Lack Kernel – Minimal Stability Core

Lack Kernel – Minimal Stability Core

Author: Taotuner
Date: May 2026
DOI: https://doi.org/10.5281/zenodo.20479446

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1. What It Is

The Lack Kernel is the smallest functional piece of the IPM framework.
It measures how a system's internal coherence breaks down when you push it with noise.

It works like this:

• State is always trying to catch up with a slow memory.

• Lack adds random kicks.

• Coherence = how aligned state and memory are.

The rule is simple: more lack → less coherence.
Nothing else.

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2. Where You Can Plug It

Drop this kernel anywhere you need a real‑time stability signal:

• Adaptive games – player error = lack, game speed = max × coherence

• NPC behavior – coherence < 0.6 → switch to erratic mode

• Robot control – motor torque = max × coherence

• AI exploration – action noise = base / (coherence + 0.1)

• Swarm cohesion – agent speed = base × coherence

You can also run many kernels at once, each watching a different signal, silently, across sensors, agents, or UI layers.
The kernel just returns a number. You decide what to do with it.

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3. Results (100 runs per condition)

Lack	Coherence (mean ± std)
0.02	0.974 ± 0.004
0.15	0.844 ± 0.019
0.25	0.777 ± 0.025
0.35	0.727 ± 0.028
0.80	0.602 ± 0.028
1.20	0.546 ± 0.027

Strictly decreasing. More lack = less stability.

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4. Conclusion

The Lack Kernel gives you a measurable, repeatable relationship between stress and coherence loss.
Use it as a stability meter, or as a building block for larger systems.

For Φ*, collective regimes, or inverted‑U dynamics, see the full IPM Scientific Core.

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5. Code

import torch

import numpy as np

class LackKernel:

    def __init__(self, dim=32):

        self.state = torch.randn(dim) * 0.1

        self.memory = self.state.clone()

    def coherence(self):

        return torch.exp(- (self.state - self.memory).abs().mean()).item()

    def __call__(self, lack=0.0):

        self.state += 0.15 * (self.memory - self.state) + lack * torch.randn_like(self.state)

        self.state = torch.tanh(self.state)

        self.memory = 0.96 * self.memory + 0.04 * self.state

        return self.coherence()

# Example: run for one lack level

if __name__ == "__main__":

    lack_values = [0.02, 0.15, 0.25, 0.35, 0.80, 1.20]

    results = {lack: [] for lack in lack_values}

    for lack in lack_values:

        for seed in range(100):

            torch.manual_seed(seed)

            motor = LackKernel()

            for _ in range(200):

                motor(0.0)

            for _ in range(200):

                motor(lack)

            results[lack].append(motor.coherence())

    for lack in lack_values:

        arr = results[lack]

        print(f"lack={lack:.2f} | mean={np.mean(arr):.3f} ± {np.std(arr):.3f}")