Informally, understanding an experiment $E$ means introducing coordinates into phase space of $E$ which are in triangular form under the action of the inputs of $E$.

- The parameter $Q(t)$ determines some obviously important aspects of the system. That is, there is a deterministic function $M(Q(t))$ which maps $Q(t)$ to "measure" some internal state of the system.
- If the values of such a parameter $Q$ is known at time $t_0$ (denoted $Q(t_0)$) and it is also known what inputs are presented to the system from time $t$ to time $t + \epsilon$(denoted $I[t_0, t_0 + \epsilon]$), then the new value of $Q$ is a deterministic function of $Q(t_0)$ and $I[t_0, t_0+ \epsilon]$.

In general, conservation principles give a first coordinate of a triangularization. In the main a large part of physics can be viewed as discovering and introducing functions $e$ of the states $q$ of the system such that under action $a$, $e(a \curvearrowright q)$ depends only on $e(q)$ and $a$, andon $q$.not

- For semidirect products, I refer you to the cutest way to write semidirect products Line of investigation to build physical intuition for semidirect products .

We first heuristically indicate the construction involved in going from the group of symmetries to the triangularization, and then precisely write it out in all pedantic detail.Let an experiment be $E \equiv (X, A, \curvearrowright)$. Then we define $\Pi$ is a

- $\Pi: X \rightarrow X$ is a permutation of $X$.
- $\Pi$ commutes with the action of each $a$: $\Pi(a \curvearrowright x) = a \curvearrowright \Pi(x)$.

- We know that the symmetries of a theory $E$ form a group.
- If $E$ is transitive, then each symmetry $\Pi$ is a regular permutation --- If there exists an $x$ such that $\Pi(x_f) = x_f$ (a fixed point), then this implies that $\Pi(x) = x$ for
*all*$x$. - Let the action split $X$ into disjoint orbits $O_1, O_2, \dots O_k$ from whom we choose representatives $x_1 \in O_1, x_2 \in O_2, \dots x_k \in O_k$. Then, if $E$ is transitive, there is
*exactly one*action that sends a particular $x_i$ to a particular $x_j$. So, on fixing*one component*of an action, we fix*all components*.

$(X, S) \leq (G, G) \wr (\{ O_1, O_2, \dots O_k\}, T)$