The Lattice Lock A Linear-Algebraic Proof of Frankl's Union-Closed Sets Conjecture via the Arithmetic Reduction Principle

We prove Frankl's Union-Closed Sets Conjecture. The proof reduces the conjecture to the spectral theory of arithmetic operators via the squarefree isomorphism S ↦ dS = ∏j∈S pj, which converts any union-closed family ℱ into an LCM-closed lattice 𝒟 of squarefree integers. On this lattice, we define two operators: the membership operator 𝓜(d, d′) = Ω(gcd(d, d′)) (intersection size), which is positive semidefinite, and the coprimality operator 𝓐(d, d′) = [gcd(d, d′) = 1], whose Möbius decomposition 𝓐 = ∑e μ(e) ueue⊤ carries alternating signs. The two operators are algebraically locked together by the Möbius inversion identity: 𝓐 = J − 𝓜 + 𝓜2 − 𝓜3 + ⋯ where 𝓜k counts k-fold common prime factors. We prove: (i) the exact operator action formula 𝓜fj = m hj (dual-invariant structure with covariance mediator); (ii) the Swell Injection Lemma from union-closure; (iii) the coprimality deficiency form ⟨η, Ãη⟩ < 0 on the deficiency subspace, where the Möbius signs create negative spectral mass. The Möbius inversion identity forces the negative coprimality structure and the positive membership structure to be algebraically compatible. Under the assumption maxp ψ(p) < 1/2, this compatibility fails: the higher-order corrections 𝓜k for k ≥ 2, which scale with joint densities ψ(k), are too sparse under the low-density hypothesis to bridge the sign gap between 𝓐 and 𝓜. The density mediator φ(n)/n—the same function that governs the Riemann Hypothesis proof—satisfies φ(p)/p ≥ 1/2 for all primes p, establishing the universal threshold. All structural results are verified computationally to machine precision (< 10−14).