AI 中文总结
研究对无三角形的Exoo-Ismailescu单位距离图EI17和EI19坐标场进行精确认证，通过分析其自由角多项式的伽罗瓦群，发现EI19可折纸构造，EI17不可解不能用根式表示，呈现可解 - 不可解二分法，还给出认证流程并关联拉曼数与伽罗瓦群。
AI 中文摘要
我们精确认证了两个相邻的无三角形Exoo-Ismailescu单位距离图（UDGs）的忠实平面实现的坐标场，并表明它们实现了可构造性层次结构的两个相反极端。17个顶点的图EI17（图之家51375）是最小的无三角形且色数为4的UDG；19个顶点的图EI19（HoG 51376）是其最先进的折纸邻图。在这两个图中，固定一条有理基边，其余顶点是单位圆的交点——每个顶点都在其两个邻点的根轴上，是自由角上的平方根塔，并且一个小的闭包系统锁定了实现。对于EI19，基位于Q(sqrt 2, sqrt 5, sqrt 7)中，一个自由角有一个不可约次数为12 = 2^2 * 3的最小多项式，其伽罗瓦群是可解传递群12T236（阶为2304 = 2^8 * 3^2）：不是尺规可构造的，而是折纸可构造的（在不可约情况下需要三次Beloch折叠O6）。对于EI17，两个自由角被两个闭包锁定，其结式是次数为20 = 2^2 * 5的不可约多项式，伽罗瓦群是全对称群S20（一个弗罗贝尼乌斯普查显示一个17 - 循环，通过乔丹定理迫使为A20，以及一个奇数20 - 循环，将其提升到S20）：不可解，所以坐标不能用根式表示——无论是任何折叠阶数的圆规还是折纸都不行。因此，最小的无三角形4 - 色UDG是一般的、最大程度奇异的情况，与其折纸邻图完全相反。我们给出了作为显式算法的完整认证流程，记录了两个方法上的陷阱，并通过拉曼数和伽罗瓦群之间的推测性桥梁来解读这一对图。
英文摘要
We certify, exactly, the coordinate fields of a faithful planar realization of two neighbouring triangle-free Exoo-Ismailescu unit-distance graphs (UDGs), and show they realize the two opposite extremes of the constructibility hierarchy. The 17-vertex graph EI17 (House of Graphs 51375) is the smallest triangle-free UDG with chromatic number 4; the 19-vertex graph EI19 (HoG 51376) is its state-of-the-art origami neighbour. In both, fixing a rational base edge, the remaining vertices are intersections of unit circles -- each on the radical axis of its two neighbours, a tower of square roots over the free angles -- and a small closure system locks the realization. For EI19 the base lies in Q(sqrt 2, sqrt 5, sqrt 7) and a single free angle has an irreducible degree 12 = 2^2*3 minimal polynomial with Galois group the solvable transitive group 12T236 (order 2304 = 2^8*3^2): not ruler-and-compass, but origami-constructible (the cubic Beloch fold O6 necessary, in casus irreducibilis). For EI17 two free angles are locked by two closures, whose resultant is irreducible of degree 20 = 2^2*5 with Galois group the full symmetric group S20 (a Frobenius census exhibits a 17-cycle, forcing A20 by Jordan, and an odd 20-cycle, raising it to S20): non-solvable, so the coordinates are not expressible by radicals -- neither compass nor origami of any fold order. Thus the smallest triangle-free 4-chromatic UDG is the generic, maximally exotic case, the exact opposite of its origami neighbour. We give the full certification pipeline as explicit algorithms, record two methodological pitfalls, and read the pair through a conjectural bridge between the Laman number and the Galois group.
Comments33 pages, 24 figures. v2: all figure labels translated to English; minor corrections
