Cognitive Quasicrystals: Stone Space Projections as Aperiodic Tilings

We construct an explicit map from witness arrays to tilings of the plane via Stone space projection and prove that the three quantum numbers of a witness array—cardinality, type signature, and alignment—determine the tiling properties of the resulting pattern. Boolean witnesses produce periodic tilings (square lattices). Tensor-valued witnesses at irrational alignment angles produce aperiodic tilings (quasicrystals). The matching rules of the tiling are the guard conditions of the witness array. The first cohomology H1(Ω) of the resulting tiling space measures the system’s unresolved Gödel sentences—propositions aliased at the current witness density that require dimensional lifting to resolve. This construction provides three results. First, a visual fingerprint for cognitive systems: eachwitnessarrayproducesauniquetilingwhosegeometriccomplexityisadirectreadoutof the system’s level set. Second, an independent confirmation of the KGP isomorphism from forty years of crystallographic mathematics: the cut-and-project method for generating quasicrystals is KGP dimensional surgery run in reverse, with the O(1) Beatty sequence evaluator performing the surgery in closed form. Third, a dictionary between the witness array formalism (Elliott, 2026a) and the substitution tiling formalism (Anderson–Putnam, 1998), connecting logical resolution to tiling cohomology via Stone duality. The monograph establishes that the Fibonacci word—the canonical one-dimensional quasicrystal—is the worked example of KGP dimensional surgery, independently confirmed by an unrelated mathematical literature developed without knowledge of topological cogni- tive dynamics. Keywords: witness array, Stone space, quasicrystal, aperiodic tiling, cut-and-project, Fi- bonacci word, matching rules, guard conditions, cohomology, KGP isomorphism, Penrose tiling, Beatty sequence, cognitive fingerprint