Does exp(+x) Mean Complete?

Correction (2026-09-13): this post called the rule below a theorem with structural proofs. None of its directions has a proof. The forward argument (T26) borrows other operators and constants, the LEX argument (T28) is refuted by explicit trees, and the EMN (T24) and exp(−x) (T27) cases are sketches. Over ℝ the rule is false for EAL: every real EAL tree is nondecreasing, so none approximates −x. Over ℂ it is an open conjecture.

Second correction (2026-09-13): this post cited the census as T25 and the softplus result as T19. In the catalog T25 is the Depth-6 Phase Transition and there is no T19, so both numbers are removed. The DEML row of the exp(−x) table gave "slope always +1" as DEML's barrier; that is T13's argument, which has a gap, since deml(x, 1) = exp(−x) decreases. The DEMN row wrote deml(x, 1) < 0 for demn(x, 1) = −exp(−x) < 0. The softplus corollary undercounted log-sum-exp and the softmax denominator.

Tier: CONJECTURE (T26–T28, structural arguments with gaps; T12 updated from Trichotomy to full characterization)

There are exactly 16 natural ways to combine $\exp(\pm x)$ with $\ln(y)$ using one arithmetic operation. Classifying all 16 for completeness — the ability to represent every elementary function as a finite tree — reveals a single structural rule that explains every case.

The Rule

Conjecture: an exp-ln operator is exactly complete if and only if its exponential term is exp(+x) and the combining operation does not introduce a domain restriction.

That’s it. One sentence. If it holds, all 16 operators follow.

The Conjectured Classification

The old “Completeness Trichotomy” (T12) described this as “1 complete / 1 approximate / 6 incomplete” because only EML was known to be complete at the time. The full census (sessions COMP-1 through COMP-5) proposes 8 complete operators — all of them structurally equivalent in the dimension that matters.

Why exp(+x) Should Work: The Forward Direction (T26)

The argument: every complete operator can construct exp(x) in 1 node and a slope-−1 linear function in 2 nodes. The slope-−1 construction is the key: it gives negation up to a constant offset. That works for EML (below), but for EXL, EDL and EPL the census used the constant e, which is not a leaf, and for LEAd it reached the identity only as a limit.

For EML specifically: eml(c, exp(x)) = exp(c) − x. At c = 0: eml(0, exp(x)) = 1 − x. That’s slope −1 with offset 1.

To get exact neg(x) = −x, we use the SuperBEST cross-family bridge (T09):

neg(x) = exl(0, deml(x,1)) = exp(0) · ln(exp(−x) − ln(1)) = 1 · (−x) = −x

Two nodes. Exact. But it uses EXL, DEML and the constant 0, not the operator being classified, so it shows nothing about what any one operator can build.

The intended mechanism: exp(+x) grows without bound as x → +∞. This unbounded upward growth, combined with ln (which maps ℝ⁺ → ℝ), gives the operator the full real line as output range. The hoped-for chain: from full range comes identity; from identity comes negation; from negation comes completeness (Ritt’s theorem). Over ℝ it breaks at EAL: every real EAL tree T = exp(A) + ln(B) has T′ = A′e^A + B′/B ≥ 0, so no EAL tree builds negation.

Why exp(−x) Should Fail: The Reverse Direction (T27)

The exp(−x) operators each appear to fail by a different mechanism:

The unifying feature: exp(−x) decays to 0 as x → +∞. A decaying term cannot provide unbounded growth in either direction. The only source of growth in h(exp(−x), ln(y)) is the ln(y) branch — which grows logarithmically, not linearly. So slope ±1 is either unreachable or achievable only with irremovable constant offsets.

DEAL is the subtlest case. It achieves slope −1 (DEAL(1, DEAL(x,1)) = e⁻¹ − x). But the offset e⁻¹ cannot be removed: doing so requires neg (circular), and using e⁻¹ − x as a right-branch input requires it to be positive, which restricts x < e⁻¹ ≈ 0.368. The domain collapses at the second level of nesting.

The Exception: LEX (T28)

LEX(x, y) = ln(exp(x) − y) has exp(+x), yet it is conjectured to be incomplete.

The reason given was a domain restriction in the combining operation: ln(exp(x) − y) requires exp(x) > y. With y = 1, this means x > 0. Nesting on the right, lex(1, lex(x, 1)) also needs x < ln(e^e + 1) ≈ 2.782, and deeper right-nested trees settle on about (3.97×10⁻⁶, 2.782): the domain does not approach the empty set under iteration.

Other LEX trees are defined on all of ℝ: lex(x, lex(lex(1,1),1)) = ln(e^x − ln(e − 2)) is defined for every x, at depth 3. So the domain argument does not show incompleteness. The statement this post gave for LEX (T28) was:

LEX is incomplete because its domain under self-composition collapses to a set of measure approaching 0 as depth increases.

Its reason is false, and whether LEX is incomplete is an open conjecture. It was meant as a different mechanism from exp(−x) incompleteness: failure at the domain level, not the growth level.

The Bridge Case: EMN

EMN(x, y) = ln(y) − exp(x) has −exp(x) (negation outside, not inside).

The difference from exp(−x) is crucial:

• exp(−x) decays: bounded above, approaches 0
• −exp(x) grows without bound (negatively): → −∞ as x → +∞

This unbounded negative growth was meant to let EMN approximate any target to arbitrary precision (T24, a conjecture). For approximate negation, the sketch used emn(0, e^(e^k)), which is e^k − 1 (not k − e^0 as first written), and said variations get arbitrarily close to −x with error of order exp(−e^k). No such family is written down; exhaustive search to 8 nodes finds neg(x) errors consistent with doubly-exponential decay.

The suggested obstruction to exact completeness: every EMN tree has an exp(·) residual in the subtracted term that cannot be made exactly zero. Neither that nor approximate completeness has a proof (T24).

The Exponential Position Theorem

Statement (a conjecture; T12 has no proof): An exp-ln operator’s completeness class is determined entirely by the position of negation relative to exp:

This is the updated T12 (was: Completeness Trichotomy with 1/1/6 split; now: 8/1/7 split with structural explanation).

What Changed from the Trichotomy

The Completeness Trichotomy post identified three classes with 1 exactly complete operator (EML), 1 approximately complete (EMN), and 6 incomplete. The full census proposes:

• 8 complete operators, not 1 — EML would not be special in the dimension that matters
• 7 incomplete operators, not 6 — LEX was added, though its domain mechanism is refuted (above)

The catalog count goes from T07 to T28 with this sprint. T12 became the Completeness Characterization (Exponential Position Theorem), a conjecture citing T26 (forward), T27 (reverse), and T28 (LEX domain).

New Result: Softplus = 1 LEAd Node

The softplus activation function ln(1 + exp(x)) is exactly LEAd(x, 1):

LEAd(x, y) = ln(exp(x) + y)
LEAd(x, 1) = ln(exp(x) + 1) = softplus(x)

One node. The entire softplus function — ubiquitous in neural networks and smooth approximations of ReLU — is a single LEAd application with constant 1.

Corollary: log-sum-exp of N terms takes at most 2N − 2 nodes: L₂ = LEAd(x₂, EML(x₁, 1)) and L_{k+1} = LEAd(x_{k+1}, EML(L_k, 1)). The denominator of a softmax over N logits, Σ exp(xᵢ), takes at most 2N − 1: one more EML node. This post said N − 1 for both, which fails already at N = 1, where the denominator exp(x₁) takes a node.

Monogate Research (2026). “Does exp(+x) Mean Complete? A Structural Conjecture for Exponential-Logarithmic Operators.” monogate research blog. https://monogate.org/blog/completeness-characterization

Session COMP-1 through COMP-5 · T12 updated, T26–T28 added · Catalog count: 28