How do I handle an $v=u$ relation in propositional modal logic? (Garson p. 102)$^\eqref{reff-1}$, in a brief discussion about the linear view of time, introduces us to the Connectedness Axiom (L): $\square(\square A\to B)\lor\square((B \land \square B)\to A).$ (L) has the following conditions for the relation, $R$. If $Rwu$ and $Rwv$, then $Ruv$ or $Rvu$ or $v=u$.
When checking the validity of (L) using a Truth Tree/Semantic Tableaux, I'm assuming we need to test $Ruv$, $Rvu$, and $v=u$ (Garson doesn't go into checking the validity of (L)).
$Rvu$ and $Ruv$ proceed as expected under the usual rules. $v=u$ also works if, instead of creating different worlds for $\lnot\square(\square A\to B)$ and $\lnot\square((B \land \square B)\to A)$ as per the $\square F$ rule, we send everything to one world (we end up getting $\lnot B$ from the first disjunct and $B$ from the conjunction in the second disjunct). In fact, under the normal tree rules, the $\color{red}{B}$ in $(\color{red}{B}\land\square B)$ seems to be unnecessary.
Questions

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*Is my approach for $v=u$ correct? Is it simply a case of sending formulas to the same world instead of separate worlds?


*Formulas that are valid under the normal rules aren't necessarily valid if $v=u$ is an option - $\square(\square A\to B)\lor\square(\square B\to A)$ is an example. What's a good source to read more about this phenomenon?
$\text{Garson, J. W. (2013). Modal logic for philosophers. Cambridge Univ. Press.} \tag{1}\label{reff-1}$
 A: This connectedness axiom only makes sense in certain connected (and possibly transitive) frames of tense logic for example, and the relation $R$ above usually means earlier than or later than with the usual set $W$ means a set of possible time instants (not worlds). It mainly describes non tree-like relations which is typical in tense/temporal logic, but not required in other alethic or deontic logics. The above frame condition of axiom L just demands whenever there's a possible fork from $w$ to $u$ and $v$, $u$ and $v$ are still under same temporal earlier or later or simultaneous relation. If you want a good source for tense logic, you can refer to its SEP first, and perhaps some related papers containing connectedness axiom like Hodkinson's paper.
Under tense logic which is a K4 system with transitivity, it's very easy to use $\square F$ rule to verify the above axiom L is 4-valid. And your extra highlighted $B$ seems not redundant since otherwise $A$ will be either strictly earlier or later than $B$. Once you add extra $B$, then $A$ and $B$ might be simultaneous since normally in tense logic $\square$ (G) strictly applies to future and thus $R$ is asymmetric.
