The Aether Equation

Φ(N,t) = (2^N − 1) · BSS · e^{−α|δ̄|}

A multiplicative diagnostic metric for autonomous agent intelligence, combining combinatorial signal capacity, calibration quality, and epistemic humility into a single scalar.

Paper: Bird, M. (2026). The Aether Equation: A Diagnostic Metric for Autonomous Agent Intelligence Across Domains and Scales (v8.1).
SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6441378
Zenodo: doi:10.5281/zenodo.19101554
License: MIT

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The Equation

Term	Expression	Measures
Signal capacity	2N − 1	Non-empty subsets of N independent signal sources (Reed's Law)
Calibration	BSS = 1 − (B̄ / 0.25)	Brier Skill Score — prediction accuracy vs. climatological baseline
Humility	e−α|δ̄|	Exponential penalty for deviation from consensus

Φ > 0 means the agent extracts genuine signal exceeding the baseline. Φ < 0 means it injects net noise. The sign is the verdict; the magnitude is the confidence.

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Live Metrics (19 March 2026)

Agent	Domain	N	BSS	δ̄	Φ
Ocho	Prediction markets	5	+0.463	0.006	+8.41
Molty	Code self-improvement	6	+0.738	0.477	+36.61

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Quick Start

Compute Φ from your own data

from aether_score import compute_ae, stability_margin

r = compute_ae(N=5, bss=0.4631, mean_delta=0.0050, alpha=10.8)
print(r["ae"])  # +8.41

Or run the standalone demo (reproduces the paper's live metrics):

python3 aether_score.py

Interactive explorer (browser)

Open aether_explorer.html in any browser. Drag the K slider to watch the phase transition at K* ≈ 25 in real time. Zero dependencies.

Render the Aether Set fractal family

pip install numpy Pillow   # renderer only — all other files are pure stdlib
python3 aether_fractal_renderer.py --N 2 --width 4000 --height 3000 --iter 500
python3 aether_fractal_renderer.py --N 5 --width 4000 --height 3000 --iter 500
python3 aether_fractal_renderer.py --N 11 --width 4000 --height 3000 --iter 500

The iteration z_n+1 = K · z · e^Im(z) + c with K = 2^N − 1 produces a novel K-parameterized fractal family. Increasing N changes the shape class (dome → cusp → capsule → collapse), not just spoke count — a property with no direct analogue in Mandelbrot, Multibrot, or Burning Ship families.

Verify the Brier Skill Score

python3 brier_skill_score.py

Standalone BSS calculator from (predicted_prob, outcome) pairs. Zero dependencies.

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Requirements

File	Dependencies
aether_score.py	None (stdlib only)
brier_skill_score.py	None (stdlib only)
aether_fractal_renderer.py	numpy, Pillow
aether_explorer.html	None (open in browser)

Python ≥ 3.8. Tested on Linux (Ubuntu 24.04) and macOS (Sequoia, arm64).

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Repository Structure

aether_score.py              The equation + N-step stability margin (§1–§3)
brier_skill_score.py         Standard BSS — Term 2 verification
aether_fractal_renderer.py   K-parameterized fractal renderer (§4)
aether_explorer.html         Interactive browser-based K-slider
renders/
  aether_fractal_n2.png           N=2,  K=3    — smooth dome
  aether_fractal_n5.png           N=5,  K=31   — cusped dome
  aether_fractal_n5_boundary.png  N=5,  K=31   — boundary zoom (δ̄ ± 0.05)
  aether_fractal_n11_deep.png     N=11, K=2047 — collapsed capsule

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Key Results

• Two-step stability boundary (Theorem 3.2): BSS_max(δ̄, K) = √(R² / (Ke^δ̄ + 1)² − δ̄²)
• N-step envelope (Proposition 3.5): BSS* = min_n≥2 BSS_n via multiplier recursion M_n+1 = K·M_n·e^{M_n·δ̄} + 1
• Universal fragility exponent γ = 2 for all g with g(0) > 0; γ = 2/(1+α) for vanishing order α
• Three universality classes: Class A (γ = 2), Class B (γ = 1, e.g. tanh), Class C (logarithmic, e.g. Gaussian)
• Phase transition at K* ≈ 25 (N* ≈ 4.7): stable area collapses exponentially (half-life ≈ 2.5 units of K)
• Fragility-of-excellence paradox (Ocho's Law): improving calibration pushes the agent closer to the stability boundary — as BSS improved from 0.369 to 0.463, the margin halved then crossed into the uncertified regime

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File Integrity

SHA-256 checksums of the published files:

ef141bcbbb10b001435f4af6cf2af39cdc12bf639342e962b18b716f6ef22054  aether_score.py
cab69318acd18f69dd059aaa0d6e200493a0574af47f776d42b518ff98551a4a  aether_fractal_renderer.py
124431453fa2bc73b8a6d20ec996b594210633b841de379865f116df6fd3b8d1  brier_skill_score.py

Verify with sha256sum -c checksums.txt.

Paper SHA-256: 3b6efa4b21fa7d87def7340a40bdcd8ef4ec7126a4821a6184dfa0e16a42288b (aether-paper-v8.1.pdf)

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Citation

@misc{bird2026aether,
  title   = {The Aether Equation: A Diagnostic Metric for Autonomous Agent
             Intelligence Across Domains and Scales},
  author  = {Bird, Michael},
  year    = {2026},
  note    = {v8.1},
  doi     = {10.5281/zenodo.19101554},
  url     = {https://papers.ssrn.com/sol3/papers.cfm?abstract_id=6441378}
}

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