The complete guide · 7 chapters · 26 lessons
Intuitions
How can simple constraints bind something that none of their parts reveal? This guide answers like a textbook: it starts from bits and rules, builds up to the knot test, then follows what knots do, with holes, choice, loops, depth and counting. Each lesson uses only what earlier lessons taught.
Constrained as a whole, unconstrained in every part.
All three rings hold together, yet no two of them are linked.
Try the knot test →Chapter 1Bits and rules
Lessons 1.1–1.7
Everything later is built from bits and from rules about them. This chapter sets up that vocabulary and its notation, reads the Game of Life as such rules, states the picture the program starts from, reads the smallest rule that binds two loops, and ends with what the guide means by a shape.
Lesson 1.1
Distinctions and states
A distinction is one bit. Three bits have eight joint states, the corners of a cube.
Meet the cube of states
Lesson 1.2
Rules and what they allow
Read ¬, ∧, ∨, → and ⊕ by the states they rule out, and learn what makes a set of variables free.
Read the symbols
Lesson 1.3
Emergence in the Game of Life
Life's famous patterns are the allowed states of simple rules. Only the grid they live on is drawn in advance.
Step a glider
Lesson 1.4
The picture
Lines only order and single loops collapse. What is real lies between bound loops.
See a binding appear
Lesson 1.5
Reading A, B → c
The atom forbids exactly one state: A and B without c. Every other state is allowed.
Find the forbidden state
Lesson 1.6
Forgetting a variable
Ignore a variable by letting it take whichever value suits. Witnesses are forgotten the same way.
Forget a variable
Lesson 1.7
What a shape is
A shape is any structure that rules make, and some run on forever. A pattern is a shape seen without its rules.
Read the definition
Subsections of chapter 1
- 1.1Distinctions and states
- 1.2Rules and what they allow
Rules and relations Symbols A warning about the arrow Clauses and CNFs Free and constrained
- 1.3Emergence in the Game of Life
Cells are distinctions The rule is a rule on ten variables A history is an allowed state The famous patterns are solutions Every rule is local, and what the rules allow is global Contradictions are objects too The grid is drawn in advance Reading
- 1.4The picture
Lines and loops, state by state A line is not enough A single loop is not enough What is real is what lies between bound loops A loop is a logical structure, not a spatial one From loops to A, B and c
- 1.5Reading A, B → c
Ways of writing it The one thing to remember is that the atom forbids exactly one state A and B The comma The arrow The direction is only a way of reading it Why it binds The AND gate The common hub
- 1.6Forgetting a variable
- 1.7What a shape is
The word Shape or pattern Rules within rules Shapes that run on forever Every bond is finite The board and true base reality Can the rule be learned? Two senses of the word Reading
Chapter 2The knot test
Lessons 2.1–2.3
Lesson 1.5 found a rule that lives in the whole and in no pair. This chapter turns that into a test, draws the test on the cube of states, and counts every knot on three variables.
Lesson 2.1
The knot test
Erase any variable. Does the binding disappear, or does a smaller constraint survive?
Erase a variable
Lesson 2.2
Knots on the cube
Every pair is free exactly when no two excluded corners of the cube are neighbours.
Mark the corners
Lesson 2.3
What a knot hides
Explore all eight states and measure the information hidden from every pair.
Build a relation
Subsections of chapter 2
- 2.1The knot test
The setting The test K1, binding K2, loss on erasure K3, coexistence Knots and non-knots
- 2.2Knots on the cube
Free pairs on the cube Every pair is free exactly when no two excluded states are neighbours Which relations are knots Independent sets
- 2.3What a knot hides
The intuition How it is measured Two loops and a contact have exactly 34 knots, in five shapes A knot hides at most one bit, and only parity hides a full bit The atom is the smallest bond Reading
Chapter 3Pairs and three places
Lessons 3.1–3.4
Which rules can build a knot? Rules on two variables never can, and rules on three variables always can. The chapter then draws what a system leaves free as a shape, and finds the knots among the shape's holes.
Lesson 3.1
Two-place rules and the majority vote
Two-place rules always allow the majority of three allowed states. Knots never do.
Take the vote
Lesson 3.2
Pairs cannot make a knot; three places can
Use a majority vote to see why pairwise rules cannot define a logical knot.
Try the majority argument
Lesson 3.3
The shape of a system
Draw what a system leaves free as points, edges and filled triangles, and look for holes.
Draw a shape
Lesson 3.4
Holes and knots
Compare an empty triangle with a square hole made entirely by pairwise ties.
Compare the shapes
Subsections of chapter 3
- 3.1Two-place rules and the majority vote
Two-place rules The vote A 2-CNF is closed under majority Witnesses do not help No knot on three variables is closed under majority
- 3.2Pairs cannot make a knot; three places can
The intuition Classical Hence no 2-CNF defines a knot Width 3 binds any number of loops Without witnesses, a knot needs full width Reading
- 3.3The shape of a system
Shapes Simplicial complexes and holes Three examples Flags and joins
- 3.4Holes and knots
Hole or knot Pairwise rules make only "flag" shapes Every knot is a hole, but not every hole is a knot Logic allows every shape Reading
Chapter 4How bindings behave
Lessons 4.1–4.5
Knots can be read as requirements or as choices, can come from rules that compete, and can be joined into rings and networks. The chapter first fixes the words for easy and hard questions, which these behaviours are measured by.
Lesson 4.1
Easy and hard questions
Linear time, polynomial time and NP-complete: the words for how hard deciding is.
Learn the classes
Lesson 4.2
Requirement and choice
Watch a requirement force one least world, while a choice leaves alternatives.
Change the starting facts
Lesson 4.3
Knots from competing pairs
Three pairwise preferences cannot all win. Their best states form a knot.
Explore a frustrated triangle
Lesson 4.4
Which knots remember a loop
Pass a bit around a ring and find out which contacts preserve its memory.
Flip the transfers
Lesson 4.5
What networks of loops remember
Free contact records each loop apart. Contact with fixed points binds loops into atoms.
Bind two loops
Subsections of chapter 4
- 4.1Easy and hard questions
Time is counted against size Deciding Expressing is not computing
- 4.2Requirement and choice
Worlds A requirement Computation is a forced world A choice Requirement and choice are one knot read two ways Knots never select Reading
- 4.3Knots from competing pairs
Conjunction and competition The intuition Exactly four knots come from competing pairs on three variables None on four or more variables With hidden variables, competing pairs make every knot Why only four Reading
- 4.4Which knots remember a loop
Rings of knots The intuition Transfers A ring excludes exactly the odd loops Only parity carries a loop at full strength Other knots fade
- 4.5What networks of loops remember
From rings to networks Free contact records each loop separately Contact with fixed points binds loops, into atoms Binding costs witness values Classical background Reading
Chapter 5Depth
Lessons 5.1–5.2
A network of knots can allow nothing while every small window of it looks consistent. This chapter asks how much of the network a proof has to hold at once to see that nothing is allowed.
Chapter 6Counting
Lessons 6.1–6.3
Deciding asks whether a network allows any world; counting asks how many worlds it allows. This chapter sets up the words for counting, then reports where counting turns hard.
Lesson 6.1
Counting worlds
Counting can be hard when deciding is easy. Planar graphs are the classical exception.
Learn the counting words
Lesson 6.2
The worlds of not-all-equal knots
Count the colourings of networks of not-all-equal knots, from polynomial to #P-hard.
See where counting turns hard
Lesson 6.3
Patterns in three dimensions
Counting maps from knot groups to solvable groups is hard, already for S₄.
Explore three character slots
Chapter 7Sources and questions
Lessons 7.1–7.2, then the glossary
What the guide takes from classical mathematics, what it adds, and what remains open.
Lesson 7.1
What is classical and what is ours
Separate classical foundations, the program’s interpretation, and its research claims.
Read the attribution map
Lesson 7.2
Open questions
What could select a geometry, supply dynamics, handle infinite limits, or admit competition?
See what remains open
Back of the book
Glossary
Every term the lessons define, from distinctions and states to refutations and depth.
Find the vocabulary
Reading the guide
All 26 lessons of knotlab’s INTUITIONS.md are included, in its 7 chapters and with its glossary, with their original explanations, sources, examples, and qualifications. The interactive illustrations add small cases you can inspect.
The original introduction and claim-status labels
The contact-first program asks whether logic can build structure, such as holes, paths and spaces, without assuming a space to start from. Its working answer begins with one object: the logical knot, two or more open distinctions bound together so that the whole is constrained but no part is.
This page states the intuitions behind that answer in plain terms. Each one is tied to what has been proved or checked in this repository, and each is labelled by what kind of claim it is.
The page is written as a textbook, to be read in order. Its seven chapters are divided into lessons, and each lesson builds on the ones before it. Chapter 1 sets up bits, rules and the notation, including how to read A, B → c, before anything is built from them.
| Label | Meaning |
|---|---|
| Proved (Lean) | Kernel-checked in lean/, core Lean 4, standard axioms only |
| Proved (note) | Proved in a note in notes/, whose numbers a script rechecks |
| Proved (here) | A short proof given on this page |
| Checked | Verified exhaustively on finite cases by tests/test_knotlab.py |
| Classical | A known theorem the intuition relies on, with its source |
| Reading | Our interpretation. It is not a theorem and nothing is proved from it |
Most of the mathematics in Chapters 1 to 4 is classical. What is ours there is the reading: which questions to ask, and what the known answers mean when asked this way. Lessons 5.2, 6.2 and 6.3 report results that are new as far as we found. Lesson 7.1 lists what is classical and what is ours.
To try anything on this page, run commands from the repository root:
PYTHONPATH=tools python3 -m knotlab demo
The logical knots in this guide are constraints among variables. The ordinary knots in the group-counting note are topological knots. The guide explains their connections and keeps the two meanings distinct. The work was prepared with AI assistance; proofs and checks are linked in the lessons.