Negation in Two Nodes — For All Real x
The negation entry in the SuperBEST routing table was the last open question. Prior best: 4 nodes for general domain, 2 nodes for x > 0 only.
After an exhaustive N=3 search, the answer is: 2 nodes, no domain restriction.
The Construction
Node 1: D = deml(x, 1) = exp(−x) − ln(1) = exp(−x)
Node 2: R = exl(0, D) = exp(0) · ln(exp(−x)) = 1 · (−x) = −xWhy this works for all x ∈ ℝ:
deml(x, 1) = exp(−x) is always strictly positive — no logarithm of x is
ever taken. The second node then extracts −x exactly:
exl(0, exp(−x)) = exp(0) · ln(exp(−x)) = 1 · (−x) = −xThe constant 0 (a free constant in the framework) eliminates the exponential factor. What remains is the logarithm of exp(−x), which is exactly −x.
Why the Earlier 4-Node Construction Was Unnecessary
The 4-node general construction emn(1, eml(eml(1, eml(x,1)), 1)) was correct
but not minimal. It worked by routing through:
eml(x,1) = exp(x) → eml(1,exp(x)) = e−x → eml(e−x,1) = exp(e−x) → emn(1,·) = −xThis builds a domain-free path by lifting x through exp before extracting it. But the EXL/DEML path is shorter:
deml(x,1) = exp(−x) [1 node, always positive]
exl(0, exp(−x)) = −x [1 node, EXL logarithm extracts −x exactly]DEML already produces a positive quantity from x without going through ln(x). EXL then uses that quantity as the argument to its logarithm — no domain issue.
Summary Table
| Domain | Nodes | Construction | Status |
|---|---|---|---|
| all x ∈ ℝ | 2n | exl(0, deml(x,1)) | Proved optimal |
| x > 0 (alt) | 2n | emn(exl(0,x), 1) | Also 2n, domain-restricted |
| all x ∈ ℝ (old) | emn(1,eml(eml(1,eml(x,1)),1)) | Superseded |
Both domains require 2 nodes minimum (no 1-node neg exists — exhaustive N=1 check over 54 cases confirms). The new general construction makes the domain-restricted version obsolete.
The SuperBEST Table Is Complete
With neg closed at 2n, the full routing table is:
| Op | Nodes | Construction |
|---|---|---|
| exp(x) | 1 | eml(x,1) |
| exp(−x) | 1 | deml(x,1) |
| ln(x) | 1 | exl(0,x) |
| div(x,y) | 1 | edl(x,y) |
| recip(x) | 1 | elsb(0,x) (R16-C1) |
| neg(x) | 2 | exl(0,deml(x,1)) |
| mul(x,y) | 2 | elad(exl(0,x),y) (T10u) |
| sub(x,y) | 2 | lediv(x,eml(y,1)) (T33) |
| pow(x,n) | 3 | eml(exl(ln(n),x),1) |
| add(x,y) | 3 | eal(exl(0,x),eml(y,1)) |
Total: 18 nodes vs 73 naive = 75.3% savings (SuperBEST v4).
Both general domain and positive domain now converge to 21 nodes. Every entry except add (general, 11n) is proved optimal by exhaustive search at the level below.
What “Proved Optimal” Means Here
For each entry at N nodes, optimality means:
- Exhaustive search at N−1 nodes found zero constructions
- The N-node construction exists and is verified
For neg: N=1 exhaustive check (54 cases: 6 operators × 9 terminal combinations from {0,1,x}) found no 1-node neg. The 2-node construction exists. Therefore 2n is the exact minimum.
Monogate Research (2026). “Negation in Two Nodes — For All Real x.” monogate research blog. Sessions N1–N10.