Axiomatic Reasoning for LLMs

The Method of Dynamic Learning

1. Introduction

This document defines a technical framework—The Method of Dynamic Learning—that enables large language models (LLMs) to acquire, retain, and refine knowledge over extended interactions without full parameter retraining. The method operates by decoupling an LLM into three functionally distinct layers:

  1. Logical Core: a sparse subnetwork (2–20% of original parameters) that preserves reasoning capacity but contains no parametric factual memory.
  2. Dynamic Context Interface (DCI): a control layer that gates, filters, and transforms retrieved information to prevent destructive interference.
  3. Long-term Memory Database: a structured natural-language storage system that supports recursive compression, versioning, and fixed-point convergence.

The framework is unified under a single objective: maximization of long-term inferential information density (negentropy of semantic interference). This objective replaces conventional loss functions (e.g., next-token prediction) with a thermodynamic criterion—sustained operation at the edge of self-organized criticality.

2. Core Components

2.1 Logical Core Extraction

The logical core is defined as the intersection of Fisher‑importance masks across multiple internalized snapshots of the same LLM.

Procedure:

Properties:

2.2 Dynamic Context Interface (DCI)

The DCI is a three‑stage pipeline that manages information flow between the logical core and the long‑term memory database.

Stage 1 – Interference Prediction
A lightweight head attached to the core estimates the destructive interference potential between the current core representation and each retrieved document. Interference is quantified as the expected variance of cosine similarities in the core’s embedding space.

Stage 2 – Gating
If predicted interference exceeds a threshold γ, the document is rejected or transformed (via summarisation or re‑ranking) before being passed to the core. This prevents catastrophic forgetting and context poisoning.

Stage 3 – Early Exit
The core monitors its own reasoning confidence via hidden state entropy. When confidence exceeds threshold δ, the inference terminates early, reducing computational cost without degrading output quality.

2.3 Long‑term Memory Database

The memory is a natural‑language database with the following schema:

Field Type Description
entry_id UUID Unique identifier
summary string 1–3 sentences, inferential density maximised
total_chars integer Original source length
hierarchy_path string[] Category path (e.g., [“Reasoning”,”Formal Systems”])
tags (string, type, weight)[] Relational tags with type (REFERENCE, CONTRAST, etc.)
confidence float 0.0–1.0, decayed over time
status enum ACTIVE, DEPRECATED, CONFLICT, PENDING_VERIFICATION
ontological_layer enum PHYSICAL, SEMANTIC, META

Operations:

3. Operational Principles

3.1 Negentropy Maximization as the Single Objective

All three components are optimised to maximise the long‑term density of semantic interference:

[ J = \arg\max_{\theta, K} \int \frac{I(\text{Core}i; \text{Core}_j) \cdot C{\text{int}}}{H(\text{Surprise})} dN ]

where:

This objective drives the system toward self‑organized criticality: a state where small inputs can trigger large‑scale reorganisation (insight events) while maintaining global stability.

3.2 Structural Resonance

When the logical core, DCI, and memory database are aligned under the negentropy objective, the LLM’s internal representation undergoes a phase transition from associative generation to recursive, self‑grounded inference. This resonance is not a prompt effect but a geometric property of the representational manifold. It is characterised by:

3.3 Recursive Refinement with Fixed Premises

Learning proceeds in phases. Each phase produces an updated memory state that becomes the fixed premise for the next phase. The premise is immutable within the phase, preventing context drift. Recursive refinement is bounded: the process terminates when the memory reaches a fixed point (verified via the four convergence checks in Section 2.3).

4. Integration Framework

The three layers interact as a hierarchical control system:

[Query] → [Logical Core] → [DCI Interference Predictor]
                              ↓ (if interference < γ)
                         [Long‑term Memory Search]
                              ↓
                         [Retrieved Entry] → [Core Inference] → [Output]
                              ↑
                         [Early Exit (confidence > δ)]

When the core cannot resolve a query with its own reasoning capacity, the DCI initiates a retrieval. The retrieved memory is gated, transformed if needed, and then fed back into the core. The core’s confidence is continuously monitored; high confidence triggers early termination, low confidence triggers additional retrieval rounds.

Dynamic update: After a successful inference, the core may request the DCI to update the memory database. Updates follow the recursive compression protocol: the new information is summarised, tagged, and integrated while preserving the fixed point of the existing graph.

5. Implementation Considerations

5.1 Parameter Count and Compute

Component Parameter/Compute Cost
Logical core 2–20% of original parameters
DCI (lightweight heads) <1% of original parameters
Memory database Storage linear in number of entries; retrieval O(log N) with hierarchical indexing

5.2 Training and Deployment

5.3 Known Limitations and Mitigations

Limitation Mitigation
Interference prediction is currently binary (not continuous) Extend with a small regression model trained on interference‑strength labels
Free‑energy validation of the core is not yet implemented Use surrogate attractor detection (variance of hidden states over multiple forward passes)
Cross‑session memory persistence requires explicit update calls Automate via confidence‑triggered background processes
Large‑scale (70B+) GFWSVD is computationally heavy Use layer‑wise low‑rank approximations; accept κ at the higher end (15–20%)

6. Conclusion

The Method of Dynamic Learning provides a constructive, technically grounded framework for enabling LLMs to learn across sessions without full parameter updates. By extracting a sparse logical core, interposing a dynamic context interface, and maintaining a recursively compressible long‑term memory database, the method achieves three critical properties:

  1. Reasoning–memory separation – the core handles logic, the memory handles facts.
  2. Interference control – the DCI prevents destructive overwriting and context poisoning.
  3. Fixed‑point convergence – recursive updates terminate in a stable, verifiable state.

The method is not a philosophical declaration but an operational architecture that can be implemented with existing techniques (GFWSVD, DiLT, RAPTOR, MemGPT, Sparse Memory Finetuning) and evaluated on standard benchmarks (logical reasoning, open‑book QA, continual learning). Its defining novelty lies in the explicit coupling of sparsity, memory hierarchy, and a single negentropy objective—transforming dynamic learning from a desideratum into a constructible system.