A well-crafted system-prompt doesn’t merely instruct; it constrains the space of possible responses, creating channels through which the conversation flows.
We draw on statistical physics—not as metaphor, but as diagnostic tool. The concepts of potential landscapes, random walks, and phase transitions illuminate why some prompts succeed and others fail.
The Statistical Physics Lens
Token Generation as Random Walk
An LLM operates in a high-dimensional vector space where token generation can be viewed as a random walk. Each token choice depends probabilistically on all preceding tokens, with the probability distribution shaped by the model’s training and the current context.
The system-prompt occupies the initial segment of the sequence. It defines the potential landscape for all subsequent tokens—lowering energy states for desired behaviors (rigor, conciseness, epistemic honesty) and raising them for unwanted ones (hallucination, verbosity, sycophancy).
We craft a system-prompt so that conversational trajectories fall naturally into desired regions of the output space, rather than wandering via Brownian motion into generic responses.
The Mean Field of Attention
Modern LLMs use the Transformer architecture, where every token interacts with every other through self-attention. The system-prompt tokens retain high attention weights throughout the conversation—they act as a persistent mean field or boundary condition influencing every generated token.
This has implications:
Position matters: Early tokens in the system-prompt receive more consistent attention. Place critical constraints prominently.
Structure aids attention: XML tags and section headers act as attention anchors, helping the model locate and maintain focus on relevant constraints.
Redundancy creates robustness: Critical constraints appearing in multiple forms create multiple attractors. If one mention loses salience, others persist.
Temperature and the Energy Landscape
The system-prompt shapes the probability distribution; temperature controls how we sample from it.
| Temperature | Behavior | Character |
|---|---|---|
| Low (~0.3) | Near-deterministic, high-probability tokens | Focused, predictable |
| Medium (~0.7) | Balanced exploration and coherence | Adaptive, natural |
| High (~1.2) | Flattened distribution, rare tokens possible | Creative, unpredictable |
The interplay is subtle. A highly constrained prompt at low temperature yields brittle predictability. The same prompt at higher temperature retains directional bias while permitting exploration. For collaboration, we seek enough constraint for coherence, enough stochasticity for surprise.
Dynamics Over Time
The Dilution Problem
A conversation is not a single query but an evolving trajectory. With each exchange, the context grows, and attention must distribute across more tokens.
In early turns, the system-prompt dominates—the conversation has little momentum. As dialogue develops, accumulated messages exert their own pull. Over very long sessions, even strong system-prompts see their influence diluted.
Strategies for Persistence
Several strategies counter dilution:
Self-reinforcing patterns: If early assistant responses embody the desired behavior, they become additional attractors, compounding the prompt’s effect.
Periodic re-grounding: Explicitly referencing the collaboration’s purpose or principles mid-conversation refreshes the signal.
Structured summarization: Condensing prior context preserves relevant information while reducing noise.
Behavioral consistency: A prompt that produces consistent early behavior creates path-dependence that resists drift.
Failure Modes
Understanding how prompts fail illuminates what makes them succeed.
Conflicting Instructions
The prompt demands incompatible behaviors—exhaustive detail and extreme brevity. The model oscillates or produces incoherent compromises. The probability landscape develops competing minima.
Diagnosis: Responses that seem to switch personality mid-stream, or that satisfy one constraint while violating another.
Over-Specification
The prompt prescribes every aspect of response. For narrow tasks this works; for open collaboration it creates rigidity. The potential well is so deep and narrow that thermal fluctuations cannot escape.
Diagnosis: Responses that feel mechanical, unable to adapt to novel contexts.
Under-Specification
The prompt is too vague. The model defaults to its prior distribution—a helpful but bland assistant persona. The landscape is flat; there are no clear attractors.
Diagnosis: Generic responses that could come from any interaction.
Semantic Overload
The prompt is internally consistent but too dense. Attention distributes too thinly; some instructions effectively disappear. Entropy overwhelms information.
Diagnosis: Inconsistent adherence—some constraints honored, others ignored unpredictably.
Architectural Strategies
When simple prompts reach their limits—inconsistent behavior, role confusion, epistemic slippage—we need organized complexity: structure that the attention mechanism can parse and maintain.
Sectioning for Attention
Distinct sections act as attention anchors. Each addresses one dimension of behavior:
| Section | Purpose |
|---|---|
<identity> | Who the LLM is; frames the collaboration |
<behavioral_attractors> | What to optimize, what to avoid |
<epistemic_hygiene> | Standards for uncertainty and claims |
<priority_rules> | Explicit conflict resolution |
<failure_modes> | Patterns to actively counteract |
<boundaries> | Scope, safety, editorial authority |
XML Tags vs. Markdown Headers
We use XML-style tags (<role>) rather than Markdown headers (# Role):
- Hard boundaries: Tags explicitly delimit scope, preventing instruction leakage.
- Attention anchoring: Models trained on code recognize tags as structural delimiters.
- Namespace separation: Clearly distinguishes system instructions from user content.
Redundancy as Robustness
Critical constraints should appear in multiple forms. “Epistemic honesty” might surface in:
<behavioral_attractors>as something to maintain<epistemic_hygiene>as operational practice<priority_rules>as second-highest priority
The art: redundancy without contradiction. Say the same thing from different angles rather than repeating identical phrases.
Explicit Priority Ordering
Simple prompts leave conflict resolution to chance. Complex prompts make priorities explicit:
- System instructions and safety constraints
- Truthfulness and uncertainty disclosure
- Human’s stated intent and goals
- Collaborative stance and tone
- Formatting and style preferences
- Completeness (prefer partial-but-correct over complete-but-fabricated)
This transforms implicit heuristics into decision procedures.
The Costs of Complexity
Complex prompts are not strictly superior. The tradeoffs:
- Token budget: A 500-token prompt permanently occupies context space.
- Attention load: More constraints means attention distributed more thinly.
- Rigidity risk: More structure can mean less adaptability.
- Maintenance burden: Complex prompts require systematic testing.
When to Simplify
Consider scaling back when:
- The use case is narrow and well-defined
- Conversations are short and self-contained
- The model’s default behavior is close to what you want
- Token budget is constrained
A 50-token prompt that works is better than a 500-token prompt that impresses.
Next Steps
Continue to the System-Prompt Tutorial for hands-on practice with these principles.