Hidden Translation Component Charge

Abstract

A hidden translation changes path component exactly when its canonical charge is nonzero.

Theorem 1.1 (Hidden translations cross components exactly at nonzero charge).

$$\forall x\in UniversalSolenoid,kappa\in Kernel\left(projection\right),\left(\neg Joined\left(x,x+kernelInclusion\left(kappa\right)\right)\right)\iff QuotientClass\left(kappa,ZMultiples\left(hiddenUnitOffset\right)\right)\ne 0$$

Proof. Machine-checked in Lean as D5/S3/Observer/MetricGeometryLaws/HiddenTranslationComponentCharge.hidden_translation_component_charge (✓ std3). ∎

Source. Repository-derived.

Commentary.

The translation is formed from a point of the universal solenoid and an element of the kernel of its visible projection.

Path connectedness in one projection fiber is classified by integer times of the real flow. The canonical hidden unit offset generates exactly those integer-flow translations.

Consequently, the translated point leaves the original path component exactly when the kernel element has nonzero class modulo the subgroup generated by the hidden unit offset.

References

• Truth anchor: D5/S3/Observer/MetricGeometryLaws/HiddenTranslationComponentCharge.hidden_translation_component_charge
• Dependency: D5/S1/Solenoid/Connectivity/SameFiberPathOrbitCriterion
• Dependency: D5/S1/Solenoid/StreamlineDecomposition