I am reading one dude's dissertation on measurements multiplexing (seems that this dude is amazed from a mere fact that he understood the Hadamard's approach to signal multiplexing).

For me, Hadamard is not only long dead, but his approach stinks when practically implemented. Long ago, people refused to use his "minus one, zero, one" multiplexing formulation to figure out which peak is which in the soup of peaks.
Any practical imlementation is doomed due to real-life cross interference between the "independent" measurements and non-linear propagation of errors (noise). My point is that in the world of MHz signals, there is always difficult to create a real "squared" ones or a flat zero.
Something is always messing around.

Now, one north-westernly scientist and water polo player obviously is getting closer to a computer screen trying to spit into this very text.
Those man-no-time-to read-your-crap, English-language-challenged bloody ones, or lost and hardened souls who are striclty perpendicular to all French, all those can skip the following non sequiturian introduction and just read the text limited by two lines of stars, or asteriscs.
Voila.

Here, we do not discuss an old dude Hadamard, but we will consider
the weighting paradox (see, the text belov, sorry below).
And BTW, Fourier rules. And fast Fourier rules faster.
Ask FT-IR spectroscopists, or intransigent enthusiasts of big and very big liquid He-cooled magnets.

By the way, a non-linear propagation of evil was a brilliant philosophical finding of my friend nick-named as Alex the Millionaire.
His "grandmaman" dug in the big wooden box stuffed with rare gems and silverware during Russian Civil war on a high right bank of the Oka river, 200 miles to the east of Moscow, and then Alex (senior engineer, rarely sober), his second cousin (dentist, always sober, and by chance my good friend), dug it out in the middle of 80-ies.

Alex verbalized propagation principle as one of Lemmas derived from both the Hegel's axiom of dialectics and Hegel' (in fact, originally Upanishada -Hindu) notion about the universalism of triangles as building blocks of any event or any imaginable/abstract thingy from material/non-material world
(e.g., one triangle is: the good, the evil, and the hard currency thing).
(in Russian can be expressed as Dobro, Zlo , and Bablo).

In fact, all Marx's Capital ( Vol. 1, particularly) is practical examples of that principle and can be derived from that Lemma about nonlinear propagation of Bablo.

Importantly, the same very dude (Alex) later and in a less than a sober state formulated a conjecture that any non-linear propagation can be terminated by cataclysmic event such as (but not limited to) a simple toilet flush.
OK...
Flush always stays for a physical removal of the subject (s), themes, or objects being taken or absorbed, subjected to a personal treatment, and then propagated or discarded for other people use. Commenting on removal of "Sub-yekts" here - one Bostonian female once shortly formulated general condition for this removal as "no money - no honey".

Other way to terminate ( to quench) the propagation chain is elimination of the propagator(s) himself.

Again, following ideology of Marx and his clevrets, the propagators of evil things can vanish into GULAG or be eliminated via skull application of an ice pike by some obscure dude called Merkador.
Then see: the "ice peak" -subject - further transforms and propagates (again non-linearly) into Mercador or Merkator which is the next propagator of evil. How?
Easy - one can see Mercator's maps (Mercator's projection) in any local office where Greenland seems larger than South America. There is no brainer to construct further propagation chain.
Modern russian writer Pelevin rests here, as any self-centered dude.

OK, back to the theme.
The multiplexing in my memory normally is associated not only with litter of hamsters zealously bred by some aspiring Karolinska U. student. She refuses to nute the main hairy propagator. A personal deck and a chair have been already promised to her by one eminent Karoinska professor, and, listen to me, professor, she has a deep memory and nasty character, really... really nasty - she eats forgetful folk alive.

OK, now really back.
Not about that kind of multiplexing I am talking about here.
Multiplexing recalls me a desperate phone call from one sharp but sometime overly finger-stretched PhD-from-MIT dude during one of his interviews held on Wall street. He wispered into a phone receiver with a barely audible voice asking
"how to find a single poisoned bottle among 1,000 using only 10 mice (or hamsters)?"

I gave him a hint: 2^10=1024

So, if each happy hamster can drink very large amount of wine (French breed of hamsters?), then one can multiplex the task and let any hairy hobo drink not from only one bottle but from different (but known, that is, numbered) bottles .
Each bottle can be numbered in a binary code.
"Ones" in this code mean the order (or position) of a hamster in a row who should drink from this bottle.

For instance, bottle number 00110 01101 nees to be tested by third, fourth, seventh, eighth, and tenth hamsters.
If all those hamsters (##3, 4, 7, 8, 10) kick the bucket, this particular bottle is poisoned .
One can convert this binary number into a decimal one to find the bottle.

OK. Now I am getting real - back to the theme.
BTW, what I am to talk about? Wait.... aha.
The dissertation dude. Really? Nice then.
Now, I'm to reveal the paradox for those who is still alive.

Read the text below (between lines of asteriscs) .
Seems, that something is wrong in the text below.
Any comment?
My comment is:
Basic law of nature is that one can not cheat on it. Imagine the situation where one measures a weight with some error. Seems from paradox below that this error can be dependent on other attempts to measure other weights or other attempts to measure the same weight combined with other weights , or both.

Or it is a some screwy play on statistics where in fact one weights is measured twice or some fractional number of attempts (say 1.5)?
Yet, in the text below the number of attempts seems to be the same for each weight.

START OF THE QUOTED TEXT
* * * * * * * * * * * * * * * * * *

We use a single pan spring balance to weigh four objects, numbered as 1, 2, 3 and 4.
We assume the balance has been well calibrated and every time it will give us the correct value (true value W) with a small random error (e). The average of e is 0, but the average of e^2 is not 0, which is the variance of e (mean squared error of each measurement = (sigma)^2).
If we simply weigh the objects one by one, we can get 4 measurements

A1, A2, A3, A4 :

A1= W1+ e1
A2= W2+ e2
A3= W3+ e3
A4= W4+ e4

The mean squared error of each measurement is (sigma)2.
On the other hand, if we measure 3 objects per weighing,
we can get another set of results:

B1= W1+W2+W3 + e1
B2= W2 +W3+W4+ e2
B3= W1 +W3+W4+ e3
B4= W1+W2 +W4+ e4

Then we can calculate the "true" weight W(n) of each object:

W 1= 1/3(B1+B3+B4-2 B2)
W 2= 1/3(B1+B2+B4-2 B3)
W 3= 1/3(B1+B2+B3-2 B4)
W 4= 1/3(B2+B3+B4-2 B1)

Subsequently, we can get the estimate weight and its error W11 (n) of each object:

W 11= W1+1/3(e1+e3+e4-2 e2)
W 22= W2+1/3(e1+e2+e4-2 e3)
W 33= W3+1/3(e1+e2+e3-2 e4)
W 44= W4+1/3(e2+e3+e4-2 e1)

Calculated mean squared error for object (piece) 1 is:
E1=(W 11-W1)2=(5/9)sigma2=E2=E3=E4

It is a considerable improvement because the variance in the measurement is reduced to 5/9 of original variance.
With different weighing method, different optimized variance can be obtained.

* * * * * * * * * * * * * * * * * *
END OF THE QUOTED TEXT

First notice is that if you square (e1 + e3 + e4 - 2*e2)
then you get
(e1)2 + (e3)2 + (e4)2 + 4*(e2)2
(cross-product terms were eliminated due to condition of mutual independence of different weight measurements)

so if e1=e2=e3=e4

the variance is 7/9 (sigma)2 but not 5/9 (sigma)2

but nevertheless....7/9<1

So what is the deal, then?