monogate

eml(x, y) = exp(x) − ln(y) — one binary operator. Open problem: can sin(x) be constructed from it?

Based on arXiv:2603.21852 (Odrzywołek, 2026).
Challenge board: monogate.dev · · ·

pip install monogate          # core — no dependencies
pip install "monogate[sympy]" # + symbolic simplification

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Quick start — expression construction

from monogate import BEST, best_optimize, CBEST, im

# BEST routing — cheapest gate per operation
BEST.pow(2.0, 10.0)   # 1024.0  (EXL, 3 nodes vs 15)
BEST.div(6.0, 2.0)    # 3.0     (EDL, 1 node  vs 15)
BEST.ln(2.718)        # ~1.0    (EXL, 1 node  vs  3)

# Complex BEST — sin and cos in 1 node via Euler path
import math
im(CBEST.sin(math.pi / 6))   # 0.5  (= sin(π/6), exact)

# Code optimizer — rewrite any Python/NumPy/PyTorch expression
r = best_optimize("torch.sin(x)**2 + torch.cos(x) * x**3")
print(r.rewritten_code)   # 72% fewer nodes

Interactive web demo (Streamlit): — Optimizer · Special Functions · PINN Demo · MCTS Explorer · Phantom Attractor

Run locally:

pip install -r requirements-streamlit.txt
streamlit run streamlit_app.py

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For Researchers

Challenge board: monogate.dev — submit a construction for sin, cos, π, or i. Get credited permanently.

Theorem catalog: monogate.dev/theorems — every result labeled honestly: theorem, conjecture, observation, or speculation.

Open problems:

• Construct sin(x) from eml(x,y) using only grammar terminals {1, x}
• Construct i (the imaginary unit) from terminal {1} alone
• Prove or disprove: the EML depth hierarchy has no level 4

Reproduce the paper results:

git clone https://github.com/agent-maestro/monogate
cd monogate
make reproduce-n11    # verify N=11 exhaustive search (~30s from cache)
make reproduce-all    # full readiness check
make paper            # compile preprint.tex (requires TeX Live)

Cite:

@misc{monogate2026,
  title  = {monogate: Universal Expression Trees from a Single Binary Operator},
  author = {[Author]},
  year   = {2026},
  eprint = {ARXIV_ID_PLACEHOLDER},
  archivePrefix = {arXiv},
}

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The sin(x) barrier

Theorem (Infinite Zeros Barrier): No finite real-valued EML tree with terminals {1, x} equals sin(x) for all x ∈ ℝ.

Proof: Every EML tree is real-analytic → finitely many zeros. sin has zeros at {kπ : k ∈ ℤ}. Contradiction.

Empirical confirmation: 208,901,719 trees evaluated (N ≤ 11, ~5 min on one CPU core). Best near-miss MSE: 1.478e-4.

Complex bypass (1 node, exact): Im(eml(ix, 1)) = Im(exp(ix)) = sin(x).

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SuperBEST v5 routing table (all 10 entries proved optimal)

Operation	Construction	SB nodes	Naive EML	Saving
exp	EML(x,1)	1	1	—
ln	EXL(0,x)	1	3	−2
e^−x	DEML(0,x)	1	5	−4
recip	ELSb(0,x)	1	5	−4
div	ELSb(ln x, y)	2	15	−13
neg	EXL(0,DEML(0,x))	2	9	−7
mul	ELAd(EXL(0,x),y)	2	13	−11
sub	LEdiv(x,EML(y,1))	2	5	−3
add	LEdiv(x,DEML(y,1))	2	8	−6
pow	EML(EXL(0,x)·n,1)	3	15	−12

Total: 18 nodes (SuperBEST v5) vs 73 (naive EML) — 75.3% fewer. (ADD-T1)

16 exp-ln operators classified: 8 exactly complete, 1 approximately complete (EMN), 7 incomplete. (T24–T28)

Research by Arturo R. Almaguer.

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When to use it

monogate is the right tool when your workload:

• Does symbolic regression or interpretable expression search
• Is dominated by pow, ln, mul, or div (all save ≥6 nodes each)
• Uses sin/cos activations (NeRF, SIREN, Fourier features, physics ML)
• Needs human-readable formula output from a differentiable tree

monogate is not a PyTorch inference accelerator. Native torch.sin is ~9,000× faster than any EML variant. The EML substrate computes in Python scalars. It is the right tool for symbolic analysis, formula construction, interpretable regression, and mathematical research.

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Install

pip install monogate                # core — no dependencies
pip install "monogate[sympy]"       # + prover (SymPy exact tier)
pip install "monogate[torch]"       # + EMLTree, EMLNetwork, HybridNetwork
pip install "monogate[llm]"         # + LLM-guided optimizer

JavaScript / Node:

npm install monogate

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Repository structure

monogate/
├── python/          # pip install monogate
│   ├── monogate/    # core library
│   ├── tests/       # 1184 tests
│   ├── notebooks/   # tutorials + prover_showcase.ipynb
│   └── docs/        # MkDocs site
├── lib/             # npm install monogate  — JS/Node library
├── explorer/        # monogate.dev          — Vite/React browser app
├── THEORY.md        # formal theorem/conjecture reference
├── Makefile         # make reproduce-all, make test, make docker-run
└── Dockerfile       # clean-room reproducibility environment

Research Notes

Detailed working documents, raw session logs, and full research context are maintained in a private repository for cleanliness and strategic reasons.

Public artifacts in this repo include:

• python/paper/preprint.tex — the authoritative arXiv preprint
• python/notebooks/ — clean, reproducible session benchmarks
• python/results/ — benchmark outputs and figures
• python/monogate/ — the full Python library
• capability_card.json — machine-readable capability profile

License

MIT. The underlying mathematics is CC BY 4.0 per the original paper.