AI 中文总结
研究哪些离散对称群可从策略互动产生，通过用双矩阵博弈支撑复形副本平铺平面，证明十七种壁纸群可作用其上，还阐述了相关计算路径及对称性结果。
AI 中文摘要
哪些离散对称群可从策略互动中产生？我们用双矩阵博弈的支撑复形的副本平铺平面，并通过受控边界规则连接，证明所有十七种壁纸群作用于所得覆盖：明确的生成元，每个都是机器验证的图自同构，每种实现都被认证为精确的环面商，其类型由精确有理算术的晶体学识别器确定并在GAP中交叉验证。一个三行引理将经典的对称/非对称区分转化为格分类：平移包含完整瓷砖格的实现恰好存在于十三种对称群中，四种非对称群在平移格指数恰好为二时实现，这是最小可能值：瓷砖是滑移的半步。构造过程伴随两条计算路径。在图路径上，用其平移对直覆盖进行商运算可精确恢复瓷砖，$\beq(M/\calT)=\beq(K)$，交换边界恰好增加$\binom m2$，与收益和覆盖大小无关。在博弈路径上，检测重复策略覆盖是线性时间收益扫描，一个瓷砖解折叠为覆盖均衡的完整平移轨道，平铺相关均衡系统的维度恰好为$r(d - q)+q$，且不可能扩展。多矩阵覆盖直接承载对称性：每个壁纸作用，包括滑移，都是真正的博弈自同构群，均衡沿任何对称子群坍缩为折叠的不动点问题，装饰细化的博弈自同构群恰好是环面壁纸群。
英文摘要
Which discrete symmetry groups can arise from strategic interaction? We tile the plane with copies of a bimatrix game's support complex, joined by controlled boundary rules, and show that all seventeen wallpaper groups act on the resulting covers: explicit generators, each a machine-verified graph automorphism, every realization certified as the exact toroidal quotient, with types identified by a crystallographic recognizer in exact rational arithmetic and cross-validated in GAP. A three-line lemma turns the classical symmorphic/non-symmorphic distinction into a lattice classification: realizations whose translations contain the full tile lattice exist precisely for the thirteen symmorphic groups, and the four non-symmorphic groups are realized at translation-lattice index exactly two, the minimum possible: the tile is the glide's half-step. Two computational tracks accompany the construction. On the graph track, quotienting a straight cover by its translations recovers the tile exactly, $\beq(M/\calT)=\beq(K)$, and swap boundaries add exactly $\binom m2$, independent of payoffs and of cover size. On the game track, detecting a duplicated-strategy cover is a linear-time payoff scan, one tile solution folds to a full translation orbit of cover equilibria, and the tiled correlated-equilibrium system has dimension exactly $r(d-q)+q$, with expansion impossible. The polymatrix cover then carries the symmetry outright: every wallpaper action, glides included, is a group of genuine game automorphisms, equilibria collapse along any symmetry subgroup to a folded fixed-point problem, and a decorated refinement has game automorphism group exactly the toroidal wallpaper group.
Comments27 pages
