Compact Root Coherence
Abstract
Compactness turns separate positive roots into one coherent rational homomorphism.
Theorem 1.1 (Positive roots and rational homomorphisms are equivalent).
Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Coding/PrimitiveLifts/CompactRootCoherence.exists_all_positive_roots_iff_exists_rat_hom (✓ std3). ∎
Source. Repository-derived.
Commentary.
For an arbitrary compact Hausdorff topological group H and h in H, the existence of a positive n-th root of h for every positive natural n is equivalent to one algebraic homomorphism from the additive rationals, represented as Multiplicative Q, sending additive rational one to h, that is, phi(ofAdd(1:Q)) = h.
The forward direction puts all H-valued functions indexed by Q into the compact product H^Q. Each finite set of rational coordinates has a nonempty closed constraint set: a single D-th root of h supplies the values, and denominator clearing makes the finite addition equations hold inside its cyclic subgroup. Compactness then supplies a point in every finite constraint set, which is a global homomorphism. Hausdorffness is used for the closed equalizers.
The reverse direction uses the root phi(ofAdd(1/n)) and applies its power map. No topology on Q, continuity, connectedness, Lie structure, faithfulness, or commutativity of H is asserted.