## 5 Point Energy Minimization Suite

This webpage has links to my work that
relates my (claimed) solution of the
N=5 case of the 1977 Melnyk-Knop-Smith Conjecture. This
conjecture predicts the power-law phase transition in 5-point energy
minimization. A special case of my main result is a solution to Thomson's
1904 problem about 5 electrons. I had already solved this some years ago.
Here is a reference:
- R. E. Schwartz, The 5-Electron Case of Thomson's Problem

Experimental Math, vol 22, issue 2, (2013) pp 157-186
pdf

companion program

Here is my monograph, which (I claim)
solves the N=5 case of the MKS conjecture:
- Divide and Conquer: A Distributed Approach to Five Point Energy Minimization

preprint (2023) 77 pages
pdf

companion programs

My various failed attempts to
get some version of this monograph published have left me
doubtful that I will ever publish it. In spite of
my efforts
to make the latest version of the monograph modular and
checkable, the monster still gives the impression of
being too difficult to referee.
I wrote the latest version with the idea that it could be
checked in a distributed way, with a team of people each
verifying small parts. I even included a complete logic tree
for the proof. (See Figure 0 in the monograph.) However, I did not quite
make the details of the distributed checking process
clear. While the
proof divides neatly into independent parts, these parts
rely on some common background material in a way I did not
make explicit. So, it is not obvious what
each member of a team of verifiers would need to read.
To make the distributed checking process clear, I have broken the
monograph into 7 papers which are all independent
from one another. This adds some length, because now some background
material appears in several different places. On the other hand,
each paper can be checked separately from the others.
Paper 0 gives the overall argument, simply assembling
the results from Papers 1 - 6.
- 5 Point Energy Minimization 0: Main

preprint (2024) 14 pages
pdf

- 5 Point Energy Minimization 1: Energy Lemma

preprint (2024) 19 pages
pdf

- 5 Point Energy Minimization 2: Big Calculation

preprint (2024) 17 pages
pdf

companion programs

- 5 Point Energy Minimization 3: Local Analysis

preprint (2024) 10 pages
pdf

companion programs

- 5 Point Energy Minimization 4: Interpolation

preprint (2024) 12 pages
pdf

companion programs

- 5 Point Energy Minimization 5: Symmetrization

preprint (2024) 18 pages
pdf

companion programs

- 5 Point Energy Minimization 6: Endgame

preprint (2024) 13 pages
pdf

companion programs

Here is a discussion of the dependence of these
papers on the computer.
Papers 3 - 6 rely on relatively straightforward
computer programs, written either in Mathematica or
Java. These programs perform their calculations with
exact integer arithmetic. The programs are relatively
short and a competent programmer (in each case) could
probably reproduce the programs in a day. The software
you can download for Paper 5 is quite extensive, but
most of it is present for the purposes of experimentation.
The proof part is quite short.
Paper 2 relies on a big Java program that runs its
calculations using interval arithmetic. The calculations
take about a day to run. The program comes with an
extensive graphical user interface and its own documentation.
Given the detail with which I have described the
calculation in Paper 2, I think that a good programmer
could reproduce the code for Paper 2 in a few weeks.