Axiomatic Reasoning for LLMs

1. Abstract

The coding principles of brevity with high semantic density, clarity through increased explicit connectivity, and changeability via non-destructive flexibility are mapped to canonical software engineering best practices. The mapping reveals partial isomorphism with established patterns such as DRY, KISS, SRP, OCP, and high-cohesion/low-coupling architecture. The principles further align with emerging information-theoretic metrics for LLM evaluation, including semantic density, f‑mutual information, and entropy‑based complexity decomposition. The convergence suggests that code quality can be formulated as a constrained optimization problem balancing information compression, structural explicitness, and evolvability preservation.

2. Introduction

The objective of this analysis is to examine three code quality directives:

These directives are compared against established coding best practices, then examined for compatibility with information‑theoretic evaluation frameworks relevant to Large Language Models.

3. Mapping to Established Coding Best Practices

3.1 Brevity and Semantic Density

Principle Conventional Counterpart Relationship
Eliminate redundancy DRY (Don’t Repeat Yourself) Direct alignment
Minimize unnecessary constructs KISS (Keep It Simple, Stupid), YAGNI Direct alignment
Maximize information per token Source Code Density (Hönel, 2023); Semantic Density Optimization (Ustynov, 2026) Information‑theoretic extension of DRY/KISS

Conventional best practices target reduction of duplicate and speculative code. The notion of semantic density reframes this target as a measurable quantity—the ratio of meaningful behavioral specification to total syntactic volume.

3.2 Clarity and Explicit Connectivity

Principle Conventional Counterpart Relationship
Single responsibility SRP (Single Responsibility Principle) Aligned
Visible dependencies Explicit dependency injection; Information Flow Visibility Aligned
High intra‑module connection High Cohesion Direct correspondence

The phrase increased connectivity superficially contradicts the low‑coupling mandate. Under the reinterpretation required, it denotes explicit and local connections that enhance understandability. In cohesive modules, element interdependencies are dense but confined, thereby supporting clarity.

3.3 Changeability and Non‑Destructive Flexibility

Principle Conventional Counterpart Relationship
Extension without modification OCP (Open/Closed Principle) Direct isomorphism
Interface stability Encapsulation, Information Hiding Direct isomorphism
Non‑breaking releases MACH architecture (Microservices, API‑first, Cloud‑native, Headless) Architectural realization
Meta‑model evolvability XDef metamodel (O(1) DSL toolchain evolution) Formalized counterpart

The prefix non‑destructive emphasizes a guarantee absent from generic maintainability claims: changes must not silently corrupt existing behaviors. OCP and MACH architectures provide design‑level mechanisms for this guarantee.

4. Technical Concept Formalization

4.1 Semantic Density as Information Compression

Semantic density is quantifiable through information‑theoretic measures:

Tools such as aieattoken (2026) and LongCodeZip (2025) implement lossless semantic compression for LLM consumption, achieving 30‑55% token reduction while preserving behavioral fidelity.

4.2 Explicit Connectivity and Structural Complexity

Explicit connectivity corresponds to maximizing the visibility of intentional dependencies while minimizing hidden coupling (global state, side effects).

4.3 Non‑Destructive Flexibility as Evolvability Preservation

Formalized by:

The property is stronger than mere modifiability; it requires that the system’s extension does not perturb verified invariants.

5. Integration with LLM Code Evaluation

5.1 Information‑Theoretic Evaluation Metrics

Metric Basis Application to Code
Semantic Density (Qiu & Miikkulainen, 2024) Uncertainty quantification in semantic space Can score code generation by response semantic concentration
f‑Mutual Information (Robertson & Koyejo, 2025) Information‑theoretic gaming resistance Distinguishes faithful vs. strategic code generation
PPLqa (Friedland et al., 2024) Unsupervised quality via perplexity and coherence Correlates with human judgment of generated text and code summaries

These metrics provide objective, non‑subjective signals for assessing LLM outputs, circumventing biases observed in LLM‑as‑Judge settings (e.g., preference for fabricated content over accurate summaries).

5.2 Code‑Specific LLM Complexity Models

5.3 Good Code as an Optimization Problem

The three principles can be restated as a multi‑objective optimization:

Objective Mathematical Proxy
Maximize semantic density Minimize token count subject to semantic equivalence constraint
Maximize explicit connectivity Maximize normalized cohesion; minimize hidden coupling
Ensure non‑destructive flexibility Satisfy OCP invariants; minimize cascade size upon change

Constraint satisfaction among these objectives requires trade‑off resolution. For instance, maximizing density via aggressive compression may increase cognitive or computational decoding cost, as observed in Ustynov (2026). The optimization space is therefore non‑trivial.

6. Discussion

The three coding directives exhibit partial isomorphism with established software engineering best practices:

The novelty resides in the unified information‑theoretic framing and the explicit shift toward machine‑readable (LLM‑oriented) code optimization. The framework also serves as a meta‑evaluation layer for LLM‑generated code, connecting disparate metrics under a common conceptual umbrella.

Empirical validation remains incomplete. Future work includes:

7. Conclusion

The three principles—brevity with semantic density, clarity via explicit connectivity, and non‑destructive flexibility—are largely isomorphic to canonical best practices but introduce an information‑theoretic and LLM‑centric perspective. They frame code quality as an optimization problem amenable to measurement and automated evaluation. The convergence with emerging LLM evaluation metrics (Semantic Density, f‑mutual information, LM‑CC) indicates a viable path toward objective, mathematically grounded assessment of code quality in both human and machine contexts.