Algebraic and dynamical structure of critically sparse discrete harmonic functions in Z^3

We study a discrete harmonic function whose support in a box of radius R has at most AR^2 points. At one radius comparable to R, we obtain a linear evaluation Hilbert profile and a covering by a surface of degree O_A(1) and a curve set of degree O_A(R), without a logarithmic loss. Disjoint private subsets of the support pay for the degrees and intrinsic persistence of the factors of a minimal vanishing polynomial. For nonpersistent factors with sufficiently large excess persistence, every coordinate propagation rule produces a quantitatively active exact translation, a boundary contribution, or a reset with bounded multiplicity. We prove finite-depth rigidity for exact curve translations and a merge-safe total degree budget for full three-color continuation depths, counted once per weakly connected curve component. First-visit forests satisfy a separate budget for their actual truncated depths; no identification of merges with nonzero cycles is used. We also prove support coercivity for noncharacteristic finite affine stacks and an offset-gap estimate with explicit common-region and scale hypotheses. Globally, a harmonic function with proper algebraic support is supported on finitely many characteristic planes. The finite-scale results do not imply a covering by boundedly many characteristic slabs of sublinear width. 32 pages, no figures. Lean 4 formalization project (in preparation): https://github.com/hxypqr/z3-sparse-harmonic-rigidity-lean