See how far it goes — and where it stops.
Every expression below is a tree built from that one operator. Some things appear surprisingly early. Some cannot appear at all.
What do Proved / Measured / Model / Play mean? →Play
Gravity Surfer is a game whose physics is the operator. The field you are flying through is the surface of an expression — steer it, break it, feel what the shape does. No mathematics required to start.
Every expression is a tree of one operator. Its depth is how many levels deep that tree goes — and depth is not decoration. New behaviour becomes possible at specific depths, and below those depths it is not merely hard but impossible. That is the interesting part: not how much the operator can express, but exactly where each new thing first appears.
Nothing has happened yet. A number, or the input itself.
One node buys growth and decay — compound interest, half-lives. The cheapest transcendental there is.
Combining the operator with itself gives sigmoids, softplus, logarithmic spirals. The compositions are richer than the parts.
The first place a logarithm can appear. Below this depth the value gap that separates exp-bounded growth from logarithmic growth cannot be crossed.
log xx > 0Proved Proved minimum depth. 2 ledger entries.
Addition of one is not cheap. No tree of depth three or less agrees with it on the positive reals — the explicit witness needs a fourth level.
x + 1(0, ∞)Proved Proved minimum depth. 2 ledger entries.
See the boundary
sin(x) over the real numbers has infinitely many zeros. A tree of depth k has at most 2k. No finite tree of this operator can ever equal it — not at any depth, not with any cleverness. You can approach the wall. You cannot cross it.
Proved Machine-checked, zero sorry.
Build
Depth is a price. The optimizer takes an expression you write and shows what it costs to say it in this language — and what the cheapest known form is.
Measured Reproducible measurement, not a theorem.
Explore everything
Visual, audio, research, engineering, tools and teaching — every one of them built on the same operator. Each carries a badge saying what kind of claim it makes.
Six ways into one equation
Fractals, domain coloring, Julia sets, Mandelbrot operators, bifurcation diagrams. See what happens when you iterate one equation in the complex plane.
Hear the math. Click a fractal and listen to its orbit. Mix operators like a DJ. Play the depth hierarchy as a keyboard. Mathematics as sound.
Type any equation. See its cost. Optimize it. Annotate your code. The expression optimizer finds cheaper routes through exp and ln.
From zero to the infinite barrier. Interactive lessons, quizzes, and games that teach EML without prerequisites. Start with addition. End with silence.
Browse 265 equations by domain and cost. Visualize the four boundaries of elementary functions. Explore the FMA staircase. The data behind the claims.
Which algorithm stays inside ELC? How many nodes does your FEA cost? Optimize engineering expressions with domain-specific patterns.
Research-derived audio + visual demos: Bessel FM synth (the math behind FM synthesis), math-as-melody chord stacks, Pfaffian phase portraits. Self-contained exhibits from the latest exploration sessions.
Proofs, papers and the research record live at monogate.org — including verified mathematics well beyond this one operator.