My 20th go book Endgame 5 - Mathematics is available for €26.50
(printed) or €13.25 (PDF). Theorems and their proofs establish endgame evaluation as always correct truths. The book studies most important
aspects of modern endgame theory.

Webpage
http://home.snafu.de/jasiek/Endgame.html
Cover
http://home.snafu.de/jasiek/Endgame_5_Cover.png
Sample
http://home.snafu.de/jasiek/Endgame_5_Sample.pdf
Table of Contents
http://home.snafu.de/jasiek/Endgame_5_TOC.pdf
Review by the Author
http://home.snafu.de/jasiek/Endgame_5_Review.html



Endgame 5 - Mathematics

Review by the Author


General Specification

* Title: Endgame 5 - Mathematics
* Author: Robert Jasiek
* Publisher: Robert Jasiek
* Edition: 2021
* Language: English
* Price: EUR 26.50 (book), EUR 13.25 (PDF)*
* Contents: endgame
* ISBN: none
* Printing: good
* Layout: good
* Editing: good
* Pages: 240
* Size: 148mm x 210mm
* Diagrams per Page on Average: 2
* Method of Teaching: truths, mathematics, methods, classification,
examples
* Read when EGF: 15 kyu - 9 pro
* Subjective Rank Improvement: -
* Subjective Topic Coverage: + (endgame in general), + to ++ (endgame evaluation)
* Subjective Aims' Achievement: ++


Endgame Theory


Introduction

This mathematical textbook with definitions, theorems and their proofs establishes endgame evaluation as always correct truths. The book
studies most important aspects of modern endgame theory.
Introductions, many remarks and examples, 20 tables, several game
trees and a few thermographs assist the reader.

Endgame 5 Mathematics justifies and verifies most of the theory of
endgame evaluation of Volumes 2 - 4 and introduces further advanced
theory. Besides excursions to combinatorial game theory and scoring,
by far the largest part of the book develops modern endgame theory as
the new field of endgame evaluation. While combinatorial game theory,
which can be applied to various games including go, could explain the microendgame and provide some very general but rough approximations
depending on the larger temperature, modern endgame theory better
describes the large late endgame by de-emphasising the last move of
the game and the early endgame by emphasising tighter approximations
depending on the smaller drops between moves.

Missing Contents

There are especially the following few exceptions of theory the book
does not explain. The chapter Microendgame only studies a few new
details of the microendgame (see Volume 2), which has already been
established by the literature on combinatorial game theory. Consult it
for difficult kos. The book shows the limitations of the method of
making a hypothesis in Volume 3 so that the advanced and tedious tools
of thermography remain necessary for the most difficult shapes of
local endgames. It remains an exercise for a future researcher to
prove the principle of extreme difference values for iterative local
endgames in Volume 4. Although the book can be read independently,
Volumes 2 - 4 provide the practical side of the theory for improving
one's playing skill.

Combinatorial Game Theory

The combinatorial game theory in chapter 2 and sections 4.3 + 4.4
describes the basic calculations involving algebraically represented
positions and their resulting scores, difference games, and
simplifying positions by ignoring dominated or reversible plays. This
low-level theory can be used directly but a few proofs elsewhere in
the book also apply it. The model of a rich environment enables the
definition of count and move value more elegantly than the
literature's older definition relying on an infinite number of
multiple copies of a position.

After this preparation, long sequences can be evaluated by T-orthodox
(worth playing successively), the orthodox forecast and accounting
theorems, the sentestrat algorithm and thermography. The iterative
algorithm of calculating a thermograph algebraically is explained in
detail together with step-by-step calculations deriving the counts,
move values and thermographs of an example position and its followers.
If you found the literature on combinatorial game theory too hard, the
careful selection and new study in this book give you a fresh access
in a simplified and go-friendly notation.

Modern Endgame Theory

The book studies the following topics of modern endgame theory:
definitions of the basic values and their relations, identification of
the types of local endgames, evaluation of local endgames with short
or long sequences, the value of starting in an environment and
modification of alternating sums. Half of the book determines the
correct move orders, and first and last moments of playing in a
particular local endgame instead of the environment.

The timing is solved for either starting player, all temperatures of
the environment and all basic types of local endgames with arbitrary
values: a local gote, an ambiguous local endgame or a local sente with
one or two simple follow-ups; a local endgame with gote and sente
options; pairwise comparison among several local endgames each with
one player's follow-up.

During the late endgame, the solutions are exact if the environment
comprises simple gotes without follow-ups. In practice, tactical
reading can stop whenever such a position is reached. During the early
endgame, the solutions are the best available approximations. The
proofs presume such an arbitrary environment or sometimes the model of
an ideal environment. In practice, such environments closely resemble
ordinary environments of quiet positions without active fights and
also allow application to local endgames with iterative follow-ups.

Of course, the early endgame cannot be solved completely yet.
Therefore, mathematical research assumes some simplified value
environment of an early endgame position. Combinatorial game theory
uses an arbitrarily dense rich environment. Modern endgame theory uses
an environment of simple gotes without follow-ups and arbitrary
values, or an ideal environment with constant drops. The assumed
environments enable the proofs of theorems but these models only
provide good approximations for more complicated ordinary
environments. Nevertheless, the great difficulty of constructing
exceptional counter-examples indicates that these approximations
describe reality very well.

We learn the basic values: count, move value, gain and net profit. The
largest value of the environment is called the temperature. We
sometimes also consider its second-largest value. During the late
endgame, it can be necessary to calculate the alternating sum of the environment's remaining simple gotes by adding the values taken by the
first moving player and subtracting those taken by the opponent. More sophisticated alternating sums accelerate calculation by ignoring the immaterial values. Conditions in theorems compare the relevant values
to describe correct decisions between playing locally or in the
environment.

We characterise each local endgame as a local gote, ambiguous or local
sente so that we calculate its correct gote or sente values. We also
learn when it is better to describe a local gote by sente values or
vice versa. The type of an ordinary local endgame is determined by
four alternative value conditions, whose equivalence we prove.
Similarly, a local endgame with gote and sente options permits two
alternative value conditions. We prove the non-existence of local
double sente.

As expected, evaluation of local endgames with long gote, sente or
reverse sente sequences is much more difficult because we must
determine for how long successive local play may proceed before
continuing in the environment. For this purpose, the book establishes
these methods: comparing the opponent's branches, comparing counts,
comparing move values, thermography and making a hypothesis.

The book does not just solve local endgame positions in environments
during the late endgame phase but even presents a fully developed
theory with different, alternative methods and proves their
equivalence. The methods of comparing either counts or net profits
replace the too complex tactical reading by only considering two
particular test sequences. The method of applying a principle
disregards sequences; instead, move decisions only rely on values.

The theory of scoring relates area to territory scoring and parity to
the winner, describes the impact of approach moves and proves the
equivalence of life defined by capturability or two eyes.


Mathematics

The book contains 40 major definitions and 149 statements of truth, of
which 139 are proved in the book and 10 are taken from the literature.
The statements of truth represent 60 theorems, 76 propositions, 11
corollaries and 2 algorithms. Besides, one lemma is embedded in a
proof. Statements of truth are labelled as theorems, propositions or
lemmas in decreasing order of relevance but mathematics does not use
these terms consistently. The book refers to major results as
theorems, intermediate results or preparations as propositions, and
helping statements of truths as lemmas. Corollaries are derived by
symmetry, trivial transformation, or as less general implications of
theorems or propositions. For comparison, the book Mathematical Go
Endgames about the microendgame and scoring has 24 statements of
truth, of which 11 are theorems, 9 propositions (called lemmas), 1
corollary and 3 algorithms.

The average level of difficulty of the mathematics in this book is
that of the first year of study at university. You can understand much
with school algebra. The rich variety of proof techniques is a great
source for learning them. Most definitions and theorems are understood
easily like principles. The mathematical proofs are as detailed and step-by-step as is common at school. This is unlike mathematical books
or journals read at university, where the student needs to think 30
minutes per line of text. The diligent reader has a chance to
understand even the advanced proofs of this book by spending half an
hour per page. A reader skipping the proofs misses half of the book's
contents but can still learn a lot.

We study the fundamentals as well as advanced theory. We prove both
the 'obvious' and the difficult. However, allegedly obvious theory may
be easy or difficult, and has required between 20 minutes and three
weeks per proof. Some proofs need one line applying theorem A to
theorem B while the longest proof studies 30 cases on 11 pages.

A professional mathematician has proofread chapters 2 to 5 and I have
proofread everything, especially the mathematical theory, several
times very carefully.


Inventions

If we compare 100% of the theory of endgame evaluation in this book,
then informal go theory has 30%, of which more than half is guesses
and partial descriptions. Besides, informal go theory contains wrong
theory. Combinatorial game theory represents 33%, of which almost one
third is partial descriptions. 56% of the theory in this book is new
and was not part of informal or combinatorial game theory. The book
clarifies prior guesses and works out previous partial descriptions.
Hence, in comparison to informal go theory, the book triples the
theory of endgame evaluation. Compared to combinatorial game theory,
the book greatly enhances theory.

The inventors of mathematical theory, such as theorems and their
proofs, in this book are as follows: the author Robert Jasiek 82%,
Bill Spight 13%, others 8%, unknown 13%. Note that 15% of the theory
has been developed by two, or in one case three, persons.

Why does the book make such a great progress? Combinatorial game
theory has evolved for more than a century. Bill Spight has done
pioneer research in modern endgame theory for decades. Eventually, I
have spent circa 15 months on full time research for this book and
years of further endgame study for my other endgame books. Informal go
theory was conceptually limited whereas modern endgame theory profits
from the power of mathematics.


Layout

The mathematical contents uses a generous layout while the
commentaries on examples are dense and sometimes refer to several
theorems. Mathematical text uses a font that eases reading and
resembles script in block letters. Clearly, the book emphasises the
theory. Nevertheless, the examples cover all important or rare cases
proclaimed in theorems or proofs.


Conclusion

Obviously, the book is not for players preferring informal go theory
to mathematics. Read the comprehensive Endgame 5 Mathematics if you
are interested in the endgame, mathematics, the most advanced new
theory at its highest level of truth and its verification.



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