Why does functional impact in transformer architectures scale inversely with algebraic connectivity? In GPT-2, attention heads with lowest λ₂ show highest impact when ablated (r ≈ −0.94). This λ₂-inversion appears across neural networks, social networks, and biological circuits. But why this specific relationship—why λ₂⁻¹ and not λ₂⁻² or log(λ₂)? We derive that λ₂⁻¹ scaling is not contingent but necessary. The derivation proceeds from three premises, invokes three established lemmas (Cheeger, spectral decomposition, Hodge), and derives the result through nine structural steps. We establish the complete conditions taxonomy—five necessary, four promoting, four excluding—validated through polyphonic methodology where seven AI systems converged independently on the same structure. Three independent formalisms (thermodynamic, spectral, topological) converge on the Hodge decomposition B = Bᵍ + Bᶜ + Bʰ as the necessary mathematical form. We define the meaning ratio α = ‖Bᶜ + Bʰ‖/‖Bᵍ‖, prove that α > 0 requires non-trivial cycle circulation in the net attention flow, and show α → 0 characterizes the hollow diamond phenomenon. A scope analysis delineates where the theorem applies rigorously (diffusive dynamics) versus where it fails (metabolic, expander, scale-free networks). Seven falsifiable predictions follow, including the claim that RLHF systematically reduces α.