## A Remark based on False Claim 1

So I was thinking that, the statistical description of a dynamical system (as one described by a generator Pasquali patch) really does give us a lot of power in computing the probable "position" of a particle (photon, electron) moving in space at different (integer) time intervals.  If the quantum mechanical supposition of time having a minimum discreteness (Planck-time) is correct, we can find the "finest" Pasquali patch generator that will give a complete description of the dynamical system.  Any Pasquali patch generator descriptive of the system which is not this "first" will generate an accurate, yet less refined ("coarser") version of the system (this is what we mean by Claim 3 of the previous post, in that such Pasquali patch will be "contained" in the finest description, yet is not the finest), and in fact either system of course converges to the same steady state (this is what is meant by Claim 4).  If we are able to find a continuous description (like the Shrodinger equation, via a "Pasqualian") of such system then we are in luck (this description would be the finest, though non-discrete, description), and I speculate though I cannot be sure yet that either discrete descriptions will be contained in such.

Whatever the description of the dynamical system via a generator Pasquali patch (or a Pasqualian), each Pasquali patch represents the transition (position) probabilities of a particle (photon, electron) moving within that system.  If we suppose that the particle moves with same velocity (take photons in vacuum as an example), then each Pasquali patch power is descriptive of the transition-position probability at equally spaced spacial or distance intervals.  Though I've remarked about this before, if it were the case that, for a particle with a particular (steady) velocity, the Pasquali patch power is not exactly equally spaced in distance intervals, it must mean that the arrow of time is bent (time is moving faster for smaller-spaced intervals, slower for longer-spaced).  We have not yet described accelerating particles but at present that is not of our interest.

However, we can tell if time is passing so long as each Pasquali patch description is different at each (equal or unequal) interval.  If a single Pasquali patch were to describe the system at EACH distance interval, there is no way to know if time is moving at all.  Take for example the False Claim 1 of the previous post.  We had the collection $\mathbb{Q} = \{Q^1, Q^2, \ldots, Q^k, \ldots\}_{k \in \mathbb{Z}^+}$ with $Q = q(x,y)$ being an explicit function of $y$ and converging to $Q^\infty = 1$.  We could track the time-distance interval via the Pasquali patch power, so that $1$ was the first distance interval from start (we take it as given that time is not being bent, so that a fixed distance implies the passage of 1 unit of time), $2$ was the second distance interval (2 units of time), and so on, and we could tell if time were bent if each power were descriptive of different distance intervals.  Furthermore, since each $Q \in \mathbb{Q}$ is different, this implies each position transition probability is different and the system is in movement.

This is definitely not the case with the collection $\mathbb{P} = \{P^1 = 1, P^2 = 1, \ldots, P^j, \ldots \}_{j \in \mathbb{Z}^+}$ which also converges to $P^\infty = 1$.  Since at each distance interval the movement probability is the same (uniform), one cannot be convinced that each power represents a distance interval equal to equally spaced time intervals or different-spaced time intervals.  Where we could with $\mathbb{Q}$ ascertain that time was moving, we cannot with $\mathbb{P}$.  The statistical description cannot tell if the system is frozen.

When a system has reached the steady state (which, is the highest entropy state!), there is no way to tell if time flows, as the statistical description is and forever will be unchanging.  Recall that the steady state for a Pasquali patch is always a function of $x$ alone, say $p(x)$.  Furthermore recall that any power of $p(x)$ is always $p(x)$ itself (see Compendium).  We reach an impasse: is time flowing normally, faster than what is conventional, slower? At such a point it is impossible to say, at least from the statistical point of view.  We would have to track particles individually in order to ascertain if they have deviated their path at all (it could be the case that they shifted to all positions with equal probability or in the shape of $p(x)$, e.g., but we cannot be sure of either situation).