Enhanced Dirichlet and Topology Risk (EDTR)

• Enhanced Dirichlet and Topology Risk (EDTR) is a framework that fuses Dirichlet uncertainty modeling with geometric and topological analysis to quantify risk across domains.
• It combines chain-of-thought embedding metrics with Dirichlet-based uncertainty to robustly calibrate confidence in large language models.
• EDTR extends classical Dirichlet methods with topological risk features, yielding enhanced performance in financial modeling, neural inference, and behavioral economics.

Enhanced Dirichlet and Topology Risk (EDTR) encompasses a family of methodologies and mathematical frameworks that combine Dirichlet-based uncertainty modeling with geometric and topological analysis of solution manifolds or reasoning processes. Originating independently in LLM confidence calibration and in behavioral risk modeling on Markowitz-Tversky-Kahneman (MTK) manifolds, EDTR provides a unified perspective on risk and uncertainty through fusion of distributional (Dirichlet) and geometric/topological (e.g., cluster structure, torsion) signals. EDTR uniquely leverages both classical potential theory (Dirichlet problems) and higher-order behavioral or manifold-theoretic corrections (topological risk, risk torsion), yielding robust, theoretically-grounded risk functionals relevant for neural inference, financial modeling, and behavioral economics.

1. Conceptual Foundations

The EDTR functional arises in two principal domains: uncertainty estimation for reasoning with deep models, and utility/value-function analysis under behavioral risk preferences.

In the context of inference-time confidence estimation with LLMs, EDTR evaluates a set of chain-of-thought (CoT) reasoning trajectories as a high-dimensional point cloud. Eight topological risk features—derived from statistics such as pairwise distances, cluster analysis (DBSCAN and KMeans), silhouette scores, and outlier quantification—quantify the geometric coherence of the distribution. Simultaneously, a Dirichlet-based uncertainty module models epistemic uncertainty in predicted answer probabilities, parameterized via an MLP mapping token-level variances and entropies onto Dirichlet concentration parameters.

For risk on MTK manifolds, EDTR combines the classical Dirichlet problem for first-exit-time value functions with enhancements arising from behavioral operations (risk aversion/seeking, torsion operators, loss-aversion gauges) on paracompact topological manifolds. The solution is the unique fixed point of a perturbed elliptic PDE, with the perturbation arising from the risk torsion operator and prospect-theoretic gauge transformation.

2. Mathematical Structure of EDTR

2.1 LLM Confidence Estimation

For a given query $q$, $k$ independent CoTs $\{o_1, \ldots, o_k\}$ are sampled by varying the model's softmax temperature. Each CoT is embedded as $e_i \in \mathbb{R}^{d}$ (typically $d=384$) by a pre-trained embedding model (e.g., all-MiniLM-L6-v2). The following features are computed:

Feature	Description	Formula
Reasoning Spread	Std. of all pairwise distances	$\sigma_{\text{dist}} = \mathrm{std}\{d_{ij}\}$
Consistency Score	Mean angular similarity deviation	$$C_{\text{cos}} = 1 - \frac{2}{k(k-1)}\sum_{i<j} \frac{e_i \cdot e_j}{\|e_i\|\|e_j\|}$$
Complexity Entropy	Relative spread	$$E_{\text{comp}} = \sigma_{\text{dist}}/\mu_{\text{dist}}$$
DBSCAN Stability	Measures cluster noise/outlier ratio	$$S_{\text{DBSCAN}} = \frac{n_{\text{noise}}}{k} + \frac{1}{n_{\text{clusters}} + 1}$$
Centroid Coherence	Dispersion around the mean embedding	$$C_{\text{centroid}} = \mathrm{std}\{r_i\}/\mathrm{mean}\{r_i\}$$
Diversity Penalty	Penalizes large mean pairwise distance	$$P_{\text{div}} = \max(0,\, 0.5(\mu_{\text{dist}}-1))$$
Outlier Risk	Fraction of offset samples by Tukey's rule	$$R_{\text{outlier}} = \frac{1}{k}\sum_{i} 1[r_i > Q_3 + 1.5 \text{IQR}]$$
Cluster Quality (Silhouette)	Cluster separation/overlap	$$Q_{\text{sil}} = 1 - \max_{2 \leq n_c \leq \min(k,5)} \text{silhouette}(K\text{Means}(n_c))$$

These are linearly combined via learned weights into a single scalar topological risk. In parallel, Dirichlet-based uncertainty is modeled by passing token-level variance and entropy from each CoT through a two-layer MLP, producing Dirichlet parameters $\alpha = (\alpha_1, \ldots, \alpha_n)$. Confidence is derived from the Dirichlet concentration, normalized entropy, and maximum class probability.

The final scalar confidence is computed by logistic regression fusion:

$$C(q) = \sigma(w_{\text{topo}} \cdot \mathrm{risk}_{\text{topo}} + w_{\text{dir}} \cdot \mathrm{conf}_{\text{dir}} + b).$$

2.2 Behavioral and Geometric Risk on MTK Manifolds

Let $M$ denote the MTK reference-point manifold, and $D \subset M$ an open domain with regular boundary $\partial D$. The enhanced (behavioral) Dirichlet problem is

$$\begin{cases} \left(\tfrac{1}{2}\Delta_M + V\right)u(x) = 0,\quad x \in D, \ u|_{\partial D} = \phi, \end{cases}$$

where the potential $V u(x)$ encodes the risk torsion operator,

$$V u(x) = (A_{\text{ra}} - A_{\text{rs}})u(x) = \sum_{i,j} c_{k.ij} \alpha_i \beta_j u(x),$$

potentially gauge-inflated via a prospect-theoretic parameter $\lambda$ as $V = (1 + \lambda)A_{\text{ra}}$. The Feynman–Kac formula provides the solution:

$$u(x) = E^x\left[\exp\left(-\int_0^{\tau_D} V(B_s)\, ds\right) \phi(B_{\tau_D})\right].$$

3. Algorithmic and Operational Procedures

3.1 LLM Inference-Time EDTR

The LLM-based EDTR workflow proceeds as follows:

1. Sampling: Query $q$ is processed by generating $k=5$ CoT trajectories via sampling at temperatures $\tau \in \{0.7, 0.8, 0.9, 1.0, 1.1\}$.
2. Embedding: Each CoT is embedded into $\mathbb{R}^{384}$.
3. Feature Extraction: Compute the eight topological risk features and combine them.
4. Dirichlet Head Computation: Token-level statistics (variance, entropy) are passed through a two-layer MLP to yield Dirichlet concentration parameters.
5. Fusion: Output confidence is calculated as a sigmoid-activated weighted sum of the topological and Dirichlet scores.
6. Selection: Return the answer with majority vote (or highest confidence among unique answers) paired with $C$.

3.2 Enhanced Dirichlet Problem in Risk-Torsion Contexts

1. Domain and Data: Specify $M$, $D$, boundary data $\phi$.
2. Risk Torsion Operator Construction: Compute $A_{\text{ra}}$ and $A_{\text{rs}}$ using Arrow-Pratt formulae and behavioral parameters; compose torsion via structure constants $c_{k.ij}$.
3. Gauge Transformation: Adjust torsion via prospect-theoretic parameter $\lambda$.
4. PDE Solution: Numerically or analytically solve the elliptic PDE with the specified potential $V$ for $u(x)$, obtaining the risk-adjusted value function.

4. Empirical Performance and Comparative Analysis

EDTR’s application to LLM confidence calibration demonstrates superior empirical results across diverse reasoning benchmarks—AIME (olympiad math), GSM8K (grade-school math), CommonsenseQA, and stock price prediction:

Metric	EDTR	Best Baseline	Relative Improvement
Accuracy (Avg., Llama-3.1-8B)	0.550	0.500
Expected Calibration Error (ECE)	0.306	0.446	31% ↓
Composite Score	0.662	0.536

Domain-wise, EDTR achieves 100% accuracy with ECE 0.432 on AIME and ECE 0.107 on GSM8K (>50% improvement), and yields the lowest Brier score (0.301) on stock prediction. Averaging across three model sizes, ECE is reduced by 41% to 0.287 with the best mean composite score of 0.672.

A plausible implication is that EDTR’s geometric-topological risk functional is particularly effective at capturing overconfidence in settings with diverse or ambiguous solution trajectories, providing robustness lacking in standard calibration approaches.

5. Theoretical and Topological Underpinnings

The risk-torsion extension positions EDTR within the rich mathematical framework of representation theory, topological manifolds, and Lie-algebraic structures. The MTK manifold—paracompact and second-countable—supports a partition of unity, allowing global solutions to be assembled from local EDTR functionals. The risk-torsion operator $A_{\text{tor}}$ is linked to the Lie algebra $\mathfrak{sl}_n$ and, in the presence of hyperbolic Gauss curvature at reference points, to the quantum group $\mathrm{SU}(n)$. This topological signature reflects the behavioral-geometric complexity of value surfaces under risk, going beyond harmonic (Dirichlet) utility to include Feynman–Kac–type corrections for endogenous risk dynamics.

In the LLM context, the embedding space geometry serves as the analog of the manifold, with reasoning clusters and dispersions acting as a proxy for the underlying solution landscape’s curvature and torsion.

6. Synthesis and Unifying Themes

Enhanced Dirichlet and Topology Risk functionals, whether applied to model uncertainty in deep networks or in the behavioral estimation of value and loss aversion, combine classical potential-theoretic notions (Dirichlet problems, harmonic functions) with topological and algebraic corrections (risk-torsion, cluster structure, gauge invariance). EDTR's defining feature is its composite nature, leveraging both distributional sharpness (via Dirichlet uncertainties) and geometric coherence (via topological risk features or risk-torsion operators), to measure and calibrate confidence or risk robustly in complex, high-dimensional settings. The framework’s adaptability to both statistical learning and behavioral economics underscores its generality and theoretical depth.