University of North Carolina at Greensboro
A formal model and empirical calibration of capability-threshold dynamics in frontier AI. Combines Yudkowsky's intelligence explosion equation, Omohundro's instrumental resource acquisition, and Lotka-Volterra competitive exclusion into one coupled ODE system; derives the conditions under which a single dominant agent emerges; and tests the model's central empirical premise against six public AI capability proxies.
Paper: paper/main.pdf (compiled from paper/main.tex)
What this is
A formal model that combines three known frameworks (Yudkowsky's recursive
self-improvement equation dS/dt = S^(1-β), Omohundro's instrumental
resource acquisition, and Lotka-Volterra competitive exclusion) into one
coupled ODE system, derives several theorems, and tests the model's central
empirical premise (Assumption A4: that β can flip negative for some reachable
capability) against the public record on AI training compute and benchmark
performance through 2026.
The paper is conditional: if A4 holds, a singleton emerges in finite time under adversarial resource dynamics. The theorems specify how. The calibration in §8 reports what the data say about A4 itself.
Main results
- Theorem 3.4 (finite-time separation): once a leading agent crosses the β-threshold under A1–A5, its capability ratio over any competitor diverges in finite time.
- Theorem 6.2 (critical α for coalition suppression): an exact transcendental condition governs when a coalition of N members can keep a singleton candidate from crossing T. Numerical solution α* ≈ 0.64.
- Theorem 7.1 (stochastic blowup probability): under multiplicative GBM-type noise, the probability of singleton emergence equals P(J ≥ c) where J is a Dufresne perpetuity. The probability lies in (0, 1) for any σ > 0; it tends to 1 as σ → 0 and 0 as σ → ∞.
- Conjecture 3.12 (N-agent generalization): supported by simulation (200/200 trials, N=10), with an acknowledged simultaneous-dynamics gap.
- Calibration (§8): on six capability proxies (frontier compute plus five public benchmarks), the curvature γ of the log-frontier is statistically nonpositive; on two benchmarks it is significantly negative. The hypothesis β(S) < 0 is not supported by the data through 2026.
The paper does not claim a singleton is inevitable. It claims a conditional inevitability under A4, and reports preliminary empirical evidence against A4 for the proxies tested.
Repository layout
paper/main.tex LaTeX source (22 pages)
paper/main.pdf Compiled PDF
paper/refs.bib Bibliography (12 entries)
theory/ Working notes (informal; not part of the paper)
derivations.md Step-by-step derivations
formal_claim.md Theorem statements and sketches
foundations.md Literature review
simulations/ Eleven Python scripts producing all figures
intelligence_explosion.py
competition.py
agents.py
run_experiments.py
beta_regimes.py
stochastic.py
late_entrant.py
timescale.py
continuous_entry.py
cooperation.py
cooperation_alpha.py
calibration.py §8 compute-based calibration
benchmark_calibration.py §8 benchmark-based calibration
make_hero_gif.py README hero animation
data/epoch/ Input data snapshot
notable_ai_models.csv
frontier_ai_models.csv
benchmarks/ Per-benchmark CSVs (GPQA, FrontierMath, ARC-AGI, ...)
figures/ All output plots (PNG)
findings.md Per-finding parameter tables (F1–F25 + addenda)
Reproduction
Tested with Python 3.10+, numpy, scipy, matplotlib. No GPU required.
# 1. Install dependencies pip install numpy scipy matplotlib # 2. Run all simulations (regenerates figures used in the paper) python simulations/competition.py # Fig 1 python simulations/stochastic.py # Fig 2 python simulations/cooperation.py # Fig 3 python simulations/cooperation_alpha.py # Fig 4 python simulations/calibration.py # Fig 5 (compute calibration) python simulations/benchmark_calibration.py # Fig 6 (benchmark calibration) # 3. Compile the paper cd paper pdflatex main.tex bibtex main pdflatex main.tex pdflatex main.tex
All simulation scripts use fixed random seeds; outputs are bit-for-bit reproducible across runs (verified against MD5 hashes).
The Epoch AI snapshot under data/epoch/ is the version used in the paper
(accessed 2026-05-13). To re-fetch the latest data:
curl -o data/epoch/notable_ai_models.csv https://epoch.ai/data/notable_ai_models.csv curl -o data/epoch/frontier_ai_models.csv https://epoch.ai/data/frontier_ai_models.csv curl -o data/epoch/benchmark_data.zip https://epoch.ai/data/benchmark_data.zip unzip data/epoch/benchmark_data.zip -d data/epoch/benchmarks
Findings
| # | Finding |
|---|---|
| F1 | Growth equation analytical solution matches numerical integration |
| F2 | Competitive exclusion holds for all initial gaps tested (1%–100%) |
| F3 | β-threshold crossing produces super- vs sub-exponential separation |
| F4 | Winner = initial leader in 200/200 trials (N=10) |
| F5 | Elimination order is strictly weakest-first (N=8 test case) |
| F6 | Separation increases monotonically with α |
| F7 | Initial leader wins in 100% of trials down to σ=0.001 initial spread |
| F8 | β-flip mechanism operates independently of resource overlap (out-of-model) |
| F9 | Minimum β-flip separation across tested params: 1,179× (snapshot at adverse params) |
| F10 | Threshold agent defeats flat agent up to ~2.9× initial disadvantage |
| F11 | In threshold race, lower-T agent compensates ~1.19× initial disadvantage |
| F12 | Threshold advantage plateaus beyond gaps of ~4 units |
| F13 | Capped simulation reaches the cap in 100% of tested noise levels (cap = 10⁸) |
| F14 | 1% initial gap is overwhelmed by σ ≥ 0.05 noise |
| F15 | Post-threshold moat grows from 3× to >10⁶× within 3 time units |
| F16 | Late entry threatens incumbent only pre-threshold |
| F17 | Empirical scaling: t_10× ≈ 2.44·N^0.96·α^(-0.30)·gap^(-0.15)·|β_low|^(-0.31) |
| F18 | Critical entry rate λ ≈ 0.25; survival → 0% at λ ≈ 6.3 |
| F19 | Heavy-tailed entry distributions are more dangerous than high-mean |
| F20 | Coalition critical size N=2 (for α ≥ α* ≈ 0.64) |
| F21 | 8-member coalition with 8× combined power loses individual race |
| F22 | Rational defection rate is zero, coalition stable, singleton wins |
| F23 | Singleton emerges in 100% of trials across three cooperation regimes |
| F24 | Critical α for coalition suppression: transition near α=0.75 |
| F25 | At α=2.0, N=4: external suppression succeeds; internal singleton forms |
See findings.md for parameter tables and post-revision interpretation
notes. Caveats and corrections to claims that overstated their support
in earlier drafts are listed in the Addenda section there.
Limitations
The paper is honest about three structural limitations:
- Scalar capability. The β-flip mechanism is stated for a scalar S crossing a single threshold T. Real optimization capability is multi-dimensional (compute, algorithmic efficiency, data, post-training, inference compute, agentic scaffolding). Whether competitive exclusion survives in vector capability space is open.
- N-agent simultaneous dynamics. The induction in Conjecture 3.12 is formally complete in the sequential limit but the simultaneous-dynamics correction is not analytically bounded.
- A4 is the load-bearing assumption. The theorems are conditional on A4. The calibration in §8 makes A4 falsifiable on specific datasets; under every proxy tested through 2026 the data do not support A4.
Citing
@misc{langley2026singleton, author = {Langley, Nathan}, title = {The Singleton Attractor: Formal Theory and Simulation of Dominant Intelligence Emergence}, year = {2026}, note = {Available at https://github.com/ninjahawk/singleton-attractor} }
License
Code and paper text are released under the MIT license. Epoch AI data
snapshots under data/epoch/ are redistributed under their original
Creative Commons Attribution license.