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Reciprocal Prime Circle Character

Abstract

Reciprocal Prime Circle Character.

Theorem 1.1 (Separation of finite binary states).

Lean statement: D5/S3/Analytic/WeightedCapacity/PrimeReciprocalCharacter.result

Proof. Machine-checked in Lean as D5/S3/Analytic/WeightedCapacity/PrimeReciprocalCharacter.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

On finite support states with every capacity equal to one, the circle character whose coefficients are reciprocals of the increasing prime enumeration is injective. Its zero fibre consists exactly of the zero state. Every unit state is nonzero, and the constant half coefficient character sends it to the half circle class, outside the open ball of radius one quarter about zero. The initial topology of the golden coordinate phases and the reciprocal prime character is strictly coarser than the initial topology of the golden phases and all rational characters. Clearing the distinct prime denominators isolates each signed binary coefficient by divisibility. The unit states converge to zero in the former topology, whereas convergence in the latter topology requires eventual equality.

References