by conflating horizontal (earth reference) with horizontal (aircraft reference). Hopefully the explanation above separates things out?

And, to bring things back to gross weight, the higher the weight, the greater the lift from the wings, so a Sin(5 degrees) (I think that's right -- LONG time since I took Trig!) component of "more lift" is a greater force to resist heading change than that same component of "less lift" with a lighter aircraft.  The other forces involved in the balance, the asymmetric thrust and the rudder force, are independent of aircraft weight.

Best wishes for your MEL checkride!