Minimal Embedding Dimension for Self-Intersection-Free Recurrent Processes

Information geometry provides a natural framework for studying inference and decision-making, endowing families of probability distributions with Riemannian structure via the Fisher information metric. A fundamental question arises when we consider dimensional collapse: what happens when a high-dimensional statistical manifold is projected onto a lower-dimensional subspace?

We focus on a specific aspect: the emergence of self-intersections between temporally distinct states. When a trajectory $\gamma(t)$ on a statistical manifold $\mathcal{M}$ is projected to a lower-dimensional space via a map $\pi_k: \mathcal{M} \to \mathbb{R}^k$, distinct times $t_1 \neq t_2$ may be mapped to the same point. Such self-intersections destroy temporal information.

Our main results establish that:

1. For cyclic processes with monotone meta-time, self-intersections are structurally unavoidable in $\mathbb{R}^2$
2. Self-intersection-free embeddings into $\mathbb{R}^3$ always exist
3. The dimension $k = 3$ is therefore a critical threshold

A recurrent process $\gamma$ is cyclic with monotone meta-time if:

1. There exists a "phase coordinate" projection $\theta: \mathcal{M} \to S^1$ such that $\theta \circ \gamma$ covers the circle $n \geq 2$ times
2. The process encodes a strictly monotone "meta-time" variable $\tau: [0,T] \to \mathbb{R}$ with $\tau'(t) > 0$

Examples:

• Biological oscillators: Neural limit cycles, circadian rhythms, cardiac pacemakers
• Recurrent neural networks: Hidden state trajectories processing sequential data
• Logical paradoxes: The Liar's paradox ("This statement is false") oscillates between TRUE and FALSE, with each cycle occurring at a distinct meta-time

Theorem (Minimal Embedding Dimension). Let $\gamma: [0, T] \to \mathcal{M}$ be a cyclic process with monotone meta-time. Then:

(i) For any continuous $\pi_2: \mathcal{M} \to \mathbb{R}^2$ preserving the cyclic structure, the self-intersection functional satisfies $E_{\cap}(\pi_2, \gamma) > 0$.

(ii) There exists a continuous $\pi_3: \mathcal{M} \to \mathbb{R}^3$ such that $E_{\cap}(\pi_3, \gamma) = 0$.

Proof of (ii). Define $\pi_3(\gamma(t)) = (\cos(2\pi \theta), \sin(2\pi \theta), \tau(t)/T)$. This maps the trajectory to a helix in $\mathbb{R}^3$. Since $\tau$ is strictly monotone, $t_1 \neq t_2$ implies $\tau(t_1) \neq \tau(t_2)$, so the third coordinates differ and the map is injective.

The self-intersection phenomenon has a natural interpretation in terms of metric degeneracy.

Proposition (Fisher Metric Singularity). Consider a statistical manifold parametrized by $(\theta, \tau) \in S^1 \times \mathbb{R}$. Assume the Fisher information matrix $G(\theta, \tau)$ has full rank 2.

Under a projection $\pi_2$ that discards the $\tau$ coordinate, the induced metric has rank at most 1. The smallest eigenvalue of the induced metric tensor vanishes.

This rank drop corresponds to non-identifiability: distinct values of $\tau$ produce identical sufficient statistics after projection. Self-intersections are thus equivalent to singularities in the accessible information geometry.

Our results formalize a fundamental dichotomy in information processing:

• Categorical regime ($k \leq 2$): Self-intersections are structurally unavoidable for cyclic processes with meta-time, forcing the system to treat temporally distinct states as equivalent. This naturally encourages discrete categories and symbolic representations.

• Continuous regime ($k \geq 3$): Self-intersection-free embeddings exist, allowing the system to maintain temporal distinctness and represent processes rather than just states.

This dichotomy suggests that the emergence of discrete symbols from continuous dynamics may be geometrically inevitable under dimensional constraints.

We have established that $k = 3$ is the minimal embedding dimension for self-intersection-free representation of cyclic processes with monotone meta-time on statistical manifolds.

This result identifies a critical threshold in information geometry: below three dimensions, temporal information is necessarily lost when the cyclic structure is preserved, forcing categorical representations; at three dimensions and above, continuous processes can be faithfully represented.

The proofs are elementary, relying on the topology of the circle and the definition of injectivity, combined with the classical Whitney embedding principle. Yet the result has implications for understanding when and why discrete structures emerge from continuous substrates.