The price of perfection

Answered Can that price be computed from a game’s rules?

Not in general.

1. The uncomputable price of precision

   Seth Douglas and Nidhal Mghirbi · version 4.0.0 · 29 October 2026 · 119 pages

   For finite rational constraint-system games of quantum value one, the positive exponents of the least local dimension needed near perfect play are exactly the positive ${\mathrm{\Delta }}_2^0$ reals, with matched distillable-entanglement realizations; the lower and upper exponent pairs are exactly the $({\mathrm{\Sigma }}_2^0,{\mathrm{\Pi }}_2^0)$ pairs.

   • Answered Can the rate at which the least dimension grows near perfect play be computed from a finite game’s rules?
   • Answered Which lower and upper exponents can one fixed game have?
   • Open Can a finite rational value-one game with no finite-dimensional perfect strategy have dimension exponent zero? posed here
   • Open Is every positive usable-entanglement coefficient of a finite rational game limit-computable? posed here
   • Open Which finite power exponents above one occur for a fixed finite rational game? posed here
   • Open Which positive local-reset heat coefficients occur? posed here

   The paperPDFDOICode

2. Uncomputable critical exponents

   Seth Douglas and Nidhal Mghirbi · version 2.1.0 · 29 October 2026 · 64 pages

   Translation-invariant nearest-neighbour quantum Hamiltonians whose critical gap exponents are exactly the nonnegative ${\mathrm{\Delta }}_3^0$ reals, already in two dimensions; and robust laws of noisy media that no effective inquiry can learn beyond ${\mathrm{\Delta }}_2^0$.

   • Answered Are the critical exponents of lattice matter computable from its Hamiltonian?
   • Answered Can the robust laws of a noisy cellular automaton be unlearnable?

   The paperPDFDOICode

3. A beginning is worth one limit: robust unlearnable laws in noisy cellular automata

   Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 29 October 2026 · 19 pages

   At equal depth, a computable law of a cellular automaton that has a beginning reaches one arithmetical level above any law on which its stationary states agree; at depth one, a rule started from cloned uniform noise realizes every ${\mathrm{\Delta }}_3^0$ value in $[0,1]$ as a robust law.

   • Answered Can the robust laws of a noisy cellular automaton be unlearnable?
   • Open Can cloned preparation be removed from the realization of robust unlearnable laws? posed here

   The paperPDFDOICode

4. An explicit nonlocal game of quantum value one that needs exponential entanglement

   Nidhal Mghirbi and Seth Douglas · version 1.1.0 · 2 November 2026 · 93 pages

   A binary linear system game with 1687 equations in 1993 variables, of quantum value one, in which winning with probability $1-\epsilon$ needs entanglement entropy ${\epsilon }^{-1+o(1)}$, hence local dimension $\exp ({\epsilon }^{-1+o(1)})$; the exponent is sharp.

   • Answered How much entanglement can an explicit game of quantum value one force?

   The paperPDFDOICode

5. The price of precision: usable entanglement in nonlocal games and the geometry of groups

   Seth Douglas and Nidhal Mghirbi · version 1.1.0 · 29 October 2026 · 160 pages

   For the game classes treated, the entanglement demand near perfect play diverges exactly when no finite-dimensional strategy is perfect; bounded demand is undecidable among games of value one; and the geometry of groups gives logarithmic, almost inverse-loss and intermediate laws.

   • Answered When does near-perfect play need unbounded entanglement, and can one tell?
   • Open Is every positive usable-entanglement coefficient of a finite rational game limit-computable? posed here

   The paperPDFDOICode

6. The price of unreachable perfection: dimension laws at every regular scale

   Seth Douglas and Nidhal Mghirbi · version 1.0.1 · 29 October 2026 · 24 pages

   The growth of the least dimension can be prescribed at every regular scale, not only along power laws. At every asymptotically flat scale the normalized endpoints are exactly the $({\mathrm{\Sigma }}_2^0,{\mathrm{\Pi }}_2^0)$ pairs, and a computable target can still lie outside every game’s profile.

   • Answered Which lower and upper exponents can one fixed game have?

   The paperPDFDOICode

7. Matter that pays the price of perfection

   Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 29 October 2026 · 27 pages

   Gapped translation-invariant lattices whose entanglement has any prescribed limit-computable logarithmic coefficient, and a referee cell that converts low energy into strategies for a nonlocal game. The game-driven material theorem is stated conditionally.

   • Answered Can gapped lattice matter carry any limit-computable amount of entanglement?

   The paperPDFDOICode

8. Perfect play in a binary-scoring nonlocal game requires infinite dimension but only finite energy

   Seth Douglas and Nidhal Mghirbi · version 2.0.0 · 29 October 2026 · 39 pages

   A game with ten and eleven questions and at most six answers is won perfectly by an infinite-dimensional strategy and by no finite one, with dimension profile $10{\log }_2(1/\epsilon )+O(\log \log (1/\epsilon ))$. Perfect play needs only finite energy: $275/18$ quanta per oscillator at best.

   • Answered Can a game with a binary score witness the separation between finite-dimensional and spatial quantum correlations?
   • Answered Can perfect play in a finite game require infinite dimension yet only finite energy?

   The paperPDFDOICode

9. The heat price of precision

   Seth Douglas and Nidhal Mghirbi · version 1.0.0 · 27 October 2026 · 21 pages

   For games whose usable-entanglement demand diverges, resetting locally before communicating forces divergent work, while bringing the systems back together before reset keeps the operating work bounded.

   • Answered Does resetting a device before its parts communicate cost unbounded heat?

   The paperPDFDOICode

• Answered Can a game with a binary score witness the separation between finite-dimensional and spatial quantum correlations?