Compact Character Modulus and Mellin Obstruction

Abstract

Continuous complex characters on compact groups have unit modulus, so a Mellin mode with nonzero real drift cannot descend through a Pontryagin character.

Theorem 1.1 (Compact characters exclude nonzero Mellin drift).

$$\begin{array}{c}\forall G,\\ \mathrm{Group}(G)\wedge \mathrm{TopologicalSpace}(G)\wedge \mathrm{IsTopologicalGroup}(G)\wedge \mathrm{CompactSpace}(G)\Rightarrow \\ (\forall chi:\mathrm{ContinuousMonoidHom}(G,\mathrm{Units}(\mathbb{C})),\forall g\in G,\Vert \mathrm{coe}(\mathrm{chi}(g))\Vert =1)\wedge \\ (\forall \delta ,\gamma \in \mathbb{R},\delta \ne 0\Rightarrow \\ \forall descent:\mathrm{ContinuousMonoidHom}(\mathrm{Multiplicative}(\mathbb{R}),G),phase:\mathrm{PontryaginDual}(G),\neg \forall t\in \mathbb{R},\mathrm{mellinCharacter}(\mathrm{coe}(\delta )+i\cdot \mathrm{coe}(\gamma ),\mathrm{ofAdd}(t))=\mathrm{coe}(\mathrm{phase}(\mathrm{descent}(\mathrm{ofAdd}(t))))).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaLinear/CompactCharacterMellinObstruction.compact_character_modulus_and_mellin_obstruction (✓ std3). ∎

Source. Repository-derived.

Commentary.

The carrier G is an arbitrary compact topological group. A continuous homomorphism from G to the units of the complex numbers is bounded on its compact source; applying the same bound to every positive power and to the inverse forces unit norm.

The second public conjunct uses the canonical repository Mellin character. If it factored through a Pontryagin character, every value would lie on the complex unit circle, contradicting the exact frozen criterion when the real drift delta is nonzero.

The no-factorization statement quantifies the descent map, phase character, drift, frequency, and time; it does not replace the source character by an abstract unitary predicate.

References

• Truth anchor: D5/S3/Weil/ZetaLinear/CompactCharacterMellinObstruction.compact_character_modulus_and_mellin_obstruction
• Dependency: D5/S3/Weil/ZetaLinear/OfflineZeroCharacter