O-Minimal Theory with Trans-Exponential Growth

An O-Minimal Trans-Exponential Expansion

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

An expansion of the ordered real field is o-minimal if each definable subset of the line, allowing parameters, is a finite union of points and intervals. A trans-exponential function $f$ eventually dominates every finite iterate of the usual exponential: for each $k\geq1$, $f(x)>\exp^{\circ k}(x)$ for all sufficiently large $x$. Seek an o-minimal expansion defining such a function. [S1] [S2]

Statement recorded in the supplied source.

Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function? [S1]

Short English statement

Can an o-minimal expansion of the real field define a function that eventually grows faster than every fixed-height exponential tower?

Research attempt: a trans-exponential candidate whose derivative singularities define the integers

Status and source scope. We construct a concrete rapidly growing function and prove its expansion cannot be o-minimal. This refutes a natural construction strategy, not the existence asked for. The inspected revised abstract of Fu [S2] reduces o-minimality of an analytic candidate to a regular-value condition; it does not announce unconditional o-minimality.

1. A continuous increasing tower interpolation

Fix $a\geq1$, set $c=e^a-a>0$, and let $h(u)=a+cu$ for $0\leq u\leq1$. For $x=n+u$, $n\in\mathbb N_0$ and $0\leq u<1$, put \[f(x)=\exp^{\circ n}(h(u)).\tag{1}\] At an integer endpoint the left limit is $\exp^{\circ(n-1)}(e^a)=\exp^{\circ n}(a)$, the value from the right. Thus $f$ is continuous and strictly increasing on $[0,\infty)$, and $f(x+1)=\exp(f(x))$.

For a fixed $k$ and $n\geq k$, applying $k$ logarithms to $1$ gives at least $\exp^{\circ(n-k)}(a)$. For $t\geq1$, $e^t\geq2t$, so this quantity is at least $2^{n-k}a$. For sufficiently large $n$ it exceeds $n+1\geq x$. Applying $k$ exponentials back proves $f(x)>\exp^{\circ k}(x)$ eventually. Thus the growth requirement itself has been verified.

2. Every positive integer leaves a definable defect

Write $E_n=\exp^{\circ n}$ and $E_0$ for the identity. On each open unit interval $f$ is analytic. At the positive integer $n$, the one-sided derivatives are \[f'_-(n)=cE'_{n-1}(e^a),\qquad f'_+(n)=cE'_n(a)=ce^a E'_{n-1}(e^a). \] They are positive and unequal, with ratio $e^a>1$. Hence the nondifferentiability set of $f$ in $(0,\infty)$ is exactly $\{1,2,3,\ldots\}$.

Differentiability is first-order definable from the field operations, order and graph of $f$: there exists a real $L$ such that for every $\epsilon>0$ there is $\delta>0$ with \[0<|t-x|<\delta\ \Longrightarrow \left|\frac{f(t)-f(x)}{t-x}-L\right|<\epsilon. \] All quantifiers range over reals. Consequently the expansion by $f$ defines its nondifferentiability set, an infinite discrete subset of the line. Such a set is neither a finite union of points nor a union containing an interval. The expansion is therefore not o-minimal. Defining $f$ arbitrarily on the negative half-line cannot remove this already definable obstruction on $(0,\infty)$.

3. What a smoother interpolation would have to change

Replacing the linear $h$ by a differentiable connector with $h(0)=a$ and $h(1)=e^a$ removes the first-derivative jump only if \[h'(1)=e^a h'(0). \] Higher smoothness imposes further endpoint identities obtained by differentiating $f(x+1)=e^{f(x)}$. Even satisfying all such identities would establish smoothness, not o-minimality: other definable sets in the expansion would still need to be controlled. Conversely, the failure of the piecewise-linear connector says nothing about an analytic candidate whose integer joins are invisible.

4. Verification and remaining gap

The diagnostic checks the endpoint and derivative-ratio formulas for initial iterates and the elementary inequality $e^t\geq2t$ on the domain needed by the symbolic growth induction. The proof does not rely on computing enormous tower values. It distinguishes the two goals explicitly: $1$ has the required domination, but its singularity set violates o-minimality. An o-minimal construction must avoid this definable discrete set and every other such obstruction; proving that for any candidate remains absent. The result here is a rigorous failure analysis of one natural tower interpolation, not a nonexistence theorem for all trans-exponential expansions.

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1465 (MOD-008), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Y. Fu, Towards Trans-Exponential O-minimal Expansion of the Real Field, arXiv:2604.03477 $2026$. https://arxiv.org/abs/2604.03477. Bibliographic information from the supplied record.

• S3 Trans-exponential o-minimal structure status page. https://www.emergentmind.com/open-problems/existence-of-transexponential-o-minimal-structure. Bibliographic information from the supplied record.