The Assembly Theorem

In this article, The Assembly Theorem is presented, a fundamental identity that related the Generalized Harmonic Number of a Partition $\mathbf{\lambda}$ and a $k$-tuple, partition-based $H_n^{\mathbf{\lambda}}$ and tuple-based $H_n^{\lambda}$ respectively, with the natural number $n$ in a very deep, essential and fundamental way. This result reveals that every $n \in \mathbb{N}$ is assembled either by partition-based or by tuple-based harmonic numbers in an infinite variety of ways, each uniquely determined by a $m \in \mathbb{N}$.The original partition-based version stated in Theorem 1 establishes that for all $n, m \in \mathbb{N}$, the following identity holds:\[\mathfrak{S}_n^{m} := \sum_{k=1}^n \sum_{\mathbf{\lambda} \in \wp_k^{k+m-1}} H_n^{\mathbf{\lambda}} = n\]where $\mathbf{\lambda} = (\lambda_1, \dots, \lambda_k) \vdash_k k+(m-1)$ be a partition of $k+(m-1)$ with length $k$, $\wp_k^{k+(m-1)}$ be the Assembly Set (the collection) of all distinct partitions $\mathbf{\lambda}$ of $k+(m-1)$ into exactly $k$ parts and $H_n^{\mathbf{\lambda}}$ is the $n$-th generalized harmonic number of order $\lambda_1+ \dots+ \lambda_k$ associated with the partition $\mathbf{\lambda} \vdash_k k+(m-1)$. The equivalent tuple-based version, using the $n$-th generalized harmonic number of order $\lambda_1 + \lambda_2 + \cdots + \lambda_k$ associated with the specific $k$-tuple $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ is proposed in Corollary 1 is given by the following identity:\[\mathfrak{S}_n^{m} = \sum_{k=1}^n \sum_{\mathbf{\lambda} \in \wp_k^{k+m-1}} \sum_{\Theta \in \Psi(\mathbf{\lambda})} H_n^{\Theta} = n.\]