发表机构
University of Ostrava; Charles University; VSB–Technical University of Ostrava（俄斯特拉发大学; 查理大学; 俄斯特拉发理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对形态演算细化等问题，构建脚本链复形的有限语义，推导映射环面单值群缺陷，应用于Sommen双向量偏差计算，得到四维、五维相关轨道与同调结果。
AI 中文摘要
形态演算通过实直线符号的类多项式表达式表示几何对象的分解，Sommen的例子揭示了两个基本难点：单个空间可能容许多个此类表达式，且标量加法与乘法会抑制重建空间所需的关联与附着映射。我们基于脚本链复形构建有限语义，其中单元计数多项式满足\\( \mathcal M_S(t)=\mathcal P_{S,\Bbbk}(t)+(1+t)\mathcal B_{S,\Bbbk}(t) \\)，\\( \mathcal P \\)为庞加莱多项式，\\( \mathcal B \\)记录边界秩。在整数环\\( \mathbb Z \\)上，史密斯标记区分单位对（此处处理的显式细分中细化开销的模型）与带挠率的非单位对。我们将笛卡尔积与丛分离，推导映射环面的精确单值群缺陷，并用脚本复形图的有限棒构造替代标量粘合。映射锥、连接与双映射柱是该图语义的简化模型。我们将此构造应用于Sommen计算中未解决的双向量偏差，采用单位球正则化后的有限CW模型。四维情形下，霍奇分解将单位双向量球等同于\\( S^2*S^2 \\)；Sommen计算中的角超额是插入秩二中点产生的可缩相对复形。五维情形下，\\( SO(5) \\)在\\( S(\Lambda^2\mathbb R^5)=S^9 \\)上的作用有主轨道\\( SO(5)/T^2 \\)与奇异轨道\\( \widetilde G_2(\mathbb R^5) \\)和\\( \mathbb{CP}^3 \\)；对于标准布吕阿胞腔结构，简化双映射柱清单与球同调相差7个单位标记对。
英文摘要
Morphological calculus represents decompositions of geometric objects by polynomial-like expressions in a symbol for the real line, but scalar operations suppress the incidence and attachment maps needed to reconstruct the space. We formulate a finite semantics using script chain complexes. The cell-count polynomial satisfies $$ \mathcal M_S(t)=\mathcal P_{S,\mathbb K}(t)+(1+t)\mathcal B_{S,\mathbb K}(t), $$ where $\mathcal P$ is the Poincar'e polynomial and $\mathcal B$ records boundary ranks. Over $\mathbb Z$, Smith labels separate unit pairs, representing refinement overhead, from non-unit pairs carrying torsion. We distinguish Cartesian products from bundles, derive the monodromy defect for mapping tori, and replace scalar gluing by a finite bar construction. Mapping cones, joins, and double mapping cylinders arise as reduced models. We apply the construction to bivector discrepancies in Sommen's calculations. In dimension four, Hodge decomposition identifies the unit bivector sphere with $S^2*S^2$; the angular excess is a contractible relative complex from inserting the rank-two midpoint. In dimension five, the $SO(5)$-action on $S(Λ^2\mathbb R^5)=S^9$ has principal orbit $SO(5)/T^2$ and singular orbits $\widetilde G_2(\mathbb R^5)$ and $\mathbb{CP}^3$. For standard Bruhat cell structures, the reduced double-mapping-cylinder inventory differs from sphere homology by seven unit-labelled pairs. We also treat Borel--Moore realizations of selected noncompact symbols, define division as an action-certified partial operation, and describe the orbit-type face diagram for unit bivectors in dimension six. There the correction polynomial has nonnegative coefficients, while compatibility of the face attachments remains open. These examples separate subdivision and attachment data from quotient actions and support conditions suppressed by scalar notation.
