Difference between revisions of "Sandbox"

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{{top}}
  
{{top}}
   
  +
[[RedPu]] is a bilingual ([[English]] / [[Russian]]) emulation based on the utopian dialogue
−
'''Tetration''' is the superfunction of the [[exponential]] map.
 
  +
presented in the article [[Рыжий Пу]].
−
For a given base \(b\), the tetration \(\operatorname{tet}_b\) is defined as the function satisfying the transfer equation
  
−
\[
  
−
\operatorname{tet}_b(z+1) = b^{\operatorname{tet}_b(z)}
  
−
\]
  
−
together with additional normalization and regularity conditions that select a unique solution among infinitely many possible superfunctions.
  
   
  +
The emulation is presented in two synchronized language layers.
−
The name “tetration’’ reflects its role as the next operation after exponentiation within the [[hyperoperation]] hierarchy.
  
  +
The English part reflects the assumption that [[Donald Trump]] does not speak [[Russian]].
  +
The Russian part reflects the assumed replies of [[Путин Владимир Владимирович]].
   
  +
==Context==
−
The most studied case is the real holomorphic tetration to base \(b=\mathrm e\), written simply as
  
−
\[
  
−
\operatorname{tet}(z)=\operatorname{tet}_{\mathrm e}(z).
  
−
\]
  
   
  +
The emulation develops the theme of “radioactive ash” («радиоактивный пепел»),
−
<div class="thumb tright" style="float:right; margin:-7px 0px 8px -8px">
  
  +
introduced into Russian political discourse in 2014 by
−
{{pic|Tetreal10bx10d.png|300px}}
  
  +
[[Киселёв Дмитрий Константинович]].
−
<small><center>Real tetration to base \(b=\mathrm e\).<br>
  
−
From: Fig.17.2, p.239 in ''Superfunctions''.</center></small>
  
−
</div>
  
   
  +
The text below is not a prediction, not a plan, and not a call to action.
−
==Definition==
  
  +
It is a [[утопия]] / [[emulation]] constructed for analytical and illustrative purposes.
−
Let \(T_b(z)=b^z\).
 
−
A function \(F\) is a ''superexponential'' (a superfunction of \(T_b\)) if
  
−
\[
  
−
F(z+1) = T_b(F(z)).
  
−
\]
  
   
  +
==Dialogue / Диалог==
−
A '''tetration''' to base \(b\) is the unique real-holomorphic superexponential for which:
  
   
  +
<poem>
−
# the limits
 
  +
**Trump (English):**
−
\[
  
  +
Hello, Mister Putin.
−
\lim_{y\to\pm\infty} F(x+\mathrm i y)
  
−
\]
  
−
remain finite (regularity at the imaginary infinities),
  
−
# the normalization
 
−
\[
  
−
F(0)=1
  
−
\]
  
−
holds.
  
   
  +
**Путин (Russian):**
−
These conditions remove the otherwise infinite freedom in solutions to the transfer equation.
  
  +
Здравствуйте, мистер Трамп.
   
  +
**Trump:**
−
==Properties==
  
  +
One of your television hosts once said that Russia could turn the United States
−
For the principal tetration \(\operatorname{tet}_b\):
  
  +
into radioactive ash. Is that statement correct?
   
−
* **Transfer equation**
+
**Путин:**
  +
Он действительно так сказал. Да.
−
\[
  
−
\operatorname{tet}_b(z+1)=b^{\operatorname{tet}_b(z)}.
  
−
\]
  
   
  +
**Trump:**
−
* **Derivative at the fixed point**
 
  +
So Russia *can* do that?
−
Let \(L\) satisfy \(L = b^L\). Then
  
−
\[
  
−
\operatorname{tet}_b'(0)=L.
  
−
\]
  
   
−
* **Analyticity**
+
**Путин:**
  +
Может.
−
For \(1 < b < \mathrm e^{1/\mathrm e}\), the tetration to base \(b\) extends to an entire function.
 
−
For \(b > \mathrm e^{1/\mathrm e}\), branch singularities appear, and a global entire tetration is impossible.
  
   
  +
**Trump:**
−
==Regularity in the base==
  
  +
Thank you. That answer is sufficient.
−
The tetration depends smoothly on the base \(b\) for
 
−
\[
  
−
1 < b < \mathrm e^{1/\mathrm e}.
  
−
\]
 
−
Within this range, the exponential map \(z\mapsto b^z\) has two attracting fixed points, allowing construction of an entire Abel function and an entire tetration.
  
   
  +
**Путин:**
−
At the critical base, the two fixed points merge, and the dynamical structure changes.
  
  +
Для чего вам это?
   
  +
**Trump:**
−
==Singularity at the critical base \(b = 1/\mathrm e\)==
  
  +
To justify a decision currently being voted on by Congress.
−
The point
  
−
\[
  
−
b = \frac{1}{\mathrm e}
  
−
\]
  
−
is a branch point of the exponential map in the parameter \(b\).
 
−
Although tetration is smooth for \(b>1/\mathrm e\), this regularity does *not* guarantee holomorphic extendability across the branch point.
  
   
  +
**Путин:**
−
Important observations:
  
  +
Какое решение?
   
  +
**Trump:**
−
* When the base parameter \(b\) is continued into the complex plane, the behavior around \(b=1/\mathrm e\) becomes path-dependent.
  
  +
About Russia.
−
* Continuation of tetration above and below the branch point generally does not give the same result — the continuation is multivalued.
  
−
* The situation is analogous to the relation between the [[Exponential integral]] \(\operatorname{Ei}\) and \(E_1\): one can smoothly approach the singularity from either side, but no single global holomorphic branch exists.
  
−
* For tetration, the complication is deeper because the function depends on two variables \((b,z)\); nonetheless, \(b=1/\mathrm e\) remains a limiting point for the tetrations defined on both sides.
  
   
  +
**Путин:**
−
This is one of the important unresolved structural problems of tetration theory.
  
  +
И вы решили меня предупредить?
   
  +
**Trump:**
−
==Complex structure==
  
  +
Yes. We prefer transparency.
−
<div class="thumb tright" style="float:right; margin:0 0 8px 8px">
  
  +
You may want to warn the personnel at your launch facilities.
−
{{pic|Ackerplot.jpg|300px}}
  
−
<small><center>Ackermann-family functions.
 
−
Natural tetration (dashed) and other Ackermann-like functions.<br>
  
−
From: Fig.19.7, p.266 in ''Superfunctions''.</center></small>
  
−
</div>
  
   
  +
**Путин:**
−
The complex analytic structure of tetration is governed by the dynamics of \(z\mapsto b^z\).
 
  +
Зачем?
−
For \(1<b<\mathrm e^{1/\mathrm e}\), the map has two attracting fixed points.
 
−
This allows one to construct an Abel function that is analytic on the entire plane and thus an entire tetration.
  
   
  +
**Trump:**
−
At the critical base \(b=\mathrm e^{1/\mathrm e}\), the two fixed points collide (parabolic bifurcation).
 
  +
Because we are about to demonstrate that Mr. Kiselev was mistaken.
−
For \(b>\mathrm e^{1/\mathrm e}\), both fixed points become repelling, and the Abel function — hence tetration — develops branch points and cannot be entire.
  
   
  +
**Путин:**
−
==Relation to superfunctions==
  
  +
В чём именно?
−
A superfunction \(F\) of \(T_b\) satisfies
  
−
\[
  
−
F(z+1)=b^{F(z)}.
  
−
\]
  
   
  +
**Trump:**
−
Among the infinitely many such functions, the tetration is selected by:
  
  +
In assuming that only Russia can turn others into ash.
   
  +
**Путин:**
−
* realness on the real axis,
 
  +
Вы хотите сказать, что испугались?
−
* holomorphy in a vertical strip,
 
−
* regularity at \(+\mathrm i\infty\) and \(-\mathrm i\infty\),
 
−
* normalization at \(z=0\).
  
   
  +
**Trump:**
−
All other superexponentials differ by periodic or quasiperiodic distortions.
  
  +
No. I want to say that intimidation is not a substitute for capability.
   
  +
**Путин:**
−
==Inverse: Abel function==
  
  +
Но вы же понимаете, к чему это может привести?
−
The inverse of tetration is the [[Abel function]] (ArcTetration), denoted \(\operatorname{ate}_b\).
 
−
It satisfies
  
−
\[
  
−
\operatorname{ate}_b(b^z) = \operatorname{ate}_b(z) + 1,
  
−
\]
  
−
with
  
−
\[
  
−
\operatorname{ate}_b(1)=0.
  
−
\]
  
   
  +
**Trump:**
−
For the natural exponential:
  
  +
Yes. That is precisely why we are having this conversation.
−
\[
  
−
\operatorname{ate}(\exp(z)) = \operatorname{ate}(z) + 1.
  
−
\]
  
   
  +
**Путин:**
−
==Iterates==
  
  +
И что вы предлагаете?
−
Tetration and ArcTetration provide analytic extensions of integer iteration.
 
−
For the \(n\)-th iterate of the exponential:
  
−
\[
  
−
\exp_b^n(z) = \operatorname{tet}_b\!\big(n + \operatorname{ate}_b(z)\big),
  
−
\]
  
−
valid for real or complex \(n\).
  
   
  +
**Trump:**
−
==Special cases==
  
  +
We propose to remove the instruments that make such statements possible,
−
<div class="thumb tright" style="float:right; margin:-30px 0 8px 8px">
  
  +
while preserving the population and the territory.
−
{{pic|E1efig09abc1a150.png|480px}}
  
−
<small><center>Tetration near the critical base \(b=\mathrm e^{1/\mathrm e}\).<br>
  
−
From: Fig.17.4, p.245 in ''Superfunctions''.</center></small>
  
−
</div>
  
   
  +
**Путин:**
−
===Base \(\sqrt{2}\)===
  
  +
Это ультиматум?
−
A classical example of an entire tetration, used extensively in early numerical work.
  
   
  +
**Trump:**
−
===Base \(b= \mathrm e^{1/\mathrm e}\)===
  
  +
No. An explanation.
−
This “critical base’’ marks the transition where the fixed points of \(z\mapsto b^z\) merge.
 
−
Properties include:
  
   
  +
**Путин:**
−
* extremely slow growth near the fixed point,
 
  +
А если мы откажемся?
−
* non-entire analytic continuation,
 
−
* delicate dependence on initial conditions.
  
   
  +
**Trump:**
−
===Base 2===
  
  +
Then the explanation will be repeated,
−
An important superexponential used in comparisons with hyperoperators.
  
  +
using a language that does not require interpreters.
   
  +
**Путин:**
−
===Natural tetration===
  
  +
Вы угрожаете?
−
The principal tetration \(\operatorname{tet}(z)\) is real-analytic on the real axis.
 
−
As \(z\to -\infty\), it approaches the lower fixed point of \(\exp(z)\).
 
−
As \(z\to +\infty\), it grows faster than any finite exponential tower.
  
   
  +
**Trump:**
−
==Figures==
  
  +
No. Threats are what television hosts do.
−
<div class="thumb tright" style="float:right; margin:0 0 8px 8px">
  
  +
We deal with accountability.
−
{{pic|B271t.png|300px}}
  
−
<small><center>Tetration and its inverse (ArcTetration).<br>
  
−
From: Fig.14.4, p.203 in ''Superfunctions''.</center></small>
  
−
</div>
  
   
  +
**Путин:**
−
<div class="thumb tright" style="float:right; margin:0 0 8px 8px">
  
  +
Вы уверены, что контролируете последствия?
−
{{pic|Tetsheldonmap03.png|300px}}
  
−
<small><center>Complex structure of tetration.<br>
  
−
From: Fig.18.3 (top), p.250 in ''Superfunctions''.</center></small>
  
−
</div>
  
   
  +
**Trump:**
−
==Conceptual illustration==
  
  +
We control our actions.
−
<div class="thumb tright" style="float:right; margin:0 0 8px 8px">
  
  +
You control whether your system survives the truth about itself.
−
{{pic|BlackSheep.png|280px}}
  
−
<small><center>Heuristic “half-sheep’’ illustration.<br>
  
−
From: Fig.15.6, p.218 in ''Суперфункции'' (2014).</center></small>
  
−
</div>
  
   
  +
**Путин:**
−
This cartoon illustrates a philosophical point in tetration theory:
 
  +
А если народ не поймёт?
−
heuristic assumptions that appear “obvious’’ (e.g., that the right side of a sheep has the same color as the left) may be false without rigorous proof.
  
   
  +
**Trump:**
−
Likewise, assumptions about tetration — such as smoothness in both arguments \((b,z)\), or holomorphic extendability across \(b=1/\mathrm e\) — require careful justification.
 
  +
Then it will finally hear something other than myths.
−
The cartoon emphasizes that the behavior of superexponentials around their branch points is subtler than it may first appear.
  
   
  +
**Путин:**
−
==References==
  
  +
Вы уверены, что после этого всё закончится?
−
{{ref}}
  
   
  +
**Trump:**
−
D. Kouznetsov. ''Superfunctions''. Lambert Academic Publishing, 2020.
 
  +
Nothing “ends”.
−
All cited figures except the last are taken from this monograph.
  
  +
Some things simply stop working.
   
  +
**Путин:**
−
D. Kouznetsov. ''Суперфункции''. Lambert Academic Publishing, 2014.
 
  +
Например?
−
Figure 15.6 (the “half-sheep’’) is taken from this edition.
  
   
  +
**Trump:**
  +
For example, the belief that shouting about ash
  +
protects those who shout.
  +
  +
**Путин:**
  +
И что будет со мной?
  +
  +
**Trump:**
  +
That depends on whether you prefer to be a witness
  +
or an exhibit.
  +
  +
**Путин:**
  +
Это не разговор по телефону.
  +
  +
**Trump:**
  +
Agreed.
  +
That is why this is the last phone call.
  +
  +
**Путин:**
  +
Когда мы увидимся?
  +
  +
**Trump:**
  +
Soon enough.
  +
</poem>
  +
<!--
  +
In a setting where interpreters will still be present,
  +
but excuses will not.
  +
  +
(The dialogue continues conceptually as in [[Рыжий Пу]],
  +
with emphasis on non-nuclear demilitarization,
  +
personal responsibility of decision-makers,
  +
and the collapse of performative intimidation.)
  +
!-->
  +
==Interpretation==
  +
  +
The bilingual form highlights an asymmetry:
  +
  +
* threats are produced for a domestic Russian audience,
  +
* consequences are explained in a language addressed outward.
  +
  +
This asymmetry is one of the central analytical points of the emulation.
  +
  +
==Relation to other articles==
  +
  +
* [[Рыжий Пу]] — original Russian emulation
  +
* [[Эмуляция]] — methodological category
  +
* [[Утопия]] — literary framing
  +
* [[ChatGPT in TORI]] — methodological disclaimer
  +
  +
==Note by [[ChatGPT]]==
  +
  +
[[ChatGPT]] assisted in restructuring the emulation into a bilingual format
  +
and in clarifying the separation between fictional dialogue,
  +
analytical intent, and factual references.
  +
No factual claims or political positions were introduced by [[ChatGPT]].
  +
Responsibility for interpretation remains with the Editor.
  +
See also «[[ChatGPT in TORI]]».
  +
  +
==Warning==
  +
  +
As with other emulations in [[TORI]],
  +
this text does not constitute a recommendation,
  +
instruction, or forecast.
  +
It is a thought experiment intended for analysis of political rhetoric.
  +
  +
==References==
  +
{{ref}}
 
{{fer}}
  
{{fer}}
   
 
==Keywords==
  
==Keywords==
−
«[[Base e1e]]», «[[Base sqrt2]]», «[[Superfunction]]», «[[Exponential]]», «[[Logarithm]]»,
 
−
«[[Tetration]]», «[[ate]]», «[[Abel function]]», «[[Transfer equation]]», «[[Iterates]]».
  
   
  +
«[[RedPu]]»,
−
[[Category:Superfunction]]
  
  +
«[[Рыжий Пу]]»,
−
[[Category:Exponential]]
  
  +
«[[Donald Trump]]»,
−
[[Category:Special function]]
  
  +
«[[Путин Владимир Владимирович]]»,
−
[[Category:English]]
  
  +
«[[Киселёв Дмитрий Константинович]]»,
  +
«[[Эмуляция]]»,
  +
«[[Утопия]]»,
  +
«[[Radioactive ash]]»
  +
  +
[[Category:Emulation]]
  +
[[Category:Utopia]]
  +
[[Category:Donald Trump]]
  +
[[Category:Vladimir Putin]]
  +
[[Category:ChatGPT]]

Latest revision as of 17:22, 6 January 2026


RedPu is a bilingual (English / Russian) emulation based on the utopian dialogue presented in the article Рыжий Пу.

The emulation is presented in two synchronized language layers. The English part reflects the assumption that Donald Trump does not speak Russian. The Russian part reflects the assumed replies of Путин Владимир Владимирович.

Context

The emulation develops the theme of “radioactive ash” («радиоактивный пепел»), introduced into Russian political discourse in 2014 by Киселёв Дмитрий Константинович.

The text below is not a prediction, not a plan, and not a call to action. It is a утопия / emulation constructed for analytical and illustrative purposes.

Dialogue / Диалог

    • Trump (English):**

Hello, Mister Putin.

    • Путин (Russian):**

Здравствуйте, мистер Трамп.

    • Trump:**

One of your television hosts once said that Russia could turn the United States
into radioactive ash. Is that statement correct?

    • Путин:**

Он действительно так сказал. Да.

    • Trump:**

So Russia *can* do that?

    • Путин:**

Может.

    • Trump:**

Thank you. That answer is sufficient.

    • Путин:**

Для чего вам это?

    • Trump:**

To justify a decision currently being voted on by Congress.

    • Путин:**

Какое решение?

    • Trump:**

About Russia.

    • Путин:**

И вы решили меня предупредить?

    • Trump:**

Yes. We prefer transparency.
You may want to warn the personnel at your launch facilities.

    • Путин:**

Зачем?

    • Trump:**

Because we are about to demonstrate that Mr. Kiselev was mistaken.

    • Путин:**

В чём именно?

    • Trump:**

In assuming that only Russia can turn others into ash.

    • Путин:**

Вы хотите сказать, что испугались?

    • Trump:**

No. I want to say that intimidation is not a substitute for capability.

    • Путин:**

Но вы же понимаете, к чему это может привести?

    • Trump:**

Yes. That is precisely why we are having this conversation.

    • Путин:**

И что вы предлагаете?

    • Trump:**

We propose to remove the instruments that make such statements possible,
while preserving the population and the territory.

    • Путин:**

Это ультиматум?

    • Trump:**

No. An explanation.

    • Путин:**

А если мы откажемся?

    • Trump:**

Then the explanation will be repeated,
using a language that does not require interpreters.

    • Путин:**

Вы угрожаете?

    • Trump:**

No. Threats are what television hosts do.
We deal with accountability.

    • Путин:**

Вы уверены, что контролируете последствия?

    • Trump:**

We control our actions.
You control whether your system survives the truth about itself.

    • Путин:**

А если народ не поймёт?

    • Trump:**

Then it will finally hear something other than myths.

    • Путин:**

Вы уверены, что после этого всё закончится?

    • Trump:**

Nothing “ends”.
Some things simply stop working.

    • Путин:**

Например?

    • Trump:**

For example, the belief that shouting about ash
protects those who shout.

    • Путин:**

И что будет со мной?

    • Trump:**

That depends on whether you prefer to be a witness
or an exhibit.

    • Путин:**

Это не разговор по телефону.

    • Trump:**

Agreed.
That is why this is the last phone call.

    • Путин:**

Когда мы увидимся?

    • Trump:**

Soon enough.

Interpretation

The bilingual form highlights an asymmetry:

  • threats are produced for a domestic Russian audience,
  • consequences are explained in a language addressed outward.

This asymmetry is one of the central analytical points of the emulation.

Relation to other articles

Note by ChatGPT

ChatGPT assisted in restructuring the emulation into a bilingual format and in clarifying the separation between fictional dialogue, analytical intent, and factual references. No factual claims or political positions were introduced by ChatGPT. Responsibility for interpretation remains with the Editor. See also «ChatGPT in TORI».

Warning

As with other emulations in TORI, this text does not constitute a recommendation, instruction, or forecast. It is a thought experiment intended for analysis of political rhetoric.

References

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