Fractal Patterns in Discrete Laplacians: Iterative Construction on 2D Square Lattices

The paper states a universal Proposition that for every binary-style figure, for arbitrary seeds and neighborhoods, the pattern at i = 8k is a spread of its seeds and at least 13-steps away from connectedness, with a proof sketch deferred to the Appendix. The formal sketch, however, is only carried out for the Diag–Neumann neighborhood and seeds inside a 3×3 square, not for arbitrary neighborhoods; the text then generalizes based on observation rather than proof. Moreover, the universal claim is false: in the translation-invariant one-sided neighborhood M = {(1,0)}, the linear operator L = I + T obeys L^8 = I + T^8 in characteristic 2, so F8 = S ⊕ (S − 8e1). Choosing S1 = {(0,0),(8,0)} shows F8 is not a disjoint union of translates of S1, contradicting the ‘spread of seeds’ claim; choosing S2 = {(0,0),(1,0)} yields two components only 7 steps apart, contradicting the claimed ≥13-step bound. The paper’s proof sketch and assertions do not address these counterexamples. See Proposition and its scope claim in the main text and the Appendix’s neighborhood/seed restrictions in the proof sketch.