Why Three Dimensions? Dimensional Uniqueness of Gauge Structure in Informational Field Theory

We consolidate five independent mathematical constraints into a single funnel theorem: n = 3 is the unique positive integer satisfying all five simultaneously. The constraints are: (C1) gauge-covariant Derrick virial balance admitting stable solitons only for n ≤ 3; (C2) the Lie algebra isomorphism so(n) ≅ su(2), required for electroweak structure, holding only for n = 3; (C3) existence of non-trivial knots, required for topological confinement, holding only for n = 3; (C4) the pre-metric Hodge constraint requiring n = 3, corroborated by the Eckmann cross-product theorem; (C5) the Cartan decomposition sl(n,ℝ) = so(n) ⊕ Sym₀(n) matching dim su(3) = 8 only for n = 3. The constraints are evidentially independent: their proofs invoke disjoint mathematical machinery. The convergence of five proof-independent results on a single dimension is the content of the funnel theorem.