Historical precedents. Euler's manipulations of divergent series, Riemann's conjectures on zeta functions, and Grothendieck's "yoga of motives" all began as heuristic programs long before rigorous justification emerged.
Modern practices. Computational experiments in number theory, probabilistic heuristics in algebraic geometry, and physics-inspired string dualities provide contemporary examples where heuristic insight guides the formulation of conjectures and even entire research programs.
Within this tradition, CAS-6 offers a systemic heuristic---an analytic lens mapping closure, stability, and emergence---to illuminate whether or not algebraic cycles can account for rational Hodge classes.
2. Heuristics as systemic metaphors
CAS-6 does not claim to prove HC, but rather to reinterpret it as a systemic balance problem:
If topology, algebra, and geometry achieve complete closure, the system is stable and the conjecture holds locally.
If closure fails---e.g., transcendental blocks remain uncovered---the system is incomplete and risks instability.
By translating HC into the language of systemic dynamics (level, structure, weight, probability, stability, and output), CAS-6 offers a metaphorical model where mathematical obstacles are reframed as imbalances in systemic interaction. This metaphor is heuristic but not arbitrary: it imposes structure and permits experiments (rank tests, candidate correspondences, closure checks).
3. The role of heuristics in guiding search
A critical function of heuristics is not to settle truth but to guide search strategy:
CAS-6 identifies where to look: e.g., transcendental blocks in K3K3K3 \times K3K3K3 that resist algebraic coverage.
It identifies what to try: e.g., correspondences generated by Fourier--Mukai kernels or automorphism graphs.
It clarifies what failure means: not the falsity of HC, but the persistence of an "incomplete system" requiring additional cycles.
Thus heuristics act as navigational tools, steering attention toward promising structures and away from blind alleys.
4. The epistemological status of heuristic confirmation
The experiments conducted here (elliptic curve products, higher products, K3K3K3 \times K3K3K3) exemplify how heuristics can produce a layered epistemology:
Strong alignment (elliptic products): supports belief in HC where closure is transparent.
Partial alignment (higher products): suggests local completeness, motivating extension.
Tension zones (K3K3K3 \times K3K3K3): illuminate precisely where the conjecture strains, making the case for deeper theories (motives, derived categories).
This layered evidence does not constitute proof, but it raises or lowers credence in HC across different settings, shaping collective mathematical judgment about plausibility.
5. Toward a philosophy of heuristic mathematics
