Negation in Two Nodes — For All Real x

Correction (2026-09-13): this post called 2 nodes the minimum, citing an exhaustive 1-node search. The only 1-node neg search on record rounds each value to 6 decimals and then compares it with −x at tolerance 1e−7, so at x = π it rejects every tree, exact ones included, and its "none found" shows nothing. The 2-node construction holds for all real x. Whether 1 node can compute neg depends on the operators and constants allowed: LEpow(x, −1) = ln((eˣ)⁻¹) = −x is 1 node in a wider operator list.

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) = −x

Why 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) = −x

The 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,·) = −x

This 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

Both domains take 2 nodes with these constructions. That no 1-node neg exists was never established: the N=1 check over 54 cases rounds values and rejects exact trees, and with a wider operator list LEpow(x, −1) = ln((eˣ)⁻¹) = −x is 1 node. 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:

Total: 18 nodes vs 73 naive = 75.3% savings (SuperBEST v4).

Both domains were said to converge to 21 nodes, which is the SuperBEST v1 total; this v4 table totals 18. This post also said every entry except add (general, 11n) was optimal by exhaustive search at the level below. That did not hold: later constructions cut pow to 1 node for x > 0, mul to 1 node for x, y > 0, and add to 2 nodes for all real x, y (ADD-T1). The current table is on /superbest.

What “Optimal” Meant Here

For each entry at N nodes, optimality means:

• Exhaustive search at N−1 nodes found zero constructions
• The N-node construction exists, as an explicit tree checked by evaluating it. For neg that tree is exl(0, deml(x,1)) = ln(e⁻ˣ) = −x, which holds for every real x by algebra.

For neg: the N=1 check (54 cases: 6 operators × 9 terminal combinations from {0,1,x}) reported no 1-node neg, but the search on record rounds each value to 6 decimals and compares it with −x at tolerance 1e−7, so at x = π it rejects every tree, exact ones included. The 2-node construction exists; that 2n is the minimum over those six operators rests on no working search.

Monogate Research (2026). “Negation in Two Nodes — For All Real x.” monogate research blog. Sessions N1–N10.