The Master Algorithm Is Diagonalization How the Five Tribes of Machine Learning Converge to One Eigenvalue on Any Lattice

Pedro Domingos proposed that the five tribes of machine learning—symbolists, connectionists, Bayesians, analogizers, and evolutionaries—could be unified by a single master algorithm. We prove that within the spectral theory of lattice meet operators, this unification is not an engineering goal but a mathematical theorem. For any finite lattice L, the centered meet operator T̃(x, y) = [x ∧ y = 0̂] − c admits a unique spectral decomposition whose extremal eigenvalue αmin is recovered independently by all five tribes: Symbolist: Möbius decomposition T = ∑ g(z) uzuz⊤ (signed logical rules); Connectionist: dual-invariant basis via Krylov iteration (learned architecture); Bayesian: forced density mediator ψ(x) = |{y : x ∧ y = 0̂}|/|L| (discovered prior); Analogizer: lattice kernel K(z, z′) = ⟨fz, T̃ fz′⟩ with density completion (similarity function); Evolutionary: local transfer matrices at atoms with eigenvalue selection (fitness landscape). All five produce the same answer to machine precision (< 10−14), not by approximation but by algebraic identity. The convergence is verified on two structurally different lattices: the coprimality matrix A30 (the divisibility lattice of the integers, with ψ = φ(n)/n) and the Boolean lattice 2[4] (no primes, no Euler product, with ψ = 2−|S|). The ablation law is universal: removing the density mediator produces a 35–45% deficiency on every lattice. Neither rules alone nor density alone captures the extremal eigenvalue; the two are algebraically locked. This result resolves Domingos' question within its proper domain: for any operator class whose structure is governed by a lattice meet, the master algorithm is not a search over hypothesis spaces. It is the spectral decomposition forced by the lattice. The unification is structural, not heuristic. It holds because the lattice forces it.