Golden Scale Helix

Abstract

Golden completion lifts to a helix whose deck step advances one scale period and reverses orientation.

Theorem 1.1 (Golden Scale Period pos).

$$(0<goldenScalePeriod).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.golden_scale_period_pos (✓ std3). ∎

Source. Repository-derived.

Commentary.

The golden logarithmic scale period is strictly positive.

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

Theorem 1.2 (Golden Scale Period eq neg Log Multiplier).

$$(goldenScalePeriod=-Real.log|goldenProjectiveMultiplier|).$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.golden_scale_period_eq_neg_log_multiplier (✓ std3). ∎

Source. Repository-derived.

Commentary.

The logarithmic scale period is exactly the negative logarithm of the absolute golden projective multiplier.

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

Theorem 1.3 (Golden Helix Step Level).

$$\begin{gathered} \forall state:GoldenHelixState, \\ ((goldenHelixStepstate).level=state.level+1). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_level (✓ std3). ∎

Source. Repository-derived.

Commentary.

This theorem establishes golden helix step level in the module’s typed setting.

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

Theorem 1.4 (Golden Helix Step Scale Lift).

$$\begin{gathered} \forall state:GoldenHelixState, \\ ((goldenHelixStepstate).scaleLift=state.scaleLift+goldenScalePeriod). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_scaleLift (✓ std3). ∎

Source. Repository-derived.

Commentary.

This theorem establishes golden helix step scale lift in the module’s typed setting.

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

Theorem 1.5 (Golden Helix Step Orientation).

$$\begin{gathered} \forall state:GoldenHelixState, \\ ((goldenHelixStepstate).orientation=!state.orientation). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_orientation (✓ std3). ∎

Source. Repository-derived.

Commentary.

This theorem establishes golden helix step orientation in the module’s typed setting.

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

Theorem 1.6 (Golden Helix Step Twice Orientation).

$$\begin{gathered} \forall state:GoldenHelixState, \\ ((goldenHelixStep(goldenHelixStepstate)).orientation=state.orientation). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_twice_orientation (✓ std3). ∎

Source. Repository-derived.

Commentary.

Two completion turns restore the orientation sheet.

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

Theorem 1.7 (Golden Helix Step Twice Scale Lift).

$$\begin{gathered} \forall state:GoldenHelixState, \\ ((goldenHelixStep(goldenHelixStepstate)).scaleLift=state.scaleLift+2\times goldenScalePeriod). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_twice_scaleLift (✓ std3). ∎

Source. Repository-derived.

Commentary.

Two completion turns add exactly two golden scale periods.

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

Theorem 1.8 (Golden Helix Step Scale Lift Strict).

$$\begin{gathered} \forall state:GoldenHelixState, \\ (state.scaleLift<(goldenHelixStepstate).scaleLift). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_scaleLift_strict (✓ std3). ∎

Source. Repository-derived.

Commentary.

Every completion turn strictly increases the lifted scale coordinate.

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/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_level
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_orientation
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_scaleLift
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_scaleLift_strict
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_twice_orientation
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.goldenHelixStep_twice_scaleLift
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.golden_scale_period_eq_neg_log_multiplier
• Truth anchor: D5/S3/CompletionDynamics/GoldenMobius/GoldenScaleHelix.golden_scale_period_pos
• Dependency: D5/S3/CompletionDynamics/GoldenMobius/GoldenProjectiveDerivative