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.
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 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.
Determining spectral gaps in the thermodynamic limit is a central challenge in quantum many-body physics. When a fixed boundary condition is imposed, this problem is undecidable: no algorithm, even with unlimited resources, can decide for every Hamiltonian whether it is gapped. Yet the thermodynamic limit is meant to capture the intrinsic properties of very large systems, which should not depend on how their boundaries are chosen. We therefore consider the bulk spectral gap, which concerns only the intrinsic excitations of the infinite system, and ask what can and cannot be computed about it. First, we give a general algorithm producing a hierarchy of certified upper bounds on the bulk gap, arbitrarily tight at the cost of more computation. Second, we prove that the spectral gap problem remains undecidable in the bulk, already for translationally invariant nearest-neighbour chains of fixed local dimension. Hence no general algorithm produces convergent lower bounds, and no convergent upper-bounding algorithm, ours included, admits a computable convergence rate. The bulk spectral gap is thus certifiable from above but uncomputable from below; more precisely, the bulk-gaplessness promise problem is RE-complete, exactly as hard as the halting problem. The same construction shows that no general algorithm can decide whether a given product state is a ground state, and that convergent bounds on ground-state expectation values admit no computable convergence rate in general. Finally, we obtain the first nontrivial certified upper bounds on the bulk gap of the spin-1/2 kagome Heisenberg antiferromagnet.
We classify the nonnegative finite-size spectral-gap exponents of translation-invariant nearest-neighbour Hamiltonians on the square lattice, with finite local dimension, integer interaction matrices and open boundaries. The gap is measured above the entire ground space on an L×L box. The existing exponents in Δ(L)=L−z+o(1) are exactly the nonnegative limit-computable (Δ20) reals. The finite nonnegative lower/upper exponent pairs are exactly the ordered pairs consisting of a left-Σ20 real and a left-Π20 real. In particular, one fixed integer-valued model has exponent equal to the halting probability of any chosen prefix-universal machine. A classical boundary preference fixes a size counter and a computation tableau; a reversible mixed-radix counter supplies the small gap. We bound every sector of the full Hilbert space. The closing modes live on a boundary segment of length O(logL), and the local dimension may depend on the prescribed pair. These are finite-box exponents, with no assertion about bulk dispersion or stability of thermodynamic critical exponents.
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 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.