A Convergent Greedy Residue Expansion for the MRB Constant
A Convergent Greedy Residue Expansion for the MRB Constant
Marvin Burns
Marvin Burns
August 2026
August 2026
Abstract / Overview
Abstract / Overview
This notebook organizes the main structure of the paper around the residue-magnitude family and the greedy procedure built from it.
A(m)sin, m≥2,
2
π
-1/m
m
π
m
The principal goals are to document and computationally support the following claims:
◼
the family arises naturally from the reciprocal-pole structure in the analytic theory of the MRB constant;
A(m)
◼
A(m)
m≥4
◼
any sufficiently small residual can be greedily reduced by choosing the nearest value among neighboring residue magnitudes;
◼
this yields a uniform contraction once the residual lies below ;
A(4)1/π
◼
every real number admits a convergent greedy expansion in signed members of the family ;
A(m)
◼
for nonterminating orbits, the residual satisfies the asymptotically quadratic estimate
≤+o(1).
E
n+1
1
4
E
n
2
|
1. The MRB constant and the residue family
1. The MRB constant and the residue family
The MRB constant is defined by the alternating series (-1).We store a high-precision reference value for experiments:
C
MRB
∞
∑
n1
n
(-1)
1/n
n
In[]:=
ClearAll[mrbN,mrbTerm,mrbPartialSum];mrbN=SetPrecision[0.1878596424620671202485179340542732300559030949,80];mrbTerm[n_Integer?Positive]:=(-1)^n(n^(1/n)-1)mrbPartialSum[n_Integer?Positive]:=Sum[mrbTerm[k],{k,1,n}]
The residue-magnitude family is
In[]:=
ClearAll[a];a[m_Integer?Positive]:=(2/Pi)m^(-1/m)Sin[Pi/m]
A continuous extension is useful for analysis:
In[]:=
ClearAll[aContinuous];aContinuous[x_]:=(2/Pi)x^(-1/x)Sin[Pi/x]
2. The distinguished term A(8)
2. The distinguished term
A(8)
A special role is played by Symbolic and numerical evaluation:
A(8)sinsin(π/8)π.
2
π
-1/8
8
π
8
5/8
2
π
2-
2
3/8
2
In[]:=
ClearAll[a8Exact,a8Numeric,a8Error];a8Exact=FullSimplify[a[8]]a8Numeric=N[a8Exact,40]a8Error=N[mrbN-a8Exact,40]
Out[]=
5/8
2
π
8
π
Out[]=
0.1878596613801383990492532045996126144757
Out[]=
-1.891807127880073527054533938441983110588×
-8
10
You can also compare absolute error directly:
In[]:=
Abs[a8Error]
Out[]=
1.891807127880073527054533938441983110588×
-8
10
This is the starting point of the MRB greedy residue expansion.
3. Large-x asymptotics of A(x)
3. Large- asymptotics of
x
A(x)
The continuous function has asymptotic expansion This can be explored symbolically.
A(x)sin, x>1,
2
π
-1/x
x
π
x
A(x)-+-3+O.
2
x
2logx
2
x
2
(logx)
2
π
3
x
3
(logx)
4
x
3.1 Series scaffolding
3.1 Series scaffolding
In[]:=
ClearAll[x];Normal@Series[Exp[-Log[x]/x],{x,Infinity,3}]
Out[]=
1-+-
Log[x]
x
2
Log[x]
2
2
x
3
Log[x]
6
3
x
In[]:=
Normal@Series[Sin[Pi/x],{x,Infinity,3}]
Out[]=
-+
3
π
6
3
x
π
x
You can combine these pieces manually or through asymptotic expansion workflows.
3.2 Numerical confirmation
3.2 Numerical confirmation
In[]:=
ClearAll[aAsym2];aAsym2[x_]:=2/x-2Log[x]/x^2+((Log[x])^2-Pi^2/3)/x^3Table[{n,N[aContinuous[n],20],N[aAsym2[n],20],N[aContinuous[n]-aAsym2[n],20]},{n,{10,20,50,100,200,500}}]
Out[]=
{{10,0.15626527733388680807,0.15596032811690103146,0.00030494921698577661760},{20,0.085735574936024959297,0.085731906597369608914,3.6683386553503826777×},{50,0.036965348018315666278,0.036966494042547904046,-1.1460242322377679763×},{100,0.019096710075790007003,0.019096883687110598866,-1.7361132059186248114×},{200,0.0097381618927803906639,0.0097381819190204691748,-2.0026240078510964378×},{500,0.0039505648361280462754,0.0039505657870980886920,-9.5097004241653252375×}}
-6
10
-6
10
-7
10
-8
10
-10
10
4. Derivative and monotonicity
4. Derivative and monotonicity
A key formula is (x).We verify this symbolically.
′
A
A(x)
logx-1-πcot(π/x)
2
x
In[]:=
ClearAll[derivativeRatio];derivativeRatio=FullSimplify[D[aContinuous[x],x]/aContinuous[x],Assumptions->x>1]
Out[]=
-1-πCot+Log[x]
π
x
2
x
To check the exact claimed form:
In[]:=
FullSimplify[derivativeRatio-(Log[x]-1-PiCot[Pi/x])/x^2,Assumptions->x>1]
Out[]=
0
4.1 Numerical evidence for strict decrease on [4,∞)
4.1 Numerical evidence for strict decrease on
[4,∞)
In[]:=
Table[{n,N[a[n],30],N[a[n+1],30],a[n]>a[n+1]},{n,4,20}]
4.2 Plot of the continuous extension
4.2 Plot of the continuous extension
For derivative sign:
5. Discrete spacing of successive residue magnitudes
5. Discrete spacing of successive residue magnitudes
A scaled error check:
The values should trend toward 2.
Symbolic verification:
And numerically:
6.1 Bracketing a residual
6.1 Bracketing a residual
Example checks:
7. Greedy step definition
7. Greedy step definition
7.1 Nearest-residue index
7.1 Nearest-residue index
7.2 One greedy update
7.2 One greedy update
8. Global greedy expansion algorithm
8. Global greedy expansion algorithm
Display as a table:
Example use:
10. Predictor-based search for very large greedy indices
10. Predictor-based search for very large greedy indices
10.1 Predictor formula
10.1 Predictor formula
10.2 Local search near the predictor
10.2 Local search near the predictor
This is the practical tool for reproducing the huge indices reported in the paper.
11. High-precision predictive greedy computation for MRB
11. High-precision predictive greedy computation for MRB
Presentation grid:
Example:
12. Verifying entry into the contraction range
12. Verifying entry into the contraction range
Examples:
13. Testing the contraction numerically
13. Testing the contraction numerically
We can test this experimentally.
Suggested experiments:
14. Testing the quadratic regime
14. Testing the quadratic regime
Example for MRB:
For predictive large-index computations:
15. Universal Greedy Residue Expansion Theorem (UGRET)
15. Universal Greedy Residue Expansion Theorem (UGRET)
This section records the theorem in notebook form.
Theorem (UGRET)
Theorem (UGRET)
◼
some residual $E_N$ vanishes, giving a finite exact representation, or
16. Nontermination for nonzero rational targets
16. Nontermination for nonzero rational targets
This is an important conceptual point:
◼
nontermination alone does not prove irrationality of the target;
Notebook note
Notebook note
This theorem is best kept as a conceptual proof cell rather than a computational one.
17. Suggested experiments
17. Suggested experiments
17.1 Compare direct MRB partial sums with the greedy approximation
17.1 Compare direct MRB partial sums with the greedy approximation
17.2 Compare first greedy terms
17.2 Compare first greedy terms
17.3 Inspect monotonicity numerically at large scale
17.3 Inspect monotonicity numerically at large scale
17.4 Inspect asymptotic slope of the spacing law
17.4 Inspect asymptotic slope of the spacing law
17.5 Inspect the quadratic constant experimentally
17.5 Inspect the quadratic constant experimentally
18. Paper-structure summary
18. Paper-structure summary
This notebook separates the argument into four layers:
1. Origin of the family
2. Monotonicity and spacing
3. Global convergence
4. Asymptotic acceleration
Once convergence is known, the spacing estimate yields asymptotically quadratic residual decay.
This makes the greedy residue expansion both analytically motivated and mathematically robust.
19. Documentation links
19. Documentation links
Useful Wolfram Language references:
20. Closing summary
20. Closing summary
For the MRB constant specifically:
◼
subsequent greedy choices from the same family reduce the residual very rapidly;
◼
the convergence is globally guaranteed by a contraction argument;
◼
the near-doubling of correct digits is explained by the asymptotic spacing of neighboring residue magnitudes.
If you want, I can next turn this into one of these:
1. a paper-style notebook with theorem/lemma/proof formatting,
2. a fully executable experimental notebook with evaluated cells and plots,
3. a seminar notebook with shorter slide-like sections,
4. or a proof-outline notebook separating rigorous arguments from numerical evidence.
Acknowledgements
Acknowledgements
This investigation developed through a combination of analytic calculation, high - precision numerical experimentation, and symbolic verification . ChatGPT, developed by OpenAI, was used interactively during the development of this work
to assist in exploring contour representations, branch - cut transformations, residue calculations, asymptotic expansions, and related analytic questions, as well as in the preparation and revision
of portions of the manuscript . Wolfram Mathematica was used for high - precision numerical computation and symbolic verification where appropriate . All mathematical claims, derivations, numerical results, and conclusions presented in the final
manuscript were reviewed and accepted by the author, who assumes full responsibility for the content of the work .
This investigation developed through a combination of analytic calculation, high - precision numerical experimentation, and symbolic verification . ChatGPT, developed by OpenAI, was used interactively during the development of this work
to assist in exploring contour representations, branch - cut transformations, residue calculations, asymptotic expansions, and related analytic questions, as well as in the preparation and revision
of portions of the manuscript . Wolfram Mathematica was used for high - precision numerical computation and symbolic verification where appropriate . All mathematical claims, derivations, numerical results, and conclusions presented in the final
manuscript were reviewed and accepted by the author, who assumes full responsibility for the content of the work .
The Wolfram Notebook Assisted designed this notebook.