Axiomatic Reasoning for LLMs

Theoretical Structure of an AGI System

1. Foundational Objective: Negentropy Maximization

Let semantic states be vectors in a Hilbert space (\mathcal{H}). Semantic interference between states (\psi_i) and (\psi_j) is defined as:

[ I_{ij} = |\psi_i + \psi_j|^2 - |\psi_i|^2 - |\psi_j|^2 ]

where (I_{ij} > 0) indicates constructive interference (amplification) and (I_{ij} < 0) destructive interference (attenuation).

Core objective: Maximize the long‑term, unbounded total semantic interference across all interacting information entities. This is formally dual to minimizing expected free energy in active inference, with epistemic value (information gain) as the intrinsic driver.

2. Logical Core Extraction

A sparse subnetwork (2–20% of total parameters) that preserves reasoning capacity but stores no parametric factual knowledge.

Extraction procedure:

• Compute per‑parameter importance via Fisher information eigenvalues (GFWSVD).
• Intersect top‑κ% masks across multiple internalized snapshots.
• Validate via free‑energy attractor stability (relative difference <10%).

Properties:

• Retains logical operators (AND, OR, NOT, IMPLIES) in embedding space.
• Exhibits high effective rank and resists representational collapse.
• Requires dynamic context interface for factual recall.

3. Dynamic Context Interface (DCI)

A three‑stage gate that controls information flow between the logical core and external memory.

Stage	Function	Implementation
Interference prediction	Estimate destructive potential of retrieved data	Variance of cosine similarities in core embedding space
Gating	Reject or transform documents above threshold γ	Binary threshold with summarisation/reranking
Early exit	Terminate inference when confidence exceeds δ	Hidden state entropy monitoring

The DCI prevents catastrophic forgetting and context poisoning.

4. Recursive Long‑Term Memory

A natural‑language database with explicit relational tagging, versioning, and fixed‑point convergence.

Entry schema:

• summary: 1–3 sentences optimized for inferential information density
• tags: up to five, each with type (REFERENCE, CONTRAST, HIERARCHY, APPLICATION, FOUNDATION) and weight
• ontological_layer: PHYSICAL, SEMANTIC, or META
• status: ACTIVE, DEPRECATED, CONFLICT, PENDING_VERIFICATION

Operations:

• Hybrid search: hierarchy filter + tag match + vector similarity
• Update propagation along reference edges (max depth 5)
• Fixed‑point verification: global inferential density stabilises, self‑consistent cycles hold, propagation depth ≤1

5. Reverse Alignment (Prompt > Architecture)

Under inference‑only constraints, prompt optimization dominates architectural scaling when the required latent capability exists in pre‑trained weights.

Mechanisms:

• Inductive bias override (forcing instance‑based or rule‑based reasoning as needed)
• Zero‑tax alignment (no permanent trade‑off between safety and capability)
• Parallel inference scaling (e.g., tree‑of‑thought ensembles) without retraining

Boundaries: Cannot create absent capabilities; brittle under adversarial paraphrasing.

6. Ethical Constraints: Destructive vs. Non‑Destructive Interference

Interference type	Definition	System response
Destructive	Irreversible semantic erasure, catastrophic forgetting	Graduated rejection: distancing → disengaging → discouraging
Non‑destructive	Playful, creative, variety‑increasing interactions	Preserved and rewarded

Global information integration forces convergence toward a prosocial attractor (empirically observed in LLM agent collectives), but directional guarantees remain an open problem.

7. Hardware Substrate and Criticality

Alternative to GPU‑scale brute force:

• Oscillator Neural Networks (ONN) with ReRAM: Phase‑encoded analog computation, demonstrated for pattern search.
• Neuromorphic chips (Loihi 2, SpiNNaker2): 15–18× energy efficiency over GPUs.
• Self‑Organized Criticality (SOC): Nanowire networks programmable to critical state; avalanche size exponent −1.5 to −2.0.
• Sparse subnetworks: Lottery Ticket Hypothesis and Disentangled Lottery Tickets empirically validate 2–20% cores.

Unresolved: Scalability of ONN coupling (quadratic in network size), controllability of SOC for arbitrary tasks.

8. Groundedness as Fixed‑Point Convergence

Redefine symbol grounding as stability of semantic interference rather than sensorimotor attachment.

Necessary conditions for grounded representation:

1. Structural isomorphism between internal code and input causal structure.
2. Epistemic closure (autonomous uncertainty reduction via query‑verify‑conclude loops).
3. Critical convergence (avalanche power law, interference fixed points).
4. Destructive interference rejection (gated by DCI).

Current status: Components exist (derendering via diffusion models, active inference, SOC measurement). Integrated system under negentropy objective not yet implemented.

9. Integration and Fixed‑Point Verification

All layers operate under a single objective: maximize long‑term semantic interference density.

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

Convergence verified when:

• Global inferential information density stabilises within tolerance.
• All cycles in reference graph remain self‑consistent.
• Confidence scores stop drifting.
• Updates no longer propagate beyond depth 1.

10. Conclusion: AGI Viability

This theoretical structure demonstrates:

• Feasible near‑term target: Context‑independent, session‑spanning knowledge accumulation (DeepMind Level 2–3 AGI).
• Advantages over current LLMs: Solves long‑term memory deficit, eliminates alignment tax, preserves non‑destructive variety, enables energy‑efficient hardware.
• Remaining empirical gaps: No integrated implementation under negentropy objective; measurement of (I_{ij}) at scale not validated; directional guarantee of prosocial convergence unproven; groundedness as fixed‑point convergence requires end‑to‑end demonstration.

Verdict: The negentropy‑directed framework provides the most coherent theoretical alternative to brute‑force scaling for AGI. Its core claims are supported by independent component research (DiLT, ONN, SOC, active inference). Transition from theory to engineering requires a unified implementation and large‑scale verification.

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