exp(−x) and the Five-Operator Barrier
The negative exponential appears in almost every physics law. It is blocked for 1-node construction in EML and four related operators. DEML is the operator that breaks through.
Why exp(−x) matters
exp(−x) is not an exotic function. It is the backbone of physics:
- Radioactive decay: N(t) = N₀ · exp(−λt)
- Boltzmann distribution: P ∝ exp(−E/kT)
- Damped oscillation: A(t) = A₀ · exp(−γt)
- Gaussian: exp(−x²/2σ²)
- Heat equation kernel: exp(−x²/4t)
Under the EML operator eml(x, y) = exp(x) − ln(y), constructing
exp(−x) in a single node requires presenting −x as input.
But −x is not available from the grammar starting at {1}
without spending nodes to negate first.
The five-operator check
We checked all five operators in the EML family for 1-node exp(−x) construction:
| Operator | Definition | 1-node exp(−x)? |
|---|---|---|
| EML | exp(x) − ln(y) | blocked |
| EDL | exp(x) / ln(y) | blocked |
| EXL | exp(x) · ln(y) | blocked |
| EAL | exp(x) + ln(y) | blocked |
| EMN | exp(x) − ln(−y) | blocked |
| DEML | exp(−x) − ln(y) | ✓ one node |
DEML: the complementary operator
deml(x, y) = exp(−x) − ln(y). Setting y = 1:
One node. Every decay law in physics expressible in a single EML-family operation.
DEML and EML are complementary operators. EML handles the forward exponential; DEML handles the decay direction. Together they cover the full exponential family without requiring negation as a separate node.
Expressiveness vs. efficiency
DEML adds efficiency, not expressiveness. At depth ≥ 3, the closure of EML alone already contains exp(−x) as a composition. DEML just gives it to you in one node instead of three. This is the motivation for the BEST routing system — choosing the cheapest operator for each primitive, rather than forcing everything through EML.
Cite this work
Monogate Research (2026). "exp(−x) and the Five-Operator Barrier." monogate research blog. https://monogate.org/blog/negative-exponent
License
CC BY 4.0 — free to share and adapt with attribution. ·
Code: pip install monogate ·
Paper: arXiv:2603.21852