I am not doing this in the manner you are Alphanumeric. I am using $$ij$$ notation as markers. They are not computing anything in any manner you are relating this to.
So no. It doesn't put a fork in anything.
Doesn't fly. You use the i,j notation as indices on the $$\sigma_{ij}$$ Pauli matrix object so you're using $$G_{ij}$$ as an array of values and they work by matrix algebra. My point stands, you haven't even done basic matrix algebra properly. Plus the index notation is still nonsense and inconsistent. It's known as the '3 eyed monster', where you repeat the same index more than twice and thus make the expression ambiguous and thus nonsense.
If the notation has put you off AN, stop thinking I am doing anything in the way of tensor calculus. They were there on the variables where the strength between the two particles becomes significant. They are doing nothing more than reminding the reader... they are not performing any kind of influence on the [basic] algebra being performed.
If that's your excuse then you use the same notation in different ways in the same sequence of expressions without explaining yourself. Furthermore, despite the fact the OP openned with default bookwork for half of it your excuse means you immediately jump into a non-standard use of a standard notation, right down to standard objects like $$\sigma_{ij}$$, without explaining yourself. That's a big no no.
Besides, it's not like your equations go anywhere. In some cases all you're doing it stating identities, akin to saying if A = XYZ then BA = BXYZ. That's all you're doing with the $$\theta_{ij}$$, $$\hat{n}\cdot \sigma_{ij}$$ objects, just writing them in terms of one another and then expanding out the second term as its matrix expression. Even if you hadn't botched all the algebraic formalism and managed to explain your notation you still accomplish nothing, It's your standard thing, you spew out a bunch of equations which amount to spinning your wheels and say "Look, I've got
Gravigyro-Magnetic Equations with the Angle Between Spin States and a Force Equation ! I is a well good physicist!". No, you're just a dishonest hack who thinks if he copies enough identifies and expressions from other sources, jam them together and perhaps changes a symbol here and there to try to hide the fact he's copying things then perhaps it'll magically convince people he's not wasting his time.
Other people have weighed into the thread, with similar less than stellar comments about you. No one buys what you're doing. If you were making an
honest effort to learn but were just thick as 2 short planks then people would have time for you but instead you're being dishonest as well as being not too bright so people get sick of it. I don't have to lobby James or anyone else to try to convince them you're dishonest, it's clear to any reader from our 'little exchanges' you're dishonest. There's no conspiracy against you, people reach their conclusions about you all by themselves. I chime in to make sure you don't sucker in any new people because clearly when you're left to your own devices you'll pile BS on BS, just as you did with your equation which you retroactively said "It's just a drunken evenings nonsense, don't pay any attention to it". You then went on to post half a dozen times about it, until you were called on it again. You're actively trying to deceive people and I'll point that out. If you put me on ignore I'll continue to point it out, you'll just have no chance to defend yourself. Though saying that even when I give you a chance you can't defend yourself or ever step up.
How about you give yourself a self imposed holiday for a fortnight? That way you can let this thread drop down the forum a bit and then you can start afresh with
another thread of you spewing nonsense about some 'result' you've come up with using remedial flawed algebra. It'll at least forgo you having a holiday imposed on you. Go on, see if you've got that self control, see if you can go a whole fortnight without telling a blatantly obvious lie about your knowledge/understanding/'work'.