Monogate — Research Blog

Every function has a minimum node count.

The neg gap is closed: exl(0, deml(x,1)) computes −x in exactly 2 nodes for all x ∈ ℝ, with no domain restriction.

# EML Meets Neural Networks

12 classical geometric primitives — hyperbolic distance, Lie group maps, curvature, conformal maps — expressed as EML operator trees.

Partition functions, time evolution, density matrices, and quantum information geometry measured in matrix EML nodes.

ELSb(0, x) = exp(0 − ln(x)) = 1/x.

Every binary combination of exp(±x) with ln(y) via arithmetic — completeness classification of all 16 operators.

Every arithmetic operation now has a proved-optimal or exhaustively-bounded node count in the exp-ln operator family.

A single transcendence fact about tan(1) is the root cause behind three separate EML results: the multiplication lower bound, the depth-3 ceiling for standard functions, and the complex density behavior.

Three completeness classes for exp-ln operators: exactly complete (EML), approximately complete (EMN), and incomplete (all others).

We enumerated EML constant trees to depth 7, tested 20 random complex targets, and tracked how close EML gets to π.

Iterating exp(z)−k is Devaney's exponential family.

f(x) = exp(x) − ln(x) satisfies f(x) > x for all real x > 0.

Each Fourier harmonic is one complex EML node.

sin(x) costs 101 nodes as a Taylor series in BEST routing.

The BEST router's mul entry drops to 3 nodes via exl(ln(x), exp(y)) = x·y.

We applied the DEML incompleteness template to seven exp-ln operators.

We proved that a depth-k EML tree has at most 2k+2 real zeros, and verified computationally that the true bound may be as low as 2 for all k ≥ 3.