Topological Kleene Field Theory

• TKFT is a geometrico-topological framework that represents partial recursive functions via smooth dynamical flows on manifolds with boundaries.
• It establishes a categorical equivalence by modeling computation using clean dynamical bordisms that simulate composition, primitive recursion, and minimization through distinct topological motifs.
• Intrinsic complexity measures such as flow-time, Betti numbers, and dimensionality bridge computational performance with fundamental topological properties in TKFT.

Topological Kleene Field Theory (TKFT) is a geometrico-topological framework for computation in which the class of partial recursive functions is realized via smooth dynamical flows on manifolds with boundary, known as bordisms. Inspired by Stephen Kleene's work on partial recursive functions, TKFT provides an alternative to the Turing machine paradigm by encoding computation as the trajectory of a point—representing an input—under a continuous vector field until it reaches a designated output boundary. Nontrivial topological features of the bordism play a central role in both computational Universality and complexity, establishing deep connections between topological invariants and classical notions from computability theory (González-Prieto et al., 20 Mar 2025).

1. Geometric and Formal Underpinnings

TKFT begins with an encoding of computational data using Cantor-set embeddings within the cube $I^n = [-1,1]^n$. Each finite one-sided tape $$t = (t_0, \ldots, t_\ell) \in \{0,1,\square\}^{\mathbb{N}}$$ is assigned a real coordinate by

$$x_t= \frac{2}{3} \sum_{i=0}^{\ell} \sigma(t_i)\cdot 3^{-i}$$

where $\sigma(0) = -1$, $\sigma(1) = +1$, $\sigma(\square) = 0$. For two-sided sequences, the encoding is $\kappa(t) = (x_{t^-}, x_{t^+}) \in I^2$. The set $\kappa(\mathbb{N}^n) \subset I^n$ forms a Cantor-type subset representing all possible tapes.

A dynamical germ is a triple $(M, X, \iota)$:

• $M$ is a compact $n$-manifold with boundary,
• $X$ is a nowhere-zero smooth vector field on $M \oplus \mathbb{R}$ with negative $\mathbb{R}$-component (ensuring flow enters $M$),
• $\iota: I^n \to M$ embeds the Cantor-set encoding.

A dynamical bordism of dimension $n+1$ from $(M_1, X_1, \iota_1)$ to $(M_2, X_2, \iota_2)$ is a compact manifold-with-corners $W$ with boundary $M_1 \sqcup M_2 \sqcup \partial'W$, equipped with a vector field $X$ such that:

• $X$ is transverse inward to $M_1$ and outward to $M_2$,
• $X$ is tangent to side faces $\partial'W$,
• $X|_{M_i}$ agrees with $\pm X_i$ so that flow enters at $M_1$ and exits at $M_2$.

A field $X$ is called tame if its zero-set consists of finitely many nondegenerate points.

The fundamental operation is the reaching map: For $c \in \mathbb{N}^n$, the orbit of $x = \iota(\kappa(c))$ under the flow $\varphi_t(x)$ of $X$ is followed. If it first intersects the outgoing boundary $M'$, then the preimage of the intersection under $\iota'$ defines $\psi_{(W,X)}(c)$; otherwise, the function is undefined. Imposing invariance on auxiliary coordinates, the reaching function $Z_0(W,X): \mathbb{N} \to \mathbb{N}$ is extracted.

A bordism is clean if it deformation retracts to a finite CW-subcomplex constructed by gluing finitely many basic bordisms, locally inducing simple computable diffeomorphisms. Complex computation arises purely from the topology of $W$.

2. Equivalence to Partial Recursive Functions

The central result of TKFT establishes a categorical equivalence between computation via clean dynamical bordisms and partial recursive functions. Specifically:

• Every partial recursive function $f: \mathbb{N} \rightharpoonup \mathbb{N}$ is represented by $Z_0(W,X)$ for some volume-preserving clean dynamical bordism $(W,X)$ between standard discs,
• Conversely, every $Z_0(W,X)$ obtained from a clean bordism is a partial recursive function.

Thus, the monoidal subcategory of clean dynamical bordisms is full and essentially surjective onto the category $\mathbf{PRF}$ of partial recursive functions. The functor

$$Z: \mathrm{Bord}_3^{c\text{–}dy} \longrightarrow \mathbf{PRF}, \quad Z(W,X) = Z_0(W,X)$$

formally realizes TKFT (González-Prieto et al., 20 Mar 2025).

The proof proceeds by showing:

• Gluing of bordisms models composition of functions,
• Basic topological patterns (cylinders, handles, pair-of-pants) encode Lehene's three computational schemata: composition, primitive recursion, minimization,
• Any partial recursive $f$ can be realized by interpreting a reversible Turing machine as a graph of local diffeomorphisms between discs, assembling a global bordism by controlled gluing along Cantor-fibers.

3. Construction of Fundamental Computable Functions

Basic computable functions are modeled explicitly:

• Successor ($f(n)=n+1$): Realized by a basic bordism corresponding to a horseshoe-type diffeomorphism $\varphi_+$ that shifts the Cantor-set encoding left, implemented as a mapping cylinder whose reaching map shifts the index.
• Addition ($f(m, n) = m+n$): Constructed by concatenating $n$ successor bordisms along their boundaries, thus topologically composing $n$ mapping cylinders in a chain.
• Multiplication ($f(m, n) = m \cdot n$): Implemented through a handle, specifically a mapping torus over the addition complex, effectively looping over the addition subroutine $m$ times, so each handle traversal contributes $n$ branches.

These constructions exploit combinatorial gluing and looping patterns—pair-of-pants for conditional branching, handles for iteration—to mirror exactly the inductive build-up of partial recursive functions in classical recursion theory.

4. Topological Motifs and Computational Phenomena

The topological structure of the bordism precisely encodes computational patterns:

• The pair-of-pants (sphere minus three discs) models a two-branch conditional, joining two incoming collars to one outgoing collar corresponding to an "if" statement.
• A handle or mapping torus provides recursion, carrying a boundary component back to itself and inducing subroutine iteration.
• Nontrivial fundamental group and higher Betti numbers directly reflect necessary loops and branching, so the first Betti number $b_1(W)$ bounds recursion depth. Proposition 4.3 states that if the shortest Turing machine computing $f$ uses $b$ nested loops, then any clean bordism computing $f$ satisfies $b_1(W)\geq b$.

These motifs tie the expressive capacity and complexity of computation to global topological properties.

5. Complexity Measures in TKFT

TKFT admits several intrinsic complexity notions:

• Flow-time complexity $T_f(n)$: The minimal time $t$ so that the trajectory $\varphi_t(x)$ reaches the output, as a function of input size $n$.
• Topological complexity: Summarized by Betti numbers, $\sum_k \mathrm{rank} H_k(W)$, providing a lower bound on the resources needed for recursion and branching.
• Dimensional complexity: Given by $\dim W$ or the count of basic bordisms (CW-complex cells) used.

Preliminary evidence indicates flow-time and topological complexity can trade off, paralleling classical time-space resource analysis. The possibility is raised that clean bordisms not corresponding to classical Turing machines—especially with "advice" diffeomorphisms as extra basic bordisms—might yield computational power beyond Turing and quantum models.

6. Perspective, Challenges, and Prospects

TKFT unifies computability, topology, and dynamics, potentially offering novel analytic tools (e.g., for ordinary/partial differential equations), providing geometric proxies for computational complexity, and facilitating the study of robustness and noise due to the continuity of flows.

Highlighted open directions include:

• Constructing explicit bordism families with optimal Betti numbers for specific functions,
• Establishing tight bounds for flow-time as a function of input size,
• Investigating whether advice-augmented basic bordisms yield genuine super-Turing power,
• Extending the theory to stochastic or symplectic dynamical systems for greater affinity to quantum computation.

A prominent unresolved problem is to algorithmically minimize the topological complexity of bordisms implementing a given function and to determine whether physically realizable dynamical systems can embody partial recursive computation with greater efficiency than any Turing machine. TKFT initiates a new interface where computation is fundamentally encoded in the topology and dynamics of manifolds, suggesting the possibility that the geometry of a space may one day serve directly as a program (González-Prieto et al., 20 Mar 2025).