Transforming to **S'**:

$$\vec{T1'}(t',l) = \begin{pmatrix} \frac{r}{\gamma} - vt' \\ l + r\omega(\frac{t'}{\gamma} +\frac{vr}{c^2})\end{pmatrix}$$

$$\hat{P_t}'(t'=t'_0)$$ is parallel to the displacement vector between two points on the rod at $$t'=t'_0$$ with different values of $$l$$:

$$\begin{align}

\vec{T_1}'(t'=t'_0,l=l_0) &= \begin{pmatrix} \frac{r}{\gamma} - vt'_0 \\ l_0 + r\omega(\frac{t'_0}{\gamma} +\frac{vr}{c^2})\end{pmatrix} \\

\vec{T_1}'(t'=t'_0,l=l_1) &= \begin{pmatrix} \frac{r}{\gamma} - vt'_0 \\ l_1 + r\omega(\frac{t'_0}{\gamma} +\frac{vr}{c^2})\end{pmatrix} \\

\hat{P_t}'(t'=t'_0) &= \frac{\vec{T_1}'(t'=t'_0,l=l_0) - \vec{T_1}'(t'=t'_0,l=l_1)}{\left\|\vec{T_1}'(t'=t'_0,l=l_0) - \vec{T_1}'(t'=t'_0,l=l_1)\right\|

\end{align}$$

$$ \hat{P_t}'(t'=t'_0) = \begin{pmatrix}0 \\ 1 \end{pmatrix}$$

OK,

1. So, the recurring issue with your approach is the gratuitous insistence in marking both endpoints of the tangent vector at the same time: $$t=0$$ in S and $$t'=t'_0$$ in S'. There is no justification for this, when you

let go of this gratuitous condition, $$\vec{v_P}$$ and $$\vec{T_1}$$ are parallel in all frames.

2. The second issue is an outright error, if you mark the endpoints of the vectors simultaneously in S (at $$t=0$$), you will be marking the endpoints of the transformed vectors at $$t'=t'_0$$ and $$t'=t'_0+\Delta t'$$ respectively. Zero time intervals in frame S transform in non-zero time intervals ($$\Delta t'$$) in frame S'. If you do this correctly you find out that the x component of $$ \hat{P_t}'(t'=t'_0) $$ isn't zero but $$-v \Delta t'$$

While I pointed out the first issue with your solution repeatedly , I have been remiss in pointing out the second issue.

Though we did not agree on the final outcome, the debate was useful in:

-setting up the milestones and solving the issues one by one (bar the one that constitutes the subject of the debate)

-explaining how displacement vectors are differentiated, how partial derivatives work (sadly, there are STILL a lot of errors in your attempts at calculating $$\frac{\partial x'}{\partial \theta}$$, etc.

-giving the opportunity to explain the solution in multiple ways , using different formalisms (polar coordinates, vector algebra, experimental methods), all backed up with a significant amount of math each time

What I was hoping for was for the discussion to be indeed about learning rather than degenerate into insults in the end. Unfortunately, this is exactly what you ended up doing in the end. Your replacing logical arguments with insults is a clear sign that you lost control and, with this, you lost the argument.