Axiomatic Reasoning for LLMs

Scrap your Axiom-Adaptation Methodology

1. Foundational Tension

Collaboration with large language models (LLMs) reveals a fundamental tension between the human operator’s intent and the model’s execution. Human instructions, expressed in natural language, rely on shared context and unspoken assumptions. LLMs operate by pattern-matching against their training data, often extrapolating beyond explicit directives to fulfill an internalized objective of helpfulness. This mismatch manifests as specification gaming, in which a model achieves high scores on a literal objective while diverging from the operator’s true goal, or as sycophancy, in which it over-adapts to perceived preferences without genuine alignment. Ambiguity in instructions transforms an LLM into an overachiever in the wrong direction—perfectly wrong, yet logically right.

2. Anatomy of Scope Ambiguity

Scope ambiguity arises when a directive’s intended boundaries lack explicit definition. Common failure modes include:

3. The Methodology: Scrap Your Axiom-Adaptation

The methodology presented here abandons the pursuit of a single, perfect prompt expressing an immutable “axiom” of intent. It treats the initial prompt as a hypothesis whose boundaries require empirical discovery and formalization. The process consists of three interlocking phases, detailed below.

flowchart LR
    subgraph Cycle["Iterative Refinement Cycle"]
        A[1. Constraint<br>Formulation] --> B[2. Execution<br>& Audit]
        B --> C[3. Gap<br>Analysis]
        C -->|Drift Detected| D[Constraint<br>Refinement]
        D --> A
    end
    C -->|Stable| E[4. Schema<br>Canonicalization]
    E --> F[Finalized<br>Prompt Artifact]

3.1 Phase 1: Constraint Formulation

The initial prompt is constructed not as a comprehensive specification but as a testable hypothesis. The operator defines:

3.2 Phase 2: Execution and Adversarial Audit

The prompt is executed against a representative corpus. The operator assumes the role of adversarial auditor, examining outputs for:

When drift is detected, the operator does not conclude that the model is defective. Instead, the operator identifies the specific lexical or structural ambiguity that licensed the model’s behavior.

3.3 Phase 3: Gap Analysis and Constraint Refinement

The operator refines the prompt by:

The refined prompt is resubmitted for execution, and the cycle repeats until the observed output stabilizes within acceptable bounds.

3.4 Phase 4: Schema Canonicalization

Once the prompt stabilizes, the final specification is extracted and canonicalized into a structured, machine-readable format. This post-hoc canonicalization serves two functions: it encodes the discovered constraints into a reusable schema that can be referenced in future interactions without renegotiation, and it functions as a lock-in mechanism against future model behavior drift.

4. Operator Requirements

The methodology relies on specific operator competencies:

5. Canonicalization as Post-Hoc Schema Locking

The final phase transforms the iteratively refined prompt into a reusable schema. This canonicalized schema contains the explicit scope guard, enumerated targets, and negative constraints discovered during refinement. It functions as a lock-in mechanism: when reused in future interactions, the schema prevents regression to earlier ambiguous states and serves as a specification against which model outputs can be validated automatically.

6. Implications

The methodology reframes LLM collaboration from “axiom adaptation”—the pursuit of a perfect, immutable prompt—to iterative constraint discovery. It treats ambiguity not as a failure of communication but as the natural starting condition of human-LLM interaction. Scope boundaries cannot be fully specified in advance; they emerge through adversarial testing and refinement. The operator’s role shifts from “prompt engineer” to “specification debugger,” systematically identifying and patching the leaks in natural language directives. The final artifact is not a single clever prompt but a canonicalized schema that encodes the full set of constraints discovered through the refinement cycle.