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

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

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},
}

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).

SuperBEST v5 routing table (all 10 entries proved optimal)

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.

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.

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

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.