AI 中文总结
该研究针对有限连通CW复形自映射，引入可见Nielsen数与观测者复杂度，推导环面自同态等的观测者复杂度公式，构造了观测者复杂度序列指数差异的两类同拓扑性质系统。
AI 中文摘要
对于有限连通CW复形的自映射(f\colon X\to X)，我们通过将Reidemeister迹推至与(f)兼容的有限正则覆盖，引入可见Nielsen数。对这类覆盖进行优化可得到可见性轮廓及三个消解阈值，其中包括观测者复杂度(\operatorname{oc}(f))，即分离所有本质Nielsen不动点类的最小覆盖次数。我们将有限观测者不可区分性与 profinite 完备化中的扭曲共轭等同，并确定相关的盲子群。我们还实现了对每个有限观测者均消失的非零Reidemeister迹，证明即使基本群与同调上的诱导映射、Lefschetz数及Nielsen数均固定，观测者复杂度在固定有限复形上仍无界。对于环面自同态，我们将完整可见性轮廓计算为精确的除数阶梯。对于主环面丛的兼容映射，在有限Reidemeister regime中，我们确定唯一的最大消解核，并将(\operatorname{oc}(F))用Nielsen数及丛特征类模纤维映射的根表示。这为积分Heisenberg幂零流形的标量伸缩提供了精确公式，同时得到观测者熵及有理观测者复杂度zeta函数。最后，我们构造了具有相同Nielsen序列、Lefschetz序列、Nielsen zeta函数、微分特征值及拓扑熵，但观测者复杂度序列呈指数差异的环面与Heisenberg系统。
英文摘要
For a self-map (f\colon X\to X) of a finite connected CW complex, we introduce visible Nielsen numbers by pushing the Reidemeister trace to finite regular covers compatible with (f). Optimizing over such covers gives a visibility profile and three resolution thresholds, including the observer complexity (\operatorname{oc}(f)), the least cover degree that separates all essential Nielsen fixed-point classes. We identify finite-observer indistinguishability with twisted conjugacy in the profinite completion and determine the associated blind subgroup. We also realize nonzero Reidemeister traces that vanish for every finite observer, and show that observer complexity is unbounded on a fixed finite complex even when the induced maps on the fundamental group and homology, the Lefschetz number, and the Nielsen number are fixed. For toral endomorphisms we compute the complete visibility profile as an exact divisor staircase. For compatible maps of principal torus bundles, in the finite-Reidemeister regime, we identify the unique maximal resolving kernel and express (\operatorname{oc}(F)) in terms of the Nielsen number and the radical of the bundle characteristic class modulo the fibre map. This yields exact formulas for scalar dilations of integral Heisenberg nilmanifolds, together with observer entropy and a rational observer-complexity zeta function. Finally, we construct toral and Heisenberg systems with identical Nielsen and Lefschetz sequences, Nielsen zeta functions, differential eigenvalues, and topological entropy, but exponentially different observer-complexity sequences.
