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.
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.
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.
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 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 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.
Volume-Uniform Finite-Volume Bounds with the Representation Cutoff Removed: a Reconstruction and Computer-Assisted Improvement of Creation-Operator Constants We study the Kogut–Susskind Hamiltonian of pure SU(2) and SU(3) lattice gauge theory on three-dimensional cubic boxes, H = aΣ_e L_e + bΣ_p (1 - F_p), with L_e the electric Casimir on link e and F_p = N⁻¹Re tr U_p, at strong coupling (small b/a). An unrefereed preprint by Gauvin adapts the creation-operator method of Bravyi, DiVincenzo and Loss to this setting and states, for SU(3), a physical gap uniform in the volume, with the representation cutoff removed, for b/a ≤ 1/648. We give a self-contained reconstruction of the finite-volume argument, identify its external analytic inputs, and reproduce its constants. We then prove a sequence of refinements of the commutator estimate: a selection rule of Wilson-loop multiplication that restricts admissible creation placements, fractional witness counting, orthogonality of output sectors, Haar-measure bounds for vacuum projections before and after the plaquette multiplication, a two-link sandwich identity, a joint vacuum-row bound, and a size-weighted norm. The finite tuple constants and final sufficient inequalities are checked in exact rational arithmetic by two author implementations and a separate verifier written from the stated definitions. The operator inequalities that justify these arithmetic definitions are proved analytically. On open boxes the physical gap is at least 3a/2 for SU(2) whenever b/a ≤ 2/95, and at least 8a/3 for SU(3) whenever b/a ≤ 2/35 (1/16 with a sharper representation-theoretic bound), uniformly in the box. These sufficient couplings are 3072/95 and 1296/35 times those obtained from the preprint's constants (for SU(2), our transfer of them). The corresponding open-box endpoints in Hamiltonian units are β_H=√(4/95) ≈ 0.205 for SU(2) and β_H=√(6/35) ≈ 0.414 for SU(3), or √(3/16) ≈ 0.433 with the strengthened SU(3) bound. The basic open-box certificates also bound the static fundamental potential below linearly: for static charges at graph distance L ≥ 6, E_{x,y} - E_vac ≥ σ L with σ = 5/16 a for SU(2) at b/a ≤ 2/95 and σ = 5/9 a for SU(3) at b/a ≤ 2/35, uniformly in the box. A variable exclusion fraction and an independent local-decoupling upper bound bracket the first physical excitation energy above the vacuum. On open boxes containing an elementary plaquette, at b/a = 1/5000 this energy lies in [2.990, 3.050] a (SU(2)) and [5.326, 5.390] a (SU(3)). These are strong-coupling statements at fixed lattice spacing, not continuum statements. Files ym_strong_coupling_gaps.pdf: the paper. zenodo_ymgap_v1.0.0_2026-10-23.zip: the certificate bundle: two independent implementations of the certified tuple constants, the verifier, the tamper-test driver, the Section 8 upper-bound program and its exact checker, the number generator, manifests and verifier outputs for all 34 certificate configurations, the complete tuple files, and the paper source with every number generated from these files. Its SHA-256 is in the adjacent .sha256 file; SHA256SUMS inside covers every file. README.md, LICENSE.md: copies of the bundle's front matter. Verification ./replay.sh (Python 3, standard library only; a few minutes) regenerates every tuple file, compares it entry by entry with the second implementation, verifies every manifest in exact rational arithmetic, runs the tamper tests and the exact check of the two-sided bracket upper bounds, and regenerates every number of the paper; it exits non-zero on any failure. Scope: strong-coupling statements for the Kogut–Susskind lattice Hamiltonian at fixed lattice spacing, uniform in the volume, with the representation cutoff removed. They are not continuum statements and do not address the continuum Yang–Mills mass gap problem. Licence: CC BY-NC 4.0 (LICENSE.md), licensed by SmartLedger Solutions. Commercial-use enquiries: labs@smartledger.solutions. No patent licence is granted by this release.
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.