Uncomputable critical exponents

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.