  P(X)_p = \frac{\dim_{\mathbb{Q}} W(X)_p}{\dim_{\mathbb{Q}} (H^{p,p}(X) \cap H^{2p}(X, \mathbb{Q}))},
  \]
where \(W(X)_p\) is the algebraic span (image of the cycle class map) and the denominator is the space of rational Hodge classes. This metric quantifies the "probabilistic alignment" between algebraic and Hodge structures: \(P(X)_p = 1\) indicates full closure (HC holds at level \(p\)), while \(P(X)_p < 1\) signals a transcendental gap. Properties: \(P\) is functorial under isomorphisms and submultiplicative for products, \(P(X \times Y)_p \leq \prod_{i+j=p} P(X)_i \cdot P(Y)_j\).
Stability Invariants: Stability \(S(X)_p\) is formalized as the dimension of the invariant subspace under the action of the deformation group on the moduli space. Precisely, let \(\mathcal{M}\) be the moduli stack of deformations of \(X\), and \(\pi: \mathcal{X} \to \mathcal{M}\) the universal family. Then \(S(X)_p = \dim \ker(\rho: \Gamma(\mathcal{M}, R^{2p} \pi_* \mathbb{Q}) \to \End(W(X)_p))\), where \(\rho\) is the monodromy representation. This invariant measures persistence: high \(S\) implies robust algebraic cycles under perturbations, aligning with Noether-Lefschetz loci in Hodge theory87cfdc. Axiom: \(S(X \times Y)_p \geq S(X)_i + S(Y)_j\) for decomposable classes.
Output Realizations: Outputs \(O(X)_p\) are realized as the cokernel of the map from stable classes to geometric cycles, quantified by the realization index \(r(X)_p = \dim O(X)_p - \dim S(X)_p\). Positive \(r > 0\) indicates emergent cycles beyond stability predictions; in HC, \(r(X)_p = 0\) when closure holds. This metric chains with prior layers: outputs emerge if \(P(X)_p = 1\) and \(S(X)_p\) spans the Hodge space.
These definitions ensure metrics are computable (via linear algebra on bases) and compatible with CAS-6 axioms, providing a heuristic yet rigorous toolkit for HC analysis.
B. Implementation in Symbolic Algebra
The metrics are implemented using symbolic algebra tools like SymPy, which handles exact computations over \(\mathbb{Q}\) for dimensions and ranks. We provide examples, with code executed in a Python environment for verification.
For Hodge dimensions in elliptic curve products, recall that for \(E^n\), the Hodge number \(h^{p,p}(E^n) = \binom{n}{p}\), as it counts ways to distribute \((1,0)\) and \((0,1)\) forms across factors.
Example Code and Execution:
import sympy as sp
