The Lattice Spectral Classification Theorem Multilinear Rank-Nullity and the General Theory of Meet Operators on Finite Lattices

We prove a general classification theorem for meet operators on finite lattices. Let L be a finite lattice with bottom element 0̂, atoms {a1, …, an}, and Möbius function μL. For any function f : L → ℝ with f(0̂) ≠ 0, the meet operator T(x, y) = f(x ∧ y) satisfies five structural properties: (C1) exact Möbius decomposition T = ∑z∈L g(z) uzuz⊤ with canonical kernel 𝒦 = {z : g(z) = 0} giving lattice rank-nullity; (C2) structural deficiency of the atom indicator subspace; (C3) a unique density mediator ψ(x) = |{y ∈ L : x ∧ y = 0̂}|/|L| discovered by the operator action, equal to the Radon–Nikodym derivative between flat and coprimality-weighted measures; (C4) iterative Krylov completion in finitely many steps, defining the spectral depth d(L) as a lattice invariant; (C5) exhaustive classification of every meet operator by the triple (𝒦, d(L), ψ). The Classification Theorem for Arithmetic Operators (Papers 1–6) is the special case L = ({1, …, N}, |) with ∧ = gcd and ψ(n) = φ(n)/n. The proof of Frankl's conjecture (Paper 9) is the restriction to LCM-closed sublattices. The underlying mechanism is the multilinear rank-nullity principle: the meet operation x ∧ 0̂ = 0̂ is the zero-argument property of multilinear maps, which forces intrinsic kernels, cross-factor images, and canonical quotient factorizations. All results are verified computationally on four lattices covering every structural class: Boolean, divisor, partition, and pentagon (non-modular).