We construct translation-invariant nearest-neighbour lattice families whose fixed-coupling thermodynamic gap stays positive while the entanglement entropy of a growing cell-anchored region has any prescribed nonnegative limit-computable logarithmic coefficient. Ordered lower and upper coefficients have the exact second-level one-sided arithmetic classes within an effective cell model. Separately, a referee history cell converts low energy into a nonlocal-game strategy with identical pure-state Schmidt coefficients and a mixed-state distillable-entanglement certificate. The extraction cost is controlled by the clock grid's corner-to-uniform resistance: linear, logarithmic, and bounded for one, two, and three clocks. We give a conditional material theorem with explicit layout and selection hypotheses, a worst-case whole-cell perturbation law, and necessary temperature, preparation-time, and work bounds. Disjoint quasi-local spin-lattice measurements yield only the closure of finite-dimensional correlations. All gap limits use factorial tori before the coupling limit; the entanglement cut grows inside a selected cell. The game-driven material construction remains conditional.
We construct translation-invariant nearest-neighbour lattice families whose fixed-coupling thermodynamic gap stays positive while the entanglement entropy of a growing cell-anchored region has any prescribed nonnegative limit-computable logarithmic coefficient. Ordered lower and upper coefficients have the exact second-level one-sided arithmetic classes within an effective cell model. Separately, a referee history cell converts low energy into a nonlocal-game strategy with identical pure-state Schmidt coefficients and a mixed-state distillable-entanglement certificate. The extraction cost is controlled by the clock grid's corner-to-uniform resistance: linear, logarithmic, and bounded for one, two, and three clocks. We give a conditional material theorem with explicit layout and selection hypotheses, a worst-case whole-cell perturbation law, and necessary temperature, preparation-time, and work bounds. Disjoint quasi-local spin-lattice measurements yield only the closure of finite-dimensional correlations. All gap limits use factorial tori before the coupling limit; the entanglement cut grows inside a selected cell. The game-driven material construction remains conditional.
We construct translation-invariant nearest-neighbour quantum Hamiltonian families with rational affine coefficient matrices whose existing finite nonnegative critical gap exponents at coupling zero are exactly the nonnegative Delta^0_3 reals, already in two dimensions. Volume limits along factorial tori precede coupling, and gaps are above the entire ground space. Separate finite registers share an energetic ruler, a framed computation and a final-time broadcast to positive-density quantum readers. The inner gap is eventually exact at fixed positive coupling without a general computable settling bound. Three-dimensional broadcast and wetting films give independent routes; tamed rulers and nested limits retain their classifications. A fixed universal noisy rule, with a computable bottom boundary, program origin and infinite cloning, has exactly Delta^0_3 density and counting laws in [0,1] under prescribed time-then-window limits, robust on an open positive-rate kernel region. At computable kernels no effective inquiry converges with positive probability to a fixed such law outside Delta^0_2; the exceptional learnable kernels are null, meager but dense for external finite-cylinder inquiry. Unique classical states of computable specifications have computable local numbers, while growing readouts and state selection permit noncomputable asymptotic content. Restricted response certificates and an exact calibrated-ruler criterion delimit robustness claims. A computable randomized adaptive inquiry reaches a fixed real with positive probability only at Delta^0_2. The beyond-Delta^0_2 gap exponents have no computably bounded relative gap-settling scale, excluding a specified uniform scaling tail. All-size convergence, unique ground states, locally nondegenerate gap identification and robust intrinsic equilibrium exponents remain unestablished.
We construct translation-invariant nearest-neighbour quantum Hamiltonian families with rational affine coefficient matrices whose existing finite nonnegative critical gap exponents at coupling zero are exactly the nonnegative Delta^0_3 reals, already in two dimensions. Volume limits along factorial tori precede coupling, and gaps are above the entire ground space. Separate finite registers share an energetic ruler, a framed computation and a final-time broadcast to positive-density quantum readers. The inner gap is eventually exact at fixed positive coupling without a general computable settling bound. Three-dimensional broadcast and wetting films give independent routes; tamed rulers and nested limits retain their classifications. A fixed universal noisy rule, with a computable bottom boundary, program origin and infinite cloning, has exactly Delta^0_3 density and counting laws in [0,1] under prescribed time-then-window limits, robust on an open positive-rate kernel region. At computable kernels no effective inquiry converges with positive probability to a fixed such law outside Delta^0_2; the exceptional learnable kernels are null, meager but dense for external finite-cylinder inquiry. Unique classical states of computable specifications have computable local numbers, while growing readouts and state selection permit noncomputable asymptotic content. Restricted response certificates and an exact calibrated-ruler criterion delimit robustness claims. A computable randomized adaptive inquiry reaches a fixed real with positive probability only at Delta^0_2. The beyond-Delta^0_2 gap exponents have no computably bounded relative gap-settling scale, excluding a specified uniform scaling tail. All-size convergence, unique ground states, locally nondegenerate gap identification and robust intrinsic equilibrium exponents remain unestablished.
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.
We construct translation-invariant nearest-neighbour quantum Hamiltonian families with rational coefficient matrices whose critical gap exponents at the computable transition coupling zero fill Delta-2 in the bulk on Z^2 and Delta-3 in the bulk on Z^3. Volume limits use cubic tori of side L=n! and precede the coupling limit; gaps are above the entire ground space. A classical energetic ruler selects pitch proportional to inverse coupling. In three dimensions a self-created frame supports an unbounded computation, whose final-time output is broadcast by nearest-neighbour classical constraints to positive-density bulk counters. Their full gaps give an extensive flat band and nonzero local spectral weight per volume at each fixed positive coupling. The inner limit is eventually exact, with no general computable settling bound. The two-dimensional ruler also realizes ordered left-Sigma-2 and left-Pi-2 endpoint pairs, including a halting probability. Surface-anchored wetting films independently attain Delta-3; fixed-thickness films and dependent nested box limits yield the stated arithmetic hierarchy. We give full-sector counter bounds, conditional ceilings for effectively controlled limits, and an explicit onsite instability of the earlier rulers. Finite analytic systems have integer exponents. No all-integer-size limit, robustness, unique-ground-state realization, correlation-length classification, or identification with locally nondegenerate or KMS-extremized gaps is established.
Magic, or nonstabilizerness, is the resource that promotes stabilizer operations to universal quantum computation. For bipartite pure states, its component intrinsic to the correlations between the subsystems, the nonlocal magic, is obtained by minimizing a magic measure over local unitaries. For the stabilizer Rényi entropy (SRE), the minimum is conjectured to be attained by the computational-basis (CB) representative, the state obtained by assigning the Schmidt coefficients in decreasing order to matching computational-basis labels. We prove this conjecture for two families of states at every system size and bipartition: states with dyadic-staircase Schmidt spectra and states of Schmidt rank at most six. For arbitrary bipartite pure states, we show that the CB value exceeds the nonlocal SRE by at most 4, fixing the leading term of any divergent scaling of nonlocal SRE. We further show that the nonlocal SRE grows at most logarithmically with the entanglement entropy. Consequently, one-dimensional area-law states have bounded nonlocal SRE, while critical states with logarithmic entanglement entropy permit at most doubly logarithmic growth with system size. We apply these results to the transverse-field Ising chain, where we tightly bound the nonlocal SRE in the gapped phases and identify its double-logarithmic growth at criticality.
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.