Static Canonical Trace Divergence (SCTD)

• Static Canonical Trace Divergence (SCTD) is a divergence measure defined on various mathematical objects that compares static structures via geometric and spectral frameworks.
• It leverages dually flat statistical manifolds and operator-theoretic formulations to extend classical measures like the Kullback–Leibler divergence and quantum relative entropy.
• In algorithmic code evaluation, SCTD quantifies diversity by analyzing normalized opcode distributions, guiding model assessment and optimization.

Static Canonical Trace Divergence (SCTD) characterizes a family of divergence measures arising in information geometry, operator theory, and, more recently, in the evaluation of algorithmic diversity among functionally correct code. SCTD is defined on a broad spectrum of mathematical objects—probability distributions, density operators, operator algebras associated to spectral triples, and multinomial distributions over program opcodes—with each instantiation grounded in a rigorous geometric or spectral formalism. In all cases, SCTD functions as a "distance-like" static comparison between objects, eschewing temporal or dynamical elements in favor of a purely state-to-state or structural measure.

1. Abstract Definitions and Geometric Frameworks

The canonical divergence underlying SCTD is constructed on dually flat statistical manifolds $(M, g, \nabla^m, \nabla^e)$ equipped with a Riemannian metric $g$ (often the Fisher–Rao metric) and two torsion-free flat, dual affine connections (mixture $\nabla^m$ and exponential $\nabla^e$) (Felice et al., 2019). In global affine coordinates:

• $\theta^i$—$\nabla^e$-affine, with potential $\psi(\theta)$,
• $\eta_i$—$\nabla^m$-affine, with dual potential $\psi^*(\eta)$, related by Legendre duality,

$$\psi^*(\eta) = \sup_{\theta}(\theta\cdot\eta - \psi(\theta)), \qquad \psi(\theta) = \sup_{\eta}(\theta\cdot\eta - \psi^*(\eta)).$$

The canonical divergence is

$$D(p, q) = \psi(\theta(p)) + \psi^*(\eta(q)) - \theta(p)\cdot\eta(q).$$

Alternatively, in geodesic form,

$$D(p, q) = \int_{0}^{1} t\,g_{\gamma^m(t)}(\dot{\gamma}^m(t),\dot{\gamma}^m(t))\,dt,$$

where $\gamma^m$ is the $\nabla^m$-geodesic from $q$ to $p$.

In operator-theoretic contexts, especially for spectral triples $(\mathcal{A},\mathcal{H},D)$ as in the Connes–Moscovici calculus (Paycha, 2010), SCTD emerges as a zeta-regularized, or spectral cutoff, trace functional on suitable operators $A$.

2. Classical and Quantum Instantiations

On the probability simplex $\Delta_{n-1}$, the canonical divergence reduces to the Kullback–Leibler divergence:

$$D_{\mathrm{KL}}(p\|q) = \sum_{i=1}^n p_i \ln \frac{p_i}{q_i},$$

obtained by specializing the convex potentials and dual coordinates to the multinomial family (Felice et al., 2019). This form quantifies the deviation from reference (usually exponential family) models.

On the space of full-rank quantum density operators,

$$\mathrm{SCTD}(\rho \| \sigma) = \mathrm{Tr}[\rho(\ln\rho-\ln\sigma)],$$

which coincides precisely with the Umegaki quantum relative entropy; the geometric structure is furnished by the Bogoliubov inner product (quantum Fisher metric), mixture/exponential connections, and corresponding convex potentials $\Psi(\Theta)=\ln\mathrm{Tr}\,e^{\Theta}$, $\Psi^*(\rho)=\mathrm{Tr}(\rho\ln\rho)$ (Felice et al., 2019).

3. Operator-Theoretic and Spectral-Analytic Formulations

For an abstract pseudodifferential setup as in noncommutative geometry (Paycha, 2010):

• Given a spectral triple $(\mathcal{A},\mathcal{H},D)$, the key analytical object is the zeta function $\zeta_A(s) = \mathrm{Tr}(A|D|^{-s})$ for a suitable operator $A$.
• The spectrum of singularities (poles) $\Sigma$ is the dimension spectrum.
• The "static" (high-energy/spectral cutoff) canonical trace divergence is:

$$\mathrm{SCTD}(A) = \lim_{N\to\infty} \left\{ \mathrm{Tr}[A\,\Pi_{|D|\le N}] - \sum_{\lambda\in\Sigma\setminus\{0\}} c_\lambda N^\lambda \right\} = \operatorname{FP}_{s=0}\mathrm{Tr}(A|D|^{-s}),$$

with $c_\lambda$ the residues at poles of $\zeta_A(s)$ and $\operatorname{FP}_{s=0}$ indicating the finite part at $s=0$. This construction generalizes the Kontsevich–Vishik canonical trace to the full spectral triple setting.

• Regularity, order, and commutator-vanishing properties are required for well-definition of SCTD; for non-singular orders, SCTD reduces to the canonical trace extending the usual operator trace.

4. Algorithmic Structure Divergence in Code Generation

In code evaluation contexts, SCTD has been adapted to quantify algorithmic diversity among LLM-generated solutions. Each code artifact is first represented by its static Python bytecode—abstracted as a multinomial probability distribution $p_{s,i}$ over opcodes. With $m$ solutions and $d$ opcodes:

• $c_{s,i}$: count of opcode $i$ in solution $s$,
• $w_i \in \{1,10,100\}$: heuristic cost per opcode,
• $p_{s,i}=c_{s,i} / (\sum_j c_{s,j})$ is the structural PMF,
• $q_{s,i}=(w_i c_{s,i}) / (\sum_j w_j c_{s,j})$ is the cost-weighted PMF.

The divergence between solutions is then computed in two variants:

a) Jensen–Shannon Version

Parameter $\alpha\in[0,1]$ interpolates between structural and cost-weighted divergence:

$$\mathrm{SCTD_{\mathrm{JSD}}} = \alpha\cdot\left[\frac{2}{m(m-1)}\sum_{s<t} \mathrm{JSD}(p_s\,\|\,p_t)\right] + (1-\alpha)\cdot \left[\frac{2}{m(m-1)}\sum_{s<t} \mathrm{JSD}(q_s\,\|\,q_t)\right]$$

where $\mathrm{JSD}$ is the Jensen–Shannon divergence between PMFs (bounded in [0,1]) (Rajput et al., 7 Nov 2025).

b) Covariance-Based Version

Define random variables $X_P$, $X_Q$ each uniformly sampling from the $m$ PMFs. Let $\mu$, $\Sigma$ be the mean and covariance of $X$ in the $d$-simplex:

$$\tau(X) = \frac{\mathrm{tr}\,\Sigma}{1 - \|\mu\|_2^2},$$

then

$$\mathrm{SCTD}_\tau = \alpha\tau(X_P)+(1-\alpha)\tau(X_Q).$$

5. Operationalization: Extraction, Preprocessing, and Computation

Opcode Extraction:

Python solutions are compiled and disassembled (using the dis module); each static opcode occurrence is tallied, mapped to a canonical index, and normalized to form PMFs.

Preprocessing:

Consistency of Python interpreter versions is assumed to maintain opcode sets. No code tokenization is required since bytecode offers a canonical, normalized representation.

Pseudocode Outline:

1. Collect opcode vocab across all solutions.
2. Build count (structural) and weighted count matrices for the $m$ solutions.
3. Normalize per-solution opcode counts to obtain $p_s$ and $q_s$.
4. Compute average pairwise divergences (JSD or total variance ratio; see formulas above).
5. Output SCTD score in $[0,1]$ interval.

Interpretation:

• $\mathrm{SCTD} = 0$: All code solutions are bytecode-identical (maximal algorithmic uniformity).
• $\mathrm{SCTD}$ close to $1$: Maximal algorithmic diversity.
• Empirical values (e.g., $0.03$–$0.05$ on real data) indicate moderate underlying diversity (Rajput et al., 7 Nov 2025).

\begin{table} \centering \begin{tabular}{l|l|l} \textbf{Context} & \textbf{SCTD Formula} & \textbf{Interpretation} \ \hline Probability simplex & $\sum_i p_i\ln(p_i/q_i)$ & KL divergence \ Density operators & $\mathrm{Tr}[\rho(\ln\rho - \ln\sigma)]$ & Quantum relative entropy \ Bytecode PMFs & See SCTD$_{\mathrm{JSD}}$ above & Opcode distributional divergence \ Spectral triples & $\operatorname{FP}_{s=0}\mathrm{Tr}(A|D|^{-s})$ & Canonical trace, noncommutative \ \end{tabular} \end{table}

6. Properties, Validation, and Comparison to Alternative Metrics

SCTD, as a canonical divergence, satisfies:

• Non-negativity.
• Vanishing if and only if arguments coincide.
• Bregman-type joint convexity.
• Data-processing (monotonicity) under appropriate structure-preserving maps (e.g., stochastic, CPTP).
• Geodesic/Pythagorean projection theorems in the geometric setting.
• Orthogonality to token-overlap and AST similarity metrics (empirically, Pearson correlations to CodeBLEU and n-gram metrics are low), confirming SCTD’s sensitivity to algorithm structure rather than surface syntax (Rajput et al., 7 Nov 2025).

In code evaluation, the counterpart dynamic divergence (DCTD) operates on runtime traces. The ratio BEF$=$DCTD/SCTD signals the degree of behavioral versus structural redundancy or instability.

7. Worked Example and Practical Implications

For two generated code artifacts, one using a set-based and one a loop-based solution, their opcodes yield distinct PMFs, and a sample computation produces $\mathrm{SCTD}\approx0.06$ under the JSD variant, quantifying their moderate algorithmic difference (Rajput et al., 7 Nov 2025). Low SCTD signifies uniform algorithm selection by the model; high SCTD indicates exploration of multiple solution strategies, which has direct implications for codebase stability, maintainability, and performance testing.

A plausible implication is that SCTD enables objective quantification of algorithmic diversity beyond surface similarity, thus supporting robust evaluation, benchmarking, and optimization in generative code systems. Furthermore, in mathematical and physical models, SCTD forms a rigorous bridge connecting noncommutative analysis, quantum information, and statistical inference through a common information-geometric machinery.