Is probability pixelated?
If A=1 has a floor — a smallest possible probability — then interference and coherence have a grain, and higher-dimensional lattices have a coarser one.
The new integer-token engine (dcl-core v0.2.1) makes A=1 exact: a session is a budget of N indistinguishable probability tokens, and conservation is integer arithmetic rather than a floating-point near-miss. That reframing forces a question the continuum never had to answer:
Is there a minimum probability — a floor?
Call it \delta p_\min. If it is zero, probability is the smooth, infinitely divisible quantity of textbook quantum mechanics. If it is not zero, then probability is pixelated — and that has teeth.
What a floor would mean
A floor says |\psi|^2 cannot be subdivided without limit. There is a smallest amount of probability the world can move, hold, or distinguish. On the lattice the natural candidate is immediate: with N tokens, the smallest change is one token out of N. Below that, there is nothing to move. Continuity would then be an excellent approximation — the way a high-resolution photograph looks continuous — but an approximation, with a pixel size.
This is not a claim that the floor exists. Whether \delta p_\min > 0 is an open question — the point of the δp_min investigation. It is a claim that the question is now decidable, and that the answer, either way, is physical. Here is why it matters.
Pixelated interference, pixelated coherence
Interference and coherence are where a probability floor would first show.
A dark fringe is a place where amplitude cancels. In the continuum it can be arbitrarily dark — cancellation to any precision. With a floor, it cannot: the darkest a fringe can get is one quantum above nothing. Fringe contrast cannot fall continuously to zero; it bottoms out. Likewise coherence carries a grain — you cannot hold a superposition whose components differ by less than \delta p_\min, because there is no such difference to hold. The smooth visibility curves of standard quantum optics would, at the finest scale, be quantised.
That is a genuine departure from continuous quantum mechanics, and it is falsifiable: sufficiently precise interferometry or weak-coherence measurement either finds a floor or it does not. If interference stays perfectly smooth below any reachable scale, then \delta p_\min is zero or below experimental reach, and the pixelated picture is — at least there — wrong. A framework that can be told it is wrong by an experiment is doing its job.
Higher dimensions pixelate more
The most distinctive consequence is structural. In the lattice family, a higher-dimensional member spreads each step’s probability across more neighbours — the same thinning the Plato’s-Cave essay described, and the same economy the six-of-eight essay traced through the lattice’s coordination.
Now add a floor. With a fixed token budget N divided across more directions, each direction receives fewer tokens — and so the dynamics hit the one-token floor sooner. The grain gets coarser. Pixelation scales with dimension: the more dimensions the lattice spends, the blockier its interference.
That turns pixelation into a measurement of dimension. The earlier scoping of the family proposed spectral dimension as the way to tell members apart; pixelation is a second, independent handle — and a more directly observable one, because it lives in interference contrast rather than in the asymptotics of a random walk. Two handles on “which lattice is this?” are far better than one.
The engine is the instrument
What makes this more than a thought experiment is that the release that raised the question can also test it. The integer-token engine does not approximate the floor — it is the floor, by construction. So you can:
- run discrete A=1 against the continuous-amplitude engine on the same interference problem and watch for where they part company — the pixelation scale; and
- run the discrete engine at different lattice dimensions and watch the grain coarsen, testing the “higher dimensions pixelate more” prediction directly, in silico, before any telescope or interferometer is involved.
Empirically falsifiable in the lab; computationally checkable on a desk.
What this is, and is not
In the spirit of the program’s Statement of Intent: none of this asserts that the universe is a token lattice with a hard probability floor. It says that if the A=1 object carries a \delta p_\min, then continuity is the coarse-grained limit, interference and coherence inherit a grain, and that grain encodes the dimension of the substrate. Those are consequences of the object, stated so they can be checked — and now, with the integer engine, checked.
If probability turns out to be pixelated, the surprising thing will not be the pixels. It will be that the size of the pixel was telling us how many dimensions we were standing in all along.
References
dcl-corev0.2.1 — the integer-token engine that makes A=1 exact and the floor testable.- Six of eight and Plato’s Cave and the DCL projector — the lattice family and the dimension question pixelation feeds into.
- Paper I — the continuous-amplitude implementation the discrete engine is tested against.