Uncomputable critical exponents

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.