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SECTIONS
Entropy’s Great...
Oversights
The Second Law of Thermodynamics
Oh, My Ergodic Hypothesis...
Brownian Motion,
“Brownian Entropy” and “Entropo-dynamics”
The “Arrow of Time”
Entropy as
“Disorder”
A Gedanken Experiment
Thermodynamic Equilibrium
“Time
Reversal”
“Hyper-Dimensional”, “Hyper-Topological” Time
Digression: Maxwell’s
Demon
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“Time
Reversal”
We are now more or less ready to open the can of worms — we can jokingly call it a “cosmos of worms” — of “time
reversal”. We can call it a can of worms, not
so much because of the number of... oversights, but because of their quality.
Science will have to soul-search and question itself deeply on this one.
This is the
unrecognized essential question. It should not have been obvious what we
meant. “Time
reversal” is a Gedanken
Convenience Concept mistaken for something else.
How did the concept come about? One of the places that it got impetus was
in the development of the concept of the “arrow
of time”. Scientists took a gedanken
situation where entropy was obviously increasing and asked the question:
what happens if we “‘reverse
time’
by reversing the velocities of all the particles?” And the answer they
gave was: “entropy decreases! So there is an arrow of time!”
From our examination of
ergodic and
Brownian considerations in
entropy (above), we have a better understanding of what these scientists
did. Look again at
Figures 2a-e.
If we picked any point on one of those graphs where entropy was
increasing, and then we looked at the trajectory while “reversing
time”
by looking at values in the decreasing instead of increasing x
direction, we would probably (but not always since we might have
hit a local minimum) see entropy decreasing... for a while, a very short
while, and then in another very short while we
would see it increase again, rapidly changing its direction of change, or
at least its rate of change. If we looked long or far enough in both
directions, we would completely lose our sense of this
“arrow
of time”.
Scientists... oversighted this because they... oversighted the essential
nature of ergodic processes in this regard, and they... oversighted also
the Brownian nature of entropy.
But there is a much more
embarrassing... oversight here. It has to do with the difference between a
model-system and the behavior of the model-system. It is a subtle variant of the
mistaking the map for the territory, or of mistaking the
output of a theoretical
machine for the operation
of that machine. And given that automata theory has progressed as far as
it has, and for as long as it has, this mistake should never have been
allowed to continue for the last 4-5 or so decades.
We model
processes (notice the emphasis), and we do this in various ways. One way
is with equations, usually differential equations in which “time”
is itself modeled by a one dimensional numeric parameter
(i.e., by a scalar quantity). We take these two qualities (differential equations and a
scalar time parameter) and their general relationship for granted, just as
we take for granted that we can reverse the arithmetic sign of this
both one dimensional and numeric parameter. But...
But, the equations, the scalar quantity, the
possibility of sign reversal, these are
all
artifacts of
our model and our modeling process(es)! I.e. they are artifacts of the map
and mapping process. What we don’t know for sure
is if they are merely
artifacts, or whether they help carry the load of what we intend to
accomplish with our modeling or mapping.
(A similar situation arose
when Newton and Leibniz developed calculus using different methods, with
different artifacts. E.g. Leibniz had a notation that made a derivative
look like an algebraic ratio of 2 infinitesimals, e.g. df/dx,
where one could separate these and.. well, you know the story. One could
write equations like df = f’(x)dx, which one could not
do with Newton’s calculus, its notation in particular, and its-their
artifacts. This artifact of notation, and the fact that its apparent
algebraic extrapolation worked so well, so much of the time, made
Leibniz’s calculus the model for our modern version. But, if we try to
extrapolate usage the same way with partial differential equations, we get
a non-algebraic mess. Artifacts of the model and modeling process.)
To notice the artifactual nature of “time
reversal” a bit more solidly,
think of the ergodic process (style of) model of thermodynamics: just how would
we “reverse
time” for an ergodic process? or for a more
general stochastic process? If we try to take a trajectory and “time
reverse”
that... we mistake “time
reversing” the
output of the process (a particular
historical instantiation of it) for “time
reversing” the
process
itself.
Quite a bad mistake.
I.e., what we looked at was, not the operation of the
model (or of the reality modeled), but a particular instantiation of the trajectory of
the output of the operation of the model. This is equivalent to looking at the
output of a computer
program and “time
reversing”
it, but not looking at the
computer or the
program or the
operation of that
computer or program
(that gave that output) and “time
reversing”
it! or (make it more inclusive)
them!! Or do we need to “time
reverse” the model (i.e. “reverse the computer
and-or program”,
what would that mean?!) as opposed to, or before, “time
reversing” the operation of that model
(computer and-or program)?!
We reversed not
general system time, or
general system operation time
(and remember e.g. that a computer has different levels of hardware
operation and different levels of software operation), but a
particular state trajectory,
i.e. an established event history.
We reversed the fixed
history and noticed that “impossible” events occurred in one direction.
(The fact that the impossible events were actually ultra-low probability
events makes the situation only a little worse, since the asymmetry is
still there.) THIS is
where the ostensible asymmetry of time comes from! This is outright specious
reasoning.
So, to try to beat it to death, we did
not change the
system
so that it functioned “backward” or with “time reversed”, but we
did make a gedanken movie of a
particular
state trajectory (with time running forward) which in our case we gedanken chose
(statistics again) because it was asymmetrical, and
when we ran that movie backwards, when we reversed it, we said
“aha! a-symmetry!
ir-re-versible!” (Why that means it is “irreversible” is
also a question that the answer to which should not be obvious.) We failed to distinguish between
reversing the
system-operation time and
reversing the
trajectory-movie time of the movie
we made while running the system in a given direction of time. If we “reversed
time” in the system proper, not the
movie, we would change the state of the system (momenta, etc), but it
tends to be obvious upon reflection that its
reversed-time trajectory would NOT in general match that of the reversed movie... uhh...
Actually, this last statement has a problem and raises
an issue: when we gedanken put the system in a state and make a movie of
the trajectory with “time running forward”,
we don’t actually have a trajectory or a movie
of that trajectory to run backwards “past”
the point in time — i.e. before
that point in time — where we started the system. We can get in trouble
trying to extrapolate in such situations, especially when we take into
account that we can easily gedanken the system starting in a state that it
could never — or not easily; statistics again —
arrive at from some other starting state.
And beyond the question of
what it would mean to “time reverse” e.g. a computer add instruction,
there is also the question of what it would mean to “time reverse” the
computer output. Would we see numbers disappear from the printed page in
reverse order back to the original first number? or we see them
cumulatively printed in reverse order? or what?
But, in any case:
-
Major... oversight: we failed to distinguish
between reversing the
system-operation-time and
reversing the
trajectory-movie-time of the
movie we made of the system trajectory while running the system forward
in time (or in a given direction of time, if we actually figure out how
to run the system with “time reversed”).
Let’s also ask, did we do a good job of “reversing
time” when we merely changed the velocities of the particles? If we “reverse
time”, shouldn’t gravity (etc.) work “backwards”?! E.g. shouldn’t objects repel each other
gravitationally (etc.)?! This sounds naïve, I know, but it points at yet another
bunch of... worms in the can-cosmos. Forces and accelerations both have a relationship
with time that can be reversed; potentials less obviously, but... (These forces-fields, e.g.
molecular, that didn’t get reversed are where we don’t get cups un-breaking.)
Do you get the idea?! Our standard scientific analysis of the “time reversal”
concept has been naïve in the extreme, and that’s about as nicely as it can be
put.
For example, the first derivative of a velocity
is an acceleration. When we “time reverse” a velocity by setting v(t’) to -v(t),
what do we do to the first derivative?! Do we sign reverse it?! Should we sign reverse it?! What
should we do to it?!
If a particle has been following a (Newtonian) parabola with its
velocity a linear function of time, when we reverse the particle trajectory, we
sign reverse the velocity, but we leave the constant acceleration
un-sign reversed (that is, if we are playing the trajectory movie backwards at
the same rate we recorded it going forwards). But if the particle has been following
an exponential curve, the acceleration will change sign when we sign reverse the
velocity.
Changing the system state is a further issue: if
we are in a given state, what state corresponds to reversing system time? Well,
we think reversing the velocities-momenta of particles, which of course gives us
an entirely different state. It does this since the momenta are state variables,
whose values collectively determine or help determine the system state. But this can
obviously be done independently of changing the direction of system time.
Indeed, it seems like we are merely putting the system in a different starting state and
letting it run forward in time.
But what if the system relates essentially, not to
keeping particle spatial position continuous as above (i.e. we think of
reversing the trajectory of particle spatial position, giving us a discontinuous
change-reversal in particle velocity), but to keeping the particle velocity continuous
and reversing its trajectory?! Then, when we “reverse time”, instead of
“leaving the particle spatial position the same but reversing its course back
along its past trajectory”, we “leave the particle velocity the same, but reverse
its course, as time flows backwards, to the values it had in its along
its past trajectory”. This gives a decidedly different dynamic to “time
reversal”.
-
It’s obvious that we can choose position, velocity,
acceleration or anything else as the essential pivot for “time reversal”.
And it is quite likely that this choice of a pivot point will then make time
seem to be either symmetrical or asymmetrical when it is “reversed”.
(Having a new concept to chew on
may be a comfort as beloved old concepts die ugly deaths from Old Yeller Fever.)
Also, are there perhaps 2 sets of states, “forward” and
“backward”, not “accessible” to each other (also in the usual state transition
sense, not the “time reverse” sense), so that when we reverse a state by
reversing the velocities, we put the state from a state in the “forward” set
into a state in the “backward” set? Are there other “equivalence classes” of
states that are accessible (also in the usual state transition sense, not the
“time reverse” sense) only to states within each class?
(Temporary end of digression-like substances.)
We noted above that a more complete analysis of Brownian entropy causes us to
lose sight of our fair-haired
boy, our “arrow of time”. When we look at the reversal
of the trajectory, it is statistically symmetrical. (YES, statistics
yet again.)
-
We can only “re”-obtain an asymmetry in time if there is a “stylistic”
difference in reality (which we are able to notice and model
successfully) between forward and backward time. (E.g., perhaps we will eventually
notice that when entropy reverses as it inevitably does, it decreases noticeably
more or less quickly than when it increases.)
Ergodicity means
that for every increase there’s a decrease (along a “continuous” trajectory),
since to re-attain a given level of entropy we must have equality of the sum(s) of the increases and the sum(s) of the decreases. And we notice this more
assuredly if we have a yet much longer trajectory. Brownian ergodicity means we
have reversals of the direction of the change of entropy “almost everywhere” in
the time dimension (the mathematical way of saying at “almost every instant” or
“almost every point in time”).
The question of another as yet unrecognized source of
asymmetry in time is an essential question, but we will bypass it here in favor
of further analyzing the source of our concept of the “arrow of time”
(attempting to further beat it to death and drive a stake through its heart):
What we had, but failed to recognize as such, was a
situation where in either direction of time (along the trajectory-movie time,
not system-operation time) we tended over the long haul to find the system in a high entropy state.
We fallaciously interpreted this as meaning that, along a gedanken
trajectory of which we made a gedanken movie going gedanken forward in time, we
would find that there was an overwhelming propensity for entropy to increase.
But we just found the opposite, that along a single continuous trajectory, one
that is sufficiently long to Brownianly-ergodically return to the initial state,
there is-are statistically just as much-many increase(s) as decrease(s),
and this with mathematical unassailability.
Statistics: we can compare this result —
paradoxical at first glance — with gedankening
what would happen if we put the system in every state at every level of
entropy and studied every trajectory that resulted. Here the statistics would
almost certainly be more what we feel is reasonable, i.e. in all likelihood almost all
would yield a
definite — even if Brownian —
increase in entropy. Note that yet again we see the statistical selection
process in action: in one case we select all possible states and all possible
entropy levels, in the other we select a starting state and its entropy level
and the states that the trajectory — that happens
in this one instance to derive from that state — transitions to and
through, and their entropy levels. Note also that the “state” may or may not be
sufficiently completely defined so as to determine the trajectory
“deterministically” as opposed to statistically; if we just group states
according to their entropy level we are stuck with statistically; it is in this
case that we find the above selection processes to seem to be paradoxically
different. I.e., if we follow one continuous trajectory, there is an equal
chance of increasing or decreasing at a given entropy level (with care to
properly handle the cases of local maxima and minima), whereas if we look
at all possible states we get our usual overwhelming propensity. (And, to
reiterate, we don’t get an “arrow of time” in either case.)
CLUES: This “tendency” to go (implicitly from low probability states)
to and to
remain in highly probable states... well, that it happens is inevitable pretty much by
definition. (There is actually more to it mathematically, but... it wouldn’t fit
in the margins, even if this were a good place to pursue it.) I.e. all we have really
said with our current concept of entropy that is scientifically correct
is that higher probability events are more highly probable. This would be merely
humorous except for the fact that this higher probability and the concept of
entropy that gives rise to it together give tiny clues to an invisibly budding
concept of “hyper-dimensional” time, and the insights that this new concept will
eventually offer. E.g. the gedanken higher probability states are a complex
reflection-function of which dimensions of time we have chosen to reverse and
which not to reverse in making and reversing our gedanken trajectories in our
(as a reminder: gedanken) models.
QUICK GEDANKEN EXPERIMENT: Assume that we have a system at its highest level of
entropy. We observe it for some time, implicitly taking that level of entropy as
the standard by which we will measure any tendency for entropy to increase.
“How to Lie With Statistics” tells us that we can now
claim that there is “NO tendency for entropy to increase”. (See also
earlier remarks on this
tendency.) If we are
careful to note the Brownian entropy decreases, we can even say that “LAW OF
THERMODYNAMICS: ENTROPY ALWAYS DECREASES”, especially if we are also “How to
Lie With Statistics”-ally careful not to notice the corresponding increases (which
we can easily do by
appropriately selecting and deselecting the sets of events of which we allow
ourselves to make observations, and then selecting and deselecting the sets of
observations from which we will take our experimental data).
Before going further, let us note that, IN REALITY, we
have never actually scientifically noticed, let alone observed, time reversing, and therefore have no
real world data from which to conjecture real world-style models of such. We are merely playing, gedankening with our models that all per force derive from forward-only time
observations. We have never done anything other than gedanken reverse the
direction of time for
trajectories of models we have made using a given model of time, and not nearly all
such models. If we
had noticed that, of course, we would much more likely have noticed that our concept(s) of time reversal were more artifactual than real. Since we have
failed to notice that we were reversing system trajectories (movies, output
histories) instead of the systems
or system operations themselves, we have also failed to start to analyze reversing systems
and their operations. E.g. when we eventually gedanken reverse gravitational accelerations
and molecular forces, we may
be closer to reversing “system time” or “system operation time”, and we will
definitely be closer to cups un-breaking.
Further, there is the problem that, IN REALITY, we are
never able to put the real world system in all possible states, as we can
gedanken for our model. We can only view the “one true trajectory” of the real
world (and only infinitesimally, at that). Somewhere in there we should look for
“real world statistics” (etc.) that correspond to the only-one-total-system-trajectory reality,
and not (necessarily) to the all-possible-states gedanken model. In almost all
cases we merely assume that entropy increases like we think it does. If entropy
decreases Brownianly-ergodically (as opposed to by energy-matter exchange with a
different, lower entropy system), as noted before we will probably interpret it as something
else.
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Scientifically, we need to start paying attention.
And
we need to pay attention to the whole process, and that includes the observer and
the means of observing, as well as paying better attention to the (usual) observed.
The situation may look bleak to some, but...
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SECTIONS
Entropy’s Great...
Oversights
The Second Law of Thermodynamics
Oh, My Ergodic Hypothesis...
Brownian Motion,
“Brownian Entropy” and “Entropo-dynamics”
The “Arrow of Time”
Entropy as
“Disorder”
A Gedanken Experiment
Thermodynamic Equilibrium
“Time
Reversal”
“Hyper-Dimensional”, “Hyper-Topological” Time
Digression: Maxwell’s
Demon
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“Hyper-Dimensional”, “Hyper-Topological” Time
But, we can let serendipity work for us here.
We actually have a situation where we can see any
number of ways that “time” can be “reversed” independently of the other ways
(velocities, forces, etc.), and the system... well, the system can still make
sense in some sense(s). Even if these are artifacts of our model, they still might
be, by serendipity as well as by design, parts of our model that model
reality... “approximately” (rarely well-defined).
The terms
“hyper-dimensional” and
“hyper-topological” are meant to indicate that it is not merely
multi-dimensional, but that it is well beyond our usual concept of
multi-dimensional, and that it must also have a topology that is extremely far
from Euclidean, one that at least in part is event dependent.
E.g., we will have to learn to distinguish carefully
between-among the general hyper-dimensional spacetime that the particular
event of our general hyper-dimensional spacetime is embedded in, and
the particular hyper-dimensional spacetime general event that
is embedded in it, and the particular hyper-dimensional events that are
embedded in it... uhh... our “hyper-dimensional description spacetimes and-or
spacetime descriptions” are still naively inadequate.
To try to make a long story short, it turns out that time is —
and must be — a
function of “clocks”, i.e. of “change” as “measured by clocks” (this last will
definitely not fit in the margins), and if we stop thinking of disagreement of
clocks as clock inadequacy or failure, i.e. if we hold all clocks to be correct
by scientific fiat, then any differences are due to time running at different
rates, which can only happen if time is at least multi-dimensional, and probably
hyper-dimensional and hyper-topological.
Think of all the clocks or other “timekeeper”-like
substances that have ever been : the Earth and Sun, the Moon and tides,
sundials, seasons, flowers blooming, crops ripening, pendulum clocks, freeway traffic, atomic clocks, etc. Each
has at least somewhat different drummer-like substances (we need to
start thinking in the plural). We think that there is an absolute
one-dimensional time that these are approximations to, and that all we need to
do is find the one that is the best. But it is the other way around: “absolute
one-dimensional time” (as we think of it) is merely an approximation to this plexus of
hyper-dimensional-time with some of that indicated by various clocks-timekeepers. Each
clock is accurate in itself. It is just not
the best approximation to what we would like, at a given time. E.g. our
wristwatch won’t tell us when someone will actually arrive for lunch.
When we want to
synchronize activities, we usually choose similar clocks: “we will meet at the
big bend in the river when the salmon start running.” Notice how this last is
much more accurate than a cesium clock could ever be for this rendezvous. Time and
clocks-timekeepers are event oriented concepts-objects. It can hardly be
otherwise. We just missed that time is not only multi-dimensional, but
hyper-dimensional (i.e. it has a topology that is much more intricately complex
than suggested by merely “multi-dimensional”). There is much more that needs to
be said, of course, but for now we will un-digress and note that:
-
Einstein overlooked that his time dilation meant that
time must be “hyper-dimensional” and “hyper-topological”. Time can only
run at different rates in different trajectories (generalized places) if there
are more dimensions to time than our usual one. And trajectories can separate
and rejoin in such a way as to have the rejoining occur with different elapsed
times along each trajectory only if we have seriously non-Euclidean
hyper-topologies for such. I.e. we must have “hyper-dimensional”,
“hyper-topological” time within relativistic systems, at least.
We can try to look at why one clock runs slower, and
fool with it till it runs at a different rate, but we are actually changing e.g.
the direction in which it is moving through the (partly) event dependent flows, tides, eddies and
whirlpools of multi-dimensional —
“hyper-dimensional”
and “hyper-topological” —
time. Time could conceivably be able to reverse
in some dimensions but not others. This last is a crucial possibility for
resurrecting the
concept of time irreversibility, and it explains (at least it helps) why it is
not obvious what reversing time in a system would actually mean. It also helps
to explain why reversing time might or might not mean that cups un-break. We can
now think of reversing the dimension of time associated with velocity without
reversing the dimension of time associated with the formation of a molecular
bond... or vice-versa.
The clues
referred to above can be better understood now. In our model we reverse time in
certain of its hyper-dimensions-topologies, and we still have a tendency for the
system to (forgive the expression) “gravitate”
to states characterized by higher values of a certain scalar function (here, of
our poorly understood state function “entropy”; that it is
scalar is a major clue to our next steps in understanding hyper-dimensional topology). This will eventually help
us to study the relationships among the reversed and non-reversed... “things”,
“entities”... “whatevers” (so poorly defined as
to warrant the tedious parentheses, as has been generally true throughout this article).
Any theory that will successfully give us e.g. a handle on anti-gravity will
probably have to await advances in (meta-) theory (unacknowledged as such, but
meta-physics all the same) along these lines.
We can also see a different kind of sense in the idea
put forward by, among others, Stephen Hawking who said that when the expansion
following the “big bang” stops, that time will reverse as the expansion reverses
toward the “big crunch”.
-
If we have hyper-dimensional, hyper-topological time, with its own
“tempodynamics” as a counterpart to hydrodynamics, the reversing of the
cosmic expansion to a cosmic contraction could cause rivers of time in many dimensions to
reverse and flow in “opposite” directions (“opposite”, like “negation”,
“reversal”, etc., is never
well-defined, except in certain extremely limited mathematical circumstances),
except that it would look more like topologically complex swirls as the
dimensions and their flows all interact, like when the incoming tide reverses
and the dynamics of the flows reverse, but only sort of.
![[Under Construction]](../../images/undercon.gif)
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SECTIONS
Entropy’s Great...
Oversights
The Second Law of Thermodynamics
Oh, My Ergodic Hypothesis...
Brownian Motion,
“Brownian Entropy” and “Entropo-dynamics”
The “Arrow of Time”
Entropy as
“Disorder”
A Gedanken Experiment
Thermodynamic Equilibrium
“Time
Reversal”
“Hyper-Dimensional”, “Hyper-Topological” Time
Digression: Maxwell’s Demon
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Digression: Maxwell’s Demon
In this context, re-analyzing Maxwell’s
Demon may seem anti-climactic, but the usual analysis of it brings up yet another
worse than merely embarrassing... oversight.
The usual analysis is that
Maxwell’s Demon runs into trouble when he tries to select the molecules to let
through his anti-entropic trap door(s) (based on how fast they are moving compared
to the average, or some such). It is often pointed out that eventually he will be
unable to distinguish which are moving faster, or be unable to let them through
his special trap door(s) even if he can distinguish them (for statistical reasons
concerning the doors being hit by too many molecules per second from the wrong
side to be able to open, or too many molecules trying to get through the door
from the wrong direction when you would like to let a molecule go through in the
right direction, etc.) The usual finding is that therefore there can be
no such thing as Maxwell’s Demon. This last... oversight.
What if someone said that because
no chemical reaction can go to absolute completion that there is no such thing
as a chemical reaction? (There is always a non-zero partial pressure-like
substance, a non-zero reverse direction reaction, most noticeable at equilibrium, so both
in theory and in fact no chemical reaction can ever go to absolute completion.)
What if someone said that there
was no such thing as engineering because no engineer could get 100% efficiency
out of a machine? (Usually 30% is considered good for a standardly
well-engineered machine.)
You get the idea. Within current
thermodynamics theory we see how a Maxwell’s Demon could get some
anti-entropic separation of fast and slow molecules, but our theory suggests
rather strongly that eventually he will (pardon the expression) “Max Out”. But,
if we then say there can be no such thing as a Maxwell’s Demon?... oversight.
And not a subtle one.
Science needs to do much
better than this.
By the way, some scientists have
reported experiments where they tried to build a Maxwell’s Demon and failed.
They give an analysis that ostensibly shows why. But... other scientists have
reported successfully constructing small Maxwell’s Demon-like substances that
work (up to a point) in violation of the usual finding. It would be very
interesting to comparatively analyze these ostensible failures and successes to
find where scientists are having their conceptual and-or other trouble.
Also, it may be
that the concept of Maxwell’s Demon offers a place to look for where Life
affects “non-sentient” matter, in fact animating it. Even though propensities
are set up by e.g. physical and chemical properties, just as in an automobile,
the Maxwell’s Demon of Life (that old “concepta-non-grata”, “vitalism”)
— with much evolution and practice — can
wind up in a driver’s seat-like substance.
-
Telekinesis, far from being
scientifically impossible, happens in Life all the time (almost precisely like “vitalism”,
which happens in Life by definition) as we consciously will
our fingers to press the buttons on our remotes. And we also forget the
scientific implications of “Honey, can you bring me a beer?!” Perhaps Maxwell’s
Demon shows up rather more in... Love.
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