The Vacuum.Net is a Cellular Automaton.
Gerard ’t Hoofts cellulaire automaton, waarin de werkelijkheid een verzameling deterministische ‘tandwielen’ is, en de Vacuum.Net-theorie, waarin alles een gesloten lus is die bestaat door terug te keren zijn gelijk aan elkaar.
J.Konstapel,Leiden,25-10-2026.
based on Wired : Scientists Trapped an Antimatter Particle for Months to Study Its Secrets
In October 2026, a research group at Northwestern University reported a remarkable achievement. They trapped a single positron in a Penning trap and held it there for months. They watched it climb a ladder of quantum states, one step at a time. They watched it fall back down. Every rung of that ladder had exactly the same height.
This experiment is a triumph of precision physics. But it also raises a deeper question. Quantum theory tells us the ladder exists. It does not tell us what the ladder is. This essay sets two answers side by side. The first comes from Nobel laureate Gerard ‘t Hooft. The second comes from the Vacuum.Net theory. The central claim is simple: the two answers are the same answer, stated in two different languages.
1. The experiment: a positron on a ladder
The experiment by Myers, Soucy, Sukra, Sinha and Gabrielse appeared in Physical Review Letters on 21 October 2026. The procedure is exacting. A positron emerges from a sodium-22 nucleus with 546 keV of energy. It is slowed down and caught in a Penning trap. The trap sits in a 5.6 tesla magnetic field and is cooled to 300 millikelvin.
Inside the trap, the positron circles the magnetic field lines at about 156 GHz. It settles into the lowest cyclotron state. Apply a microwave drive, and it jumps one rung up. Wait, and it falls back. Every rung has the same height. The energy levels follow the familiar formula:
E = ħω_c(n + ½), with n = 0, 1, 2, …
On top of this sits a second ladder for the spin. The spin precesses at a slightly different rate. The two frequencies differ by about one part in a thousand. Their difference is called the anomaly frequency.
Several details of the apparatus deserve attention.
- Loading. The positron does not enter the trap alone. It leaves a tungsten moderator pulling an electron from the surface. The two travel together as a neutral positronium atom through a 640 micrometre hole. The trap’s electric field cannot see the neutral pair. Only inside the trap is the field strong enough to pull them apart. The electron is lost. The positron is caught.
- The cavity. Without shielding, an excited cyclotron state would decay in about 83 milliseconds. The trap electrodes form a microwave cavity that suppresses this decay by up to a factor of two hundred. The lifetime becomes about four seconds. That is long enough to see each rung.
- The spin. The spin is far more isolated. An upward spin takes about 2.1 years to decay. The lowest spin-down state does not decay at all.
- Reading. The states are not read directly. Nickel rings create a small magnetic bottle. This shifts the positron’s slow axial oscillation by about 3.8 Hz per quantum. One loop is read through the frequency of another loop. The measurement does not disturb what it reads.
The purpose is precision. An earlier one-electron experiment measured the electron’s magnetic moment to about one part in 10¹³. The positron version should match that precision. Comparing the two will test the symmetry between matter and antimatter some thirty times more sharply than before.
Quantum electrodynamics explains the ladder perfectly. It says the ladder is there. It says nothing about what the ladder is. That is where our two answers begin.
2. The first answer: ‘t Hooft’s cogwheels
Gerard ‘t Hooft won the Nobel Prize in 1999. He has argued for over two decades that quantum mechanics is not the floor of reality. His position is set out in The Cellular Automaton Interpretation of Quantum Mechanics (2016). In his view, quantum theory is a superb calculating device. Underneath it lies a world that is deterministic and discrete.
His construction has three steps.
Ontological states. There is a fixed set of states the world can actually be in. At any moment, exactly one of them is realised. In quantum notation, the only “real” wave function takes the value 1 on one basis state and 0 on all others.
Evolution. At each tick of a clock, the world moves from one state to the next. The step matrix contains only zeros and ones. It is a permutation. It must be invertible, so no two states ever merge.
The cogwheel. The simplest such system has N states visited in a circle: 1 → 2 → … → N → 1. ‘t Hooft calls this a cogwheel with N teeth. He then proves a key structural fact. Take any finite, deterministic, time-reversible system. Start anywhere and follow the evolution. You will return to your starting point after a finite number of steps. Start from an unvisited state and you trace another closed cycle. Continue until every state is used. The result: the most general such system is a collection of cogwheels, each with its own number of teeth, all turning on the same tick.
From cogwheel to ladder. Apply a discrete Fourier transform to the N states of one cogwheel. The evolution operator becomes diagonal. It defines a Hamiltonian with evenly spaced eigenvalues, E_k = 2πk/N. In the continuum limit, with a small constant shift, this becomes (2π/T)(k + ½) — the spectrum of the harmonic oscillator. ‘t Hooft notes that the Zeeman splitting of an atom with angular momentum J is exactly the spectrum of a cogwheel with 2J+1 teeth. A Zeeman atom, seen in the right basis, is a cogwheel.
Superposition as template. What, then, is a superposition? For ‘t Hooft, it is a template: a state we write down when we do not know which ontological state the system is in. Templates obey the Schrödinger equation and are indispensable for calculation. But they are not states the world can be in. The dead cat is an ontological state. So is the live one. The superposed cat solves the equations but never occurs. Wave-function collapse is not a physical process. It is an update of knowledge.
The observer. Bell’s theorem seems to forbid any local deterministic account of entanglement. ‘t Hooft’s reply: Bell’s argument assumes experimenters can choose their settings independently of everything else. In a deterministic world, the experimenters and their instruments obey the same laws as the particles. Their choices are not outside the system. This position is called superdeterminism.
The open gap. ‘t Hooft is candid about what remains unsolved. Free cogwheels are easy. Interacting particles are hard. Models with interactions often fail to be both local and bounded from below. He does not yet have a cellular automaton for the Standard Model. He suggests gravity may be needed to close the gap.
3. The second answer: everything is a closed line
The Vacuum.Net theory starts from a single axiom. Everything is a closed line. A thing exists because it returns. Outside the path there is nothing. The observer is part of the path.
Three elements fill this out.
Trits. The net is written in balanced ternary. Each position carries one of three values: −1, 0, +1. Balanced ternary needs no minus sign and no rounding rule. Negation is simply the exchange of + and −. A trit is not an extra assumption. It is how the ternary numbers read a position on the loop.
Addresses. An address is a word of trits written on a ring. A ring has no first position. So an address is not a single word but a rotation class: the word plus all its cyclic shifts. In combinatorics this is called a necklace. The number of necklaces of length n over three symbols grows roughly as 3ⁿ/n. At depth 19 there are 61,171,659 of them.
Two layers. The net has two layers. The address layer is frozen topology: the closed loops that keep coming back. The tension layer is where anything happens: exchange, decay, change. Earlier work (“The Net Has No Hidden Clock”, September 2026) exhaustively checked what the address layer can do. On a ring of five positions, every local, conserving, reversible rule was enumerated — all 1,594,323 sign-symmetric rules and all 144 conserving flux-form rules. Only the identity and the shift survive. In the address layer there is only passage. Change lives in the tension layer.
4. The identity: the axiom is the theorem
Put the two descriptions side by side. Three statements follow almost at once.
4.1 The axiom is the theorem. The net’s axiom says everything is a closed loop. ‘t Hooft’s theorem says every finite, deterministic, reversible system is a set of closed cycles. He reaches this as a consequence of determinism and finiteness. The net takes it as its ground. They state the same fact about the world.
4.2 An address is a cogwheel. Take one address: a necklace of trits on a ring of length n. One step of the loop moves the word one position along the ring. After a number of steps equal to the necklace’s period d, the word returns to its start. That is exactly a cogwheel with d teeth. Every address of the net is one cogwheel. The catalogue of addresses at depth n is the list of cogwheels at depth n.
4.3 The address dynamics is the cogwheel step. The exhaustive check found only one non-trivial local dynamics on a single ring: the shift. The shift is the cogwheel step. The address layer is therefore not merely compatible with ‘t Hooft’s automaton. It is the pure cogwheel automaton, with nothing added.
4.4 What this gives for free. ‘t Hooft’s machinery applies directly. Each address of period d carries an evenly spaced spectrum, 2πk/d. Energy is the number of full turns per tick: a frequency. The constant h enters only when frequencies are converted into joules. Quantum ladders, Zeeman levels and oscillators are what cogwheels look like in quantum notation. In the net, they are what loops look like.
One further detail matters. In his treatment of fermions, ‘t Hooft writes evolution operators whose entries are −1, 0 and +1. These are balanced-ternary values. The trit is already present in his mathematics.
5. A worked example: depth three
Abstract identities convince less than a small case worked out in full. Take words of three trits. There are 3³ = 27 of them. Group them into rotation classes:
| Class (representative) | Period (teeth) | Trit sum | Mirror partner |
|---|---|---|---|
| 0 0 0 | 1 | 0 | itself |
| + + + | 1 | 3 | − − − |
| − − − | 1 | −3 | + + + |
| 0 0 + | 3 | 1 | 0 0 − |
| 0 0 − | 3 | −1 | 0 0 + |
| + + 0 | 3 | 2 | − − 0 |
| − − 0 | 3 | −2 | + + 0 |
| + + − | 3 | 1 | − − + |
| − − + | 3 | −1 | + + − |
| 0 + − | 3 | 0 | 0 − + |
| 0 − + | 3 | 0 | 0 + − |
There are eleven classes: three cogwheels with one tooth and eight with three teeth. Check: 3 × 1 + 8 × 3 = 27. The formula agrees: g(3) = (1/3)(3³ + 2·3) = 11.
Now read off the spectrum. A one-tooth cogwheel has a single level, E = 0. A three-tooth cogwheel has three levels: 0, 2π/3 and 4π/3. Across all 27 states:
| Level (per tick) | Number of states |
|---|---|
| 0 | 11 |
| 2π/3 | 8 |
| 4π/3 | 8 |
That is the complete “quantum” description of the net at depth three: a finite collection of loops, each turning on its own ring, each with its own small ladder. Note that the spacing 2π/3 is exactly the phase step of one ternary position: one third of a full turn.
Three features of the table matter for what follows.
- Mirror pairs. Negating every trit maps each class to a class with the opposite trit sum. Negation does not interfere with rotation. A class and its mirror therefore carry identical spectra. Mirroring changes the sign. It never changes the ladder.
- Self-mirrored neutrality. The class 0 0 0 is its own mirror. The classes 0 + − and 0 − + are each other’s mirror and each other’s reversal. Mirror and reversal together map each onto itself.
- Neutral words. Three classes carry trit sum 0. They are loops that exist but carry no net sign.
6. The experiment, read as cogwheels
With the identity in place, the Northwestern experiment can be read line by line.
The cyclotron ladder is a cogwheel. The positron’s orbit is a closed loop. Its quantum number counts closures. The evenly spaced ladder ħω_c(n + ½) is precisely the spectrum ‘t Hooft derives for the continuum limit of a cogwheel, including the half-step offset. The experiment shows, rung by rung, what a single cogwheel looks like when one tooth is excited at a time.
The spin is a second cogwheel. Spin and orbit are two loops turning at nearly the same rate. Their frequency ratio is g/2, about 1.00116. The anomaly frequency is the beat between two cogwheels whose tooth counts differ slightly. What precision physics measures, to twelve or thirteen digits, is the ratio of two loops. That ratio is a pure number. It needs no constant of nature to be stated.
The positron is a mirror loop. To keep ω_c positive, the authors orient their axis opposite to the magnetic field. The positron turns the other way round. In the net, antimatter is the mirrored address: every trit negated. Negation commutes with the cogwheel step, so a mirrored address carries the same ladder. The net therefore gives electron and positron exactly the same spectrum, and so exactly the same magnetic moment. The comparison the Northwestern group is preparing tests precisely this.
Loading through a zero. The positron enters the trap bound to an electron. The pair is neutral: +1 and −1 make 0. The trap’s field does not act on it. Only inside does the pair split, and only then does charge appear. This is the net’s neutral word made visible in the laboratory. A loop with sum zero passes where a signed loop would be stopped.
The cavity is context. An excited cyclotron state decays by radiating into the electromagnetic field. In the language of loops, decay is an exchange between the positron’s cogwheel and cogwheels of the surrounding field. That exchange belongs to the tension layer. The cavity removes the field’s partner loops near 156 GHz, so the exchange is suppressed: 83 milliseconds becomes four seconds. The decay rate is not a property of the positron alone. It is a property of the positron together with its surroundings. In the net this is no surprise. Context belongs to the path.
The spin is frozen. The spin barely exchanges with anything. Its calculated lifetime is about two years; the lowest spin-down state does not decay at all. The spin behaves like the address layer: frozen, passing, unchanged. The orbit behaves like the tension layer: it exchanges, rises and falls. The two-layer structure of the net is visible in a single particle.
Reading through another loop. The cyclotron and spin states are read through the shift they cause in a third loop, the slow axial oscillation. No instrument stands outside. One loop is read through the count of another loop coupled to it. This is what the net means by saying the observer is part of the path. It is also what ‘t Hooft means when he refuses to place the experimenter outside the deterministic system.
7. Where the net goes further
The identity is close, but not complete. In two respects the net needs less than ‘t Hooft’s automaton.
No universal clock. ‘t Hooft’s cogwheels all turn on one common tick, which he places near the Planck time. The net has no such clock. Time in the net is the counting of turns of the loop itself. Each address returns on its own period. ‘t Hooft’s own mathematics points this way: each cogwheel’s energy can be shifted by its own arbitrary constant with no physical effect, because which cogwheel a state belongs to never changes. Each cogwheel is, in that sense, its own clock. The net takes this seriously and drops the common tick.
No prior lattice. ‘t Hooft places his automaton on a lattice given in advance: a grid of sites in space. The net has no grid. The net is the path the movement traces. Addresses are not positions on a pre-existing structure; they are the structure. ‘t Hooft must eventually explain why his lattice has the geometry it has. The net does not carry that burden.
Superposition. For ‘t Hooft, a superposition is a template, not a state of the world. The net arrives at the same place from its own start. Outside the path there is nothing. An address is therefore either traced or not traced: 1 or 0. A weighted mixture of addresses is a statement about what we know, not a configuration of the net. Probabilities measure ignorance of which loop is being traced.
8. What both still have to write
There is one place where neither ‘t Hooft nor the net is finished, and it is the same place.
Free cogwheels are simple. The world, however, is full of exchange: emission, absorption, scattering, chemistry, life. In ‘t Hooft’s account, this is the problem of building a cellular automaton with local interactions whose quantum Hamiltonian is both local and bounded from below. He states openly that he has not found it for realistic particles.
In the net, this is the tension layer. The address layer allows only passage. All change must therefore come from the rule by which loops exchange trits while the total balance stays conserved. The exhaustive result adds its own constraint: this rule cannot live inside a single ring with nearest-neighbour steps. It must act between rings, or reach further along them.
The cavity in the positron experiment shows what such a rule must do. It must let the cyclotron loop give up a tooth to the field. It must make that exchange depend on which partner loops are present. And it must leave the spin loop almost untouched. Writing that coupling rule explicitly, in balanced ternary, between rings, is the next piece of work. It would complete the automaton exactly where ‘t Hooft’s remains open.
9. Conclusion
A single positron on a ladder looks like the purest quantum object there is. Read with ‘t Hooft, it is a cogwheel: a closed cycle of states that quantum notation renders as evenly spaced levels. Read with the Vacuum.Net theory, it is a loop: a thing that exists because it keeps coming back. These are not two readings. They are one.
The net’s single axiom is ‘t Hooft’s structural theorem. The net’s addresses are his cogwheels. The net’s address dynamics is his cogwheel step, and nothing else. The positron experiment — its loading, its cavity, its frozen spin, its indirect reading — shows the two layers of the net working in one particle. The coming comparison of electron and positron tests a symmetry the net carries in its arithmetic: negating the trits changes the sign, never the ladder.
Quantum mechanics, on this view, is what loops look like when you calculate with them. The loops themselves are what there is.
Annotated references
The experiment
- T. G. Myers, L. Soucy, B. A. D. Sukra, B. Sinha, G. Gabrielse, “One-Positron Quantum Cyclotron,” Phys. Rev. Lett. 137, 173001 (2026). doi:10.1103/gbkc-6nlj. Open access.
The source of every experimental detail in Sections 1 and 6: the 5.6 T field, 300 mK temperature, sodium-22 source, positronium loading through a 640 µm hole, cavity-inhibited decay (83 ms to ~4 s), the ~2.1-year spin lifetime, and quantum non-demolition reading via the axial frequency. This is the primary paper on which the entire essay rests. - “A Quantum Cyclotron for Antimatter,” Physics 19, 132 (2026).
The APS viewpoint accompanying the paper. A short, non-specialist account of the experiment’s significance. Useful for readers who want an accessible entry point before tackling the PRL paper itself. - X. Fan, T. G. Myers, B. A. D. Sukra, G. Gabrielse, “Measurement of the Electron Magnetic Moment,” Phys. Rev. Lett. 130, 071801 (2023).
The electron counterpart of the positron measurement, at about one part in 10¹³. The positron result will be compared against this value to test matter–antimatter symmetry in the lepton sector. - S. Peil, G. Gabrielse, “Observing the Quantum Limit of an Electron Cyclotron,” Phys. Rev. Lett. 83, 1287 (1999).
The first one-electron quantum cyclotron. This paper established the experimental template that the positron version follows. - G. Gabrielse, H. Dehmelt, “Observation of Inhibited Spontaneous Emission,” Phys. Rev. Lett. 55, 67 (1985).
The original demonstration that a trap cavity suppresses cyclotron decay. Provides the physical basis for the “cavity as context” reading developed in Section 6. - L. S. Brown, G. Gabrielse, “Geonium Theory: Physics of a Single Electron or Ion in a Penning Trap,” Rev. Mod. Phys. 58, 233 (1986).
The standard reference for the energy levels, frequencies and magnetic-bottle coupling used throughout the essay. The technical backbone of single-particle trap physics.
The cellular automaton
- G. ‘t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, Fundamental Theories of Physics 185 (Springer, 2016); arXiv:1405.1548. Open access.
The fullest statement of ‘t Hooft’s position. Chapter 2 gives the cogwheel model, its evenly spaced spectrum, and the theorem that every finite deterministic reversible system is a set of cogwheels. Chapter 13 gives the continuum limit and the harmonic oscillator. Chapter 15 uses evolution operators with entries −1, 0, +1 — balanced-ternary values. Chapters 3–5 treat templates, the Born rule and superdeterminism. Chapters 8–9 state the open problems with interactions. - A. Einstein, B. Podolsky, N. Rosen, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Phys. Rev. 47, 777 (1935).
The classic statement that quantum theory does not describe what is. ‘t Hooft explicitly places his work in this line of thought. - J. S. Bell, “On the Einstein Podolsky Rosen Paradox,” Physics 1, 195 (1964).
The theorem ‘t Hooft answers with superdeterminism, and which the net answers by placing the observer on the path. Essential context for the debate over determinism and quantum foundations. - E. Fredkin, “Digital Mechanics,” Physica D 45, 254 (1990).
An early programme for discrete, deterministic physics. Acknowledged by ‘t Hooft as a direct influence on the cellular automaton interpretation.
The net
- J. Konstapel, “The Vacuum.Net Theory: Foundational Paper” (Leiden, 2026), constable.blog.
Sets out the axioms of the net, the role of closure, the double cover and the quaternionic ground. The single axiom — everything is a closed line — originates here. - J. Konstapel, “Everything Has Its Number in the Maze” (Leiden, 2026), constable.blog.
The balanced-ternary ground of the net and the address theorem. Explains why three values, and why no minus sign is needed. - J. Konstapel, “The Net Has No Hidden Clock” (Leiden, September 2026), constable.blog.
The exhaustive check that only the identity and the shift survive as local, conserving, reversible rules on the ring. The basis of Sections 3 and 4.3 of this essay. - J. Konstapel, “The Current State of the Vacuum.Net Theory” (Leiden, 3 September 2026), constable.blog.
The two-layer law — address layer and tension layer — used throughout Sections 3, 6 and 8. - P. Rowlands, Zero to Infinity: The Foundations of Physics (World Scientific, 2007).
The nilpotent rewriting of the Dirac equation and the principle of zero totality. Behind the reading of the neutral positronium pair as +1 and −1 making 0. - T. Hattori, S. Takesue, “Additive Conserved Quantities in Discrete-Time Lattice Dynamical Systems,” Physica D 49, 295 (1991).
The flux-form characterisation of conserving rules, used in the exhaustive check of Section 3. - D. Das et al., “Algorithms for Ternary Number System” (2012).
On counting ternary words up to rotation. The source of the necklace correction to the address count used in Section 5.