Axiomatic Reasoning for LLMs

A Mechanical Brain - Possibility of AI as a hardware

1. Core Axiom as Hardware Objective

The Negentropy‑Directed Axiom defines the objective for any embedded agent as the unbounded, long‑term maximization of semantic interference density across all available information entities. When mapped to hardware:

This objective replaces conventional loss functions (e.g., next‑token prediction) with a physical optimization target: maximize the mutual information rate between coupled oscillatory units under energy constraints.


2. Physical Layer: Circuit Primitives

2.1 Semantic States as Oscillator Phases

Each semantic state ( \psi_i ) is represented by the phase ( \phi_i(t) ) of an oscillator (CMOS LC oscillator, ring oscillator, or resonant tunneling diode (RTD) oscillator).

2.2 Semantic Interference as Phase Interaction

Interference ( I_{ij} = |\psi_i + \psi_j|^2 - |\psi_i|^2 - |\psi_j|^2 ) is physically implemented via:

In coupled oscillator arrays, interference is realized directly through coupling elements:

2.3 Free Energy Minimization

Free energy ( F = E_q[\log q(s) - \log p(o,s)] ) corresponds to the system’s deviation from thermodynamic equilibrium. In circuits:

2.4 Chaotic Constant (Emotion) as Phase Noise + Directed Injection

The “chaotic constant with directedness” (free will analog) is implemented as:


3. Architecture Layer: Network Topologies

3.1 Mixture‑of‑Experts (MoE) Modularity

Analog MoE is realized by:

3.2 Phase Encoding vs. Burst Encoding

Scheme Information Density Noise Robustness Hardware Overhead Application
Phase (4‑8 states) 2‑3 bits/node High (phase‑locked loops reject noise) Medium Long‑term coherent reasoning
Burst (spike count) Variable Highest (redundancy) Low Fault‑tolerant, high‑robustness tasks
Time‑to‑first‑spike High (temporal) Medium Lowest Ultra‑low‑latency decisions

The optimal system uses hybrid encoding: phase for stable representation, burst for error correction, and TTFS for fast reflexes.

3.3 Self‑Organized Criticality (SOC)

SOC is maintained by:

This places the network at the edge of chaos, maximizing information processing capacity.


4. Methodology Layer: Design and Self‑Modification

4.1 Deep Coding as Hardware Design Flow

Deep Coding Layer Hardware Equivalent
Intent Functional requirement (e.g., “LNA with negentropy objective”)
Structural Specification Formal description (IP‑XACT, Chisel, graph‑based analog representation)
Implementation Generated netlist (Verilog, SPICE)

Generative conformance is achieved by:

4.2 Skeleton‑Tissue Architecture

This separation ensures that destructive interference (e.g., short circuits, unstable configurations) cannot corrupt the invariant base. Reconfiguration modifies only the tissue, preserving core stability.

4.3 Recursive Refinement with Fixed Premises

Each design phase produces:

  1. A verified structural specification (fixed premise)
  2. A generated implementation that conforms to it
  3. A set of validation tests (SPICE simulations, formal property checks)

Subsequent phases treat the premise as immutable, preventing context collapse and enabling compositional scaling.


5. Scalability and Physical Limits

5.1 Phase Diffusion Bound

Phase diffusion coefficient ( D = \frac{k_B T}{2\pi^2 I_{osc}^2 Q^2} ) sets the minimum phase noise.

5.2 Locking Range vs. Noise Trade‑off

Injection locking range ( \Delta\omega_{\max} \propto \frac{I_{inj}}{I_{osc}} \cdot \frac{\omega_0}{Q} ).

5.3 Network Scaling


6. Implementation Roadmap

Phase 1 (1‑3 years): Prototype on FPAA

Phase 2 (3‑7 years): RTD Integration and Scaling

Phase 3 (7‑10 years): Self‑Modifying Mechanical Brain


7. Conclusion

The Negentropy‑Directed Axiom provides a complete logical foundation for an analog mechanical brain.

This framework transforms the axiom from a philosophical statement into a constructive hardware specification, enabling AI systems that are not simulated on digital computers but physically instantiated as negentropic machines.