Field Notes · Quantum Traction Theory

Hamiltonian: Another step of the Turtle.

A theory has to say who built the stage before it can run the play. Notes on assembling the QTT Hamiltonian from the scattered pieces of the book — and on the no-nonsense physicist who made me ask the question.

Ali Attar · Colombes, France · June 2026 · 9 min read · DOI ↗

There is an old story. A scientist finishes a public lecture on the cosmos, and an elderly woman stands up to tell him he has it all wrong — the world, she says, rests on the back of a giant turtle. And what does the turtle stand on? The woman smiles. “You’re very clever, young man. But it’s turtles all the way down.”

Physicists tell that joke to mock an infinite regress. I have come to think it is the most honest picture of theoretical physics we have. Every law sits on a deeper law; every deeper law on a deeper one still. The only question a foundational theory is really allowed to ask is whether the stack ever bottoms out — whether, somewhere down there, there is a last turtle.

This is a note about the day I went looking for one particular turtle: the Hamiltonian. It is another step down the stack — another step of the turtle.

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I · The outsider’s guide

Learning physics without the gobbledygook

I came to this from outside the academy. A great deal of my physics I learned the way many people now do — late at night, from people generous enough to explain the real thing without the gobbledygook. Chief among them, for me, is Sabine Hossenfelder.

What I value is the straightness of the voice. She does not sell wonder; she names the pain — the places where the field has spent decades and billions and produced theories that cannot be tested, the habit of mistaking elegance for evidence. Her book Lost in Math makes that case directly: that physics has too often fallen in love with how its equations look and confused the feeling with the truth. For someone outside the guild, that honesty was a map of where the real problems live.

One of her videos walks through the strangest numerical coincidences in nature — the proton-to-electron mass ratio sitting suspiciously near 6π⁵, and the one that simply refuses to go away: Koide’s formula for the charged leptons,

(m_e + m_μ + m_τ) ⁄ (√m_e + √m_μ + √m_τ)² = 2⁄3

— still true to under a standard deviation more than forty years after Koide wrote it down, and a better fit now than the day it was proposed. That formula lodged in me and would not leave. It is exactly the kind of thing a substrate ought to explain rather than shrug at, and the instinct behind it ran straight into the book I was already writing.

And then there was a video where she simply explained the Hamiltonian — what it is, why physicists reach for it before almost anything else. I followed every step. And at the end I was left holding a question she had not asked, because nobody asks it.

II · The question the textbooks skip

Who built the stage?

Here is the standard answer, and it is a good one. The Hamiltonian is the energy of a system written as an operator. Hand it a state, and it tells you how that state turns in time. It is the engine of motion, the thing from which Schrödinger’s equation and Hamilton’s equations and half of physics unspool.

But notice what every textbook quietly assumes before the first symbol. It assumes a system. It assumes a laboratory. It assumes a clock already ticking and coordinates already painted on the floor — a stage already built, with the lights already on. The Hamiltonian runs the play beautifully. It never asks who built the stage.

For most of physics that is the correct division of labour; you take the stage as given and get to work. But QTT is not a theory of the play. It is a theory of the stage. And so the question stops being rhetorical and becomes the entire job.

The laboratory Hamiltonian is not the source object. It is the access image of a deeper ledger.

III · The inversion

Source first, access second, audit last

In Quantum Traction Theory there is no stage to begin with. There is a substrate that keeps a ledger — a running account of events that have actually finished closing. An address, a completed fact, written into the world. Particles, fields, clocks, coordinates: in this picture they are not the starting ingredients. They are what the ledger looks like once you have chosen a way to read it.

Which means the Hamiltonian a physicist writes on the board is not the bottom of anything. It is a reading. And before QTT is allowed to generate a single tick of time, its Hamiltonian has to answer four questions a textbook never poses: which completed events are being funded; which clock is being used; which bundle has closed; and which access channel is permitted to read the result.

That is the discipline that became the spine of the paper: source first, access second, audit last. Write the substrate object honestly. Then map it to what a laboratory sees. Then — the part most theories skip — keep the books, so that nothing observed can be smuggled back in and quietly renamed as a prediction.

IV · The construction

Gathering the pieces

Here is the honest part of the story. The Hamiltonian was not sitting in the book, finished, waiting to be photographed. It was scattered — a fragment in the chapter on addresses, another in the chapter on the real rotor, a third buried inside the derivation of gravity. What I did, with the help of my AI assistants, was gather the fragments and check, line by line, whether they snapped together into a single object without one new adjustable number.

They did. That was the good day. Seven moves, axioms A1–A7, in order:

Not one of those moves carries a knob. Every prefactor is forced. That was the rule of the day, and the day held to it.

V · The payoff

Do the old sentences still parse?

The only honest test of a new grammar is whether the old sentences still parse inside it. So you take the standard sectors of physics — free particles, photons, fermions, the Higgs, QCD, thermal reservoirs, measurement, gravity — and you drop each one into the same machine. The claim is not that a new set of equations replaces the old ones. It is that the familiar ones are recovered as readouts rather than assumed as inputs:

Schrödinger
Dirac
Maxwell
the Boltzmann weight
the path integral

Each is what the ledger looks like through one particular access window. The difference is in what you are allowed to say about it.

And nowhere along the chain is there a tuned number. The half-angle cos(π/8) ≈ 0.92388 is forced. The bundle closes at 2π because it is the same universe on both sides. The neutrino mass-squared ratio is put forward as 4π²·cos²(π/8) ≈ 33.7 — a sharp comparison target against global-fit oscillation data, with no tuning parameter introduced here.

A residual, here, is never a fudge factor. It is the ledger telling you which rail you left out.

VI · The referee

The substrate grades the work

Sabine’s sharpest lesson is the warning in Lost in Math: physics goes astray when it falls in love with how its equations look and mistakes the feeling for proof. A theory that draws itself as a golden mandala with an angle of π/8 at its heart could be exactly that trap — so QTT is built the other way around. The discipline is not is it beautiful. The discipline is: every load-bearing number is forced by the axioms, nothing is fitted, and a measurement can kill it. The neutrino ratio is a number an experiment can move. The CP phase the theory points to is a long-baseline target for experiments such as DUNE to constrain or falsify as exposure accumulates. If the prettiness is real, it is a consequence of the structure, never the method that chose it.

The approval of QTT does not come from me, and it does not come from any authority. It comes from one place only: whether the equations are proper representatives of the dynamics of Artian’s universe. The substrate is the referee. Every load-bearing number is forced, and every one is exposed to measurement — so measurement, not authority, does the confirming or rejecting. That is the standard those late-night videos taught me to hold.

VII · The next turtle

What the turtle does when you ask it the time

So — is the Hamiltonian the last turtle? No. The Hamiltonian is what the turtle does when you ask it the time. The turtle itself is the ledger: the substrate that keeps its accounts in completed events and pays for existence one tick at a time. This day was one more step down the stack — another step of the turtle — and the energy of the world now stands as one object, written from the axioms upward, with no knobs attached.

The full construction — every sector, every recovery, every audit — is in the paper below. Read it with a skeptic’s eye, then take it to the substrate. That is the only referee, and it does not grade on beauty.

Quantum Traction Theory · Ali Attar · quantumtraction.org · ORCID 0009-0008-9931-2691