We exhibit non-closedness at perfect play for normal injective hosts: an explicit finite two-player game with 0/1 scoring has a perfect commuting-operator correlation approached by finite-dimensional quantum correlations, while no perfect strategy has a normal marginal on an injective von Neumann algebra of either player. The game has 561 Alice questions, 157 Bob questions, 3,696 sampled pairs and at most 20 answers per question. Perfect play is realized on a matrix amplification of the Bellaterra critical-KMS factor of type III with parameter 1/2. After restricting to the support of the chosen normal marginal, every perfect strategy yields a modular Cuntz pair and six self-similar relations that force the canonical trace on the free product of three groups of order two. The resulting noninjective group von Neumann algebra is an expected subalgebra of that support corner, with a faithful normal state-preserving expectation. A two-sided POVM lift through a matrix ultraproduct proves approximation by finite systems. Consequently correlations admitting normal injective hosts are not closed. Under the explicit hypotheses of Halvorson's Proposition 2.31, locally normal quantum-field-theory strategies cannot win exactly. We also formulate a KMS-face criterion for complete modular presentations. Exact arithmetic certificates check every supported losing event; the forcing proof applies to arbitrary normal strategies.
We exhibit non-closedness at perfect play for normal injective hosts: an explicit finite two-player game with 0/1 scoring has a perfect commuting-operator correlation approached by finite-dimensional quantum correlations, while no perfect strategy has a normal marginal on an injective von Neumann algebra of either player. The game has 561 Alice questions, 157 Bob questions, 3,696 sampled pairs and at most 20 answers per question. Perfect play is realized on a matrix amplification of the Bellaterra critical-KMS factor of type III with parameter 1/2. After restricting to the support of the chosen normal marginal, every perfect strategy yields a modular Cuntz pair and six self-similar relations that force the canonical trace on the free product of three groups of order two. The resulting noninjective group von Neumann algebra is an expected subalgebra of that support corner, with a faithful normal state-preserving expectation. A two-sided POVM lift through a matrix ultraproduct proves approximation by finite systems. Consequently correlations admitting normal injective hosts are not closed. Under the explicit hypotheses of Halvorson's Proposition 2.31, locally normal quantum-field-theory strategies cannot win exactly. We also formulate a KMS-face criterion for complete modular presentations. Exact arithmetic certificates check every supported losing event; the forcing proof applies to arbitrary normal strategies.
We construct finite two-player games with binary scoring whose normal strategies distinguish operator-algebraic type. A 243-pair game has perfect play in type III and none when either ambient algebra is semifinite. For factors with separable predual, hosting it is equivalent to an invariant normal state on the flow of weights, or to an expected hyperfinite type III1 subfactor. Prepared-state hosting is therefore insensitive to the ratio invariant throughout the injective flow-finite class. A different quantifier recovers that invariant: every faithful normal state has value one for the flagged magic-square game exactly when 1/2 belongs to the Connes invariant; a nine-question conference-basis game gives the corresponding criterion for 1/3. Together these two response properties characterize type III1. One prepared state already wins both games in dimension 25. We prove that every positive rational is forced as a modular ratio by some finite game, and identify the exact rational orthogonal-pair single-bridge parameters. For the type-III-only game, a robust sorted spectral law holds for every normal semifinite strategy. It yields matching entropy, deficit, dimension and variance exponents, including an exponential inverse-square-root dimension law. No validated numerical constant is given; the sharp constant is conjectural. We give explicit finite templates, retain their boundary penalties, and distinguish sorted trace position from log density. The specified normal semifinite perturbative gravitational models cannot attain perfect play, whereas suitable type III quantum-field-theory hosts can. This conditional algebraic consequence gives no preparation or energy bound.
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.
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.
Quantum non-local games with quantum inputs or outputs have been formulated in several different languages, including rank-one and quantum XOR games, support maps between projection lattices, probabilistic quantum hypergraphs, and Frobenius-algebraic rule operators on finite quantum sets. We give a unified operator-algebraic comparison of these models by assigning to each rule its winning transformation space: the operator space of transformations accepted with certainty. We introduce R-projection-test quantum games with finite-dimensional input and output von Neumann algebras and an arbitrary, possibly infinite-dimensional, referee von Neumann algebra R. Their winning transformation spaces are precisely operator spaces with a natural bimodule structure, equivalently rectangular quantum relations. We compare this formalism with projection-lattice games, hypergraph quantum games, and the graphical rule-operator definition. Projection-lattice games capture exactly the reflexive winning bimodules. Hypergraph quantum games admit value-preserving projection-test realisations, and, after passing to perfect transformations, describe the same reflexive part as projection-lattice games. Using Daws' technique, we also translate rule operators to projections in tensor products of finite-dimensional von Neumann algebras. This identification places graphical rules in the same operator-bimodule framework and preserves both values and perfectness. Finally, we compare concurrency with the synchronicity conditions of Goldberg and of Bochniak--Kasprzak--Sołtan. The framework is illustrated by classical, rank-one, quantum XOR, colouring, and quantum graph homomorphism and isomorphism games.
We construct a finite rational two-player nonlocal game with a binary winning predicate that has a perfect strategy on ℓ²⊗ℓ² and no perfect finite-dimensional strategy. The game has ten questions for Alice, eleven for Bob, at most six answers per question, and a uniform distribution on fifty-two question pairs. We glue two copies of Mančinska's Flagged Magic-Square game along complementary labels, adding a boundary answer to the second copy. An inverse-free sector argument forces every perfect strategy to contain a Schmidt-coefficient chain closed under halving, which rules out finite rank. An explicit normalized spatial strategy has twenty coefficients at each geometric level. Its compression to 20K dimensions has loss exactly 1/[624(4^K−1)], giving D(ε) ≤ 10 log2(1/ε) + 20 for general local POVMs and 0 < ε < 1. We prove a weaker algebraic lower bound and state a matching logarithmic lower bound as an open problem. Thus a binary-scoring game distinguishes spatial from finite-dimensional perfect attainment even though both values are one.