(Paper014)The Boolean Lock Classical Sunflower Conjecture via Coprimality on the Boolean Lattice

I'm still never forgiving you mfers for saying I ought to work on the Sunflower Scheisseshow as the next thing btw.We develop a spectral framework for the classical Erdős–Rado sunflower conjecture by identifying set disjointness as the coprimality operator on the Boolean lattice. The disjointness indicator [A ∩ B = ∅] admits an exact Möbius decomposition ∑S(−1)|S| 1S⊆A · 1S⊆B, the Boolean analogue of the arithmetic identity [gcd(m, n) = 1] = ∑d μ(d)[d | m][d | n] that drives Papers 1–7. A k-sunflower with core Y requires k pairwise disjoint sets above Y; sunflower-freedom at each layer r imposes the spectral moment constraint ∑ λi3 = 0 on the disjointness adjacency matrix restricted to the petal family. Combined with eigenvalue interlacing from the Johnson scheme, this yields a spectral recurrence across the layer decomposition whose convergence determines the sunflower constant. We prove that the Erdős–Rado conjecture is equivalent to a uniform spectral bound in the Johnson association scheme, and that the Alweiss–Lovett–Wu–Zhang breakthrough bound f(s) ≤ (C log s)s corresponds to a specific decay profile of spectral ratios across layers. This reframes the classical conjecture as a purely linear-algebraic question about triangle-free subgraphs of disjointness graphs, completing the passage from the trilinear entropy barrier of Paper 13 to the bilinear spectral territory where algebraic locks operate.