Automorphism-Invariant ℤ₂ Symbol Freedom in Graphs: The `rank_hol` Invariant, a Proof of Its Non-Reducibility, and an Application to the Emergence of Charge Sign

Every edge of a graph can be assigned a symbol +1 or −1. Walking once around the graph and multiplying the symbols of the edges encountered, one obtains a "traversal value" (holonomy). If two edge-symbol assignments differ only by the part arising from "flipping all edges incident to some vertex set", then their traversal values agree everywhere, so they may be regarded as the same "flat labelling". The set of all flat labellings forms the space $\mathrm H^1(G;\mathbb Z_2)$, whose dimension is the graph's cycle rank $\beta_1$. Now add a symmetry constraint: consider only those edge-symbol assignments that are invariant under the graph's automorphisms (taking a constant value on each edge orbit). Such assignments are determined solely by "assigning 0 or 1 to each edge orbit", giving $2^{n_{\rm orb}}$ of them, where $n_{\rm orb}$ is the number of edge orbits. Their image in $\mathrm H^1(G;\mathbb Z_2)$ is a subspace, and we define $$\operatorname{rank\_hol}(G) \;:=\; \dim_{\mathbb Z_2}\operatorname{im}\Phi_G \in \{0,1,2,\dots\},$$ where $\Phi_G$ is the above linear map. This quantity has three immediately verifiable properties: - (Label-independent) It depends only on the isomorphism class of the graph. Renumbering vertices or changing the storage order of edges does not change it. - (Zero minimality principle) The all-zero assignment always maps to 0, so 0 is always in the image; $\operatorname{rank\_hol}=0$ means the symbol is uniquely locked by the topology and symmetry of the graph. - (Exact formula) $\operatorname{rank\_hol}(G) = n_{\rm orb} - \dim(W\cap\mathrm{Cut})$, where $W$ is the space of automorphism-invariant assignments and $\mathrm{Cut}$ is the cut space. The main result of this paper is an irreducibility theorem: $\operatorname{rank\_hol}$ cannot be determined by any combination of any of the following classical quantities: cycle rank $\beta_1$, bipartiteness, number of edge orbits $n_{\rm orb}$, number of vertex orbits, order of the graph's automorphism group, cycle rank of the quotient graph, triangle count, degree sequence. In particular, we give three 12-vertex 3-regular graphs for which all of the above classical quantities agree item by item, while `rank_hol` takes the values $3,4,4$ respectively. We also give a smaller counterexample pair, on 7 vertices with a hand-checkable edge list. At the level of application, this paper points out that `rank_hol` physically measures how many external inputs are needed to pin down one bit of symbol freedom. Under this measure, the electron's skeleton is completely locked (`rank_hol = 0`), the proton's skeleton has exactly one bit not locked (`rank_hol = 1`), and the neutron has two bits (`rank_hol = 2`). This provides a purely graph-theoretic quantitative answer to "why can the charge signs of the electron and the proton be distinguished, while the absolute positive/negative polarity cannot be derived". 一个图的每条边可以被赋予一个符号 +1 或 −1。绕图走一圈,把沿途边的符号相乘,得到一个"绕行值"(holonomy)。两种边符号赋值若只相差"翻转某个顶点集的所有关联边"造成的部分,则它们的绕行值处处相同,于是可以视为同一个"平直标号"。所有平直标号构成的空间是 $\mathrm H^1(G;\mathbb Z_2)$,其维数是图的循环秩 $\beta_1$。 现在加上对称性约束:只考虑那些在图的自同构下保持不变的边符号赋值(每条边轨道上取常值)。这类赋值只由"每条边轨道取 0 或 1"决定,共 $2^{n_{\rm orb}}$ 个,$n_{\rm orb}$ 是边轨道数。它们在 $\mathrm H^1(G;\mathbb Z_2)$ 中的像是一个子空间,我们定义 $$\operatorname{rank\_hol}(G) \;:=\; \dim_{\mathbb Z_2}\operatorname{im}\Phi_G \in \{0,1,2,\dots\},$$ 其中 $\Phi_G$ 是上述线性映射。这个量有三个立刻可验证的性质: - (标号无关) 它只依赖图的同构类。给顶点重新编号、改变边的存储顺序,都不改变它。 - (零极小原理) 全 0 赋值恒映射到 0,所以 0 永远在像中;$\operatorname{rank\_hol}=0$ 意味着符号被图的拓扑与对称性唯一锁定。 - (精确公式) $\operatorname{rank\_hol}(G) = n_{\rm orb} - \dim(W\cap\mathrm{Cut})$,其中 $W$ 是自同构不变赋值子空间,$\mathrm{Cut}$ 是割空间。 本文的主要结果是一个不可还原性定理:$\operatorname{rank\_hol}$ 不能由任何下述经典量的任何组合决定:循环秩 $\beta_1$、二分性、边轨道数 $n_{\rm orb}$、顶点轨道数、图的自同构群的大小、商图循环秩、三角形数、度数序列。特别是,本文给出三张 12 顶点的 3-正则图,它们上述全部经典量逐一相同,而 `rank_hol` 分别取 $3,4,4$。本文同时给出一组 7 顶点、边表可手算的更小反例对。 在应用层面,本文指出 `rank_hol` 在物理上度量的是一个符号自由度需要多少个外部输入才能被钉死。按此度量,电子的骨架完全锁定(`rank_hol = 0`),质子的骨架恰有一比特未被锁定(`rank_hol = 1`),中子有两比特(`rank_hol = 2`)。这为"为什么电子与质子的电荷符号可以被区分,而绝对正负极性不能被推导出来"提供了一个纯图论的量化答案。