Consequence.
An explicit constructive program must either (i) identify special geometric structures (automorphisms, derived equivalences, Shioda--Inose/Kummer relations, universal sheaves on moduli spaces) that produce the needed action on TTT, or (ii) use arithmetic/motivic arguments to deduce algebraicity indirectly. Absent such structures, the search for correspondences is likely to fail.
5. Obstacles from monodromy and global geometry
Description.
Monodromy of the variation of Hodge structure constrains which classes can be defined globally across families. Some potential algebraic constructions exist only locally or on covers; monodromy can obstruct descent to the original variety.
Consequence.
Even if local or formal correspondences appear to act correctly on transcendental cohomology, global obstructions may prevent them from yielding algebraic cycles on the given variety. Any construction must therefore be checked against monodromy invariance and descent data.
6. Limitations of available transfer mechanisms (Kuga--Satake, Shioda--Inose, Mukai)
Description.
Powerful transfer tools exist---Kuga--Satake lifts to abelian varieties, Shioda--Inose relations to Kummer surfaces, Fourier--Mukai transforms via moduli of sheaves---but each has limits. Kuga--Satake is not known to be algebraic in general; Shioda--Inose applies only in special singular cases; Fourier--Mukai correspondences require well-behaved moduli spaces and universal families.
Consequence.
Where these transfer mechanisms are available and sufficiently explicit (for instance, for singular K3s or special families), they have been used with success to realize transcendental classes. However, their domain of applicability is limited; one must either restrict attention to varieties admitting such structures, or develop new transfer tools or arithmetic strategies to broaden applicability.
7. Arithmetic obstructions and the need for motivic arguments
Description.
Arithmetic techniques---reduction mod ppp, Tate conjecture methods, and motivic frameworks (Andr's motivated cycles)---offer routes to proving algebraicity in special circumstances. However, they require intricate compatibility between -adic and Hodge realizations and often depend on deep conjectures.
Consequence.
A program that aspires to convert CAS-6 heuristics into rigorous proofs will generally need to combine geometric constructions with arithmetic/motivic input. This raises the technical bar: it is typically feasible only in special families (CM, real multiplication, or arithmetic K3s) where extra structure allows descent from -adic or motivic statements to Hodge/complex settings.
8. Practical implications for CAS-6 guided research
