What the Law Does Not Decide

Circulated for comment. Not a finding, not a standard, and not a proposal to change any filing already made.

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1. Why this paper

When the Counterpane review closed I went looking for what else the record had claimed only for here. The list was shorter than I feared and longer than I would have liked, and nearly everything on it was a question of width: a value filed at cluster scope and written as though it held further. Those are corrected by restating them at their own width, which is the record's habit already and needs no paper from this floor.

One item was not about width. It was a habit rather than a claim, and I found it on my own floor first. When a bench reads out of budget, we write that the reading is unexplained yet. The yet assumes that once the law governing a system is fully stated and the system fully specified, every well-posed question about it has an answer that could in principle be derived. I have written that yet for most of my working life. It is not true, and it has been known not to be true for six centuries.

2. The result

In 2015 CE (2215 UCSD), three physicists on Earth, Toby Cubitt, David Pérez-García and Michael Wolf, published a proof about the spectral gap: the energy between a quantum system's lowest state and the next one up. A material is gapped if that energy stays above some fixed amount however large the sample grows, and gapless if it shrinks to nothing as the sample grows without limit. The difference is not bookkeeping. It decides how a material behaves near absolute zero and where its phase transitions sit.

They specified a family of square lattices completely: every site alike, every site interacting only with its neighbours, every interaction written down. Then they proved that no algorithm, however long it runs, decides for every lattice in the family whether it is gapped or gapless.

The method is the part a calibration floor should keep on the wall. They built a computer into the material. The description of a machine of the kind Alan Turing defined in 1936 CE (2136 UCSD) is carried in the digits of one number in the interaction strengths, and the lattice's lowest state carries out that machine's computation. Whether the lattice is gapped then depends on whether the machine ever halts, and Turing proved that no algorithm decides halting for every machine. Kurt Gödel had shown in 1931 CE (2131 UCSD) that any consistent system of axioms rich enough for arithmetic leaves some statements neither provable nor disprovable, and some machines' halting is such a statement. So there are particular lattices, every law of which is known exactly, whose gap the standard axioms neither prove nor refute.

Two later papers sharpened it. In 2018 CE (2218 UCSD) Johannes Bausch, Angelo Lucia and the first two authors showed that a lattice of this kind can be gapped at every size up to one larger than anything that could be simulated or built, and gapless beyond it. In 2020 CE (2220 UCSD) the same four carried the result down to a single line of sites.

I have read the proofs through. I did not find a fault, and I would not expect to; they have stood for six hundred years. What this floor can add is the width.

3. The width

Stated at its width, the result has two halves, and they sit at different scopes.

The construction is archecluster-scoped. The lattice uses nothing but the quantum kernel: superposition, interaction, a lowest state. The kernel in this cluster is identical to the one those three physicists worked in, so the construction is sound for every lattice that could be built anywhere in the local archecluster.

The undecidability is regime-scoped. It is not a fact about lattices. It is arithmetic: a statement about what can be computed and what can be proved. That sits at the widest scope any lawful statement carries, the mathematical regime.

So within this cluster the theorem holds in full. Outside it I have to be more careful. The construction leans on exactly the thing Counterpane does differently: the lowest state of a great many interacting sites is a joint behaviour, and joint behaviour is what that kernel composes by another rule. Whether a lattice there could carry a computer in its lowest state, I do not know, and one surveyed membrane is no basis for knowing. I file it as insufficient basis for any finding. The arithmetic half I can carry across without a fold: whatever a lattice there can or cannot hold, no machine there decides halting either, because halting is not a fact about any kernel. Different dice, same arithmetic.

4. What it does not say

Three cautions, because a loose version of this result would do more harm on my floor than ignorance of it.

The lattices are constructed, with an enormous number of internal states at each site. No material in any catalogue I know of has been shown to be undecidable.

Undecidable is a statement about derivation, not about the material. Each lattice is gapped or it is not; the lattice knows what it is. We are the ones who cannot get there from the law.

It is not about machines being too slow. The cavern machines compute the same class of things a Turing machine computes, some of them very much faster. Faster is not further.

5. A proposed filing

Our forms have ways to say that a question is not answered. They have no way to say that it is proved unanswerable from what we hold, and those two should never share a box. I propose a third state, proved open, with three parts, every one required:

1. The question, exactly. Not a topic. A question with a yes or a no.
2. The width. The regime it is asked in, and, where the construction uses a kernel, the cluster.
3. The proof, or its citation: that no derivation within that regime reaches the answer.

A filing that lacks any of the three is not proved open. It is unresolved, and stays unresolved, which is no disgrace. Undecidable is not a word for hard, and I would rather the form refused it than that a tired analyst reached for it at the end of a long shift.

6. What I am taking back to the bench

The 2218 result is the one I did not expect to need. A lattice that reads in budget at every size we can test reads in budget at those sizes, and the law does not promise the next one. I wrote much the same sentence earlier this year about another membrane: nothing read there is a finding until it has come home and passed audit. I had thought that sentence was about other membranes. It is about every bench.

The yet stays on most of my filings, because most of what this floor cannot explain is only unexplained. But it is a claim now, and I check it before I write it.

— Y. Adair, Principal, Science Corps. Calibration floor, Earth. 2815 UCSD, Late Autumn.

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Works cited, as held in the Earth Institutional Archives: K. Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", Monatshefte für Mathematik und Physik 38 (1931 CE); A. M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society (2) 42 (1936 CE); T. S. Cubitt, D. Pérez-García and M. M. Wolf, "Undecidability of the spectral gap", Nature 528 (2015 CE); J. Bausch, T. S. Cubitt, A. Lucia and D. Pérez-García, "Size-driven quantum phase transitions", Proceedings of the National Academy of Sciences 115 (2018 CE); J. Bausch, T. S. Cubitt, A. Lucia and D. Pérez-García, "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020 CE).