Three Prime-Observer Completions

Abstract

Prime exponents, realizability mass, and operator phase are distinct completions.

Theorem 1.1 (Deterministic, probabilistic, and operator completion).

$$PrimeExponentBijection\wedge \exists globalLaw⟺s>1\wedge samePrimeDiagonal\wedge differentOperator\wedge PrimeDiagonalLanguage\subset OperatorLanguage.$$

Proof. Machine-checked in Lean as D5/S3/PrimeObserver/ThreeCompletionFinalProposition.prime_observer_three_completion_final_proposition (✓ std3). ∎

Source. Repository-derived.

Commentary.

Positive integers are reconstructed bijectively from finitely supported prime-exponent profiles.

The independent geometric profile is globally realizable precisely when the zeta parameter is greater than one.

A named pair of qubit density states has one prime-diagonal image but distinct operators, so commuting diagonal observations do not give operator tomography.

References

• Truth anchor: D5/S3/PrimeObserver/ThreeCompletionFinalProposition.prime_observer_three_completion_final_proposition
• Dependency: D5/S3/Analytic/PrimeProducts/GlobalPrimeExponentRealizability
• Dependency: D5/S3/ConceptDynamics/ObservationOrder/TypedPrimeLanguageHierarchy
• Dependency: D5/S3/Factorization/ExponentCoordinates/PrimeExponentBijection