A standard noosphere is a construction in computational memetics.

Let $X$ be a regular cell complex with raw data $\mathbf{X}$, where $\mathbf{x}_{\sigma}$ is a raw data vector at cell $\sigma \in X$ and $\mathbf{X}$ is simply all of data at all cells. $X$ represents a social network. Let $E_\phi^\Sigma : \mathcal{X} \to \mathcal{Z}$ be an encoder (for example an autoencoder) from some dataset $\mathcal{X}$ broken into tokens, that is a formal language alphabet $\Sigma = \{x_i\}_{i = 1,\cdots,N}$. Let $L$ be some hypothetical true global syntactic structure of the dataset with respect to gradation into the alphabet $\Sigma$, and let $\hat{L}$ be the inferred syntactic structure by by choice of weights $\phi$ in encoder $E_\phi^\Sigma$. $\hat{L}$ is the token-coupling structure across each unit vector space that preserves the encoder-decoder coherence (with decoder $D^\Sigma_{\phi'} : \mathcal{Z} \to \mathcal{X}$) up to some statistical divergence (loss) across all pairs.

For each cell $\sigma \in X$ encode a stalk $\mathcal{F}(\sigma) = E_\phi^\Sigma(\sigma)$ and construct a weighted cellular sheaf $\mathcal{F}(X)$ with restriction maps in the orthogonal group $\mathrm{O}(n)$. The sheaf becomes a discrete bundle in $\mathrm{O}(n)$. This discrete bundle is a tangent bundle on a manifold, and the sheaf Laplacian of the $\mathrm{O}(n)$-bundle is equivalent to connection Laplacian, that is the orthogonal restriction maps describe how vectors are rotated when transported between stalks, in a way analogous to the transportation tangent vectors on a manifold. By the manifold assumption, the encoded data across sheaf is sampled from some underlying manifold $\mathcal{M}$, geometry of which evolves through time, so at time $t = \tau$ we write $\mathcal{M}_\tau$. That manifold with fiber-structure across all of base space of consideration is a section of the global noosphere or simply, for brevity, a noosphere sectioned in semantic alphabet $\Sigma$ through weights $\phi$ over base space $X$, which we write $\mathcal{M}^{E^\Sigma_\phi(\mathbf{X})}({X})$.

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