AI 中文总结
该研究识别出三元Γ-乘积编码的交替五元核心，完成其范畴重构，建立二元坍缩与原子相关性质的等价关系，给出有限核心的相关算法，并构造无二元坍缩的指标敏感示例。
AI 中文摘要
我们识别出由三元Γ-乘积编码的双类原子运算——交替五元核心，其幂集复形是完全加性有序三元Γ-半环。反之，每个加性归约为完全原子布尔代数的原子全结构，典范地为这类复形，从而得到范畴重构。对对角核心，二元可归约性等价于保原子的并分配结合二元坍缩。有限核心具有精确的见证反链代数，以及多项式等价与不等式有效性的终止过程。一个四点不可约对称五元带给出16个元素的指标敏感示例，无二元坍缩，其中一个不等式恢复原子上非平凡的奇偶商。
英文摘要
We study congruences and binary reducibility in completely additive ternary $Γ$-semirings through their two-sorted atomic cores. For an odd $m\geq3$, an abelian group $G$ of exponent dividing $m-1$, and $u,v\in G$, we construct a two-branch symmetric $m$-ary band $F_m(G;u,v)$. We prove \[ \Con(F_m(G;u,v))\cong\operatorname{Sub}(G)\times B_2 \] and classify every carrier--index congruence pair: $(θ,ϕ)$ is typed exactly when $θ$ is an ordinary congruence and $ϕ$ refines $θ$. We also give a quotient-by-quotient reducibility criterion. In the finite case, $F_m(G;u,v)$ is quotient-critically irreducible exactly when $G$ is a cyclic $p$-group and $u-v$ has order $p$. The specialization $G=\mathbb Z_2$ yields a four-point family $H_m$ extending the irreducible ternary example of Devillet and Mathonet. Its congruence lattice is $B_3$, its alternating core has exactly $36$ typed congruence pairs, and its automorphism group is $C_2$. Its seven proper quotients are reducible and form five isomorphism types; in arity five their exact reduction counts are determined. The powerset lift of $H_5$ is a $16$-element atom-total complete atomic Boolean ternary $Γ$-semiring with no atomic binary collapse, whereas each proper diagonal strong atom-saturated quotient has one. Its parity congruence is recovered on atoms by one explicitly specified depth-one polynomial inequation.
