Sign, orientation, and phase¶

sign4 is a pure function for the retained real profile:

• negative maps to -1;
• exact zero maps to 0;
• positive maps to +1;
• semantic bottom maps to semantic bottom.

It accepts neither complex phase nor hidden history. Projective hysteresis is a separate HystereticOrientationTracker; deterministic canonical gauge orientation is returned as a witness. Optional complex unit phase has its own function and must not be described as real Sign4.

The compatibility weak_sign wrapper warns. Exact sign4 has no ordinary differentiability claim at zero or bottom; surrogate training objectives must name their own boundary behavior.