Rigidity Descent for the Tate Projector

We analyze the passage from motivated to algebraic Tate fixed parts using rigidity, finite–field specialization, and a spiral obstruction mechanism. We prove intrinsic probe completeness (CP2) inside André motivated motives, establish a precise equivalence between Frobenius semisimplicity at eigenvalue 1 and the absence of Frobenius–fixed inverse–limit spiral obstructions, and show that algebraicity of bounded Frobenius projectors at infinitely many finite–field reductions forces global algebraicity by rigidity. The remaining obstruction to the Tate fixed–part statement is isolated in a single explicit realization–completeness condition, which we verify in abelian type and modular–curve settings.