How expensive is an optimum that a finite quantum system can approach but never attain? For a fixed finite rational nonlocal game of quantum value one, write A_G(n)=log2 D_G(2^(−n)), where D_G is the least local dimension at loss 2^(−n). We prescribe log2 A_G(n) at every logarithmic-space, nondecreasing unit-increment scale F with F(n)/log n → ∞, up to an additive O(log n+log²(F(n)+2)) error, and match the one-way distillable entanglement of the original mixed strategy state. At every asymptotically flat such scale, the positive finite lower and upper normalized dimension endpoints are exactly the ordered pairs with lower endpoint in Σ⁰₂ and upper endpoint in Π⁰₂; existing positive limits are exactly Δ⁰₂. A single uniform spectral-tail construction supplies the density-free profile law behind these results. It also classifies stretched-scale pairs in the unit square and gives quasi-polynomial dimension windows of width three in the exponent. A computable unit-increment target outside every rational game's profile shows that computability alone is insufficient. We locate the boundaries through a quantitative locality cap, an honest-linear compression cap, and an explicitly conditional route to finite exponents above one. A separate exact-marginal correlation theorem identifies a Farey law and its Diophantine endpoint curve. The dimension arithmetic necessity applies to all rational games; the operational realization statements concern the constructed games.
How expensive is an optimum that a finite quantum system can approach but never attain? For a fixed finite rational nonlocal game of quantum value one, write A_G(n)=log2 D_G(2^(−n)), where D_G is the least local dimension at loss 2^(−n). We prescribe log2 A_G(n) at every logarithmic-space, nondecreasing unit-increment scale F with F(n)/log n → ∞, up to an additive O(log n+log²(F(n)+2)) error, and match the one-way distillable entanglement of the original mixed strategy state. At every asymptotically flat such scale, the positive finite lower and upper normalized dimension endpoints are exactly the ordered pairs with lower endpoint in Σ⁰₂ and upper endpoint in Π⁰₂; existing positive limits are exactly Δ⁰₂. A single uniform spectral-tail construction supplies the density-free profile law behind these results. It also classifies stretched-scale pairs in the unit square and gives quasi-polynomial dimension windows of width three in the exponent. A computable unit-increment target outside every rational game's profile shows that computability alone is insufficient. We locate the boundaries through a quantitative locality cap, an honest-linear compression cap, and an explicitly conditional route to finite exponents above one. A separate exact-marginal correlation theorem identifies a Farey law and its Diophantine endpoint curve. The dimension arithmetic necessity applies to all rational games; the operational realization statements concern the constructed games.
In a nonlocal game of quantum value one that no finite-dimensional strategy wins, near-perfect play needs growing dimension, which separable states can have. We ask how much usable entanglement it needs: the demand at loss ε is the least one-way distillable entanglement of the original state of a strategy of loss at most ε. For synchronous games with tested diagonals and for constraint-variable consistency games, demand diverges exactly when no finite-dimensional strategy is perfect; then, for any number of Bell pairs and any fidelity below one, every sufficiently good mixed strategy yields that many at that fidelity by local isometries. For rational synchronous or binary constraint games, a loss guaranteeing this is computable under a promise, but boundedness of demand is undecidable among games of value one. Demand is at least logarithmic if near-winners have a sign relation and an even dilation mirrored by both players, as in a game from Abels' group. For agreement games the directions agree up to a factor three in loss, optimal among Bob-fixed deterministic local symmetrizations. The geometry of the group behind a game, not its presentation, sets the rates. Fixed games demand αlog(1/ε) for every α=21∑ilogmiqi with qi≥2 a power of two dividing mi, a dense set containing all positive rationals and some transcendentals, such as 21log62; a configuration-lamp game demands ε−1+o(1); Pauli lamps on the points of Houghton's three rays give demand between ε−0.0690 and ε−1/4. If group relations hold exactly and Alice matches Bob to error e, demand counts qubits addressable by short words: Θ(1/e), Θ(e−k) for integers k≥1, or 2Θ(1/e). Conditional on our attainment and tail bounds, the I3322 optimum needs bounded usable entanglement.
Can uncomputability occur in the resource law of one fixed finite nonlocal game, rather than in a problem whose input is a game? We construct a finite two-player Boolean constraint-system game with rational question probabilities, quantum value one, and no finite-dimensional perfect strategy. If D(ε) is the least local dimension needed to win with probability at least 1−ε, then limn→∞log2D(2−n)/n=(3+θK)/4, where θK=∑e∈K4−(e+1) and K is the halting set. This exponent has the Turing degree of K. The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.
We construct a finite rational two-player nonlocal game with ten Alice questions, eleven Bob questions, at most six answers per question and 52 equally likely supported pairs. It has a perfect spatial strategy and no perfect finite-dimensional strategy. For arbitrary finite-dimensional local POVMs and mixed states, its dimension profile is D_Gamma(epsilon) = 10 log2(1/epsilon) + O(log log(1/epsilon)). The lower bound is 10 log2(1/epsilon) - 10 log2 log2(1/epsilon) - O(1). A direct square-root magic-square extraction, robust flagged sector law and approximate gluing force twenty Schmidt coefficients per appreciable halving level. The explicit strategy compressed to 20K dimensions has loss exactly 1/[624(4^K-1)], yielding the matching leading upper coefficient 10.
A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile DΓ(ε)=10log2(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.362740278498934 ebits, and least mean occupation, 275/18 per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II1 factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit Δ∗=ln(7/6)−ln(2)/7. Type-II∞ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.
A finite rational binary-scoring game can require infinite dimension for perfect play while admitting finite energy and finite entanglement entropy. Our game has ten Alice questions, eleven Bob questions, at most six answers, and 52 uniform question pairs. It is spatially perfect and never finitely perfect, with sharp dimension profile DΓ(ε)=10log2(1/ε)+O(loglog(1/ε)). Among pure perfect strategies, the displayed strategy attains the least entanglement entropy, 5.362740278498934 ebits, and least mean occupation, 275/18 per oscillator. We classify all perfect normal type-I marginal spectra and prove that this spectrum is the unique majorization maximum. These constants are also the small-loss limits of the entropy and one-sided oscillator energy prices. In the massive free scalar field we construct a pure perfect state in two strictly separated bounded regions, equal to the vacuum outside a bounded containing region and with every polynomial energy moment finite. In contrast, every fixed faithful split state has a positive loss floor; the complementary-wedge vacuum has supremum one without attainment. Energy-analytic states, bounded-region invertible vacuum preparations and product-Kraus preparations cannot be perfect. Every perfect pure spatial strategy has infinite Schmidt rank; finite entropy is attained but not forced. In every type-II1 factor the normalized trace has a loss floor, while bounded faithful prepared densities win perfectly with sharp trace-entropy deficit Δ∗=ln(7/6)−ln(2)/7. Type-II∞ permits every finite trace entropy. A graded modular ladder gives a normal semifinite hosting criterion. Its scalar fingerprint has sharp capacity and entropy-gap bounds; the capacity floor is approached but not attained in type I, and attained in type II.
Does playing a nonlocal game more precisely require more heat? We study devices that start and finish in pure product blanks, receive thermal answer carriers, and release classical answers. The answer depends on when the devices may communicate again. If they reset locally first, their minimum cyclic work differs from the game's minimum mutual information by a bounded alphabet cost and is at least twice its one-way distillable entanglement demand. Published entanglement bounds therefore give fixed finite games with local work 2αlog2(1/ε)+O(loglog(1/ε)), for a dense set of coefficients α, including transcendental values. Quantum reunion before reset keeps the work bounded. With coherent block recovery allowing q qubits and c dephased bits per play, the leading cost is (αL−q)++(αL−q−c)+, where L=log2(1/ε). A relative-entropy potential proves the lower bound; compression proves the upper bound. An exact entropy-production identity records arbitrary schedules without solving their optimization. For exactly matched controls, a sector allocation theorem evaluates the entanglement and correlation costs. A W-state example shows why knowledge of accessible marginals cannot replace operational access. Perfect magic-square play has cyclic local work exactly four units. Work and bath heat agree at the zero-energy endpoints used here; one unit is kBTln2.