In CAS-6, stability corresponds to the persistence of a closed algebraic--topological system under perturbations:
For divisors and low-dimensional products, algebraic cycles are stable objects. Their classes survive deformations of the variety, consistent with Lefschetz-type theorems. Stability here mirrors structural rigidity: once the interaction is closed, it remains closed across families.
For transcendental candidates (e.g. in K3K3K3 \times K3K3K3), stability is absent. The interaction loop does not close, so perturbations may alter the rational Hodge structure in ways that algebraic cycles cannot capture. CAS-6 therefore diagnoses such settings as unstable systems, at risk of "breaking" under even small moduli variations.
Thus, CAS-6 reveals that HC implicitly requires stability: for the conjecture to hold globally, every rational Hodge class must correspond to a cycle that persists as the system evolves.
2. Adaptability as a measure of deformational resilience
Adaptability in CAS-6 refers to a system's ability to absorb changes in input parameters (complex structures, Picard rank, polarization choices) while maintaining output consistency:
In complete systems, adaptability is high: deformations may alter coefficients but not the fundamental algebraic span.
In incomplete systems, adaptability is low: the presence of transcendental subspaces means that deformation often amplifies instability, enlarging the gap between algebraic and topological dimensions.
In this way, HC can be reframed as a claim that adaptability is universal: all rational Hodge classes are deformationally adaptable, with hidden algebraic representatives ensuring system-wide resilience.
3. Emergent geometry from algebraic--topological closure
CAS-6 views the output layer (geometry) not as pre-given but as emergent: it arises from the successful closure of topology (nodes, structures) and algebra (weights, probabilities). In contexts where closure holds (e.g. abelian varieties, toroidal products):
The geometry emerges naturally as a stable configuration of cycles realizing all Hodge classes.
This emergent geometry corresponds to complete systemic harmony, where every algebraic cycle can be seen as the geometric manifestation of a closed CAS-6 loop.
Where closure fails (e.g. transcendental contributions in K3K3K3 \times K3K3K3):
Emergent geometry is partial: only a subset of Hodge classes are geometrically realized.
The missing cycles correspond to "phantom geometries," predicted by topology but unmanifest in algebraic reality.
Thus, the CAS-6 framework interprets the Hodge Conjecture as the hypothesis that geometry always fully emerges from algebraic--topological interactions---never partially, never incompletely.
4. Broader implication
