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 full field articles, dated, and honest about whether the line of thinking led somewhere or did not. Anyone is welcome to read along.
Dates on entries are when the work happened. Hunches stay labeled hunches. Small results stay labeled small. Nothing here is a proof of Collatz.
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.
Four readings
The story next to the math.
These are my philosophical kit for Going Home. They are not lemmas. They are how I stay human inside a problem that can eat people. The dated notebook below is the field record. The essays for each reading land under /going-home/essays on their dates, not as a same-day dump.
- 01Essay live
Sufficient awareness guarantees return
If you can see far enough ahead, you know you are going home.
In Collatz: enough look-ahead and the path back to 1 becomes visible in the window you can actually compute.
In life: enough clarity and you stop mistaking motion for progress. You see the route, not just the next reaction.
Slogan, not a theorem. The Santana Principle is a bet underneath the math.
- 02Essay live
The future redeems the present
A bad moment now can be paid for by good steps later.
A number that grows for a stretch can still collapse afterward. Look-ahead exists because the next step is not the whole story.
Ugly weeks do not cancel a life aimed at home. They ask whether you budgeted enough horizon to see the redemption queued ahead.
Emotional core of look-ahead. Not a license to ignore harm in the present.
- 03Essay dated 2026-08-21
1 as home / observer beyond 1
1 is not just the end of the sequence. It is home.
The cycle 4-2-1 is the attractor everyone tests. Beyond 1 is the vantage that sees every trajectory at once, not an extra calculation step.
Home is the center you return to when the map finally makes sense. The observer is meaning, God, the whole picture: identity that can hold Brazil and Las Vegas, builder and student, debt and devotion, without splitting.
Metaphysical reading. Keep it out of the lemma line.
Essay lands 2026-08-21
- 04Essay dated 2026-08-28
Look-ahead depth and consciousness
How far you can see is a metaphor for how awake you are.
Shallow look-ahead is reactive. Deeper look-ahead is planning across a window. Infinite look-ahead would be seeing the whole orbit.
Maps awareness onto horizon. Fixed depth fails in math and in life when the ugly streak is longer than the dashboard you built.
Poetry next to math. Not a proof. The obstruction to fixed-k energy made this distinction expensive and real.
Essay lands 2026-08-28
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 by when the work happened. None of them are a proof. Some are hunches. Some are dead ends or small results I am publishing anyway because the ruling-out is the work. New entries appear here as I clear them for the public record.
The notebook
Entries, newest work first.
- Worked 2026-07-31Small result
Beyond the two coefficients
The May obstruction killed a two-parameter energy. Digging further: any fixed look-ahead window reward fails on long enough ugly streaks. The open road is adaptive depth with a real contraction test.
- Worked 2026-05-15Small result
What died, what remains
A clean obstruction to bounded look-ahead Lyapunov energies. Mersenne odds as witnesses. The plan dies. Honesty and an adaptive reformulation remain. Not a Collatz proof.
- Worked 2025-11-30Note
A heavy day in the notebook
A long day of edits in the private Notion page. Volume went up. Clarity did not always follow. More toggles is not more proof.
- Worked 2025-02-24Note
Pressure-testing
If an argument only works when numbers behave, it is not an argument for all n. Two failure modes: unbounded growth, and cycles that are not 1-4-2-1.
- Worked 2025-02-17Hunch
Energy and structure
Scaffolding for a score: log size minus rewards for two-adic collapse now and ahead. Promising on nice numbers. Shaky on ugly ones. Not yet a theorem.
- Worked 2025-02-09Hunch
Look-ahead intuition
Seeing more than the next step. Collapse queued across a short window. The future might redeem a present that looks like it is growing. Intuition with a name forming.
- Worked 2025-01-10Note
Collatz in plain language
Even, halve. Odd, triple and add one. Always to 1, says the conjecture. Easy to state. Wildly hard to prove. I treat Erdős as a warning, not a dare.
- Worked 2025-01-07Note
Why this problem
Not glory. The cleanliness of the question. Homecoming if every integer returns. A rewritten story if one does not. A record of thinking before the thinking was good.
- Worked 2024-12-31Note
Jetlag and a problem that would not leave
Back from China, sleep broken, a little lost. Looked up conjectures. Collatz would not leave. An itch and a notebook. Private obsession, not a public claim.
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.