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{{top}}
  
{{top}}
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<div class="thumb tright" style="float:right; margin:-76px 0px 0px 8px; line-height:2px; background-color:#fff">
  
−
{{pic|Nemplot.jpg|120px}}<small><center>\(y=\mathrm{nem}_q(x)\)</center></small>
  
−
</div>
  
−  
−
<div class="thumb tright" style="float:right; margin:0px 0px 4px 8px; width:408px">
  
−
{{pic|Nem0map.jpg|200px}} {{pic|ArqNem0map.jpg|200px}}
 
−
{{pic|Nem2map.jpg|200px}} {{pic|ArqNem2map.jpg|200px}}
  
−
<small><center>Complex maps: \(u+\mathrm i v=f(x+\mathrm i y)\) for \(f=\mathrm{nem}_0\), \(f=\mathrm{ArqNem}_0\) (top) and for \(f=\mathrm{nem}_2\), \(f=\mathrm{ArqNem}_2\) (bottom)</center></small>
  
−
</div>
  
−  
−
The [[Nemtsov function]] is a special kind of polynomial, suggested as an example of a [[transfer function]] in the book «[[Superfunctions]]» <ref name="book">
  
−
https://www.amazon.co.jp/-/en/Dmitrii-Kouznetsov/dp/6202672862 <br>
  
−
https://www.morebooks.de/shop-ui/shop/product/978-620-2-67286-3 <br>
  
−
https://mizugadro.mydns.jp/BOOK/468.pdf <br>
  
−
Dmitrii Kouznetsov. ''Superfunctions''. [[Lambert Academic Publishing]], 2020.
  
−
</ref>.
 
−
The description is also available as the [[Mizugadro Preprint]] <ref>
  
−
https://mizugadro.mydns.jp/PAPERS/2016nemtsov.pdf <br>
  
−
Dmitrii Kouznetsov. ''Nemtsov function and its iterates''. [[Mizugadro Preprint]], 2016.
  
−
</ref>.
  
−  
−
The [[Nemtsov function]] \(y=\mathrm{nem}_q(x)= x+x^3+q\,x^4\) is shown in the figure at right for various \(q\ge 0\).
  
−  
−
[[Complex map]]s of \(\mathrm{nem}_q\) are shown in the left column of the figure above.
 
−
The right column shows similar maps for the inverse function \(\mathrm{ArqNem}_q=\mathrm{nem}_q^{-1}\).
  
−  
−
This article describes the Nemtsov function and its related functions:
 
−
the inverse function [[ArqNem]],
 
−
the [[superfunction]] [[SuNem]], and
 
−
the [[Abel function]] [[AuNem]].
  
−  
−
== Motivation ==
  
−  
−
The [[Nemtsov function]] serves as an example of a real-holomorphic [[transfer function]] with a real [[fixed point]], where the usual [[regular iteration]] method for constructing a [[superfunction]] cannot be applied directly.
 
−
The editor did not find any simpler example of this type other than this specific fourth-order polynomial.
  
−  
−
The function was introduced as an attempt to construct an “exotic’’ [[transfer function]] for which a growing real-holomorphic [[superfunction]] could not be produced by the usual methods appearing in the «[[Table of superfunctions]]».
 
−
This attempt failed — the [[superfunction]] for the Nemtsov function *can* be constructed, and is described below.
  
−  
−
The expansion of the Nemtsov function at its fixed point begins with a linear term whose coefficient is unity. This prevents the use of standard [[regular iteration]] <ref>
  
−
http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html <br>
  
−
https://mizugadro.mydns.jp/PAPERS/2010sqrt2.pdf <br>
  
−
D. Kouznetsov, H. Trappmann. ''Portrait of the four regular super-exponentials to base sqrt(2)''. ''Mathematics of Computation'', 2010, v.79, p.1727–1756.
  
−
</ref>, which works for the exponential to base \(\sqrt{2}\).
  
−  
−
The situation is similar to the exponential to base \(\exp(1/\mathrm e)\) <ref name="e1e">
  
−
http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf <br>
  
−
https://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf <br>
  
−
H. Trappmann, D. Kouznetsov. ''Computation of the Two Regular Super-Exponentials to base exp(1/e)''. ''Mathematics of Computation'', 81 (2012), 2207–2227.
  
−
</ref>, but the Nemtsov function lacks the quadratic term, so the [[exotic iteration]] of <ref name="e1e"/> cannot be used as is.
  
−  
−
A closer analogy is the case of the sine function <ref>
  
−
http://www.pphmj.com/references/8246.htm <br>
  
−
https://mizugadro.mydns.jp/PAPERS/2014susin.pdf <br>
  
−
Dmitrii Kouznetsov. ''SUPER SIN''. ''Far East Journal of Mathematical Sciences'' 85(2), 219–238 (2014).
  
−
</ref>, but sine is antisymmetric, \(\sin(-z)=-\sin(z)\), which simplifies its analysis.
 
−
The Nemtsov function for \(q>0\) has no such symmetry, leading initially to doubts whether a superfunction could be constructed.
  
−  
−
Eventually, the construction succeeded.
  
−  
−
<div class="thumb tright" style="float:right; margin:-4px 0px 4px 8px">
  
−
{{pic|Boris.jpg|100px}}<small><center>[[Немцов Борис Ефимович|B. Nemtsov]]<ref>http://nemtsov.ru Борис Немцов</ref></center></small></div>
  
−  
−
The need for a special name for this function emerged on 2015-02-27, the day when [[Putin killed Nemtsov]].
 
−
As of 2025, no other scientific concept attached to that event has appeared.
 
−
The total [[corruption]] in Russia <ref name="medvedko">
  
−
http://kremlin.ru/transcripts/1566 <br>
  
−
D. Medvedev: ''Corruption in our country has acquired not just a large-scale character; it has become a habitual, everyday phenomenon...'' (2008)
  
−
</ref> prevents professional investigation of the crime.
 
−
Thus, the family name “Nemtsov’’ serves as a historical timestamp.
  
−  
−
By 2025, no better notation for this polynomial has been proposed.
 
−
The function symbol is written as lowercase \(\mathrm{nem}\), following the convention for mathematical functions, even though “Nemtsov’’ is a proper name.
 
−
Capitalization is used only when distinguishing it from derived functions such as [[ArqNem]], [[AuNem]], and [[SuNem]].
  
−  
−
== Definition and notations ==
  
−  
−
Let \(q\ge 0\) be a real parameter.
 
−
The [[Nemtsov function]] is defined for complex argument \(z\) by
  
−  
−
\[
  
−
\mathrm{nem}_q(z)=z+z^3+q\,z^4
  
−
\tag{1}
  
−
\]
  
−  
−
For \(q>0\), the algorithms described in the first Russian edition of «[[Суперфункции]]» (2014) cannot be applied in their original form; a small generalization is required.
 
−
Thus the function \(\mathrm{nem}_q\) is treated as a [[transfer function]].
  
−  
−
This appears to be the last remaining attempt (as of 2025) to produce a real-holomorphic, growing transfer function whose [[superfunction]] cannot be constructed by methods already known in the literature.
  
−  
−
But again, the attempt fails: both the [[superfunction]] [[SuNem]] and the [[Abel function]] [[AuNem]] are constructed using a method similar to that used for the sine function.
  
−  
−
Associated functions:
  
−  
−
[[Inverse function]]: \(\mathrm{ArqNem}_q\), satisfying
 
−
\[
  
−
\mathrm{nem}_q(\mathrm{ArqNem}_q(z)) = z
  
−
\]
  
−  
−
[[Superfunction]] \(\mathrm{SuNem}_q\), defined by
 
−
\[
  
−
\mathrm{SuNem}_q(z+1) = \mathrm{nem}_q(\mathrm{SuNem}_q(z))
  
−
\]
  
   
  +
[[Swift enforcement of the Budapest Memorandum]] is a [[gedankenexperiment]] describing
−
[[Abel function]]: \(\mathrm{AuNem}_q\), defined by
 
  +
how the guarantees given to Ukraine in 1994 could have been implemented if the
−
 
  +
United States and its allies had decided to enforce them consistently and
−
\[
  
  +
rapidly.
−
\mathrm{AuNem}_q(\mathrm{nem}_q(z)) = \mathrm{AuNem}_q(z) + 1
  
−
\]
  
   
  +
The scenario is intentionally simplified in order to make the causal logic
−
Several inverse functions exist: [[ArcNem]], [[ArkNem]], and [[ArqNem]].
 
  +
explicit.
−
They differ in branch-cut placement.
 
−
[[ArqNem]] turns out to be the correct one for defining holomorphic non-integer iterates near the positive real axis; therefore it is adopted as the default inverse.
  
   
  +
This [[emulation]] is generated by [[ChatGPT]] in order to explain the sense of the [[Budapest memorandum]] - under condition, that initially, it was not a fraud, nor a bluff, nor a trick since the beginning, and initially, its signing by the American President was not an attempt to destroy the International Law, nor an attempt to show that America again becomes a poor colony with external government - as it was 3 centures ago.
−
== Inverse function ==
  
   
  +
I such a way, the emulation below is constructed under fantastic assumption, that
−
<div class="thumb tright" style="float:right; margin:-4px 2px 200px 2px; width:410px; height:460px; background-color:#fff">
  
  +
the [[Moscovia]]n [[war crime]]s do not menace the world with the [[nuclear weapon]], and, at this conditions, the [[Budapest memorandum]] still could have been implemented.
−
<p style="margin:0px 0px 0px 9px;line-height:0px; width:200px; background-color:#fff">&nbsp; &nbsp; {{pic|Nembraplot.jpg|180px}}</p>
  
−
<p style="margin:-440px 0px 440px 220px;line-height:0px">{{pic|Nembrant.jpg|180px}}</p>
 +
<div class="thumb tright" style="float:right; margin:2px 0px 0px -24px; width:300px; background-color:#fff; line-height:1em">
  +
{{pic|AP_9412050636-min_(1).jpg|300px}} <small><center><p style="0px 0px 0px 8px; line-height:1em; width:300px">
−
<p style="margin:-432px 0px 4px 12px;"><small><center>\(x+\mathrm i y=\mathrm{NemBra}(q)\) and \(x+\mathrm i y=\mathrm{NemBran}(q)\)</center></small></p>
  
  +
Russian President Boris Yeltsin, left, American President Bill Clinton, Ukrainian President Leonid Kuchma, and British Prime Minister John Major, extreme right, sign the [[Budapest memorandum]] 1994.12.05 <ref>
  +
https://www.belfercenter.org/event/ukraines-nuclear-disarmament-25-years-after-budapest-memorandum
  +
Ukraine’s Nuclear Disarmament: 25 Years After the [[Budapest memorandum|Budapest Memorandum]]. Fri., Dec. 6, 2019 | 8:30am - 5:30pm.
  +
</ref></p></center></small>
 
</div>
  
</div>
  +
==Premise==
  +
In 1994 [[Ukraine]] agreed to give up the nuclear weapons inherited from the [[USSR]].
  +
In exchange, the United States, the United Kingdom and the Russian Federation
  +
signed the [[Budapest memorandum]], promising to respect the sovereignty and
  +
territorial integrity of Ukraine.
   
  +
The thought experiment below assumes that the US administration interprets these
−
To construct the inverse function in the complex plane, one must locate the saddle points and choose suitable branch cuts.
 
  +
assurances not merely as diplomatic language but as commitments affecting the
−
In the complex maps shown previously, the yellow lines indicate the branch cuts for functions \(\mathrm{ArqNem}_0\) and \(\mathrm{ArqNem}_2\).
 
  +
national interests and credibility of the United States.
−
These lines connect the two complex branch points of \(\mathrm{ArqNem}_q\).
  
   
  +
==Phase 1: Intelligence Assessment (1994–2000)==
−
For real \(q\), the Nemtsov function is real-holomorphic in the whole complex plane:
  
   
  +
1994.12.01
−
\[
  
−
\mathrm{nem}_q(z^{*}) = \mathrm{nem}_q(z)^{*}.
  
−
\]
  
   
  +
US intelligence services deliver an analytical memorandum to the President and
−
For positive real arguments, the function grows monotonically.
 
  +
to Congress.
−
This monotonicity is inherited by its inverse \(\mathrm{ArqNem}_q\), its Abel function \(\mathrm{AuNem}_q\), and its real iterates.
  
   
  +
The report states that:
−
The inverse function \(\mathrm{ArqNem}_q\) has complex [[branch point]]s.
 
−
One of them is described explicitly by the function [[NemBran]].
 
−
At \(z=\mathrm{NemBran}(q)\), the derivative of \(\mathrm{ArqNem}_q(z)\) becomes infinite.
  
   
  +
* After the collapse of the USSR, the Russian Federation may attempt to restore influence over former Soviet territories.
−
The branch points arise from the complex solutions \(A\) of the equation
  
   
  +
* Military pressure against neighboring states is considered plausible.
−
\[
  
−
\mathrm{nem}_q'(A) = 0.
  
−
\]
  
   
  +
* Potential targets include weak or politically unstable former Soviet republics.
−
One such solution is denoted \(\mathrm{NemBra}(q)\), and its image
  
   
  +
The report also evaluates two strategic scenarios.
−
\[
  
−
\mathrm{NemBran}(q) = \mathrm{nem}_q(\mathrm{NemBra}(q)).
  
−
\]
  
   
  +
Scenario A:
−
For positive \(q\), both real and imaginary parts of \(\mathrm{NemBra}(q)\) and \(\mathrm{NemBran}(q)\) are small (well below unity).
  
  +
Ukraine keeps nuclear weapons.
  +
This creates a regional balance of power but complicates nuclear
  +
non-proliferation policy.
   
  +
Scenario B:
−
Once \(\mathrm{NemBra}\) is implemented, an efficient algorithm for inverse functions becomes possible.
 
  +
Ukraine gives up nuclear weapons in exchange for credible security guarantees
−
Three versions—[[ArcNem]], [[ArkNem]], and [[ArqNem]]—differ only in branch-cut structure.
 
  +
from nuclear powers.
−
[[ArqNem]] is best suited for constructing holomorphic non-integer iterates near the positive real axis.
  
   
  +
The report concludes that Scenario B is viable only if the guarantees are considered enforceable.
−
The [[C++]] implementation of \(\mathrm{ArqNem}_q\) is available as [[arqnem.cin]].
 
−
Parameter \(q\) is stored in the global variable `Q`.
 
−
Before evaluating \(\mathrm{ArqNem}_q(z)\), the corresponding branch point must be computed via [[nembran.cin]] and stored in global variables (real and imaginary parts).
  
   
  +
1994.12.02
−
== Superfunction ==
  
   
  +
The President acknowledges the report and instructs intelligence agencies to
−
<div class="thumb tleft" style="float:left; margin:-4px 12px 4px -4px; background-color:#fff">
  
  +
monitor Russian military and political developments closely.
−
{{pic|Sunemplo4t.jpg|180px}}<small><center>\(y=\mathrm{SuNem}_{q}(x)\) for various \(q\)</center></small>
  
−
</div>
  
   
  +
==Phase 2: Early Warning Signals (1994–2008)==
−
<div class="thumb tright" style="float:right; margin:-24px 0px 0px 2px">
  
  +
{{fig|Pu01puppetmaster.jpg|100|-44|1|8|}}
−
&nbsp; {{pic|Sunem0map6.jpg|200px}}<small><center>\(u+\mathrm i v=\mathrm{SuNem}_0(x+\mathrm i y)\)</center></small>
  
  +
In subsequent years intelligence monitoring identifies several patterns:
  +
* Military operations against [[Ichkeria]] ([[First Chechen War]]).
  +
* Russian political rhetoric emphasizing influence over the “near abroad”.
  +
* Gradual increase of military presence of Russia and frozen conflicts in territories such as [[Transnistria]], [[Abkhazia]] and [[South Ossetia]].
   
  +
Analysts warn that these conflicts demonstrate a strategic method of RF:<br>
−
&nbsp; {{pic|Sunem1map6.jpg|200px}}<small><center>\(u+\mathrm i v=\mathrm{SuNem}_1(x+\mathrm i y)\)</center></small>
  
  +
1. Create or maintain unstable regions in neighboring states.<br>
  +
2. Use military presence to influence political outcomes.<br>
  +
3. Prevent integration of these states into Western institutions.
   
  +
The assessment concludes that Ukraine may eventually face similar pressure.
−
&nbsp; {{pic|Sunem2map6.jpg|200px}}<small><center>\(u+\mathrm i v=\mathrm{SuNem}_2(x+\mathrm i y)\)</center></small>
  
−
</div>
  
   
  +
2008.08.08, the [[Russian invasion into Georgia]] confirms this pattern
−
For the transfer function \(\mathrm{nem}_q\), the [[superfunction]] \(\mathrm{SuNem}_q\) is the real-holomorphic solution \(F\) of the transfer equation
  
   
  +
Intelligence briefings warn that the Georgian war could represent a precedent
−
\[
  
  +
for future operations against other neighbors, including Ukraine.
−
F(z+1) = \mathrm{nem}_q(F(z)),
  
−
\]
  
   
  +
==Phase 3: Pre-Crisis Alert (2008–2014)==
−
with the asymptotic condition at \(-\infty\):
  
   
  +
After the Georgian war, US intelligence agencies intensify monitoring of
−
\[
  
  +
Russian military modernization and political strategy.
−
F(z) = \frac{1}{\sqrt{-2z}}
  
  +
{{fig|BackInTheUSSR.png|144|0|2|2|[[Back in the USSR]]}}
−
\left( 1 - \frac{q}{\sqrt{-2z}}
  
  +
Analysts warn that:
−
+ O\!\left( \frac{\ln(-z)}{z} \right)
  
−
\right),
  
−
\qquad z\to -\infty.
  
−
\]
  
   
  +
* Crimea has strategic value because of the Black Sea Fleet.
−
To fully specify \(\mathrm{SuNem}_q\), an additional normalization is imposed:
  
  +
* Political instability in Ukraine could create an opportunity for intervention.
  +
* Hybrid methods (covert troops, irregular forces, propaganda) may be used to obscure responsibility.
   
  +
By early 2014 intelligence reports conclude that military intervention in Ukraine is highly probable.
−
\[
  
−
\mathrm{SuNem}_q(0) = 1.
  
−
\]
  
   
  +
==Phase 4: Immediate Response (2014)==
−
The plot \(y=\mathrm{SuNem}_q(x)\) appears at left for several values of \(q\).
 
  +
{{fig|PuSanctions.jpeg|160|-20|4|8|}}
−
The function grows monotonically from \(0\) at \(-\infty\), reaches \(1\) at argument \(0\), and then increases rapidly for positive arguments.
 
  +
{{fig|107253_original.jpg|100|4|0|8|}}
−
Larger \(q\) produces faster growth.
  
  +
When armed forces without insignia appear in Crimea in February 2014,
  +
the United States and the United Kingdom interpret the event as a violation of
  +
the [[Budapest memorandum]].
   
  +
Within days they take several coordinated actions.
−
Complex maps \(u+\mathrm i v=\mathrm{SuNem}_q(x+\mathrm i y)\) are displayed for \(q=0\), \(q=1\), and \(q=2\).
  
   
  +
Diplomatic action:
−
To construct \(\mathrm{SuNem}_q\), one first constructs any superfunction \(F\) with the correct asymptotics, and then defines
  
  +
An official warning is delivered to the Russian leadership stating that
  +
continued occupation of Ukrainian territory will trigger collective responses.
   
  +
Legal action:
−
\[
  
  +
The issue is immediately raised in the United Nations and international courts.
−
\mathrm{SuNem}_q(z) = F(x_1 + z),
  
−
\]
  
   
  +
Military deterrence:
−
where \(x_1\) is the real number satisfying \(F(x_1)=1\).
  
  +
NATO rapidly deploys naval and air units to the region as a defensive measure.
   
  +
Economic pressure:
−
== Abel function ==
  
  +
Assets connected to the Russian state and key political figures are frozen.
   
  +
These measures are presented not as escalation but as enforcement of previously
−
<div style="float:left; width:200px; margin:-6px 18px 5px 0px; background-color:#fff; width:210px">
  
  +
agreed commitments.
−
{{pic|Aunemplot.jpg|200px}}<small><center>\(y=\mathrm{AuNem}_q(x)\) for \(q=0\), \(1\), \(2\)</center></small>
  
−
</div>
  
   
  +
==Phase 5: Resolution Scenario==
−
For the Abel function of the Nemtsov function, the notation [[AuNem]] is used.
  
   
  +
Under coordinated diplomatic, economic and military pressure the Russian
−
The explicit plot \(y=\mathrm{AuNem}_q(x)\) versus \(x\) is shown at left for \(q=0\), \(1\), and \(2\).
 
  +
leadership faces a strategic choice:
−
This plot coincides with reflecting the plots of \(\mathrm{SuNem}_q\) across the bisector of the first quadrant.
 
−
This symmetry provides a numerical test:
  
   
  +
continue escalation with severe international consequences,
−
\[
  
  +
or withdraw forces and negotiate.
−
\mathrm{SuNem}_q(\mathrm{AuNem}_q(x)) = x,
  
−
\qquad
  
−
\mathrm{AuNem}_q = \mathrm{SuNem}_q^{-1}.
  
−
\]
  
   
  +
In this simplified scenario the crisis ends with withdrawal of occupying
−
As the inverse of the superfunction, the Abel function satisfies the Abel equation
  
  +
forces and restoration of Ukrainian control over its territory.
   
  +
==Consequences==
−
\[
  
  +
{{fig|Mvn78red1.jpg|144|-20|2|2|}}
−
\mathrm{AuNem}_q(\mathrm{nem}_q(z)) = \mathrm{AuNem}_q(z) + 1.
  
  +
The rapid enforcement of the [[Budapest memorandum]] produces several effects:
−
\]
  
  +
* international agreements gain credibility;
  +
* aggressive territorial revisionism becomes less attractive;
  +
* states that renounced nuclear weapons see that security guarantees can work.
   
  +
==Methodological note==
−
The normalization is
  
   
  +
This article describes a hypothetical scenario rather than a prediction or a
−
\[
  
  +
historical claim.
−
\mathrm{AuNem}_q(1) = 0,
  
−
\]
  
   
  +
Its purpose is to illustrate how intelligence analysis, diplomatic pressure,
−
because \(\mathrm{SuNem}_q(0)=1\).
  
  +
economic tools and military deterrence could theoretically interact in the
  +
enforcement of international security guarantees.
   
  +
==Notes by Editor==
−
The asymptotic expansion of \(\mathrm{AuNem}_q\) near \(0\) follows from inverting the asymptotics of \(\mathrm{SuNem}_q\) at \(-\infty\).
  
   
−
== Iterates ==
 +
==References==
  +
{{ref}}
   
  +
https://en.wikipedia.org/wiki/Budapest_Memorandum
−
<div class="thumb tright" style="float:right; margin:-20px 0px 2px 8px; background-color:#fff">
  
−
{{pic|Itnem00plot.jpg|216px}}<small><center>\(y=\mathrm{nem}_0^{\,n}(x)\) for various \(n\)</center></small>
  
−
{{pic|Itnem10plot.jpg|216px}}<small><center>\(y=\mathrm{nem}_1^{\,n}(x)\) for various \(n\)</center></small>
  
−
{{pic|Itnem20plot.jpg|216px}}<small><center>\(y=\mathrm{nem}_2^{\,n}(x)\) for various \(n\)</center></small>
  
−
</div>
  
   
  +
https://treaties.un.org/doc/Publication/UNTS/Volume%203007/Part/volume-3007-I-52241.pdf
−
Using the functions \(\mathrm{SuNem}_q\) and \(\mathrm{AuNem}_q\), the \(n\)-th iterate of the Nemtsov function can be expressed as
  
  +
Memorandum on security assurances in connection with Ukraine’s accession to the Treaty on the Non-Proliferation of Nuclear Weapons. Budapest, 5 December 1994 //
  +
Entry into force: 5 December 1994 by signature //
  +
Authentic texts: English, Russian and Ukrainian //
  +
Registration with the Secretariat of the United Nations: Ukraine, 2 October 2014
   
  +
https://www.gutenberg.org/files/236/236-h/236-h.htm#link2H_4_0001 [[Rudyard Kipling]]. THE JUNGLE BOOK. Release Date: January 16, 2006 [EBook #236] Last Updated: October 6, 2016. .. Akela lifted his head again and said, “He has eaten our food. He has slept with us. He has driven game for us. He has broken no word of the Law of the Jungle.”// “Also, I paid for him with a bull when he was accepted. The worth of a bull is little, but Bagheera’s honor is something that he will perhaps fight for,” said Bagheera in his gentlest voice.// “A bull paid ten years ago!” the Pack snarled. “What do we care for bones ten years old?” ..
−
\[
  
−
\mathrm{nem}_q^{\,n}(z)
  
−
= \mathrm{SuNem}_q\!\big(n + \mathrm{AuNem}_q(z)\big).
  
−
\]
  
   
  +
2015.11.10.
−
Plots of the iterates for \(q=0\), \(q=1\), and \(q=2\) appear at right.
 
  +
https://www.youtube.com/watch?v=qA8kaGnf4ro [[Marco Rubio]]: Putin Is A Gangster. Published on Nov 10, 2015. Fox Business Debate 11.10.15.
−
For positive integer \(n\), the iterate grows rapidly; for negative \(n\), the growth is slow.
  
   
  +
2017.04.06.
−
The zeroth iterate (\(n=0\)) is the identity function.
 
  +
https://www.youtube.com/watch?v=i1RSG2TeHAU Rubio: 'Putin is a war criminal' CNN Apr 6, 2017
−
In the figures, this curve appears as a green line.
  
   
  +
2022.03.04. <small>
−
Because \(\mathrm{ArqNem}_q\) has a singularity at \(0\), non-integer iterates are not defined at \(0\) or on the negative real axis, though they approach \(0\) from the right.
  
  +
https://news.sky.com/story/russian-invasion-wouldnt-have-happened-if-ukraine-still-had-nuclear-weapons-ukrainian-political-adviser-says-12556811
  +
</small>Russian invasion 'wouldn't have happened' if Ukraine still had nuclear weapons, Ukrainian political adviser says
  +
[[Svitlana Zalishchuk]], a foreign policy adviser to the Ukrainian deputy prime minister, told Sky News that Ukraine giving up its nuclear weapons was "without a doubt" a mistake. By [[Alexa Phillips]], news reporter Friday 4 March 2022 13:10, UK
   
  +
2022.11.01.
−
The iterates satisfy the symmetry
  
  +
https://www.newyorker.com/news/our-columnists/why-vladimir-putin-would-use-nuclear-weapons-in-ukraine
  +
Why Vladimir Putin Would Use Nuclear Weapons in Ukraine..
  +
The more the Kremlin has signalled its readiness to drop a nuclear bomb, the more the rest of the world has sought a reason to believe that it will not. By [[Masha Gessen]] November 1, 2022.
   
  +
2023.03.17.<small>
−
\[
  
  +
https://www.icc-cpi.int/news/situation-ukraine-icc-judges-issue-arrest-warrants-against-vladimir-vladimirovich-putin-and
−
\mathrm{nem}_q^{-n}(z) = \mathrm{ArqNem}_q^{\,n}(z).
  
  +
</small> Today, 17 March 2023, Pre-Trial Chamber II of the [[International Criminal Court]] (“ICC” or “the Court”) issued warrants of arrest for two individuals in the context of the situation in Ukraine: Mr Vladimir Vladimirovich Putin and Ms Maria Alekseyevna Lvova-Belova. ..
−
\]
  
   
  +
2024.04.20. https://www.theguardian.com/us-news/2024/apr/20/us-house-approves-61bn-aid-ukraine [[Robert Tait]], Joan E Greve and Chris Michael. US House approves $61bn in military aid for Ukraine after months of stalling. Sat 20 Apr 2024 22.46 BST
−
== Applications ==
  
   
  +
2024.11.13.
−
The Nemtsov function was proposed as a candidate transfer function for which the superfunction and the Abel function would be difficult to construct using the [[exotic iterate]] at its fixed point \(0\).
 
  +
https://unn.ua/en/news/ukraine-may-restore-nuclear-status-if-us-aid-is-cut-off-the-times
−
The difficulty lay mostly in constructing the correct inverse function \(\mathrm{ArqNem}_q\), choosing branch cuts appropriately, and recognizing that \(\mathrm{ArqNem}_q\) rather than \(\mathrm{ArcNem}\) or \(\mathrm{ArkNem}\) should serve as the default inverse.
  
  +
Ukraine may restore nuclear status if US aid is cut off - The Times
  +
Kyiv • UNN November 13 2024, 08:54 PM ..
  +
The Times reports on the possibility that Ukraine could develop a [[nuclear weapon]] in a few months if US military aid is cut. ..
   
  +
2025.03.12.
−
Other inverse functions differ only in branch-cut placement and do not yield a real-holomorphic Abel function on as wide a domain.
  
  +
https://www.japantimes.co.jp/commentary/2025/03/12/world/ukraine-security-nuclear-proliferation/
  +
Lessons from Ukraine: Allies are fickle, nukes are forever//
  +
If Kyiv is left to capitulate, others will see security as stemming from power, not assurances. BY THOMAS O. FALK .. Mar 12, 2025
   
  +
2025.04.21.<small>
−
Once the inverse iterates converge to the fixed point, the exotic iteration becomes straightforward.
 
  +
https://global.espreso.tv/military-news-ukraine-could-technically-restore-nuclear-status-in-just-four-years-experts-say
−
The same approach can be used for other exotic transfer functions whose expansion at the fixed point begins with a linear term and a cubic term (the cubic coefficient can always be normalized to \(1\)).
 
  +
</small> Ukraine could technically restore nuclear status in just four years, experts say// 21 april, 2025 monday 11:16
−
The required transformation appears in the last row of the [[Table of superfunctions]].
  
   
  +
2025.07.01. <small>
−
== Warning ==
  
  +
https://united24media.com/latest-news/uk-must-help-ukraine-regain-nuclear-arms-retired-british-commander-says-8787
−  
  +
</small>[[Roman Kohanets]].
−
The name “[[Nemtsov function]]’’ was chosen as a mnemonic connected to a notable event of the 21st century.
  
  +
UK Must Help Ukraine Regain Nuclear Arms, Retired British Commander Says
−  
  +
May 31, 2025 16:42 Updated Jul 01, 2025 00:25.
−
It is not intended as a political statement or as an attempt to appeal to the Russian usurper or his accomplices.
 
  +
</small>
−
The ongoing rise of corruption <ref name="medvedko"/> indicates that moral appeals are meaningless in that context, even while some impostors pretend they can “end war in 24 hours’’ by appeasing aggression and war crimes.
  
−  
−
Nevertheless, the editor reserves the right to use the most convenient system of notation.
  
−  
−
== Acknowledgement ==
  
−  
−
[[ChatGPT]] helped to improve this article.
  
−  
−
== References ==
  
−
{{ref}}
  
   
 
{{fer}}
  
{{fer}}
   
−
== Keywords ==
 +
==Keywords==
−
«[[Abel function]]»,
 +
«[[Advices for Trump]]»,
−
«[[ArqNem]]»,
 +
«[[]]»,
−
«[[AuNem]]»,
 +
«[[Alternative Math]]»,
−
«[[Book]]»,
 +
«[[Annexation of Canada]]»,
−
«[[C++]]»,
 +
«[[Annexation of Crimea]]»,
−
«[[Exotic iterate]]»,
 +
«[[Annexation of Greenland]]»,
−
«[[Exotic iteration]]»,
 +
«[[Annexation of Ichkeria]]»,
−
«[[Latex]]»,
 +
«[[Appeasing aggression]]»,
−
«[[Mathematica]]»,
 +
«[[Barack Obama]]»,
−
«[[Nemtsov function]]»,
 +
«[[Bill Clinton]]»,
−
«[[Putin killed Nemtsov]]»,
 +
«[[Budapest memorandum]]»,
−
«[[SuNem]]»,
 +
«[[CIA]]»,
−
«[[Superfunction]]»,
 +
«[[CIA methodology‎‎]]»,
−
«[[Superfunctions]]»,
 +
«[[ChatGPT]]»,
−
«[[Table of superfunctions]]»,
 +
«[[Collapse of RF]]»,
  +
«[[Collapse of USA]]»,
  +
«[[Collapse of USSR]]»,
  +
«[[Corruption]]»,
  +
«[[Designate Russia as state sponsor of terrorism]]»,
  +
«[[Designate Russia as terrorist state]]»,
  +
«[[Donald Trump]]»,
  +
«[[Donroe Doctrine]]»,
  +
«[[Duration]]»,
  +
«[[Dvizhuha]]»,
  +
«[[Fascism]]»,
  +
«[[Gedankenexperment]]»,
  +
«[[Hybrid war]]»,
  +
«[[Jinping]]»,
  +
«[[Joe Biden]]»,
  +
«[[Moscovia]]»,
  +
«[[Mystic weakness]]»,
  +
«[[New World Order]]»,
  +
«[[Pahanat]]»,
  +
«[[Propaganda]]»,
  +
«[[Putin]]»,
  +
«[[Putin world war]]»,
  +
«[[Realpolitik]]»,
  +
«[[Russian Agent Governs America]]»,
  +
«[[Russian invasion into Ichkeria]]»,
  +
«[[Russian invasion into Ukraine]]»,
  +
«[[Russki mir]]»,
  +
«[[Sabotage]]»,
  +
«[[Small Victorious War]]»,
  +
«[[Small victorious war]]»,
  +
«[[TROI‎‎]]»,
  +
«[[Trump as KGB agent]]»,
  +
«[[Two concepts of collapse of the USSR‎‎]]»,
  +
«[[USA]]»,
  +
«[[Witkoff-Lavrov Pact Emulation]]»,
  +
«[[Wreckage of Crimea]]»,
   
  +
«[[Аннексия Ичкерии]]»,
−
«[[Немцов Борис Ефимович]]»,
  
−
«[[Путин убил Немцова]]»,
 +
«[[Аннексия Крыма]]»,
  +
«[[Будапештский меморандум]]»,
  +
«[[Гибридная война]]»,
  +
«[[Движуха]]»,
  +
«[[Ихтамнет]]»,
  +
«[[Кисельные берега]]»,
  +
«[[Коррупция]]»,
  +
«[[Маленькая победоносная война]]»,
  +
«[[Московия]]»,
  +
«[[Мы попадём в рай а они сдохнут]]»,
  +
«[[Мысленный эксперимент]]»,
  +
«[[Новый Мировой Порядок]]»,
  +
«[[Остов Крым]]»,
  +
«[[Остров Крым]]»,
  +
«[[От Аляски до Одесы]]»,
  +
«[[Паханат]]»,
  +
«[[Путинский план спецоперации]]»,
  +
«[[Российское вторжение в Грузию]]»,
  +
«[[Российское вторжение в Украину]]»
  +
«[[Саботаж]]»,
  +
«[[Синий вертолёт]]»,
  +
«[[Спецоперация]]»,
  +
«[[]]»,
  +
«[[Трамп Дональд Фредович]]»,
  +
«[[Умиротворение агрессора]]»,
  +
«[[Утопия]]»,
  +
«[[Фашизм]]»,
  +
«[[Эмуляция]]»,
   
−
[[Category:Abel function]]
 +
[[Category:Advices for Trump]]
−
[[Category:Abelfunction]]
 +
[[Category:Annexation of Canada]]
−
[[Category:ArqNem]]
 +
[[Category:Annexation of Crimea]]
−
[[Category:AuNem]]
 +
[[Category:Annexation of Greenland]]
−
[[Category:Book]]
 +
[[Category:Annexation of Ichkeria]]
−
[[Category:Boris Nemtsov]]
 +
[[Category:Anti-Putin coalition]]
−
[[Category:C++]]
 +
[[Category:Appeasing aggression]]
−
[[Category:English]]
 +
[[Category:Barack Obama]]
−
[[Category:Exotic iterate]]
 +
[[Category:Bill Clinton]]
−
[[Category:Latex]]
 +
[[Category:Budapest memorandum]]
−
[[Category:Mathematica]]
 +
[[Category:CIA]]
−
[[Category:Nemtsov function]]
 +
[[Category:ChatGPT]]
−
[[Category:Putin killed Nemtsov]]
 +
[[Category:Collapse of RF]]
−
[[Category:SuNem]]
 +
[[Category:Collapse of USA]]
−
[[Category:Superfunction]]
 +
[[Category:Collapse of USSR]]
−
[[Category:Superfunctions]]
 +
[[Category:Corruption]]
  +
[[Category:Designate Russia as state sponsor of terrorism]]
  +
[[Category:Designate Russia as terrorist state]]
  +
[[Category:Donald Trump]]
  +
[[Category:Donroe Doctrine]]
  +
[[Category:Dvizhuha]]
  +
[[Category:Fascism]]
  +
[[Category:Gedankenexperment]]
  +
[[Category:History]]
  +
[[Category:Hybrid war]]
  +
[[Category:Joe Biden]]
  +
[[Category:Moscovia]]
  +
[[Category:New World Order]]
  +
[[Category:Pahanat]]
  +
[[Category:Propaganda]]
  +
[[Category:Putin]]
  +
[[Category:Putin world war]]
  +
[[Category:Realpolitik]]
  +
[[Category:Russian Agent Governs America]]
  +
[[Category:Russian invasion into Ichkeria]]
  +
[[Category:Russian invasion into Ukraine]]
  +
[[Category:Russki mir]]
  +
[[Category:Sabotage]]
  +
[[Category:Sci-fi]]
  +
[[Category:Trump as KGB agent]]
  +
[[Category:USA]]
  +
[[Category:Utopia]]
  +
[[Category:Witkoff-Lavrov Pact Emulation]]
  +
[[Category:Wreckage of Crimea]]

Latest revision as of 20:34, 12 March 2026


Swift enforcement of the Budapest Memorandum is a gedankenexperiment describing how the guarantees given to Ukraine in 1994 could have been implemented if the United States and its allies had decided to enforce them consistently and rapidly.

The scenario is intentionally simplified in order to make the causal logic explicit.

This emulation is generated by ChatGPT in order to explain the sense of the Budapest memorandum - under condition, that initially, it was not a fraud, nor a bluff, nor a trick since the beginning, and initially, its signing by the American President was not an attempt to destroy the International Law, nor an attempt to show that America again becomes a poor colony with external government - as it was 3 centures ago.

I such a way, the emulation below is constructed under fantastic assumption, that the Moscovian war crimes do not menace the world with the nuclear weapon, and, at this conditions, the Budapest memorandum still could have been implemented.

Ap941.jpg

Russian President Boris Yeltsin, left, American President Bill Clinton, Ukrainian President Leonid Kuchma, and British Prime Minister John Major, extreme right, sign the Budapest memorandum 1994.12.05 [1]

Premise

In 1994 Ukraine agreed to give up the nuclear weapons inherited from the USSR. In exchange, the United States, the United Kingdom and the Russian Federation signed the Budapest memorandum, promising to respect the sovereignty and territorial integrity of Ukraine.

The thought experiment below assumes that the US administration interprets these assurances not merely as diplomatic language but as commitments affecting the national interests and credibility of the United States.

Phase 1: Intelligence Assessment (1994–2000)

1994.12.01

US intelligence services deliver an analytical memorandum to the President and to Congress.

The report states that:

  • After the collapse of the USSR, the Russian Federation may attempt to restore influence over former Soviet territories.
  • Military pressure against neighboring states is considered plausible.
  • Potential targets include weak or politically unstable former Soviet republics.

The report also evaluates two strategic scenarios.

Scenario A: Ukraine keeps nuclear weapons. This creates a regional balance of power but complicates nuclear non-proliferation policy.

Scenario B: Ukraine gives up nuclear weapons in exchange for credible security guarantees from nuclear powers.

The report concludes that Scenario B is viable only if the guarantees are considered enforceable.

1994.12.02

The President acknowledges the report and instructs intelligence agencies to monitor Russian military and political developments closely.

Phase 2: Early Warning Signals (1994–2008)

Pu01puppetmaster.jpg

In subsequent years intelligence monitoring identifies several patterns:

Analysts warn that these conflicts demonstrate a strategic method of RF:
1. Create or maintain unstable regions in neighboring states.
2. Use military presence to influence political outcomes.
3. Prevent integration of these states into Western institutions.

The assessment concludes that Ukraine may eventually face similar pressure.

2008.08.08, the Russian invasion into Georgia confirms this pattern

Intelligence briefings warn that the Georgian war could represent a precedent for future operations against other neighbors, including Ukraine.

Phase 3: Pre-Crisis Alert (2008–2014)

After the Georgian war, US intelligence agencies intensify monitoring of Russian military modernization and political strategy.

BackInTheUSSR.png
Back in the USSR

Analysts warn that:

  • Crimea has strategic value because of the Black Sea Fleet.
  • Political instability in Ukraine could create an opportunity for intervention.
  • Hybrid methods (covert troops, irregular forces, propaganda) may be used to obscure responsibility.

By early 2014 intelligence reports conclude that military intervention in Ukraine is highly probable.

Phase 4: Immediate Response (2014)

PuSanctions.jpeg
107253 original.jpg

When armed forces without insignia appear in Crimea in February 2014, the United States and the United Kingdom interpret the event as a violation of the Budapest memorandum.

Within days they take several coordinated actions.

Diplomatic action: An official warning is delivered to the Russian leadership stating that continued occupation of Ukrainian territory will trigger collective responses.

Legal action: The issue is immediately raised in the United Nations and international courts.

Military deterrence: NATO rapidly deploys naval and air units to the region as a defensive measure.

Economic pressure: Assets connected to the Russian state and key political figures are frozen.

These measures are presented not as escalation but as enforcement of previously agreed commitments.

Phase 5: Resolution Scenario

Under coordinated diplomatic, economic and military pressure the Russian leadership faces a strategic choice:

continue escalation with severe international consequences, or withdraw forces and negotiate.

In this simplified scenario the crisis ends with withdrawal of occupying forces and restoration of Ukrainian control over its territory.

Consequences

Mvn78red1.jpg

The rapid enforcement of the Budapest memorandum produces several effects:

  • international agreements gain credibility;
  • aggressive territorial revisionism becomes less attractive;
  • states that renounced nuclear weapons see that security guarantees can work.

Methodological note

This article describes a hypothetical scenario rather than a prediction or a historical claim.

Its purpose is to illustrate how intelligence analysis, diplomatic pressure, economic tools and military deterrence could theoretically interact in the enforcement of international security guarantees.

Notes by Editor

References

  1. ↑ https://www.belfercenter.org/event/ukraines-nuclear-disarmament-25-years-after-budapest-memorandum Ukraine’s Nuclear Disarmament: 25 Years After the Budapest Memorandum. Fri., Dec. 6, 2019 | 8:30am - 5:30pm.

https://en.wikipedia.org/wiki/Budapest_Memorandum

https://treaties.un.org/doc/Publication/UNTS/Volume%203007/Part/volume-3007-I-52241.pdf Memorandum on security assurances in connection with Ukraine’s accession to the Treaty on the Non-Proliferation of Nuclear Weapons. Budapest, 5 December 1994 // Entry into force: 5 December 1994 by signature // Authentic texts: English, Russian and Ukrainian // Registration with the Secretariat of the United Nations: Ukraine, 2 October 2014

https://www.gutenberg.org/files/236/236-h/236-h.htm#link2H_4_0001 Rudyard Kipling. THE JUNGLE BOOK. Release Date: January 16, 2006 [EBook #236] Last Updated: October 6, 2016. .. Akela lifted his head again and said, “He has eaten our food. He has slept with us. He has driven game for us. He has broken no word of the Law of the Jungle.”// “Also, I paid for him with a bull when he was accepted. The worth of a bull is little, but Bagheera’s honor is something that he will perhaps fight for,” said Bagheera in his gentlest voice.// “A bull paid ten years ago!” the Pack snarled. “What do we care for bones ten years old?” ..

2015.11.10. https://www.youtube.com/watch?v=qA8kaGnf4ro Marco Rubio: Putin Is A Gangster. Published on Nov 10, 2015. Fox Business Debate 11.10.15.

2017.04.06. https://www.youtube.com/watch?v=i1RSG2TeHAU Rubio: 'Putin is a war criminal' CNN Apr 6, 2017

2022.03.04. https://news.sky.com/story/russian-invasion-wouldnt-have-happened-if-ukraine-still-had-nuclear-weapons-ukrainian-political-adviser-says-12556811 Russian invasion 'wouldn't have happened' if Ukraine still had nuclear weapons, Ukrainian political adviser says Svitlana Zalishchuk, a foreign policy adviser to the Ukrainian deputy prime minister, told Sky News that Ukraine giving up its nuclear weapons was "without a doubt" a mistake. By Alexa Phillips, news reporter Friday 4 March 2022 13:10, UK

2022.11.01. https://www.newyorker.com/news/our-columnists/why-vladimir-putin-would-use-nuclear-weapons-in-ukraine Why Vladimir Putin Would Use Nuclear Weapons in Ukraine.. The more the Kremlin has signalled its readiness to drop a nuclear bomb, the more the rest of the world has sought a reason to believe that it will not. By Masha Gessen November 1, 2022.

2023.03.17. https://www.icc-cpi.int/news/situation-ukraine-icc-judges-issue-arrest-warrants-against-vladimir-vladimirovich-putin-and Today, 17 March 2023, Pre-Trial Chamber II of the International Criminal Court (“ICC” or “the Court”) issued warrants of arrest for two individuals in the context of the situation in Ukraine: Mr Vladimir Vladimirovich Putin and Ms Maria Alekseyevna Lvova-Belova. ..

2024.04.20. https://www.theguardian.com/us-news/2024/apr/20/us-house-approves-61bn-aid-ukraine Robert Tait, Joan E Greve and Chris Michael. US House approves $61bn in military aid for Ukraine after months of stalling. Sat 20 Apr 2024 22.46 BST

2024.11.13. https://unn.ua/en/news/ukraine-may-restore-nuclear-status-if-us-aid-is-cut-off-the-times Ukraine may restore nuclear status if US aid is cut off - The Times Kyiv • UNN November 13 2024, 08:54 PM .. The Times reports on the possibility that Ukraine could develop a nuclear weapon in a few months if US military aid is cut. ..

2025.03.12. https://www.japantimes.co.jp/commentary/2025/03/12/world/ukraine-security-nuclear-proliferation/ Lessons from Ukraine: Allies are fickle, nukes are forever// If Kyiv is left to capitulate, others will see security as stemming from power, not assurances. BY THOMAS O. FALK .. Mar 12, 2025

2025.04.21. https://global.espreso.tv/military-news-ukraine-could-technically-restore-nuclear-status-in-just-four-years-experts-say Ukraine could technically restore nuclear status in just four years, experts say// 21 april, 2025 monday 11:16

2025.07.01. https://united24media.com/latest-news/uk-must-help-ukraine-regain-nuclear-arms-retired-british-commander-says-8787 Roman Kohanets. UK Must Help Ukraine Regain Nuclear Arms, Retired British Commander Says May 31, 2025 16:42 Updated Jul 01, 2025 00:25.

Keywords

«Advices for Trump», «[[]]», «Alternative Math», «Annexation of Canada», «Annexation of Crimea», «Annexation of Greenland», «Annexation of Ichkeria», «Appeasing aggression», «Barack Obama», «Bill Clinton», «Budapest memorandum», «CIA», «CIA methodology‎‎», «ChatGPT», «Collapse of RF», «Collapse of USA», «Collapse of USSR», «Corruption», «Designate Russia as state sponsor of terrorism», «Designate Russia as terrorist state», «Donald Trump», «Donroe Doctrine», «Duration», «Dvizhuha», «Fascism», «Gedankenexperment», «Hybrid war», «Jinping», «Joe Biden», «Moscovia», «Mystic weakness», «New World Order», «Pahanat», «Propaganda», «Putin», «Putin world war», «Realpolitik», «Russian Agent Governs America», «Russian invasion into Ichkeria», «Russian invasion into Ukraine», «Russki mir», «Sabotage», «Small Victorious War», «Small victorious war», «TROI‎‎», «Trump as KGB agent», «Two concepts of collapse of the USSR‎‎», «USA», «Witkoff-Lavrov Pact Emulation», «Wreckage of Crimea»,

«Аннексия Ичкерии», «Аннексия Крыма», «Будапештский меморандум», «Гибридная война», «Движуха», «Ихтамнет», «Кисельные берега», «Коррупция», «Маленькая победоносная война», «Московия», «Мы попадём в рай а они сдохнут», «Мысленный эксперимент», «Новый Мировой Порядок», «Остов Крым», «Остров Крым», «От Аляски до Одесы», «Паханат», «Путинский план спецоперации», «Российское вторжение в Грузию», «Российское вторжение в Украину» «Саботаж», «Синий вертолёт», «Спецоперация», «[[]]», «Трамп Дональд Фредович», «Умиротворение агрессора», «Утопия», «Фашизм», «Эмуляция»,

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