Sidling Sequences in Combinatorial Games

• Sidling sequences are canonical constructs in combinatorial game topology that form convergent Cauchy sequences of short games approximating loopy games.
• They are constructed through diminishing-weight edit distances which establish a rigorous metric closure and classification of canonical and loopy game forms.
• Their analytic and algorithmic applications include similarity search in CGT databases and automated game-solving, enabling precise optimization based on minimal perturbations.

Sidling sequences are a canonical construct in the metric topology of combinatorial games. Arising from the theory of diminishing-weight edit distances on canonical short games, they formalize the process of approaching loopy games via chains of well-ordered finite approximants. Sidling sequences underpin the analytic closure of the canonical game space, provide Cauchy foundations respecting the $wd$-metric, and enable rigorous classification of loopy and infinite games according to their metric proximity to short, canonical forms.

1. The Diminishing-Weight Metric and Canonical Game Space

Consider the space $\mathcal{C}$ of short two-player combinatorial games in canonical form—excluding dominated or reversible options and repeated positions—each with a directed acyclic game graph $D(G)$. The diminishing-weight distance $wd$ is determined via edit sequences on $D(G)$: for any $G, H \in \mathcal{C}$,

$$wd(G,H) = \min_{\substack{\text{edit sequences}\ D(G)\rightarrow D(H)}} \sum_{e\in\text{sequence}} \left(\frac{1}{2}\right)^{d(e)}$$

where $d(e)$ is the graph-theoretic distance from the source of $D(G)$ to the tail of edge $e$. Each edit either adds or removes an edge (color-preserving by player), possibly dropping isolated vertices. The metric structure $\mathcal{C}$0 is proven in (Burke et al., 15 Jan 2026).

2. Formal Definition of Sidling Sequences

Fix a loopy combinatorial game $\mathcal{C}$1 (i.e., one with cycles or an infinite DAG). A sidling sequence from above is a (possibly infinite) sequence of short games $\mathcal{C}$2 satisfying

• $\mathcal{C}$3, and
• $\mathcal{C}$4 is the simplest canonical game with $\mathcal{C}$5.

Dually, sidling from below yields $\mathcal{C}$6 with

• $\mathcal{C}$7
• $\mathcal{C}$8

Typically, the initial term is a trivial bound (such as a number or ) and each step takes the simplest dyadic rational or fuzzy game strictly between the predecessor and $\mathcal{C}$9. Both $D(G)$0 and $D(G)$1 converge to $D(G)$2 under the $D(G)$3-metric and are termed *canonical sidling sequences (Burke et al., 15 Jan 2026).

3. Explicit Construction: The “Over” Example

Let $D(G)$4 denote a loopy infinitesimal game. Siding constructions are as follows:

• From Above:
  • $D(G)$5 (since $D(G)$6).
  • Recurrence: $D(G)$7, giving the sequence $D(G)$8.
• From Below:
  • $D(G)$9.
  • Recurrence: $wd$0, yielding $wd$1.

Both sequences strictly bound $wd$2 and approach it in $wd$3 (Burke et al., 15 Jan 2026). Their construction generalizes: for any loopy $wd$4, one generates $wd$5 and $wd$6 as Cauchy sequences, converging in the metric topology to $wd$7.

4. Cauchy Sequence Properties and Metric Closure

The convergence of sidling sequences is explicit. For $wd$8, embedding $wd$9 inside $D(G)$0 allows computation:

$D(G)$1

As $D(G)$2, $D(G)$3, confirming Cauchyness. A parallel calculation applies to $D(G)$4. The mutual convergence $D(G)$5 ensures both sequences share the same limit in the metric completion (Burke et al., 15 Jan 2026).

The closure $D(G)$6, defined as the set of all such Cauchy limits, includes loopy/infinite games reachable at finite $D(G)$7-distance. For example, $D(G)$8. Other limit points include “on,” “upon,” and further loopy stoppers as exhibited in (Burke et al., 15 Jan 2026). Notably, certain loopy configurations, such as Bach’s Carousel and the games "tis", "tisn," are inaccessible via sidling and hence lie outside $D(G)$9.

5. Proof Sketches for Sidling Convergence and Discreteness

Direct estimation yields:

• For $G, H \in \mathcal{C}$0, $G, H \in \mathcal{C}$1.
• For any $G, H \in \mathcal{C}$2, distinct short games satisfy $G, H \in \mathcal{C}$3, where $G, H \in \mathcal{C}$4 is the larger birthday, establishing metric discreteness of canonical games.

Thus, metric closure via sidling sequences strictly extends $G, H \in \mathcal{C}$5 by countably many loopy or infinite DAG games, precisely those approachable in $G, H \in \mathcal{C}$6 from canonical short games (Burke et al., 15 Jan 2026).

6. Topological and Algorithmic Significance

Sidling sequences formalize the analytic process of “squeezing” dyadic or infinitesimal canonical forms toward a loopy boundary in $G, H \in \mathcal{C}$7. This enables:

• Metrizable topology on combinatorial games, unifying finite and infinite forms.
• Rigorous topological classification of loopy games (stoppers, non-stoppers, and their sides).
• Quantitative metric-based similarity search in large-scale combinatorial game theory (CGT) databases.
• Application frameworks for automated game-solving and heuristic optimization, where small $G, H \in \mathcal{C}$8-perturbations provide systematic tuning.

Sidling sequences thus provide both a concrete construction tool and the analytic infrastructure for the real-valued topology in combinatorial games (Burke et al., 15 Jan 2026).