  columns.append(tvec)
M = Matrix(QQ, 4, len(columns), columns)
rankM = M.rank()
compute_cohomology_class(Z) depends on Z-type:
Diagonal: straightforward identity tensor.
Graph of : use matrix representation M_sigma and convert to tensor coordinates.
FM kernel: compute push--pull via Chern character formulas (requires symbolic algebra).
5. Practical notes, pitfalls & remedies
Rational arithmetic vs floating point. Always do exact rational linear algebra when possible (Sage/Magma). Floating approximations risk mis-detecting rank due to numerical noise. If using numeric periods, use high precision and rational reconstruction.
Twisted universal families. If only twisted universals exist, you must work with twisted Chern characters --- the computational code must support Brauer classes.
Normalization/scalars. Cycle classes may differ by nonzero scalars depending on conventions. Rank tests are invariant under nonzero scalings of columns.
Independence from basis choices. The rank of the projected set is basis-independent; however, numerical conditioning can vary --- orthonormalize or use rational Smith normal form for robust detection.
Verification with literature. If an experiment yields rank 4 for a set of FM kernels / automorphism graphs, check the literature: often these correspondences are known and algebraicity can be corroborated (Mukai, Huybrechts, Shioda--Inose).
6. Expected outputs and interpretation
Primary numeric outputs:
NS_basis.csv, T_basis.csv, Gram matrices;
For each candidate Z_k: Zk_Tproj.csv (4-vector);
Matrix M (4 m), rank r.
Interpretation:
r = 4: heuristically successful --- candidates span TTT\otimes TTT. Next step: try to find or cite rigorous algebraicity arguments for the exact cycles used.
r < 4: not enough; need additional independent correspondences or a different K3.
D. Philosophical Reflection: Heuristics as Guides for Conjectural Mathematics
The Hodge Conjecture (HC) sits at the boundary between what mathematics can rigorously establish and what intuition continues to suggest. Unlike problems with clear constructive formulations, HC resists direct computational or algebraic assault, largely because its essence concerns the hidden relationship between formal structures (Hodge decompositions) and geometric realizations (algebraic cycles). In this liminal space, heuristics---when carefully formalized---play a decisive role.
1. The legitimacy of heuristics in mathematics
While mathematics prides itself on rigor, it has always depended on heuristic methods for guidance:
