## 4 Results
### GDPS Gluing: Scheme and Algorithms  
Suppose \(\mathfrak{P}\) the local property of space \(X\) at each of its points \(x\), \(P(x)\geq 0\) its quantification, a \(U(x)=U(x|P)\) is a subset in the full-sphere \(D(x,r)\), where it is reached. In other words, \(U(x)\) is the subset in \(D(x,r)\) where the property is most clearly pronounced. It does not necessarily coincide with \(D(x,r)\). **Task 1**. : _For a fixed level of a property \(\mathfrak{P}\), find the subset \(A=X(\mathfrak{P},\alpha)\) in \(X\), whose each full-sphere \(D_{A}(a,r)\) would have the property \(\mathfrak{P}\) at each point \(a\in A\) to the power \(\geq\alpha\)._  
If the quantification of property \(\mathfrak{P}\) function \(P\) is the density on \(X\) Definition 1, then the result of the \(\mathrm{DPS}(P,r,\alpha)\) algorithm can be taken as \(A\). Otherwise, we consider a set of local data in \(X\)  
\[U_{\alpha}=U_{\alpha}(P)=\{U(x|P):P(x)\geq\alpha\}\]  
and try to comprehensively fit into \(U_{\alpha}\) the global subset \(A\subset X\). Let us formulate the requirements for \(A\):  
\[(P(a)\geq\alpha)\wedge(D_{A}(a,r)\subseteq U(a))\wedge(\text{difference }U(a)\smallsetminus D_{A}(a,r)\text{ is minimal})\ \forall a\in A \tag{8}\]  
Under the natural assumption of \"continuity\" of the property \(\mathfrak{P}\) we can expect its level of occurrence on \(D_{A}(a,r)\) to be close to \(\alpha\). The mismatch \(D_{A}(a,r)\) with \(U(a)\) in (8) can be understood differently. Some variants of it make it possible to find a solution of (8) using the DPS scheme (Definition 6-Section 3.3.3) with respect to densities specifically constructed by covering \(U_{\alpha}\). Let us focus on one of them. The initial space \(Y\) will carry the covering \(U_{\alpha}\):  
\[Y=\text{Supp}\ U_{\alpha}=\{x\in X:P(x)\geq\alpha\}\]  
The difference in (8) is expressed through the intersection and induces a density \(G\) on \(Y\):  
\[G_{B}(y)=\frac{|D_{B}(y,r)\cap U(y)|}{|U(y)|}\ y\in Y,B\subseteq Y \tag{9}\]  
The density \(G\) is normalized: \(G_{B}(y)\in[0,1]\), and its level \(\gamma\) is the proximity index in (9). The dependence \(P\to G(P)\) (9), connecting with the DPS scheme (Definition 6-Section 3.3.3), leads to another dependence \(P\to\mathrm{DPS}(G(P))\), which we designate as GDPS and will be understood like \(\mathrm{DPS}\) Remark 2 and DPS Remark 3 ambiguously:  
* as a schema if we are talking about a dependency given above;
* as the GDPS algorithm when it comes to the operation of the DPS scheme (Definition 6-Section 3.3.3) on density \(G(P)\) with parameter \(\gamma\): \(\mathrm{GDPS}=\mathrm{DPS}(G,\gamma)\). Its \(\mathrm{GDPS}(Y)\) result solves problem Task 1 by \"gluing\" local data \(U_{\alpha}\) in a certain way. This explains the name of the algorithm (gluing). If the quantification of \(P\) property \(\mathfrak{P}\) is the density on \(X\), then the space \(Y\) is the first \(\alpha\)-hull \(X^{1}(\alpha)\) of space \(X\). The first solution Task 1 (let us call it \"a strong one\") is the operation of SDPS on \(X\) with respect to \(P\) with level \(\alpha\). It may differ from the second solution Task 1 (let us call it \"a weak one\"), which represents the operation of GDPS. The point is that the SDPS algorithm in its interception on \(X^{1}(\alpha)\) is guided primarily by preserving the density level \(\alpha\), while the GDPS algorithm seeks to preserve the proximity level \(\gamma\) on \(X^{1}(\alpha)\). The weak version is more versatile, since the quantification of \(P\) property \(\mathfrak{P}\) shall not be necessarily the density on \(X\) and it is also more \"save\" when carving. Therefore, it is the GDPS algorithm that will be part of the DPS-scheme Definition 6-Section 3.3.3 operation when the property \(\mathfrak{P}\) represents a local linearity in \(X\). Let us explain the above on two examples relating to the density \(S=S(q,p)\). In this case, the property \(\mathfrak{P}\) will be the \(r\)-local \"space count\", and \(S\) will be its formal expression.