How the work fits together.

All 48 works appear once below: in one of six steps, or among the works that stand apart for now. Each step states a finding in plain words, quotes a paper that states it, and names the work to read first. The grouping is our reading; the connections drawn further down are not, since each arrow names the theorem that makes it and each arc the citation behind it.

1. Some perfect behaviour is reachable only by infinite quantum systems.

   Finite quantum systems can come as close as you like to certain perfect scores and still never reach them. Only infinite systems do, and for some games only infinite systems of a particular algebraic type.

   “How a Bell supremum can be approached in finite dimension yet attained only by an infinite-dimensional spatial strategy.”

   Start withNo finite quantum system reaches the largest violation of the I₃₃₂₂ Bell inequality

   • The same for a whole class of Bell functionals, by Nidhal Mghirbi, whose exact certification of the I₃₃₂₂ bounds came first: exact bounds, the proof-carrying release, Bellman–Jacobi duality
   • A finite game, scored right or wrong, that only infinite dimension wins perfectly, with finite energy
   • A finite game that type III algebras win perfectly and semifinite ones never do
   • A finite game that normal hyperfinite strategies approach but never win

   FieldsNonlocal games, operator algebras Bridge6

2. Coming close has a price, and the price follows exact laws.

   How large a quantum system it takes to come within $\epsilon$ of a perfect score can be pinned down exactly. It grows like $\log (1/\epsilon )$ for I₃₃₂₂ and like $\exp ({\epsilon }^{-1+o(1)})$ for one explicit game, and for broad classes of games the entanglement needed is unbounded exactly when no finite system plays perfectly.

   “How much entanglement does near-optimal play of a two-player nonlocal game require?”

   Start withAn explicit game whose near-perfect play needs exponentially large quantum systems

   • The general law, with the geometry of groups setting the rate
   • Approximating a quantum correlation that no finite system attains costs dimension governed by a Diophantine law for √5
   • The same price, paid in heat by a device that resets before its parts communicate

   FieldsNonlocal games, group theory, number theory, thermodynamics Bridges7, 10, 12

3. For some games and materials, no algorithm can compute the price.

   Finite, simple rules can still have an uncomputable price: the possible growth rates of the price of perfection are exactly the limit-computable numbers, which include numbers no algorithm can compute. Uncomputable prices appear again in the critical exponents of model quantum materials, in the laws of noisy cellular automata, and in the memory a repeated quantum channel needs.

   “A computable finite rulebook can have a noncomputable limiting resource rate.”

   Start withFor some fixed games, the entanglement needed for near-perfect play grows at a rate no algorithm can compute

   • The same classification at every regular scale, not only for power laws
   • Translation-invariant lattice models whose critical exponents no algorithm computes, already in two dimensions
   • Gapped lattices that carry any limit-computable amount of entanglement
   • In noisy cellular automata, a law read from a definite start can sit one level of uncomputability above any law its steady states share
   • No algorithm can tell how much memory a repeated quantum channel needs

   FieldsComputability, nonlocal games, lattice matter, cellular automata, quantum channels Bridges1, 2, 3, 4, 8, 9

4. Arithmetic decides whether a wave spreads.

   In the Aubry–André model of a quasi-periodic crystal, usually called localized when its potential is strong, whether a wave packet spreads, and how fast at its farthest, is set exactly, for almost every phase, by how well the frequency is approximated by fractions. The same law governs every supercritical quasi-periodic system with a real-analytic potential of type I, and for some computable frequencies its exponent is a number no algorithm can compute.

   “Transport thus switches on at the spectral transition $\lambda =e^{\beta }$ of Avila, You and Zhou.”

   Start withSupercritical almost Mathieu operator: at almost every phase, every moment has one upper transport exponent, set by the arithmetic of the frequency

   • The same law for every supercritical analytic quasi-periodic operator of type I
   • At the worst phase the localization strength is divided by at most twice the acceleration, and light cones in the XY chain follow both laws

   FieldsSpectral theory, number theory, computability Bridges5, 13

5. Infinite groups, studied through their finite pieces.

   Many questions about an infinite group come down to finite data: a finite presentation, a few elements of its group ring, a set of permutations that imitates part of it. Here that view settles Kaplansky’s group-ring problems for hyperbolic groups in positive characteristic, and two theorems carry it into nonlocal games and the memory of quantum channels.

   “Every finite piece of an amenable group can be imitated by permutations of a finite set … How many points does such an imitation need?”

   Start withAll four of Kaplansky’s group-ring conjectures fail for a hyperbolic group, in every prime characteristic

   • OpenAI’s hyperbolic group has no good imitation by finite permutations: it is not sofic
   • Finitely presented infinite p-groups for every prime, concurrently with Kenta Kitamura
   • Houghton’s group needs more than polynomially many points
   • An amenable group that needs nearly exponentially many, and channels that need nearly linear memory

   FieldsGroup theory, nonlocal games, quantum channels Bridges10, 11

6. Acting in time costs work: order, not speed, sets the price.

   A device that must let an output go before a later input arrives pays in work for every correlation its record keeps with what has left, however slowly it runs. One law gives that price for every schedule, and Landauer’s erasure bound is one case of it. Around it sit exact results on the memory, purity and apparatus a device needs to apply one quantum channel again and again.

   “The cost survives arbitrarily slow operation: order, not speed, sets it.”

   Start withOne law for the work cost of acting on any schedule

   • The four companion papers, The price of time order, Only quantum channels save work by waiting, The work cost of answering in real time and The price of keeping time, with the letter that summarizes them
   • The memory and purity cost of immediate delivery, which answers Rybár and Ziman’s question on repeatable channels
   • Memory beyond entropy: same spectrum, same entropies, different memory
   • Bath dimension and initial entropy for closed repeated use, rational channels beyond finite baths and compression of Schur channels

   FieldsQuantum channels, thermodynamics, operator algebras Bridges14, 15

Standing apart, for now

• GraphsThe Lovász number is not alone: a continuum of new points in the entanglement-assisted spectrum of graphs, and Duan and Winter’s pentagon conjecture settled. Bridge 16.
• Real quantum networksThe resource theory of complexness references and the frame bus.
• ContextualityA repair price for contextuality, which shows first that its own quantity is not new.
• Quantum computationThe clean two-query threshold for unitary synthesis, part of a question of Aaronson and Kuperberg, and quantum recovery from finitely many sharp questions.
• Quantum foundationsNidhal Mghirbi’s single-author papers: Galilean covariance almost fixes the quantum probability current, observable quantum equilibrium and the relativistic necessity of the Born rule, published in a journal.

A field is a box, and in it is the field’s headline result, which the box links to. An arrow is a theorem that takes a quantity defined in one field and turns it into a quantity of another; each is numbered, and named below with its source. A dashed arrow is a theorem stated conditionally.

The 39 works released together from August to October 2026, by line of research. An arc joins two works when one cites the other in its README, its Zenodo record or its bibliography. No link is inferred from content.

Five works cite none of the others and are cited by none: they stand alone. Arcs show a link, not its direction; the table below gives the direction and the record behind each one. Titles are shortened; each links to its result.

Nidhal Mghirbi’s single-author papers are not drawn; their citation links to the works above and to each other are listed here.