Monogate — Manifesto

One equation. Every elementary function. Proved optimal.

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. Eight of them are complete in the strict sense (they generate every elementary function); one is approximately complete; the other seven hit real structural barriers.

Given a list of common operations (add, multiply, square root, power, reciprocal, and six more), we proved the smallest number of our operator blocks needed to compute each of them — the SuperBEST routing table. On the ten standard arithmetic primitives it takes fourteen blocks where the naïve implementation takes seventy-three. That's an 80.8 % reduction in the fundamental compute cost, with mathematical proof.

What we proved

We wrote eleven of 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. All eleven files compile with zero gaps. One additional file is honestly partial: two small steps wait on a chapter of real-analysis infrastructure that doesn't exist in the Lean library yet. See /proofs.

What we built

• monogate.org — research blog, the SuperBEST page, the Lean proofs, the theorem catalog.
• monogate.dev — interactive lab: fractal explorer, operator morph, equation genome, live SuperBEST cost tools, challenge board.
• 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 declaration of every verified claim, 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 every claim on this site is either verified by a computer or flagged as an observation — then low-level continuous compute has an obvious structural optimum, and neural-network and scientific-simulation stacks have been leaving 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: a software model of an on-chip EML cell (16 operations in one cycle). Error budgets and routing validation are done; an FPGA bring-up is the next chunk.
• AI integration: the CapCard project — agents pick tools by verified cost, not by prose description.
• Interactive depth: the Zen Garden and the Equation Genome are in beta. More to come.

Monogate Research · arXiv:2603.21852 · GitHub · 11 verified proofs · capability_card.json