3. Structural consequences and necessary algebraic motifs
The failure of closure forces an explicit requirement for additional algebraic motifs: correspondences ZCH2(YY)Z\in \mathrm{CH}^2(Y\times Y)ZCH2(YY) whose induced endomorphisms Z:H2(Y)H2(Y)\Phi_Z:H^2(Y)\to H^2(Y)Z:H2(Y)H2(Y) act nontrivially on T(Y)T(Y)T(Y) (so that their classes have nonzero projection onto TTT\otimes TTT). In linear-algebra terms, letting {t1,t2}\{t_1,t_2\}{t1,t2} be a Q\mathbb QQ-basis of T(Y)T(Y)T(Y), one seeks correspondences Z1,...,ZmZ_1,\dots,Z_mZ1,...,Zm such that the collection of symmetric tensors {(Zkti)tj}k,i,j\{(\Phi_{Z_k}t_i)\otimes t_j\}_{k,i,j}{(Zkti)tj}k,i,j spans TTT\otimes TTT (or, equivalently, such that the images of the restricted maps ZkT\Phi_{Z_k}|_{T}ZkT generate End(T)\operatorname{End}(T)End(T) sufficiently to produce four independent symmetric combinations).
Concretely, restoration of CAS-6 closure requires at least the following algebraic data:
one or more correspondences whose projection to TTT\otimes TTT is nonzero (so that \Delta is nontrivial), and
a set of such correspondences whose TTT-restricted actions are linearly independent in End(T)\operatorname{End}(T)End(T), so that their symmetric tensor images span a four-dimensional subspace.
Absent such correspondences, the CAS-6 system remains incomplete.
4. Heuristic diagnostics and linear-algebra testing
The CAS-6 viewpoint suggests a pragmatic testing pipeline:
a. Construct explicit NS and T bases for the chosen YYY (using published lattice data for singular K3s or explicit computational methods).
b. Model candidate correspondences ZZZ by computing (or approximating numerically) the matrix of Z\Phi_ZZ in the NST\operatorname{NS}\oplus TNST basis. Practically, one computes pairings
(Zei,ej)=p2((p1ei)[Z]),ej,(\Phi_Z e_i,e_j) = \langle p_{2*}((p_1^* e_i)\cup [Z]), e_j\rangle,(Zei,ej)=p2((p1ei)[Z]),ej,
for basis elements ei,eje_i,e_jei,ej.
c. Extract the TTT-block of Z\Phi_ZZ, i.e. the restriction ZT\Phi_Z|_TZT. Form the induced vectors in TTT\otimes TTT corresponding to the cohomology class of ZZZ.
d. Test linear independence of the obtained TTT\otimes TTT vectors. If a set of correspondences yields rank 444, the CAS-6 algebraic layer is heuristically restored.
Our earlier numerical experiment was a linear implementation of this pipeline in a simplified model; the experiment failed to increase rank because the chosen candidate vectors had zero components in the modeled TT coordinates. The proper application of this diagnostic requires explicit NS/T data and genuine geometric correspondences (e.g. graphs of automorphisms, Fourier--Mukai kernels, Shioda--Inose transfers) whose action on TTT can be computed.
5. Consequences for the Hodge Conjecture (heuristic vs. formal)
Two logical possibilities remain:
(HC holds for this XXX). Then there must exist algebraic cycles (possibly non-obvious) whose classes project onto a basis of TTT\otimes TTT; the CAS-6 deficit is resolvable by adding appropriate correspondences and the algebraic layer AAA will equal TXT_XTX. The most promising sources for such correspondences are (a) Fourier--Mukai kernels associated with moduli of sheaves on YYY, (b) Shioda--Inose/Kummer correspondences transported from abelian surfaces, (c) graphs of automorphisms in cases where YYY has nontrivial automorphism group, or (d) motivated cycles arising from arithmetic specializations (CM/Kuga--Satake techniques). Establishing algebraicity in these cases typically requires substantial geometry and arithmetic input (e.g. modulispaces, universal families, or reduction arguments).
(HC fails for this XXX). Then the four-dimensional TTT\otimes TTT contains genuine rational Hodge classes that are not algebraic; in CAS-6 terms the system is intrinsically incomplete at this level, and no family of algebraic correspondences can fill the orphan nodes. Such a counterexample would have deep consequences and would necessitate a fundamental revision of the idea that topology must always be interpretable algebraically.
At present, the literature contains instances where specialized techniques (Kuga--Satake, Shioda--Inose, Mukai) yield algebraicity for powers of K3 surfaces in restricted families; these confirm that CAS-6 closure can often be achieved under extra structure (CM, special correspondences). Hence the CAS-6 diagnosis is constructive: it localizes the obstruction and points to precisely the kinds of data that could remove it.
6. Recommended constructive program (CAS-6 guided)
