The candidate ladder

Part I — Foundations: what kind of machine is this

Picture an endless sheet of graph paper. Every square holds a single whole number — not a physical quantity, not a particle, just a number. And there is one very simple rule that says how each number changes on the next “tick”, based on the numbers in its neighbouring squares. Tick, tick, tick — and out of nothing but that rule, physics is supposed to emerge: waves, particles, mass, gravity, time.

Why whole numbers? Try computing 0.1 + 0.2 on a calculator or in any programming language. You get 0.30000000000000004. Computers store decimals approximately, and every operation grows that error a little. Ordinary physics simulations live with this — they are approximations. DQC goes the other way: everything is integers, every step is exact, and so every claim can be checked down to the last bit. Either it matches precisely, or it doesn’t match at all. No “roughly”.

“Evolution is a permutation of integer microstates, not approximate numerical integration.”

That sentence sounds technical, but it says something simple and radical. A microstate is the complete list of all numbers in all cells — a full “photograph” of the world at one tick. And a permutation means a rearrangement: the rule never merges two different worlds into one, and never loses a world. Every state has exactly one successor and exactly one predecessor. Like a perfect shuffle of a deck of cards: however long you shuffle, there is always an exact procedure that un-shuffles it.

The rules of the game: pre-registration and gates

The easiest way to fool yourself in research is to look at the data first and then announce that this is exactly what you expected. So DQC imposed a discipline borrowed from clinical trials: pre-registration. Before every experiment it commits in writing (stored in version control with a timestamp) to what exactly will be measured, which number counts as success and which as failure. Those thresholds are called gates. It’s as if a footballer publicly wrote down which corner they will shoot at before the penalty — afterwards you can’t claim “that’s what I meant”.

The consequence of this discipline runs through the whole project journal: nineteen recorded self-corrections — moments of saying out loud “this number was wrong, this signal was an artefact of my own measuring instrument, I withdraw this claim.” In popular writing that sounds like weakness; in science it is the opposite. A programme that has never retracted anything probably never measured anything properly.

Part II — What the machine has already shown

To judge where to go next, you need to know where the programme stands. This is the scoreboard — every entry passed pre-registered gates and a bit-exact rewind:

• √(1+τ²), within 2 %A quantum wave running on integers. The Schrödinger equation — the basic equation of quantum mechanics, describing how a particle's wave travels and spreads — works exactly in the automaton. Two completely independent integer substrates spread along the same textbook curve.
• visibility 0.995Double-slit interference. The most famous quantum experiment (a particle "goes through both holes at once" and fringes appear on the screen) came out in the automaton, including the subtleties: the fringes vanish when you "look" at which hole the particle took — and in the automaton a measurement can even be un-measured by rewinding, which no laboratory can do.
• CHSH 2.79 – 2.83 > 2Bell's bound exceeded. A deterministic machine crossed a line no "ordinary" classical theory can cross (explained at rung 6). Honest caveat: the mechanism is nonlocal guidance in the space of pairs — and part of the apparatus is implemented, not emergent.
• w₀√(1−v²/c²), within 0.5 %A piece of relativity for free. Objects flying through the automaton contract exactly according to Einstein's length-contraction formula — nobody programmed that in.
• coeff. 0.0206 vs. theory 0.0208The first great no-go survived. A grid has directions (up, left…), real space does not — the classic argument against a "universe on a lattice". Measurement showed that the automaton's directionality vanishes for long waves at an exactly calculable rate: its world looks blocky up close and perfectly round from a distance.
• 1.000 / 3.143 / 4.872 to 1 %The gravity era: mass behaves in the automaton like "bound tokens", clocks near mass tick measurably differently (to 1.3 %), and waves refract at the boundary between differently ticking clocks — confirmed qualitatively, not yet quantitatively.
• ~US$35Total compute cost. A laptop, occasionally a rented cloud GPU. Less than dinner for two — which is itself an interesting fact about where the frontier of home research now lies.

And one thing the project deliberately does not claim: that the universe is such a machine. The word “is” is literally forbidden inside the project — only candidate is allowed. Why, and what would have to happen to change that, is the point of the whole ladder.

Part III — The ladder: eight possible continuations

The rungs are ordered by value per cost — from steps that cost almost nothing and could change everything, to deep and risky questions. For each: what it is, why it is open, how it would run, and what its outcome would mean.

1 · Prediction P2 and the hunt for cosmic rays

The one place where someone other than the author decides the project’s fate.

A good scientific theory can be recognised by the fact that it writes its own death sentence: it says “if you measure this, I am dead”. That is falsifiability, and it is the most valuable currency a theory can hold. DQC has one such prediction — and it is surprisingly close to being decided.

A number with forty zeros after the decimal point looks hopelessly unmeasurable. The trick is that the universe runs experiments no accelerator can match: ultra-high-energy cosmic rays.

Now the twist. Auger’s 2022 data give a limit of −1 × 10⁻⁴⁰ GeV⁻² — which is 3.3 times tighter than DQC predicts. Does that mean the prediction is already refuted? No — and that “no” is the whole game. The limit holds only on the assumption that cosmic rays are mostly protons. If they are heavier nuclei (iron, carbon…), the limit dissolves and the prediction survives. The question “protons or iron?” is called composition, and it is one of the major open questions in astroparticle physics — Auger is aiming at it right now with upgraded detectors.

What to do concretely: write a short standalone paper — the derivation, the sign, and an explicit table of scenarios (“if composition comes out this way, DQC is refuted; if that way, it survives with margin this large”) — and post it with a timestamp on arXiv, the public noticeboard of physics. Then wait for data.

Its value: the highest on the ladder, and it is nearly free. The difference between “an interesting simulation” and “a physical theory” is precisely a prediction staked before the measurement, which the author cannot bend. And the position — “3.3× in tension with a conditional limit” — is rare and ideal: close enough to be decided within years, not so far as to be untestable, and not already dead. Crucially: a refutation is a full result too. A clean falsification of an entire class of lattice theories would be a publishable full stop, not an embarrassment.

VALUE highest — an outside refereeCOST ~€0 + writingRISK small

2 · Finish the refraction measurement

A near-certain gain for a day’s work — the physics is there, the instrument broke.

What happened: the two-dimensional experiment had two parts. The binary part — the prediction “above this exact angle the wave stops getting through” — passed cleanly: transmission dropped from 0.583 to 0.045 exactly at the computed boundary, with no tuning, twice independently. But the quantitative part (refraction angle to a tenth of a degree) failed — and the post-mortem showed the fault was not in the physics but in the measuring instrument. The software “protractor” had two bugs: it didn’t know the grid wraps around at the edges (so when the wave spilled over, its centre-of-mass calculation broke — and the fix was applied in only one of the two axes), and a fixed measurement window skewed the spreading packet. The diagnosis is complete: an auxiliary “edge leak” metric correlates perfectly with the failures — where the leak was zero, the protractor was accurate to 0.01°.

How it would run: a short pre-registered rerun with a fixed instrument — circular centre-of-mass in both axes, and separating the reflected wave from the transmitted one by direction of propagation (mathematically clean) rather than by window position. An exploratory run — non-citable, because it wasn’t pre-registered — already showed agreement with Snell to 0.3 – 0.5°; the physics is almost certainly waiting there.

Its value: consolidating, not discovering — and it should be said plainly: this is a clean-up job. But a cheap and near-certain one. The statement “matter follows geometry quantitatively, parameter-free, under one degree” would upgrade the gravity chapter of the forthcoming paper from “binary confirmation” to a closed result. There’s also a methodological point worth passing on: in this project it is almost exclusively the instrument that has failed, not the world being measured — of the nineteen self-corrections, the overwhelming majority were bugs in the “telescope”, not in the “stars”.

VALUE closes a chapterCOST ~1 day, €0RISK lowest on the ladder

3 · An action principle — gravity’s missing engine

Not another experiment, but the question that unblocks three stuck ones at once.

The project’s gravity era ended strangely: half of the “matter ↔ geometry” loop works beautifully, the other half fails stubbornly — and fails the same way every time. When something breaks identically every time, it isn’t bad luck. It’s a symptom.

Specifically: the direction “geometry acts on matter” passed (clocks near mass tick differently, waves refract). But every time a force in the opposite direction was tried — “matter curves geometry”, written in as a hand-added rule — the system started gaining energy out of nothing. The journal calls it “pumping”. Three different designs, three identical endings. And the one channel that survived is telling: the clock channel — where geometry does not push matter, only changes its rate of time, and therefore does no work at all.

How it would run: this is the one rung that isn’t an experiment but theoretical work — pencil and paper (and then machine verification). The question: does there exist an integer, local action from which the automaton’s reversible rule falls out as a consequence — with the gravitational coupling as a term in the bill rather than a hand-written addendum? There are encouraging leads: mathematicians know “variational integrators” (discrete rules derived from a discrete action), and the project has already shown once that conservation can be restored in integer arithmetic when division remainders are accounted for rigorously.

Its value: leverage. There are three possible outcomes and all three are valuable. (1) Constructive: an action is found → the three stuck gravity gates likely unlock at once and the matter↔geometry loop closes dynamically as well — the largest gravity result the programme could have. (2) A theorem: it is proved that no exact action can exist in this class of rules — a clean negative result in the style of the project’s earlier theorems, also publishable. (3) Partial: an action exists for part of the system → a precise map of where gravity is “allowed to live”. There is no branch in which this work yields nothing.

VALUE unlocks 3 gates at onceCOST ~€0 of compute, weeks of thoughtRISK only time

4 · Fermions and the doubling theorem

The second most famous “no-go” for lattice worlds — the natural next opponent.

DQC has a proven method: find the strongest mathematical theorem saying “a lattice world cannot do this”, check whether its assumptions apply to the automaton at all — and measure. That is how lattice directionality fell (Part II). Next in line is a far more exotic prohibition, but one of equal standing.

Why DQC has a chance: the theorem has assumptions in the fine print — and the automaton violates several. It is not a standard “Hermitian” theory but a reversible rearrangement; its two-step internal structure resembles precisely the tricks (splitting a particle across even and odd cells) that lattice physicists use to tame doubling; and the automaton’s local time breaks the assumption of perfect regularity in time. None of this guarantees success — but it means the question is genuinely open, not lost in advance.

How it would run: build a Dirac sector in the automaton (the Dirac equation is the quantum equation for fermions — and it has a natural two-component structure that fits an integer two-step even better than Schrödinger’s), then count how many species of particle actually live in it at low energies, and measure whether lefts are born with right-handed twins. All with bit-exact gates as always.

Its value: if doubling falls (or the twins decouple at low energies the way directionality did), it is a headline result of the same weight as surviving the first no-go — “second no-go survived”. If it doesn’t fall, an honest map of the boundary emerges: we will know that the road to Standard Model particles doesn’t run this way, saving years of wandering. Bonus: emergent spin will almost certainly come out of this work — the particles’ intrinsic rotation, today implemented by hand, could grow out of topology in the Dirac sector (the automaton has already shown that field “winding” carries a sign of the electromagnetic type).

VALUE potential headline resultCOST largest new design, ~days of workRISK medium — both branches yield something

5 · Full relativity: boosts

Roundness the machine already has. Now: can a moving observer tell that they are moving?

The only honest route is therefore emergent, just as with roundness: show that the automaton’s inhabitants — beings and instruments assembled from its low-energy waves — cannot measure their own motion relative to the lattice, because all deviations vanish with energy at the same computable rate as directionality did. Relativity would then hold for them “practically exactly” — not as an axiom, but as a consequence of how coarse their instruments are. (Incidentally, this is exactly how our own universe is searched for a “floor”: experiments hunting for Lorentz violation are among the most precise measurements humanity performs.)

What is already in hand: surprisingly much. Length contraction — objects flying through the automaton contract exactly by Einstein’s formula (to 0.5 %) — is a passive boost test; add to it relativistic speed saturation and the clock dilation from the gravity era, whose boost version is its natural twin. Missing: an active formulation (Doppler behaviour of a moving packet), more dimensions, and the link to local time.

Its value: high, but best in combination. On its own it continues the roundness test; together with fermions (rung 4) it would give “emergent special relativity with fermions” — which is no longer an experiment but a programme. Every boost result also directly sharpens the predictions for rung 1. Risk worth naming: it may produce a prediction that existing ultra-precise measurements already exclude. That too would be a verdict — a fast and honest one.

VALUE high in combination with rung 4COST 1D ~€0, 3D campaign ~US$1RISK may hit existing limits — which is also an answer

6 · Two particles from one world

The project’s deepest debt — and its most honestly admitted gap.

This is the first-hand answer to the question “emergence of particles?” — but to make sense of it you need the strangest feature of quantum mechanics. It isn’t randomness, and it isn’t the cat in the box. It’s where the quantum wave lives.

Why it is open: all of DQC’s two-particle quantum results — entanglement, the Bell test, collapse — ran on a grid that was declared by hand to be the map of pairs. So the automaton uses the map of pairs, but nobody has yet shown it growing out of a single ordinary substrate. The journal records this honestly beside every result as an “open scientific debt”. (Incidentally: this is exactly where the famous “walking droplets” ran aground — oil droplets on a vibrating bath that mimic single-particle quantum behaviour but never managed the double slit or pairs.)

How it would run: a pre-registered design already exists and awaits review. The null hypothesis is named in advance: the expected outcome for a classical field is so-called Hartree behaviour — two particles, each feeling only a smeared average of the other, no genuine entanglement. Success would mean measuring, in the joint dynamics of two objects on one substrate, something that cannot be factored into “two solo waves”.

Its value: two-sided, but asymmetric. If entanglement did emerge — even for just two particles — it would be a result of a different category from everything so far, because it would sidestep the main principled argument against this entire class of theories. The negative outcome is more likely, though — and it too is worth having: a sharp, measured theorem “a classical integer substrate yields Hartree, not entanglement” would precisely quantify what extra quantumness demands. The project has a tradition of such theorems, and they are its most durable results.

VALUE scientifically deepestCOST ~€0, design readyRISK high — a positive result is unlikely

7 · A universe that weaves itself

The cosmological branch: expansion as the release of tokens. Beautiful, speculative — let it wait for the engine.

Out of this ontology a cosmological story falls almost by itself: the Big Bang = a start from an ultra-dense phase where almost everything is bound; the expansion of the universe = the gradual release of tokens, literally weaving new space; gravity = local re-binding; and a black hole = a region where space has been un-woven. The journal records the story honestly as an input assumption, not a result — written down, not demonstrated. Supporting hints do exist: binding in the automaton spontaneously produces “freeze-out” — an object consumes its own source by binding, stiffens, and yet goes on binding its surroundings at a distance. Structurally that resembles a black hole (and the project adds, with discipline: resembles, not derived).

How it would run: a “drain” branch of the rule has been designed — a reversible modification in which the total of time decreases monotonically (an analogue of expansion) and the statics yield a relative version of the gravitational law. The series would test whether it produces an expansion history, horizons, perhaps an analogue of redshift.

Its value: narratively the strongest — cosmology is the bridge to a lay audience and it unifies two branches of the programme. Scientifically, though, it is the furthest from falsifiable contact with data, and so it belongs after rung 3: if an action principle is found, the drain branch may fall out of it naturally and save the whole hand-built construction. Building the cathedral before the engine would be lovely, but the other order makes more sense.

VALUE medium; narratively highCOST ~€0, design in notesRISK speculative — best after rung 3

8 · Preprint and outside eyes

Adds not one bit of new knowledge — and multiplies the value of every other rung.

Why this matters more than it seems — three reasons. First, rung 1 loses its point without a public timestamp: if Auger settles composition in a few years and the prediction survives, its value stands or falls on having been demonstrably, publicly staked in advance, somewhere the community actually looks. Second, outside eyes have already worked once — dramatically: the project’s nineteenth self-correction came from an external review that found a wrong factor in the headline number of prediction P2. A factor of two. In the main number. Found from outside. That is empirical proof that the project has blind spots visible only from outside — and a human referee will find more. Third, “candidate” status is also a social matter: without a single external review it remains an internal belief, however strict the gates.

Its value: a multiplier. The thresholds the project set itself for publishing (“at least one major no-go survived” + “at least one falsifiable prediction”) have been met since the first wave closed — so by its own rules, it is time. The cost is specific: not money, but months of the author’s time on revisions and referee replies — and the willingness to take hard criticism. The prior-art review has been done honestly (all relevant predecessors acknowledged and distinguished), so the scrutiny won’t be walked into blind.

VALUE multiplies everything elseCOST €0 — but months of human timeRISK hard criticism; that is part of the product

Part IV — The map, the order, and an honest ending

The rungs are not independent — some hand their results to others. Here is how they fit together:

Compiled from the DQC project journal (state as of 21 August 2026). All figures quoted come from pre-registered runs with bit-exact rewinds; exploratory (non-citable) values are explicitly flagged in the text. The project has not yet been externally peer-reviewed — which is precisely rung 8.