Monogate — Manifesto

One equation. Every elementary function.

What it is

Nearly every number that matters in science can be computed with exponentials and logarithms. Monogate shows that one equation combines the two into a single building block that is enough to construct every elementary function — sines, Gaussians, growth rates, decay laws, everything — as a finite tree of applications of that one block.

Think of it as the NAND gate of continuous mathematics. NAND is one binary operator that, combined with itself, computes every boolean function. Our operator does the same thing for elementary real and complex functions.

What we found

There are exactly sixteen natural variants of this operator — the full family of two-input functions built from signed exp and signed log. We mapped all sixteen. By that census, which is checked computationally rather than proved, eight of them generate every elementary function and one is approximately complete. The other seven appear to hit structural barriers, but that count is a conjecture: the arguments are on paper, and the one for DEML has a gap (see /theorems, T12, T13 and T27).

Given a list of common operations (add, multiply, square root, power, reciprocal, and six more), we built the SuperBEST routing table: the fewest operator blocks we know of for each. On the ten standard arithmetic primitives (positive domain) it takes fifteen blocks from the sixteen operators on /framework where the naïve implementation takes seventy-three, a 79.5 % reduction. Every entry has an explicit construction; only some have a proved lower bound, and /superbest gives each entry's status.

What we proved

We wrote the core results as machine-checked Lean 4 proofs. A program — not a reviewer, not a reputation — verifies every logical step from the axioms of mathematics. Every Lean claim on this site is re-checked before each deploy: its file is compiled at a pinned commit and its #print axioms is read. A few files in the library still carry documented sorrys; /proofs gives the sorry count of every file it shows, checked against the pinned commit.

What we built

• monogate.org — research blog, the SuperBEST page, the Lean proofs, the theorem catalog.
• monogate.dev — EML lessons, the Electronics Lab and a page for reproducing their commands. Its earlier interactive lab (fractal explorer, operator morph, equation genome, SuperBEST cost tools) and challenge board were archived on 2026-09-12.
• pypi.org/project/monogate — open-source Python package with the full cost theory, 315-equation catalog, neural autodiff, and EML tree trainer.
• capability_card.json — a machine-readable, agent-queryable list of the site's claims, each with its status, served at a public URL so tools can consume it without human intervention.
• arXiv:2603.21852 — the original paper by Odrzywołek.

What it means

If the analysis is right — and /theorems says, claim by claim, whether it has a Lean proof, an argument on paper, or is still a conjecture — then low-level continuous compute has a structural optimum, and neural-network and scientific-simulation stacks have been leaving roughly eighty percent of that optimum on the table. We believe the optimum is real on silicon, not just on paper. That's the next build.

What's next

• Hardware: an on-chip EML cell (16 operations in one cycle), so far a software model; an FPGA bring-up of that cell is next.
• AI integration: the CapCard project — agents pick tools by a stated cost, not by prose description.
• Interactive depth: the Zen Garden is on 1op.io. The Equation Genome, which was on monogate.dev, was archived on 2026-09-12.

Monogate Research · arXiv:2603.21852 · GitHub · the Lean proofs · capability_card.json