Every Law Known, One Question Open

The real-world notes behind one phrase on The Scope of Physical Law, where the widest ground a lawful statement can stand on is a mathematical regime's foundations "and what can be proved in it." The Archive speaks in its own voice there and names nobody from our century; the names are here. This page adds nothing to canon.

The conversation this came from was about the limits of theoretical physics, and most of those limits turned out to be practical. The energy at which quantum gravity should matter is about fifteen orders of magnitude beyond any collider. Cosmology has one universe and cannot run its experiment twice. The vacuum energy predicted by quantum field theory misses the measured value by at least fifty-five orders of magnitude. Each of these could in principle fall to a larger machine or a better idea. One limit is different in kind, and it is the one worth a page: a question that stays open after every law is known.

What a spectral gap is. A quantum system has a lowest-energy state, its ground state. The spectral gap is the energy between that state and the next one up. A material is gapped if the gap stays above some fixed positive amount however large the sample grows, and gapless if the gap shrinks to nothing as the sample grows without limit. The difference is physical, not bookkeeping: it governs how a material behaves near absolute zero, how far correlations reach through it, and where its quantum phase transitions sit, since a phase transition is where a gap closes. One of the Clay Mathematics Institute's Millennium Prize problems, the Yang–Mills mass gap, asks for a proof that the theory of the strong force is gapped.

The result. In 2015 Toby Cubitt, David Pérez-García and Michael Wolf published "Undecidability of the spectral gap" in Nature. They built a family of two-dimensional square lattices of quantum particles in which each particle interacts only with its nearest neighbours, by the same rule at every site. The rule is written down completely. They then proved that the question is this lattice gapped or gapless? is undecidable: no algorithm, however long it runs, answers it correctly for every lattice in the family.

The proof works by hiding a computer in the material. The description of a Turing machine, the abstract model of a computer Alan Turing defined in 1936, is carried in the digits of one number in the interaction strengths, and the lattice's ground state is arranged to carry out the machine's computation. Whether the lattice is gapped or gapless then depends on whether the machine ever halts. Turing proved that no algorithm can decide halting for every machine, so no algorithm can decide the gap for every lattice.

The result goes one step further. Kurt Gödel showed in 1931 that any consistent system of axioms rich enough for arithmetic leaves some statements neither provable nor disprovable, and there are particular machines whose halting is such a statement. So, as the authors note, there are particular lattices in the family whose gap is independent of the axioms of mathematics: every law is known exactly, and whether the material is gapped can be neither proved nor disproved from the standard foundations.

Two later papers sharpened the picture. In 2018 Johannes Bausch, Cubitt, Angelo Lucia and Pérez-García showed that a lattice of this kind can be gapped at every size up to some size larger than anything that could be simulated or built, and gapless beyond it. In 2020 the same four extended undecidability to one-dimensional chains.

What it does not say. A loosely true claim is a misunderstanding waiting to happen, so three cautions. First, the lattices are constructed, with an enormous number of internal states at each site; no material found in nature has been shown to be undecidable. Second, undecidable is a statement about algorithms and proofs, not about the material. Each lattice is gapped or it is not. What fails is the derivation from law to answer. Third, it is not a statement about computers being too slow. A quantum computer computes the same class of functions as a Turing machine, some of them faster, so it cannot decide halting either. The record says the same of its own machines when it calls them fast machines rather than oracles.

The reverse limit: effective theories. The more familiar limit runs the other way. Kenneth Wilson's renormalisation group, recognised by the 1982 Nobel Prize in Physics, and the decoupling theorem of Thomas Appelquist and James Carazzone (1975) explain why physics at one scale barely depends on physics far below it: the effects of heavy, unseen particles at low energy are suppressed, or absorbed into a handful of measured constants. That is why chemistry does not wait on quantum gravity, and it is also why the foundations are so hard to see. Every theory we have is effective: valid across a stated range of scales, with the range part of the claim. Put the two limits side by side and they bracket the subject. Looking down, the deeper layers are screened off. Looking up, even a fully known microscopic law does not always yield its large-scale consequence.

What the record takes from this. The Scope of Physical Law page already files every physical claim at a stated range and no wider, which is the effective-theory habit given its own institution. Its widest level, the mathematical regime and what can be proved in it, is where the undecidability result applies. Provability is a boundary inside one regime, not only between regimes, and the page's phrasing already allows for that.

The more useful lesson is for mystery. This setting keeps its mysteries open because they cannot be closed, never because the author has not decided, and the usual gaps in the record are distance, a boundary, or a document that did not survive. The spectral gap shows that a hard-SF record has another kind available, honestly: a question whose laws are fully stated and whose answer is out of reach for a reason that can itself be proved. The 2018 result is a laboratory case of the setting's rule that familiarity is not evidence. A system that has looked the same at every scale anyone has tested is not thereby promised to stay that way.

The borrowing comes with a fence, the same one Marked Absences keeps on the record's side. An undecidable question in physics is exactly stated, and its openness is proved, not asserted. A page in this record that called a question undecidable would owe the same three things: what the question is, in which regime it is asked, and why no derivation reaches it. Without them, undecidable is only a shrug in borrowed clothes, and this record does not shrug.

References, for anyone who wants the originals: K. Gödel, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I", Monatshefte für Mathematik und Physik 38 (1931); A. M. Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem", Proceedings of the London Mathematical Society (2) 42 (1936); T. Appelquist and J. Carazzone, "Infrared singularities and massive fields", Physical Review D 11 (1975); T. S. Cubitt, D. Pérez-García and M. M. Wolf, "Undecidability of the spectral gap", Nature 528 (2015); 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); J. Bausch, T. S. Cubitt, A. Lucia and D. Pérez-García, "Undecidability of the spectral gap in one dimension", Physical Review X 10 (2020).