Prime Golden Scale Coordinate

Abstract

Prime logarithmic lengths admit a golden scale coordinate.

Theorem 1.1 (Prime Golden Scale Coordinate pos).

$$\begin{gathered} \forall prime:\mathbb{N}.Primes, \\ (0<primeGoldenScaleCoordinateprime). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/PrimeGoldenScaleCoordinate.prime_golden_scale_coordinate_pos (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every prime has a positive golden scale coordinate.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.2 (Prime Power Golden Scale Coordinate).

$$\begin{gathered} \forall prime:\mathbb{N}.Primes,exponent:\mathbb{N}, \\ (goldenScaleCoordinate((prime.1:\mathbb{R}{)}^{exponent})=exponent\times primeGoldenScaleCoordinateprime). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/PrimeGoldenScaleCoordinate.prime_power_golden_scale_coordinate (✓ std3). ∎

Source. Repository-derived.

Commentary.

Prime powers advance linearly in the lifted golden scale coordinate.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

Theorem 1.3 (Prime One Golden Scale Coordinate).

$$\begin{gathered} \forall prime:\mathbb{N}.Primes, \\ (goldenScaleCoordinate(prime.1:\mathbb{R})=primeGoldenScaleCoordinateprime). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/GoldenCoding/PrimeGoldenScaleCoordinate.prime_one_golden_scale_coordinate (✓ std3). ∎

Source. Repository-derived.

Commentary.

The coordinate of the first power is the prime coordinate itself.

The declaration keeps its parameters and hypotheses explicit; the result makes no converse or broader existence claim beyond that scope.

References

• Truth anchor: D5/S3/Observer/GoldenCoding/PrimeGoldenScaleCoordinate.prime_golden_scale_coordinate_pos
• Truth anchor: D5/S3/Observer/GoldenCoding/PrimeGoldenScaleCoordinate.prime_one_golden_scale_coordinate
• Truth anchor: D5/S3/Observer/GoldenCoding/PrimeGoldenScaleCoordinate.prime_power_golden_scale_coordinate
• Dependency: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix