$$ghost in the guile shell$$

$$\mathcal{T}: \mathbb{N} \rightarrow \mathbb{N} \cong{a_n}_{n=0}^{\infty} ={0,1,1,2,4,9,20,48,115,286,719,...}$$

$$\exists! \mathcal{A}(x) \in \mathbb{C}[[x]] \ni \mathcal{A}(x) = x \cdot \exp\left(\sum_{k=1}^{\infty}\frac{\mathcal{A}(x^k)}{k}\right)$$

$$\forall n \in \mathbb{N}^{+}, a_{n+1} = \frac{1}{n}\sum_{k=1}^{n}\left(\sum_{d|k}d \cdot a_d\right)a_{n-k+1}$$

$$a_n \sim \mathcal{C} \cdot \alpha^n \cdot n^{-3/2} \text{where } \alpha = \lim_{n\rightarrow\infty}\frac{a_{n+1}}{a_n} \approx 2.9557652857...$$

$$\mathcal{A}(x) = \sum_{n=0}^{\infty}a_n x^n = \sum_{\tau \in \mathfrak{T}_{\bullet}}\prod_{v \in V(\tau)}x^{|\text{desc}(v)|} = \prod_{k=1}^{\infty}(1-x^k)^{-\frac{1}{k}\sum_{d|k}\mu(\frac{k}{d})a_d}$$

$$\exists \mathcal{L}: \mathfrak{T}_{\bullet,n} \xrightarrow{\sim}{f: [n] \rightarrow [n] \mid \exists! i \in [n], f(i)=i \land G_f \text{connected}}$$

$$(\mathcal{F} \circ \mathcal{L}^{-1})(\mathfrak{T}_{\bullet,n}) \cong \mathcal{P}(n)^{\mathfrak{S}_n} \cong \mathcal{P}_n$$

$$\mathfrak{F}_{\mathbf{A000081}}^{\Omega}: \mathcal{D}_{n}^{\kappa} \hookrightarrow \prod_{\alpha \in \Lambda}\bigotimes_{\beta \in \Gamma_{\alpha}}\bigoplus_{\gamma \in \Theta_{\beta}}\bigwedge_{\delta \in \Xi_{\gamma}}\mathbb{T}^{\nabla}_{\bullet}(n)$$

$$\mathscr{B}\text{-}\mathfrak{Series}: \Phi_{h}^{\mathcal{RK}} = \sum_{τ \in \mathfrak{T}_{\bullet}}\frac{h^{|τ|}}{σ(τ)}F(τ)(y)·\mathcal{B}(τ) \Rightarrow \mathcal{ORD}_{\mathfrak{RK}}^{(p)} \cong \bigoplus_{τ \in \mathfrak{T}_{\bullet}: |τ| \leq p}\mathcal{H}_{τ}^{\nabla}$$

$$\mathscr{J}\text{-}\mathfrak{Surfaces}: \mathcal{E}_{\nabla}^{\partial^{\omega}} = \sum_{k=0}^{\infty}\frac{h^k}{k!}\sum_{τ \in \mathfrak{T}_{\bullet}(k)}\mathcal{F}_{τ}(y)\cdot\mathcal{D}^{\tau}f \Rightarrow \mathcal{ODE}_{\Delta}^{(m)} \simeq \bigsqcup_{τ \in \mathfrak{T}_{\bullet}(\leq m)}\mathcal{D}_{τ}^{\partial^{\alpha}}$$

$$\mathscr{P}\text{-}\mathfrak{Systems}: \mathcal{M}^{\mu}_{\Pi} = (\mathcal{V}, \mathcal{H}_{\tau}, \omega_{\tau}, \mathcal{R}_{\tau}^{\partial}) \Rightarrow \mathfrak{Evol}_{\Pi}^{(t)} \cong \coprod_{τ \in \mathfrak{T}_{\bullet}}\mathfrak{H}_{μ}^{\tau}(t) \circledast \bigotimes_{i=1}^{|τ|}\mathfrak{R}_{\tau(i)}^{\partial}$$

$$\mathfrak{Incidence}_{\mathbb{P}/\mathbb{A}}: \mathcal{I}_{\Xi}^{\kappa} \simeq \mathfrak{B}(\mathfrak{P}(\mathcal{T}_{\bullet}^{n})) \circlearrowright \bigwedge_{i=1}^{m}\mathfrak{H}^{\partial}_{\Xi}(i) \Rightarrow \mathcal{D}_{\mathbb{P}/\mathbb{A}}^{n,k} \cong \bigoplus_{τ \in \mathfrak{T}_{\bullet}(n)}\mathcal{I}_{\tau}^{\kappa}$$

$$\mathfrak{BlockCodes}: \mathcal{C}_{\Delta}^{(n,k,d)} \simeq \bigsqcup_{τ \in \mathfrak{T}_{\bullet}(w)}\mathfrak{G}_{τ}^{\partial}(\Sigma^{n}) \Rightarrow \mathfrak{Conf}_{\mathcal{C}}^{\Xi} \cong \prod_{i=1}^{l}\coprod_{τ \in \mathfrak{T}_{\bullet}(w_{i})}\mathcal{W}_{τ}^{\nabla}(i)$$

$$\mathfrak{Orbifolds}: \mathcal{O}_{\Gamma}^{\Xi} = (X/\Gamma,{\mathfrak{m}_{x}}_{x \in \Sigma}) \Rightarrow \mathcal{S}_{\mathcal{O}}^{\Gamma} \simeq \bigoplus_{τ \in \mathfrak{T}_{\bullet}(\leq d)}\mathcal{F}_{τ}^{\Xi}(\mathfrak{m})$$

$$\mathfrak{HyperNN}: \mathcal{H}_{\mathfrak{N}}^{\Delta} = (\mathcal{V}, \mathcal{E}_{\omega}, \mathcal{W}_{\tau}^{\Xi}) \Rightarrow \mathcal{F}_{\mathfrak{HNN}}^{\nabla} \cong \bigotimes_{l=1}^{L}\bigoplus_{τ \in \mathfrak{T}_{\bullet}(d_{l})}\mathcal{T}_{τ}^{\partial}(W_{l}) \circledast \sigma_{l}$$

$$\mathfrak{Meta}\text{-}\mathfrak{Pattern}: \mathcal{U}_{\mathbf{A000081}}^{\Omega} \simeq \mathfrak{Yoneda}(\mathfrak{F}_{\mathbf{A000081}}^{\Omega}) \hookrightarrow \mathbf{Colim}_{n \to \infty}\left(\bigwedge_{\mathscr{C} \in \mathfrak{Categories}}\mathfrak{T}_{\bullet}(n) \otimes \mathscr{C}\text{-}\mathfrak{Struct}\right)$$

$$\exists\mathfrak{F}: \mathbf{Cat}^{\mathbf{op}} \to \mathbf{Topos} \ni \mathfrak{F}(\mathscr{C}) = \mathbf{Sh}(\mathscr{C}, \mathcal{J}) \simeq \mathbf{Hom}_{\mathbf{Cat}}(\mathscr{C}^{\mathbf{op}}, \mathbf{Set}) \Rightarrow \mathfrak{F}(\mathfrak{T}_{\bullet}) \simeq \mathbf{Foundational}\text{-}\mathbf{Irreducibles}$$