Hex-Realizable 3-Terminal Game Values

• Hex-realizable 3-terminal values are combinatorial game values arising from Hex regions defined by three black and three white terminals, characterized by an outcome poset.
• The framework decomposes every Hex position into these 3-terminal regions using operations like concatenation and juxtaposition to analyze connectivity and local game structure.
• Explicit constructions such as the superswitch and tripleswitch families establish an infinite, operationally-rich value space that aids solver-assisted verification and template analysis.

Hex-realizable 3-terminal values are combinatorial game values arising from 3-terminal regions in Hex: regions completely surrounded by black and white stones whose boundary contains three black terminal components and three white terminal components. In this setting, the local game inside the region is evaluated by which black terminals can ultimately be connected, yielding a value over an outcome poset rather than a simple win–loss dichotomy. Recent work places these values at the center of local Hex structure: Hex is characterized as the universal planar Shannon game of degree $3$, every Hex position can therefore be decomposed into 3-terminal pieces, and the space of realizable values is shown to be infinite (Demer et al., 11 Jul 2025). In a broader combinatorial-game-theoretic sense, these values sit within the theory of passable and monotone set coloring games, where passability is the abstract condition underlying realizability by Hex-like monotone coloring mechanisms (Demer et al., 2021).

1. Definition of 3-terminal regions and their outcome poset

A 3-terminal region is a Hex region whose boundary is completely surrounded by black and white stones and has exactly three black boundary components and three white boundary components; the black components are the black terminals, customarily labeled $1,2,3$ (Demer et al., 11 Jul 2025). The local outcome relevant for Black is the terminal-connectivity pattern achieved when the region is filled.

For a generic 3-terminal region, the black outcome poset is

$P_3=\{\bot,a,b,c,\top\},$

where $\bot$ denotes that none of the black terminals are connected, $a$ denotes that terminals $2$ and $3$ are connected but not $1$, $b$ denotes that terminals $1$ and $1,2,3$0 are connected but not $1,2,3$1, $1,2,3$2 denotes that terminals $1,2,3$3 and $1,2,3$4 are connected but not $1,2,3$5, and $1,2,3$6 denotes that all three black terminals are connected (Demer et al., 11 Jul 2025). The order relation is the natural connectivity order: $1,2,3$7 lies below $1,2,3$8, each of $1,2,3$9 lies below $P_3=\{\bot,a,b,c,\top\},$0, and $P_3=\{\bot,a,b,c,\top\},$1 are pairwise incomparable.

The same paper also identifies important quotient cases. In a corner, one has the additional relation $P_3=\{\bot,a,b,c,\top\},$2. In a fork, one has $P_3=\{\bot,a,b,c,\top\},$3 (Demer et al., 11 Jul 2025). These degenerations are important because many local Hex templates are naturally corners or forks rather than generic 3-terminal regions.

Within this formalism, a Hex-realizable 3-terminal value is a canonical combinatorial game value over $P_3=\{\bot,a,b,c,\top\},$4 that is realized by some actual 3-terminal Hex region (Demer et al., 11 Jul 2025). This shifts the object of study from isolated local patterns to an order-theoretic classification problem: which abstract short values over $P_3=\{\bot,a,b,c,\top\},$5 are geometrically attainable in Hex?

2. Universality of Hex and the centrality of degree-$P_3=\{\bot,a,b,c,\top\},$6 local structure

The structural reason 3-terminal values are fundamental is the theorem that Hex is the universal planar Shannon game of degree $P_3=\{\bot,a,b,c,\top\},$7: every 3-planar Shannon game is isomorphic to a Hex position, and every Hex position is isomorphic to a 3-planar Shannon game (Demer et al., 11 Jul 2025). Here a vertex Shannon game is played on a hypergraph with two distinguished terminals, Black winning by connecting them through black vertices, and the degree bound requires each vertex to lie in at most three hyperedges.

This universality theorem implies that every Hex position can be decomposed into 3-terminal regions (Demer et al., 11 Jul 2025). Conceptually, the theorem identifies degree $P_3=\{\bot,a,b,c,\top\},$8 as the local combinatorial complexity of Hex. A plausible implication is that 3-terminal values are not merely a convenient vocabulary for endgame fragments, but the natural local state space for arbitrary Hex positions.

The topological background clarifies why such local connectivity descriptions are robust. Hex and Y are equivalent non-draw phenomena on triangulations of a disk, and the three-terminal Y game can be doubled across a side to produce a four-sided Hex instance (Prytuła, 2021). In Y, a player wins by a monochromatic chain touching all three boundary sides; after doubling, this becomes a Hex connection across opposite sides. The paper’s formulation makes the obstruction topological: a no-winner coloring would induce a retraction $P_3=\{\bot,a,b,c,\top\},$9 of a disk onto its boundary circle, which is impossible, while simultaneous winners would force a planar embedding of $\bot$0, also impossible (Prytuła, 2021).

This topological equivalence does not classify 3-terminal values in the combinatorial-game-theoretic sense, but it shows that 3-terminal connection objectives are not ad hoc variants of Hex. They encode the same disk-and-boundary obstruction as standard Hex, only with a three-terminal boundary decomposition rather than a four-terminal one (Prytuła, 2021).

3. Combinatorial game semantics, monotonicity, and realizability

The formal value theory used for 3-terminal Hex positions is a short-game theory over a poset $\bot$1. A game is either atomic, written $\bot$2 for $\bot$3, or composite, written $\bot$4, with order relations and equivalence extended recursively; equivalence is defined by

$\bot$5

(Demer et al., 2021). In the 3-terminal setting, $\bot$6 is typically $\bot$7 or one of its quotients (Demer et al., 11 Jul 2025).

The general realizability framework comes from monotone set coloring games. Such a game over a poset $\bot$8 consists of a finite carrier $\bot$9 and a monotone payoff function

$a$0

where $a$1, positions are partial colorings with empty cells marked $a$2, and the associated combinatorial game is defined recursively from the resulting move tree (Demer et al., 2021). Hex is an example: the payoff is monotone because adding black stones or removing white stones cannot hurt Black, and in the $a$3 case the value is

$a$4

a first-player win (Demer et al., 2021).

The key abstract notion is passability. A left option $a$5 is good if $a$6, a right option $a$7 is good if $a$8; a game is locally monotone if all options are good, locally passable if it is atomic or has at least one good left or right option, monotone if every position is locally monotone, and passable if every position is locally passable (Demer et al., 2021). The paper states the “Fundamental theorem of passable games”: every passable game is equivalent to a monotone game (Demer et al., 2021).

Its main theorem is stronger for Hex-like realizability: all finite passable games are realizable as monotone set coloring games (Demer et al., 2021). Thus every finite passable value is equivalent to the value of some monotone board-coloring mechanism, and every monotone set coloring game is itself passable. For Hex-realizable 3-terminal values, this identifies passability as the exact abstract condition for realizability in the broad monotone-set-coloring sense, although not every such realization is a literal Hex board (Demer et al., 2021).

The 3-terminal Hex paper specializes this general perspective. It recalls that for short games over a poset with top and bottom, monotonicity is equivalent to passability up to canonical form, and defines a value over $a$9 to be Hex-realizable precisely when it is the canonical value of some Hex 3-terminal region (Demer et al., 11 Jul 2025). This leaves a geometric realizability question strictly sharper than abstract passability: Hex imposes planar, degree-$2$0, and locality constraints in addition to monotone-set-coloring realizability.

4. Infinite families and the superswitch construction

A central theorem is that there are infinitely many distinct Hex-realizable 3-terminal values (Demer et al., 11 Jul 2025). The principal witness family is the superswitch sequence

$2$1

These are passable values previously known to form a strictly increasing sequence, and the paper proves that every member of the family is realizable by an explicit 3-terminal Hex construction (Demer et al., 11 Jul 2025).

The realization argument uses a 3-terminal concatenation operation $2$2, defined by gluing terminal $2$3 of $2$4 to terminal $2$5 of $2$6 and terminal $2$7 of $2$8 to terminal $2$9 of $3$0, thereby producing a new 3-terminal position (Demer et al., 11 Jul 2025). On atoms, the operation satisfies

$3$1

The paper proves the recurrence

$3$2

and since $3$3 is explicitly realizable, induction yields realizability of all superswitches (Demer et al., 11 Jul 2025).

The paper also defines the dual superswitch family

$3$4

obtained by interchanging the Left and Right roles (Demer et al., 11 Jul 2025). These duals are used in value separation and in several template-verification arguments.

A further family is given by ideal tripleswitches,

$3$5

and their Hex realizations

$3$6

formed by a pinwheel composition of three superswitch realizations (Demer et al., 11 Jul 2025). The paper states that the Hex tripleswitches have the same cofinality as the ideal tripleswitches, which is significant because cofinal families serve as universal test contexts in later applications.

These constructions establish more than mere infinitude. They exhibit explicit, recursively generated, geometrically realized value families inside $3$7-valued CGT. This suggests that the 3-terminal value space is structurally rich even before one invokes arbitrary passable-game realizability.

5. Operations, database infrastructure, and decision procedures

The 3-terminal theory is organized around a small set of concrete operations and test contexts. Besides concatenation, the paper introduces a juxtaposition operation $3$8 combining two 3-terminal positions into a 2-terminal position (Demer et al., 11 Jul 2025). For superswitches it proves

$3$9

This proposition turns the superswitch hierarchy into a calibrated family of comparison gadgets (Demer et al., 11 Jul 2025).

The paper also presents a database of Hex-realizable 3-terminal values. It contains $1$0 primary positions stored in primaries.txt, and a larger file realizable-3terminal.txt containing over a million realizable values (Demer et al., 11 Jul 2025). Each entry records the number of empty cells, the canonical value, and a decomposition into primary positions. The database was generated by starting from a small seed set, closing under symmetries, taking duals, applying the pinwheel composition, and canonicalizing while discarding overly complicated values (Demer et al., 11 Jul 2025).

This database is not described as an exhaustive enumeration of all small regions; rather, it is a constructive library of realizable local values. The paper notes that, by the 3-planar universality theorem, every Hex position can in principle be decomposed using just the single-cell primaries $1$1 and $1$2, but the enlarged primary set makes the database smaller and more usable (Demer et al., 11 Jul 2025). A plausible implication is that realizability theory is already operational enough to support solver-assisted local reasoning rather than merely abstract classification.

The practical strength of this framework appears in several equivalence criteria. For connects-both templates, the paper proves that a 3-terminal position $1$3 has the connects-both property if and only if

$1$4

for all dual simpleswitches $1$5 (Demer et al., 11 Jul 2025). For pivoting forks, it proves

$1$6

and also the equivalent intrinsic condition

$1$7

(Demer et al., 11 Jul 2025). For sente pivoting, the paper gives

$1$8

(Demer et al., 11 Jul 2025). These are decision procedures phrased entirely in the value language: a geometric template property is reduced to inequality against a fixed realizable context.

6. Applications, limitations, and related directions

The local-value framework is used for several concrete applications. The database and its accompanying theory support automated verification of connects-both templates, automated verification of pivoting templates, construction of a new winning opening for Black with a 1-move handicap on $1$9 Hex, and a method for constructing witnesses for the non-inferiority of probes in templates (Demer et al., 11 Jul 2025). The same machinery is then used to disprove a conjecture of Henderson and Hayward by showing that none of the ziggurat probes are inferior (Demer et al., 11 Jul 2025).

These applications depend on the fact that local template properties can be encoded by 3-terminal values and tested by juxtaposition with suitable contexts. In that sense, Hex-realizable 3-terminal values function as a computational interface between local geometry and global solver verification.

At the same time, the scope of the theory has clear limits. The realizability theorem for passable games shows that every finite passable value is realizable as a monotone set coloring game, but it does not show that every such value is realizable specifically as a Hex board or even as a planar degree-$b$0 region (Demer et al., 2021). Likewise, the topological equivalence between Hex and Y identifies shared non-draw behavior, not a full equivalence of their local value theories (Prytuła, 2021). The geometric Hex-realizability problem therefore remains more restrictive than general monotone-game realizability.

A different but related limitation appears in infinite settings. In Infinite Hex, the empty position is a draw, and although finite values can be realized by bridge or virtual-link constructions, no infinite ordinal value is realizable in an open Infinite Hex position with defined value; the paper states

$b$1

(Leonessi, 2021). This result concerns ordinal open-game values rather than 3-terminal poset values, but it delineates a boundary: Hex-like play admits substantial local complexity, yet in the open Infinite Hex framework the realizable transfinite spectrum collapses to finite ordinals.

There is also an enumerative direction associated with “Hex-trees.” Hex-trees are identified with weighted unary-binary trees of weight $b$2, with generating function

$b$3

and the resulting counting sequence is A002212 (Prodinger, 2021). The paper states that Hex-trees are the combinatorial model for certain 3-terminal Hex values (Prodinger, 2021). This suggests a complementary perspective in which some subclasses of Hex-realizable 3-terminal values can be studied through tree encodings, Horton-Strahler strata, and Schröder-type bijections, although the cited work is enumerative rather than a full CGT classification.

Taken together, these developments identify 3-terminal values as the natural local language of Hex. Topologically, they are aligned with the Y–Hex disk obstruction; abstractly, they sit inside the class of passable monotone-game values; geometrically, they are constrained by planar degree-$b$4 realizability; and computationally, they form a sufficiently rich and explicit toolkit to support solver-based theorem checking and template analysis (Demer et al., 11 Jul 2025).