The open notebook
Going Home. A long study of one math problem and what it might mean.
This is the public surface of the work I keep returning to. The Collatz Conjecture is a ninety-year-old open problem in number theory. The Santana Principle is the philosophical claim hiding inside it. I am not on the doorstep of a proof. I am inside the problem as a careful student, and this is where I publish what I learn as I learn it.
Most mathematical work lives behind walls until it is finished. This page is the opposite. An open notebook. Entries are short, dated, and honest about whether the line of thinking led somewhere or did not. Anyone is welcome to read along.
The conjecture
The simplest problem nobody has cracked.
Take any positive integer. If it is even, halve it. If it is odd, triple it and add one. Repeat.
The conjecture says every positive integer eventually reaches 1. Every number ever tested.. up into the high quintillions.. does. No one has proved why this must be true for every positive integer, including the ones too large to ever test.
Erdős said about it: “Mathematics may not be ready for such problems.” Terence Tao has called it one of the most dangerous problems in mathematics for what it does to the people who try to solve it.
Try a number, watch it walk
Run a number · the Collatz Conjecture
Pick any positive integer, up to a trillion. Halve if even, triple-plus-one if odd, repeat. Watch it go home.
- Steps to home
- …
- Peak
- 9,232
- Verdict
- In flight
Every positive integer ever tested falls eventually to 1. Nobody has proved why it must, including me.. and the gap between “every number ever tested” and “every positive integer” is the gap I am trying to live inside long enough to learn what it teaches.
The Santana Principle
“Sufficient awareness guarantees return.”
This is my philosophical reading of what the conjecture might be saying. It is not the proof. It is the shape of the bet underneath the math.
If a system has enough self-observation built into its dynamics, every starting point eventually finds its way back to the system's center. The Collatz dynamics may be one cleanest formal instance of that claim. The conjecture would not just be true.. it would be true for a reason that extends past number theory.
This is the bet. It may turn out to be wrong. The point of the notebook is to find out, in public, slowly, and honestly.
Where the work is
The state of the investigation.
I am studying the conjecture in a few directions at once. The mainstream ones (density arguments, stopping-time analysis, connections to ergodic theory) and a few personal ones (the self-similarity of the orbits, the shape of the divergence envelope on a log axis, the question of what kind of dynamical system the iteration is when you stop thinking of it as arithmetic).
I am not claiming to be ahead of the field. The field includes Terence Tao. The honest description is that I am at the bottom of a very tall mountain, learning to climb it well enough that the climb itself teaches me something worth saying out loud.
Every entry below is dated. None of them are a proof. Some are hunches. Some are dead ends I am publishing anyway because the ruling-out is the work. New entries appear here as I write them.
The notebook
No entries yet.
The notebook is currently empty.
I am collecting and editing the research notes worth publishing. The first entries will land here in the coming weeks. If you want a note when they do, the newsletter sign-up at the bottom of the home page is the way.
I would rather publish nothing than publish a claim I cannot defend.
Why this work, why in public
The point is not to be the one who proves it. The point is to be the one who studies it well enough to be changed by it.
Most of my work pays for itself in dollars or in distribution. This work does not. The companies fund this. The talks share the philosophical edge of it. The math itself.. I do because it is the cleanest mirror I have found for the question of whether the universe bends toward home or whether I am mistaking my own attention for a property of the world.
Publishing the notebook in public is a forcing function. Pages do not love hand-waving. The honest record makes me honest.