The set of real-time CRN-computable numbers $\mathbb{R}\_{\mathrm{RTCRN}}$ turns out to be more than just a field — it is an ordered exponential field, closed under $a^b$. This places it squarely in the territory of Schanuel’s conjecture and Zilber’s program in model theory. Here I explore what this connection means and what questions it opens.

What is $\mathbb{R}\_{\mathrm{RTCRN}}$?

A real number $\alpha$ is real-time CRN-computable if there exists a bounded polynomial initial value problem (PIVP)

with a distinguished output variable $y_1(t)$ converging to $\alpha$ exponentially fast:

The set of all such numbers is denoted $\mathbb{R}\_{\mathrm{RTCRN}}$. It was shown in [Huang, Klinge, Lathrop, Li 2019] that $\mathbb{R}\_{\mathrm{RTCRN}}$ coincides with $\mathbb{R}\_{\mathrm{RTGPAC}}$, the numbers computable in real time by Shannon’s General Purpose Analog Computer.

$\mathbb{R}\_{\mathrm{RTCRN}}$ is a field

The basic algebraic closure properties were established in [Huang, Klinge, Lathrop, Li 2019]:

Theorem. $\mathbb{R}\_{\mathrm{RTCRN}}$ is a subfield of $\mathbb{R}$.

The proofs are constructive: given PIVPs computing $\alpha$ and $\beta$, one builds new PIVPs computing $\alpha + \beta$, $\alpha \cdot \beta$, $-\alpha$, and $1/\alpha$ (for $\alpha \ne 0$). Each construction introduces a fresh species $Z$ whose ODE is designed so that $z(t)$ converges to the desired result. For instance, subtraction uses

so $z(t) \to 1/(\alpha - \beta)$ when $x(t) \to \alpha$ and $y(t) \to \beta$, and then $\alpha - \beta = 1/z(\infty)$ by the reciprocal closure.

Beyond a field: $a^b$ closure

In [Chen, Huang 2026], the Bounded Analog Complexity paper, it is shown that $\mathbb{R}\_{\mathrm{RTCRN}}$ is additionally closed under exponentiation:

Theorem. If $\alpha, \beta \in \mathbb{R}\_{\mathrm{RTCRN}}$ and $\alpha > 0$, then $\alpha^\beta \in \mathbb{R}\_{\mathrm{RTCRN}}$.

This follows because $\alpha^\beta = \exp(\beta \ln \alpha)$, and both $\exp$ and $\ln$ (on positive reals) are real-time computable by simple PIVPs:

• $\exp$: The PIVP $x' = -x$, $y' = -xy$ with $x(0) = y(0) = 1$ gives $y(t) = e^{1-e^{-t}} \to e$.
• $\ln$: The PIVP $r' = -r^2$, $u' = -u + rv$, $v' = -v$ computes $e^{-t}\log(1+t) \to 0$, but with appropriate readout, $\ln(\alpha)$ is extracted.

Combined with field closure, the $a^b$ operation is real-time computable.

An ordered exponential field

These closure properties give $\mathbb{R}\_{\mathrm{RTCRN}}$ the structure of an ordered exponential field: an ordered field $(F, +, \cdot, <)$ equipped with a map $\exp \colon F \to F^{>0}$ satisfying

In our case, $\exp$ is the restriction of the real exponential function to $\mathbb{R}\_{\mathrm{RTCRN}}$. Since $\ln$ is also available, we have the full “EL-field” (exponential-logarithmic field) structure.

This places $\mathbb{R}\_{\mathrm{RTCRN}}$ in the setting studied by Wilkie, Macintyre, and Zilber.

Wilkie’s theorem

In a landmark result, Wilkie [1996] proved:

Theorem (Wilkie). The theory of $(\mathbb{R}, +, \cdot, <, \exp)$ is model complete.

Model completeness means: if $\phi(\bar{x})$ is a first-order formula and $(\mathbb{R}, \exp) \models \exists \bar{x}\, \phi(\bar{x})$, then this can be witnessed by a quantifier-free formula (after allowing existential quantifiers). Combined with o-minimality, this gives the real exponential field a very tame structure.

Question. Is $\mathbb{R}\_{\mathrm{RTCRN}}$ an elementary substructure of $(\mathbb{R}, +, \cdot, <, \exp)$?

If yes, then every first-order property of the real exponential field holds in $\mathbb{R}\_{\mathrm{RTCRN}}$. For this, we would need $\mathbb{R}\_{\mathrm{RTCRN}}$ to be existentially closed: whenever an exponential-polynomial equation $f(x, \bar{a}) = 0$ with parameters $\bar{a} \in \mathbb{R}\_{\mathrm{RTCRN}}$ has a real solution, it has a solution in $\mathbb{R}\_{\mathrm{RTCRN}}$.

This is plausible:

• For purely polynomial equations, all real algebraic numbers are in $\mathbb{R}\_{\mathrm{RTCRN}}$ (Newton’s method converges exponentially).
• For $e^x = a$ with $a \in \mathbb{R}\_{\mathrm{RTCRN}}$, $a > 0$: the solution $x = \ln a$ is in $\mathbb{R}\_{\mathrm{RTCRN}}$ by $\ln$-closure.
• For mixed exponential-polynomial equations: the solutions should be computable by appropriate PIVPs, though proving real-time convergence in general is open.

Schanuel’s conjecture

Schanuel’s Conjecture (1960s). If $\alpha_1, \ldots, \alpha_n \in \mathbb{C}$ are linearly independent over $\mathbb{Q}$, then

This is one of the central open problems in transcendental number theory. It implies:

• The Lindemann–Weierstrass theorem (proved): $e^{\alpha_1}, \ldots, e^{\alpha_n}$ are algebraically independent when $\alpha_1, \ldots, \alpha_n$ are algebraic and $\mathbb{Q}$-linearly independent.
• The Gelfond–Schneider theorem (proved): $\alpha^\beta$ is transcendental for algebraic $\alpha \ne 0, 1$ and irrational algebraic $\beta$.
• $e + \pi$ is transcendental (open!).
• $e$ and $\pi$ are algebraically independent (open!).

The connection

Here is where $\mathbb{R}\_{\mathrm{RTCRN}}$ enters the picture.

Observation 1: $\mathbb{R}\_{\mathrm{RTCRN}}$ is a naturally occurring countable real exponential field.

It is countable because each element is specified by a finite PIVP (a finite list of polynomial coefficients and integer initial conditions). It contains all algebraic numbers, $e$, $\pi$, $\ln 2$, $\gamma$ (Euler–Mascheroni), and is closed under $\exp$ and $\ln$.

Observation 2: Schanuel’s conjecture constrains $\mathbb{R}\_{\mathrm{RTCRN}}$.

If Schanuel’s conjecture holds, then for any $\mathbb{Q}$-linearly independent $\alpha_1, \ldots, \alpha_n \in \mathbb{R}\_{\mathrm{RTCRN}}$, the $2n$ numbers $\alpha_1, \ldots, \alpha_n, e^{\alpha_1}, \ldots, e^{\alpha_n}$ must contain at least $n$ that are algebraically independent over $\mathbb{Q}$. This is an a priori constraint on which algebraic relations can hold among PIVP-computable numbers.

For example, taking $\alpha_1 = 1, \alpha_2 = i\pi$ (extending to $\mathbb{C}$), Schanuel implies $\operatorname{trdeg}\_\mathbb{Q} \mathbb{Q}(1, i\pi, e, -1) = \operatorname{trdeg}\_\mathbb{Q} \mathbb{Q}(\pi, e) \ge 2$, which would establish the algebraic independence of $e$ and $\pi$. Since both are in $\mathbb{R}\_{\mathrm{RTCRN}}$, this is a statement about the internal structure of our field.

Observation 3: The PIVP structure could provide evidence for Schanuel.

Every element of $\mathbb{R}\_{\mathrm{RTCRN}}$ is a limit of a solution to a polynomial ODE. Algebraic relations among such limits correspond to algebraic relations among solutions of polynomial ODEs — the domain of differential algebra (Ritt, Kolchin). If one could show that the “differential-algebraic degree of freedom” of a PIVP system with $n$ independent exponential components is at least $n$, this would verify the Schanuel property for $\mathbb{R}\_{\mathrm{RTCRN}}$.

Zilber’s program

Zilber [2005] took a bold model-theoretic approach. He constructed pseudo-exponential fields — structures $(K, +, \cdot, \exp)$ satisfying:

1. Schanuel property (the conclusion of Schanuel’s conjecture),
2. Strong exponential closure (exponential-polynomial equations have “enough” solutions),
3. Countable closure property (a technical condition ensuring uniqueness).

Zilber’s Conjecture. The complex exponential field $(\mathbb{C}, +, \cdot, \exp)$ is isomorphic to Zilber’s pseudo-exponential field.

This conjecture implies Schanuel’s conjecture. It predicts that the complex exponential field is uniquely determined by a few structural axioms.

Question. Does $\mathbb{R}\_{\mathrm{RTCRN}}$ satisfy the Schanuel property?

If so, it would provide evidence for Schanuel’s conjecture from an unexpected direction — analog computation theory. The PIVP framework gives a concrete, constructive handle on exponential-field elements that the abstract model-theoretic approach lacks.

Is $\mathbb{R}\_{\mathrm{RTCRN}}$ real closed?

A field is real closed if every positive element has a square root and every odd-degree polynomial has a root. For $\mathbb{R}\_{\mathrm{RTCRN}}$:

• Square roots: $\sqrt{a} = a^{1/2} \in \mathbb{R}\_{\mathrm{RTCRN}}$ by $a^b$ closure. $\checkmark$
• Odd-degree polynomials: If $p(x) = a_n x^n + \cdots + a_0$ with $a_i \in \mathbb{R}\_{\mathrm{RTCRN}}$ and $n$ odd, then $p$ has a real root. Newton’s method, started from a sufficiently good initial approximation, converges quadratically (exponentially fast in the number of correct bits). The iteration can be implemented as a PIVP, suggesting the root is real-time computable.

Conjecture. $\mathbb{R}\_{\mathrm{RTCRN}}$ is real closed.

If true, then by Tarski’s theorem, $\mathbb{R}\_{\mathrm{RTCRN}}$ is elementarily equivalent to $\mathbb{R}$ in the language of ordered fields. Combined with the exponential structure and Wilkie’s theorem, this would make $\mathbb{R}\_{\mathrm{RTCRN}}$ a very natural “computable core” of the real exponential field.

Summary

The real-time CRN-computable numbers sit at an unexpected intersection of analog computation, differential algebra, and model theory. The PIVP framework — polynomial ODEs with integer initial conditions — provides a concrete, constructive description of a countable exponential subfield of $\mathbb{R}$. Whether this constructive structure can shed light on the deep open problems of transcendental number theory remains to be seen, but the connections are suggestive.

This post is part of a series on bounded analog complexity and the Ripple formalization project. The formal proofs that $e$, $\pi$, $\ln 2$, and $\gamma$ are real-time computable are verified in Lean 4.

References

• A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 1051–1094.
• B. Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Ann. Pure Appl. Logic 132 (2005), 67–95.
• X. Huang, T.H. Klinge, J.I. Lathrop, X. Li, Real-time equivalence of chemical reaction networks and analog computers, DNA 25, 2019.
• H.-L. Chen, X. Huang, Bounded Analog Complexity, DNA 32, 2026.
• E.A. Karatsuba, On the computation of the Euler constant $\gamma$, Numer. Algorithms 24 (2000), 83–97.