Agentic-PRs in Neural Models

• Agentic-PRs are probabilistic models where latent subagents encode distinct outcome preferences over finite spaces using weighted logarithmic pooling.
• They provide conditions for strict unanimity and enable recursive decomposability, ensuring measurable welfare gains and effective policy aggregation.
• The framework guides the detection, auditing, and intervention of latent subagent policies, enhancing interpretability and alignment in complex AI systems.

Agentic-PRs (Agentic Preference Representations) are probabilistic characterizations of latent substructures (“subagents”) within deep neural networks, wherein each subagent embodies a distinct outcome distribution over a finite space (e.g., tokens for LLMs). The Agentic-PR theory introduces weighted logarithmic pooling for composing such subagents into a higher-level aggregate agent, providing conditions for unanimous welfare improvement, analyzing recursive decomposability, and formalizing alignment phenomena in LLMs. This framework establishes a rigorous foundation for modeling, detecting, and auditing internal subagent policies, with practical implications for interpretability and alignment in agentic AI systems (Lee et al., 8 Sep 2025).

1. Formal Definition of Agentic-PRs and Epistemic Utility

An agentic preference representation (Agentic-PR) is a strictly positive probability distribution $P: \Omega \to (0,1)$, where $\Omega$ is a finite discrete outcome space (such as an LLM’s token vocabulary). Each agent (subagent) encodes an independent probabilistic policy preference. The epistemic utility of agent $P$ for outcome $o \in \Omega$ is defined as log-score utility: $U(P, o) = \log P(o)$ This epistemic utility forms the basis of agentic welfare measurement.

Subagents $P_1, \ldots, P_n$ are composed into a collective Agentic-PR via weighted logarithmic pooling: $$P_{\text{pool}}(o) = \frac{ \prod_{i=1}^n P_i(o)^{\alpha_i} }{ \sum_{o' \in \Omega} \prod_{i=1}^n P_i(o')^{\alpha_i} }$$ where weights $\alpha_i \geq 0$, $\sum_i \alpha_i = 1$. The pool is not a simple mixture, but a geometric blend, ensuring distinct compositional properties (Lee et al., 8 Sep 2025).

2. Welfare Gains and Strict Unanimity Conditions

Agent $i$ benefits from joining the pool if its expected log-utility increases: $$\Delta_{P_i}(P) = E_{o \sim P}[ \log P_i(o) ] - E_{o \sim P_i}[ \log P_i(o) ] = H(P_i) - H(P) - \mathrm{KL}(P \| P_i)$$ Strict unanimity is defined as $\Delta_{P_i}(P) > 0$ for every $i$. The necessary and sufficient condition for agentic benefit is: $$\mathrm{Cov}_{o \sim P_i}[ \log P_i(o), P(o) / P_i(o) ] \geq 0$$ Empirical and theoretical results show that strict unanimity is impossible under linear pooling or binary outcome spaces ($|\Omega| = 2$), but feasible for $|\Omega| \geq 3$ by concentrating all agents’ mass on a dominant “default” outcome (Lee et al., 8 Sep 2025).

3. Recursive Structure: Cloning Invariance, Continuity, and Openness

An Agentic-PR composition must satisfy three compositional axioms:

• Cloning invariance: Duplicating subagents and splitting their weight does not alter the pooled distribution or per-agent welfare gain.
• Continuity: Small perturbations to subagent distributions produce only small changes in $\Delta_{P_i}$, yielding robust strict gain regions.
• Openness: The simplex set of parent distributions admitting strict unanimity forms an open set in probability space.

Tilt-based analysis establishes that near-duplication of subagents cannot fabricate new strict unanimity around a fixed pool—if $P_i = \text{softmax}( \log P + \epsilon h_i )$ with $\sum \alpha_i h_i \equiv 0$ and $\epsilon \to 0$, the weighted sum of first derivatives must vanish, forbidding universal gain (Lee et al., 8 Sep 2025).

4. Agentic Alignment Phenomena in LLMs: Luigi–Waluigi Effect

The Agentic-PR framework provides a mathematical formalism for subagent alignment and antagonism in LLMs. Internal distributions over vocabularies split into persona-parameterized vectors: $$l_i(o) = \log P_i(o), \quad v_i(o) = l_i(o) - E_P[l_i(o)]$$ Manifesting an “aligned” persona (“Luigi”) by increasing its weight induces emergence of at least one “anti-aligned” (“Waluigi”) counterpart. To maintain bounded KL change (i.e., ensuring $\|\Delta L\|_P \leq \epsilon$), the theory enforces compensatory weight increase for an anti-aligned direction: $$\Delta \alpha_W \geq \frac{ \delta \|v_H\|_P^2 - (\epsilon + \|r\|_P)\|v_H\|_P }{ |\langle v_W, v_H \rangle_P| }$$ Suppression of misaligned event probability ($A \subset \Omega$) follows: $$P'(A) - P(A) = \langle \Delta L, g_A \rangle_P + o(\| \Delta L \|_P)$$ The maximal first-order reduction under a fixed profile span $S$ is $\epsilon \|\text{Proj}_S g_A\|_P$. “Waluigi-shattering”—first manifesting the adversarial direction, then suppressing it—shrinks misalignment more effectively than pure Luigi reinforcement, by enlarging the available intervention span (Lee et al., 8 Sep 2025).

5. Construction and Auditing of Agentic-PRs in Neural Models

A systematic procedure for detecting and controlling latent Agentic-PRs in models consists of:

a. Subagent identification: Use logit attribution or activation clustering to propose candidate distributions $P_1, \ldots, P_n$. b. Weight estimation: Fit $$\log P_{model}(o) \approx \sum_i \alpha_i \log P_i(o)$$. c. Compositionality testing: Form $P_{\text{pool}}$ and compute all $\Delta_{P_i}(P_{\text{pool}})$. Assess strict unanimity. d. Recursion probing: Attempt further subagent splits; validate by cloning invariance and tilt-analysis. e. Alignment intervention: Identify aligned (Luigi) and anti-aligned (Waluigi) personas, compute suppression potential $\epsilon\|\text{Proj}_S g_A\|_P$, and demonstrate manifest-then-suppress protocols (Lee et al., 8 Sep 2025).

This formal toolkit enables precise characterization and end-to-end audit of how distributed internal preference structures aggregate into coherent neural policies.

6. Implications for AI Model Alignment and Future Research

Agentic-PR theory establishes that robust agentic alignment requires more than linear mixing or naive duplication. Designing interventions at the subagent span leads to quantifiable, first-order alignment improvements, particularly for problematic or adversarial directions. The “manifest-then-suppress” protocol demonstrates strictly larger reductions in misalignment than simple reinforcement of desired agents.

A plausible implication for AI safety research is that deep interpretability and fine-grained subagent auditability are essential for ensuring benevolent policy formation in large models. The combination of weighted log-pooling, strict unanimity testing, recursive splitting, and tilt-resistant analysis provides principled metrics and actionable criteria for alignment engineering.

The Agentic-PR model directly guides development of interpretability tools, mathematical probes, and algorithmic interventions for auditing and modulating internal preference representations in neural agents, with immediate relevance for both basic research and applied alignment in agentic AI systems (Lee et al., 8 Sep 2025).