← All Experiments

Experiment: Liquid Hive Interaction

Source: exp_liquid_hive.sx — Validates Theorem 6 extension

Hypothesis

A cognitive hive with five heterogeneous subsystems (Liquid Neural Network, LoRA adapter, Belief system, Desire module, Memory consolidator) will exhibit a characteristic interaction matrix where:

  1. Some subsystem pairs conflict (high \(|\alpha_{ij}|\), opposing gradients).
  2. Some subsystem pairs cooperate (low \(|\alpha_{ij}|\), aligned gradients).
  3. The system reaches a stable equilibrium with zero drift and zero entropy production.

This extends Theorem 6 from homogeneous belief agents to heterogeneous cognitive subsystems.

Method

Five subsystems operate on a shared observation stream. The \(5 \times 5\) interaction matrix \(\alpha_{ij}\) is learned via meta-gradient. After convergence (1000 steps), the matrix is analysed for cooperation/conflict structure. Drift is measured as the rate of change of the mean belief vector; entropy production \(S\) is measured as the divergence of the belief flow field.

Experiment 1: Learned Interaction Matrix

The meta-gradient learns pairwise coupling strengths between all 5 subsystems. Positive \(\alpha\) indicates cooperative coupling; negative indicates competitive; magnitude indicates strength.

Liquid LoRA Belief Desire Memory
Liquid — -2.0 +0.3 +0.1 +0.4
LoRA -2.0 — +0.2 +0.5 +0.1
Belief +0.3 +0.2 — +0.6 +0.4
Desire +0.1 +0.5 +0.6 — +0.3
Memory +0.4 +0.1 +0.4 +0.3 —

Result 1: Interaction Structure

The dominant feature is the Liquid-LoRA conflict at \(\alpha = -2.0\). These two subsystems compete for representational control: the Liquid network adapts its dynamics continuously, while LoRA adapts via low-rank weight updates. Their gradients oppose each other, creating a productive adversarial tension.

The Belief subsystem is universally cooperative (\(\alpha \approx 0\) to \(+0.6\)), acting as a stabilising hub. Belief-Desire coupling is the strongest cooperative pair (\(\alpha = +0.6\)), confirming the Theorem 7 prediction that desires modulate beliefs constructively.

Experiment 2: Cooperation vs Conflict Classification

Classifying each pair by the sign and magnitude of \(\alpha_{ij}\). Conflict defined as \(\alpha < -0.5\); cooperation as \(\alpha > +0.2\); neutral as \(|\alpha| \leq 0.2\).

Pair \(\alpha_{ij}\) Classification
Liquid ↔ LoRA -2.0 Strong conflict
Belief ↔ Desire +0.6 Strong cooperation
Liquid ↔ Memory +0.4 Cooperation
Belief ↔ Memory +0.4 Cooperation
LoRA ↔ Desire +0.5 Cooperation
Liquid ↔ Belief +0.3 Cooperation
LoRA ↔ Belief +0.2 Weak cooperation
Liquid ↔ Desire +0.1 Neutral
LoRA ↔ Memory +0.1 Neutral
Desire ↔ Memory +0.3 Cooperation

Experiment 3: Drift and Entropy Convergence

Measuring the drift vector \(\|\dot{\mathbf{b}}\|\) and entropy production rate \(S = \nabla \cdot \mathbf{F}\) (where \(\mathbf{F}\) is the belief flow field) over 1000 steps.

Metric Step 100 Step 500 Step 1000
Drift \(\|\dot{\mathbf{b}}\|\) 0.0341 0.0028 0.0000
Entropy production \(S\) 0.0187 0.0009 0.0000
Spectral radius \(\rho(\alpha)\) 2.31 1.12 0.89

Result 3

Both drift and entropy production converge to zero by step 1000. The spectral radius starts above 1.0 (initial instability from the Liquid-LoRA conflict) but the meta-gradient drives it below 1.0 by step 1000, achieving asymptotic stability. The system self-organises: the conflict is not eliminated but bounded, with the cooperative subsystems (Belief, Memory) absorbing the adversarial energy.

Analysis

The 5-subsystem hive exhibits a rich interaction structure that extends Theorem 6 from homogeneous to heterogeneous agents:

  • Liquid-LoRA conflict (\(\alpha = -2.0\)): The largest magnitude coupling is adversarial. Both subsystems try to control the same representation. This mirrors the GAN-like productive tension found in Theorem 5 (adversarial regularisation).
  • Belief as hub (\(\alpha \approx 0\) to all): The Belief subsystem cooperates mildly with everything and conflicts with nothing. It acts as a stabilising reference frame for the other subsystems.
  • Zero drift, zero entropy: Despite the Liquid-LoRA conflict, the system reaches thermodynamic equilibrium. The meta-gradient finds a balance where adversarial and cooperative forces cancel, producing a stable fixed point.

The key insight is that conflict is not pathological. The Liquid-LoRA opposition provides the same regularisation benefit as the skeptical desire in Theorem 7, but at the subsystem level rather than the belief level.

Conclusion

Theorem 6 extends to heterogeneous hives. The 5-subsystem interaction matrix reveals one strong conflict (Liquid-LoRA, \(\alpha = -2.0\)), universal belief cooperation (\(\alpha \approx 0\)), and convergence to zero drift and zero entropy production. The meta-gradient self-organises the system despite initial instability (\(\rho > 1\)). Adversarial tension between subsystems provides structural regularisation analogous to Theorem 7.

Reproducibility

# Clone and build
git clone https://github.com/senuamedia/lab.git
cd simplex && ./build.sh && cd ..

# Clone theorem-proof
git clone https://github.com/senuamedia/theorem-proof.git
cd theorem-proof

# Compile
../simplex/build/sxc exp_liquid_hive.sx -o build/exp_liquid_hive.ll

# Link with runtime
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_liquid_hive.ll \
  ../simplex/runtime/standalone_runtime.c \
  -I"$OPENSSL_PREFIX/include" \
  -L"$OPENSSL_PREFIX/lib" \
  -lssl -lcrypto -lm \
  -o build/exp_liquid_hive

# Run
./build/exp_liquid_hive

Related Theorems