Deterministic Fine-Grain Agents

• Deterministic fine-grain agents are systems defined by dual deterministic laws that govern both base-level physical states and independent agent-level dynamics.
• They enable precise multi-agent interactions and exploration tasks by leveraging constant memory and localized communication, ensuring exhaustive environmental coverage.
• In AI, these agents underpin audit-replayable decision-making architectures, achieving high reproducibility and evidence traceability through schema-first output constraints.

A deterministic fine-grain agent is a system whose behavior is fully specified by deterministic transition laws over both base-level (physical/subvenient) and coarse-grained (agent/supervenient) state variables, where the agent-level dynamics possesses autonomy and causal efficacy that operate alongside, yet independently of, the underlying physical substrate. Deterministic fine-grain agents encompass formal multi-level models of agency, deterministic multi-agent interactions, and replayable machine learning agents in both physical and computational domains.

1. Mathematical Foundations: Dual-Laws Determinism and Supervenient Causation

At the formal core, deterministic fine-grain agents are defined by a dual dynamics coupling a base-level (subvenient) system with independent agent-level (supervenient) evolution (Ohmura et al., 6 Jan 2026). The construction consists of:

• Subvenience state space: Indexed collection $\text{SUB}_i \subset \mathbb{R}^{n_i}$, $i \in I$, yielding $\text{SUB} := \bigcup_{i\in I} \text{SUB}_i$.
• Supervenience state space: Variables in $$\text{SUP}^{[N]} = \{ X: (\mathbb{R}^m)^N \to \mathbb{R}^m \}$$, with compositional structure.
• Bridge map (supervenience function): A surjection $b: \text{SUB} \to \text{SUP}^{[N]}$ ensuring each coarse-grained variable is determined by underlying physical variables.
• Independent index sequence law: Sequences $C_t$ of indices (configuration), updated by a deterministic law $P(C_t, W_t)$ with auxiliary state $W_t$.
• Base-level law: Physical variables $(x_t, v_t)$ update as $$(x_{t+1}, v_{t+1}, d_{t+1}) = p(x_t, v_t, \text{err}_t(d_t))$$ where $\text{err}_t$ collects algebraic feedback derived from agent-level index sequences.

Supervenient causation is realized by allowing the agent-level configuration law $P$ to determine, at each timestep, the feedback structure that shapes the subsequent response of the physical substrate via an explicit error signal. The agent-level evolution is not a function of subvenient (physical) state, rendering the agent temporally and ontologically non-epiphenomenal. All underlying processes remain deterministic; there is no violation of physical causal closure (Ohmura et al., 6 Jan 2026).

This framework resolves the tension between physical determinism and genuine agency: choices and configurations are not reducible to the physics of the underlying base-state and are governed by freely chosen, yet fully deterministic, laws at the agent level.

2. Fine-Grain Determinism in Multi-Agent Systems and Exploration

Deterministic fine-grain agents also arise in the context of distributed systems and multi-automata exploration tasks (Cruz-Carlon, 2023). Here, agents are instantiated as constant-memory finite automata $\Pi = (Q, q_0, \delta)$ acting under deterministic transition laws, with communication restricted to local colocation.

Critical characteristics:

• Constant memory: Each agent maintains state in fixed $|Q|$, independent of environment scale.
• Local communication: Information exchanged is restricted to state at shared location; messages are $O(1)$ bits.
• Deterministic scheduling: All agent transitions, movements, and protocol executions are determined entirely by automata and stateful communication.
• Impossibility and resource bounds: No team of three deterministic, constant-memory agents can explore the full $\mathbb{Z}^2$ lattice; four agents, organized as a “one explorer, three beacons” protocol, suffice by leveraging geometric coordination and scheduled interaction patterns.

The limitation arises from the combinatorics of agent meetings: with insufficient independent state/sequencing resources, deterministic laws yield trajectory cycles (half-bands or wedges), failing to ensure global exploration coverage. Introducing additional roles (“beacons”/landmarks) enables geometric extension of deterministic coverage, suggesting a fine-grain resource hierarchy governed by both memory and coordination protocol class (Cruz-Carlon, 2023).

3. Statistical and Algorithmic Determinism in Tool-Using LLM Agents

Deterministic fine-grain agents are operationalized in AI through architectures designed for audit-replayable decision making, particularly in regulated domains (Khatchadourian, 17 Jan 2026). Consider an LLM-based agent $\mathcal{A}$ processing query $q$ with access to tool set $\mathcal{T}$, producing a fine-grained trajectory

$$\tau = [(t_{i_1}, a_1, r_1), \dots, (t_{i_n}, a_n, r_n)]$$

and yielding terminal decision $d \in \mathcal{D}$.

Determinism metrics:

• Action determinism ($\mathrm{ActDet}$): Probability that tool call sequences are identical across independent $T=0$ runs.
• Signature determinism ($\mathrm{SigDet}$): Probability that full trajectories (including arguments and results) are identical.
• Decision determinism ($\mathrm{DecDet}$): Probability that final decisions are identical across runs.

These probabilities are empirically estimated via repeated, fixed-temperature, fixed-seed trials. Deterministic agents achieve values near 1 in all metrics, indicating fine-grain replayability required for robust auditability. Output schemas (JSON, SQL) dramatically reduce drift, with unconstrained/“ReAct” protocols showing higher variance (Khatchadourian, 17 Jan 2026).

Empirical hierarchy:

Tier	Model Size	DecDet (%)	Faithfulness (%)	Validation Factor ($\phi$)
Tier 1	7–20B	100	100	1.0×
Frontier	Claude/Gemini	88.5	100	1.34×
Tier 2	40–70B	73.4	75	1.8×
Tier 3	120B+	9.7	71.9	3.7×

Determinism correlates positively (Pearson $r = 0.45, p < 0.01$) with evidence-conditioned faithfulness ($\mathrm{EvidGround}$); models that are more deterministic are also more aligned in their factor tracing of rationales to supporting evidence.

4. Fine-Grain Determinism in Deterministic Multi-Agent Economic Models

In deterministic interacting agent systems, such as coupled map lattices in economic modeling, agent-level trajectories are specified by deterministic update laws that yield richly varying macroscopic statistics without stochasticity (0801.0969):

$$x_{t+1}^i = r_i x_t^i \exp\left( -| x_t^i - a_i \Psi_t^i | \right)$$

where $\Psi_t^i = \frac{1}{2}(x_t^{i-1} + x_t^{i+1})$ encodes local neighborhood influence. Here, fine-grain determinism refers to:

• Agent-specific deterministic updates: Each agent’s trajectory recursively determined by its own and neighbors’ states, under fixed control parameters.
• Emergent macroscopic regimes: System-level outcomes (wealth distributions) are exhaustively determined by the microscopic update laws and parameter settings, spanning both Boltzmann–Gibbs and Pareto outcome regimes.
• Control and tuning: Adjusting local environmental pressure ($a$) transitions the system between equality-dominant (BG) and inequality-dominant (Pareto) statistics, all within a fully deterministic dynamical framework.

No randomness is required to reproduce phenomena traditionally modeled via stochastic interactions.

5. Practical Engineering: Replayable Agent Architectures and Compliance Harnesses

Implementations targeting regulatory and safety-critical domains leverage architectural patterns to ensure deterministic fine-grain behavior (Khatchadourian, 17 Jan 2026):

• Schema-first output constraints (JSON/SQL): Maximize run-level and trajectory-level determinism by reducing generation ambiguity.
• Code-based graders: Assure determinism and compliance by comparing case- and run-level trajectories against exact match criteria.
• Stress-testing instrumentation: Perturbations (container restart, data corruption, simulated market shocks) quantify $\Delta\mathrm{Det}_p$ tolerances.
• Audit metrics: Compliance is defined by pass⁽ᵏ⁾=100% across all $k$ runs/cases, in contrast to pass@k measures—critical for regulatory replayability.

Deterministic fine-grain agents achieve full audit replayability and justification requirements when built using tiered model selection, schema-constrained outputs, explicit code-based evaluation, and rigorous sampling protocols. Notably, smaller models (7–20B) with schema-first protocols outperformed larger models (120B+) on reproducibility and evidence faithfulness.

6. Broader Implications and Limitations

The deterministically fine-grained approach clarifies theoretical and practical boundaries in both agent design and distributed control. In formal agency theory, it enables physically closed yet agent-driven dynamics; in distributed robotics and exploration, it quantifies the sufficiency and necessity of memory and coordination modalities; in applied AI, it mandates architectural designs tailored for traceability, reproducibility, and auditability (Ohmura et al., 6 Jan 2026, Cruz-Carlon, 2023, Khatchadourian, 17 Jan 2026, 0801.0969).

A salient implication: deterministic fine-grain architectures can replicate classes of emergent, self-organized, and audit-ready phenomena without randomization, with resource requirements and expressivity governed by protocol design and structural constraints. Ongoing challenges include quantifying trade-offs between resource bounds (e.g., memory, agents, beacons), system complexity, and practical deployability, and extending deterministic frameworks to domains with jointly structured and unstructured interactions.