The Theory of Plafales: P vs NP Problem Solution

1. Introduction

The first edition of [1] is published in March 2011. The publication’s goal is the creation of a new theory in mathematics, where the central object is a plafal1. The second edition of [2] is published in February 2013. After the report at the 42nd Polish Conference on Mathematics Applications [3], it is necessary to create applications based on the theory of plafales. Here we will provide an appropriate overview of the research and development process.

Finite element method. There are created mathematical models of serendipity finite elements: a new approach to construction basis and field functions. A quadruple role of the basis functions of serendipity finite elements is shown [4,5].

IT (finite element method as an algorithmic support). Due to solving the non-standard Dirichlet boundary value problem [4], using the components of the theory of plafales (to obtain the surface of the temperature field in three-dimensional space), there is developed a software for testing non-stationary temperature fields [6].

Cryptography. In September 2014 there is created a symmetric-key algorithm “ECLECTIC-DT-1” [7]. Algorithm’s characteristics: block length is 128 bits, key length is 256 bits, 14 rounds. Let us show the algorithm’s indicators. Upper bounds of practical security: $EDP\le 2^{-714}$ (against differential cryptanalysis [8]), $ELP\le 2^{-714}$ (against linear cryptanalysis [9]). Upper bounds of provable security (for the first four rounds): $d^{\Im }\le 2^{-96}$ and $l^{\Im }\le 2^{-96}$.

In December 2018 there is created a symmetric-key algorithm “STEEL” [10]. Algorithm’s characteristics: block length is 128 bits, key length is 256 bits, 14 rounds. Algorithm’s indicators: $EDP\le 2^{-595},ELP\le 2^{-595},d^{\Im }\le 2^{-80}$ and $l^{\Im }\le 2^{-80}$.

Note. Editions [1,2] are the bases of the theory of plafales. The paper presents the “linear passage”: all objects are introduced to prove P = NP as soon as possible. The theory has to be taken into consideration starting from this paper.

*Using the deterministic machine, P = NP is performed with pre-prepared tape.

2. Concept of the Plafal

We will use a framework “Standard By Default” [11]. A mathematical entity is described as $\mathcal{U}$-standard according to the following rules: a set, monoid, topological space, poset, family, graph, diagram, cardinal, ordinal etc. is standard when it is small; a category, groupoid, multicategory, locally ordered category etc. is standard when it is light; for $2\le n<\infty$, an n-category is standard when it is n-moderate. A mathematical entity is described as ${\mathcal{U}}^{*}$-standard according to the following rules: an ∞-groupoid is standard when it is small; an $(\infty ,1)$-category is standard when it is light; for $2\le n<\infty$, an $(\infty ,n)$-category is standard when it is n-moderate. A function, relation, subset, functor, natural transformation etc. is always standard.

Plafal

For any simple graph $G(V,E)$2 let there be a correspondence between each vertex and $\tilde{\mathcal{V}}=\mathcal{U}$-standard, and also between each edge and $\tilde{\mathcal{E}}=\mathcal{U}$-standard. The mentioned $\tilde{\mathcal{V}}$ and $\tilde{\mathcal{E}}$ can be the same for V and for E respectively. Thus we obtain an object, which is called a plafal and is denoted by $PF$ or $P{F}_j^i$, where i is a number of graph’s vertices, j is a number of graph’s edges. Generally, $\langle v,\tilde{\mathcal{V}}\rangle$ is called a vertex of the plafal and is denoted by $\mathcal{V}$; $\langle e,\tilde{\mathcal{E}}\rangle$ is called an edge of the plafal and is denoted by $\mathcal{E}$. By $V(PF)$ we denote a collection of vertices, by $E(PF)$ we denote a collection of edges. Between $\mathcal{E}$ and ${\mathcal{V}}_1,{\mathcal{V}}_2$, which are connected by $\mathcal{E}$, there is not necessarily a logical relation. If $\forall \mathcal{V}\in V(PF):\mathcal{V}=\langle v,v\rangle$ and $\forall \mathcal{E}\in E(PF):\mathcal{E}=\langle e,e\rangle$, then we say that $PF=G(V,E)$ is a trivial plafal and write _*$PF$. For example, the representation of $G(V,E)$ on a vector space [12] is a type of $PF$. Finally, $PF=\langle V(PF),E(PF)\rangle$ or $PF=\langle G(V,E),\diamond PF=\langle S(\tilde{\mathcal{V}}),S(\tilde{\mathcal{E}})\rangle \rangle$, where $S(\tilde{\mathcal{V}})$ is a collection of $\tilde{\mathcal{V}}$ and $S(\tilde{\mathcal{E}})$ is a collection of $\tilde{\mathcal{E}}$.

*-plafal

For any simple graph $G(V,E)$ let there be a correspondence between each vertex and ${\tilde{\mathcal{V}}}^{*}={\mathcal{U}}^{*}-\mathrm{standard}$ or ${\tilde{\mathcal{V}}}^{*}=v$, and also between each edge and ${\tilde{\mathcal{E}}}^{*}={\mathcal{U}}^{*}-\mathrm{standard}$ or ${\tilde{\mathcal{E}}}^{*}=e$. The mentioned ${\tilde{\mathcal{V}}}^{*}$ and ${\tilde{\mathcal{E}}}^{*}$ can be the same for V and for E respectively. Thus we obtain an object, which is called an *-plafal and is denoted by $P{F}^{*}$ or _*$P{F}_j^i$, where i is a number of graph’s vertices, j is a number of graph’s edges. Generally, $\langle v,{\tilde{\mathcal{V}}}^{*}\rangle$ is called a vertex of the *-plafal and is denoted by ${\mathcal{V}}^{*}$; $\langle e,{\tilde{\mathcal{E}}}^{*}\rangle$ is called an edge of the *-plafal and is denoted by ${\mathcal{E}}^{*}$. By $V(P{F}^{*})$ we denote a collection of vertices, by $E(P{F}^{*})$ we denote a collection of edges. Between ${\mathcal{E}}^{*}$ and ${\mathcal{V}}_1^{*},{\mathcal{V}}_2^{*}$, which are connected by ${\mathcal{E}}^{*}$, there is not necessarily a logical relation. Finally, $P{F}^{*}=\langle V(P{F}^{*}),E(P{F}^{*})\rangle$ or $P{F}^{*}=\langle G(V,E),\diamond P{F}^{*}=\langle S({\tilde{\mathcal{V}}}^{*}),S({\tilde{\mathcal{E}}}^{*})\rangle \rangle$, where $S({\tilde{\mathcal{V}}}^{*})$ is a collection of ${\tilde{\mathcal{V}}}^{*}$ and $S({\tilde{\mathcal{E}}}^{*})$ is a collection of ${\tilde{\mathcal{E}}}^{*}$.

Notation

We claim that $G(V,E)=G(PF)$ and $G(V,E)=G(P{F}^{*})$.

$S(PF)$ is a collection of plafales.

$S(G(PF))$ is a collection of simple graphs as supports of $S(PF)$.

$S(P{F}^{*})$ is a collection of *-plafales.

$S(G(P{F}^{*}))$ is a collection of simple graphs as supports of $S(P{F}^{*})$.

Definition 2.1.

Labeled plafal is a type of the plafal, where all or some of vertices and/or edges are enumerated. We have $V^{•}(PF)\stackrel{{\pi }_1}{\to }\left\{\overline{1,i}\right\}$, here $V^{•}(PF)\subseteq V(PF)$ and $|V^{•}(PF)|=k$; $E^{•}(PF)\stackrel{{\pi }_2}{\to }\left\{\overline{1,j}\right\}$, here $E^{•}(PF)\subseteq E(PF)$ and $|E^{•}(PF)|=l$; ${\pi }_1,{\pi }_2$ are substitutions3. By $P{F}_{j_l}^{i_k}$ we denote a labeled plafal, k is a quantity of enumerated vertices; l is a quantity of enumerated edges. In the case of $k=0$, the plafal does not contain enumerated vertices4. In the case of $l=0$, the plafal does not contain enumerated edges5. In the case of $k=l=0$, we claim that $P{F}_{j_0}^{i_0}=P{F}_j^i$. By ^τ$\mathcal{V}$ we denote a vertex with assigned number ($1\le \tau \le i$), by ^ς$\mathcal{E}$ we denote an edge with assigned number ($1\le \varsigma \le j$).

Definition 2.1 is performed for the *-plafal.

In fact, $P{F}_{j_l}^{i_k}=\langle P{F}_j^i,\langle \left\{\overline{1,i}\right\},\left\{\overline{1,j}\right\}\rangle \rangle$ and *$P{F}_{j_l}^{i_k}={\langle }^{*}P{F}_j^i,\langle \left\{\overline{1,i}\right\},\left\{\overline{1,j}\right\}\rangle \rangle$. Let us remark that a graph labeling is a type of $P{F}_{j_0}^{i_0}$ such that $\tilde{\mathcal{V}},\tilde{\mathcal{E}}\in \{v,e,\left\{\mathrm{labeling}\right\}\}$. Evidently, we can use the following configuration $\tilde{\mathcal{V}}(\tilde{\mathcal{E}})=\langle \left\{\mathrm{labeling}\right\},\mathcal{U}-\mathrm{standard}\rangle$. Note that ^τ$\mathcal{V}{(}^{\varsigma }\mathcal{E})$ does not necessarily coincide with a number of a graph labeling.

The illustrations of plafales and graphs are given for the reader’s perception.

Figure 1. $P{F}_{4_1}^{4_3}$. Vertices: $\mathcal{V}=\langle v,v\rangle$ (for three vertices), the category of small sets and small multirelations. Edges: the category of sets, the set of real numbers, the set of rational numbers, $\mathcal{E}=\langle e,e\rangle$. Three vertices are enumerated, one edge is enumerated. *$P{F}_{3_1}^{3_1}$. Vertices: ${\mathcal{V}}^{*}=\langle v,v\rangle$ (for two vertices), $(\infty ,1)$-category. Edges: ∞-groupoid, ${\mathcal{E}}^{*}=\langle e,e\rangle$ (for two edges). One vertex is enumerated, one edge is enumerated.

Figure 1. $P{F}_{4_1}^{4_3}$. Vertices: $\mathcal{V}=\langle v,v\rangle$ (for three vertices), the category of small sets and small multirelations. Edges: the category of sets, the set of real numbers, the set of rational numbers, $\mathcal{E}=\langle e,e\rangle$. Three vertices are enumerated, one edge is enumerated. *$P{F}_{3_1}^{3_1}$. Vertices: ${\mathcal{V}}^{*}=\langle v,v\rangle$ (for two vertices), $(\infty ,1)$-category. Edges: ∞-groupoid, ${\mathcal{E}}^{*}=\langle e,e\rangle$ (for two edges). One vertex is enumerated, one edge is enumerated.

Let us give an example in cryptography [7,10]. For a byte $\overline{b_7\dots b_0}$ we get configurations: ^ς$\mathcal{E}=\langle e,\left\{b_{8-\varsigma }\right\}\rangle$ (for eight edges) and $\mathcal{V}=\langle v,v\rangle$ (for eight vertices).

Figure 2. $P{F}_{8_8}^{8_0}$. Eight edges are enumerated. The plafal does not contain enumerated vertices.

Figure 2. $P{F}_{8_8}^{8_0}$. Eight edges are enumerated. The plafal does not contain enumerated vertices.

Management systems

The correspondence between each vertex and $\tilde{\mathcal{V}}$ (as abovementioned) is called a v-camoufleur and is denoted by $P{F}_{cam}^v$. The correspondence between each edge and $\tilde{\mathcal{E}}$ (as abovementioned) is called an e-camoufleur and is denoted by $P{F}_{cam}^e$. By $S(P{F}_{cam})=\langle S(P{F}_{cam}^v),S(P{F}_{cam}^e)\rangle$ we denote a collection of camoufleurs such that $G(PF)\stackrel{S(P{F}_{cam})}{\to }\diamond PF$, here $S(P{F}_{cam}^v)$ is a collection of v-camoufleurs and $S(P{F}_{cam}^e)$ is a collection of e-camoufleurs. The correspondence between each vertex and ${\tilde{\mathcal{V}}}^{*}$ (as mentioned earlier) is said to be an $*v$-camoufleur and is denoted by $P{F}_{cam}^{*v}$. The correspondence between each edge and ${\tilde{\mathcal{E}}}^{*}$ (as noted above) is said to be an $*e$-camoufleur and is denoted by $P{F}_{cam}^{*e}$. By $S(P{F}_{cam}^{*})=\langle S(P{F}_{cam}^{*v}),S(P{F}_{cam}^{*e})\rangle$ we denote a collection of *-camoufleurs such that $G(P{F}^{*})\stackrel{S(P{F}_{cam}^{*})}{\to }\diamond P{F}^{*}.$

Claim 2.2.

$\mathcal{V},\mathcal{E},{\mathcal{V}}^{*},{\mathcal{E}}^{*}$ are categories.

Proof.

Let us show for $\mathcal{V}$. $i{d}_v\circ i{d}_v=i{d}_v$, $i{d}_{\tilde{\mathcal{V}}}\circ i{d}_{\tilde{\mathcal{V}}}=i{d}_{\tilde{\mathcal{V}}}$, $P{F}_{cam}^v\circ i{d}_v=P{F}_{cam}^v$, $i{d}_{\tilde{\mathcal{V}}}\circ P{F}_{cam}^v=P{F}_{cam}^v$. □

Definition 2.3.

The correspondence between each vertex and $\tilde{\mathcal{V}}$, which is changing over time, is called a dynamical v-camoufleur and is denoted by $P{F}_{cam}^{v(t)}$. The correspondence between each edge and $\tilde{\mathcal{E}}$, which is changing over time, is called a dynamical e-camoufleur and is denoted by $P{F}_{cam}^{e(t)}$. We have the following systems

$$P{F}_{cam}^{v(t)}=\left\{\begin{array}{c}v\stackrel{P{F}_{cam}^{v_i(t)}}{\to }{\tilde{\mathcal{V}}}_i,t\in [T_i,T_{i+1}),\hfill \\ \dots ,\hfill \\ v\stackrel{P{F}_{cam}^{v_j(t)}}{\to }{\tilde{\mathcal{V}}}_j,t\in [T_j,T_{j+1}).\hfill \end{array}\right.P{F}_{cam}^{e(t)}=\left\{\begin{array}{c}e\stackrel{P{F}_{cam}^{e_k(t)}}{\to }{\tilde{\mathcal{E}}}_k,t\in [T_k,T_{k+1}),\hfill \\ \dots ,\hfill \\ e\stackrel{P{F}_{cam}^{e_l(t)}}{\to }{\tilde{\mathcal{E}}}_l,t\in [T_l,T_{l+1}).\hfill \end{array}\right.$$

Each of $P{F}_{cam}^{v(t)}$ ($P{F}_{cam}^{e(t)}$) has a unique time interval of changes. Let us remark that we may consider $P{F}_{cam}^v$ ($P{F}_{cam}^e$) as $P{F}_{cam}^{v(t)}$ ($P{F}_{cam}^{e(t)}$) in the case of $t\in (-\infty ,\infty )$.

Definition 2.3 is performed for the *-plafal. By $S(P{F}_{cam}^{(t)})$ we denote a collection of dynamical camoufleurs such that $G(PF)\stackrel{S(P{F}_{cam}^{(t)})}{\to }\diamond P{F}^{(t)}$, here $P{F}^{(t)}$ is called a flickering plafal. By $S(P{F}_{cam}^{*(t)})$ we denote a collection of dynamical *-camoufleurs such that $G(P{F}^{*})\stackrel{S(P{F}_{cam}^{*(t)})}{\to }\diamond P{F}^{*(t)}$, here $P{F}^{*(t)}$ is said to be a flickering *-plafal.

Figure 3. Management systems.

Figure 3. Management systems.

Remark 1.

*-plafales and their properties will be the object of another paper.

3. The Category of Plafales and Operations

Definition 3.1.

A plafal morphism $PF\stackrel{\Lambda =\langle f,\Theta \rangle }{\to }P{F}^{\prime }$ is a tuple of maps’s collections such that

(1) $G(PF)\stackrel{f}{\to }G(P{F}^{\prime })$, f is a graph morphism (an abstract simplicial map);

(2) $\diamond PF\stackrel{\Theta =\langle {\Theta }^{*},{\Theta }^{**}\rangle }{\to }\diamond P{F}^{\prime }$: $S(\tilde{\mathcal{V}})\stackrel{{\Theta }^{*}}{\to }S({\tilde{\mathcal{V}}}^{\prime })$ and $S(\tilde{\mathcal{E}})\stackrel{{\Theta }^{**}}{\to }S({\tilde{\mathcal{E}}}^{\prime })$.

Note that (2) is performed in accordance with (1).

The example is given in Figure 4.

Claim 3.2.

Plafales and plafal morphisms form the category Plafales, together with the componentwise compositions $\langle f,\Theta \rangle \circ \langle f^{\prime },{\Theta }^{\prime }\rangle =\langle f\circ f^{\prime },\Theta \circ {\Theta }^{\prime }\rangle$ and identities $i{d}_{PF}=\langle i{d}_{G(PF)},i{d}_{\diamond PF}=\langle i{d}_{S(\tilde{\mathcal{V}})},i{d}_{S(\tilde{\mathcal{E}})}\rangle \rangle$. The proof is left to the reader.

Figure 4. Condition 1. The image of $G(P{F}_5^4)$ under the strict homomorphism is $G(P{F}_3^3)$. Condition 2. U is a functor; g is a homomorphism of groups.

Figure 4. Condition 1. The image of $G(P{F}_5^4)$ under the strict homomorphism is $G(P{F}_3^3)$. Condition 2. U is a functor; g is a homomorphism of groups.

Remark 2.

We may consider $P{F}^{(t)}$ as a collection of $\Lambda$.

Definition 3.3.

$\langle {\widehat{0}}_{G(P{F}_j^i)}=\langle ⌀,⌀\rangle ,\diamond PF=\langle ⌀,⌀\rangle \rangle$ is called an empty plafal and is denoted by $P{F}^{emp}$, here ${\widehat{0}}_{G(P{F}_j^i)}$ is the empty graph (the initial object in the categories of simple graphs (SimpGph)).

Claim 3.4.

In Plafales, there’re no initial and terminal objects.

Proof. 

Initial. Omitted. Terminal. In SimpGph, there’s no terminal object. □

Definition 3.5.

A product of two plafales $P{F}^{prod}=PF\times P{F}^{\prime }$ is defined by the graph product $\prod =G(PF)\times G(P{F}^{\prime })$ and the following condition holds:

(i) $\forall {\tilde{\mathcal{V}}}^{prod}\in S({\tilde{\mathcal{V}}}^{prod}):{\tilde{\mathcal{V}}}^{prod}=\tilde{\mathcal{V}}\times {\tilde{\mathcal{V}}}^{\prime }$ and $\forall {\tilde{\mathcal{E}}}^{prod}\in S({\tilde{\mathcal{E}}}^{prod}):{\tilde{\mathcal{E}}}^{prod}=e$.

Notice that (i) is realized in accordance with ∏.

Definition 3.6.

A coproduct of two plafales $P{F}^{coprod}=PF+P{F}^{\prime }$ is defined by the graph coproduct $\coprod =G(PF)+G(P{F}^{\prime })$ and the following condition holds:

(i) in the general case, $\diamond PF+\diamond P{F}^{\prime }$ is a coproduct of categories.

Let us remark that (i) is performed in accordance with ∐.

Claim 3.7.

InPlafales, the (co)-product of any two plafales does not always exist.

Proof. 

It is sufficient to consider $S(\tilde{\mathcal{V}}),S({\tilde{\mathcal{V}}}^{\prime })\subset \mathrm{Ob}(\mathrm{Field})$. □

Corollary 3.8.

Plafalesis not a (quasi)-topos. The proof is streightforward.

Agreement. Moreover, we will refer a mathematical entity as an object.

3.1. Basic Operations

Elementary operations

1. Vertex deletion $P{F}_{j_l}^{i_k}-\mathcal{V}$. 2. Edge deletion $P{F}_{j_l}^{i_k}-\mathcal{E}$. 3. Edge addition $P{F}_{j_l}^{i_k}+\mathcal{E}$.

Figure 5. Elementary operations.

Figure 5. Elementary operations.

Advanced operations

Let us introduce the following notation. $P{F}_{\rho }$ is a plafal with assigned number. In the general case, ${\tilde{\mathcal{V}}}_{\alpha }$ and ${\tilde{\mathcal{E}}}_{\beta }$ are the components of $P{F}_{\alpha }$ and $P{F}_{\beta }$ respectively. Let us remark that ${\tilde{\mathcal{V}}}_{\alpha }$ and ${\tilde{\mathcal{E}}}_{\beta }$ are not the entities with the specified numbers.

In definitions 3.9 – 3.14, we will use $I\subseteq \left\{\overline{1,n}\right\}$. In definitions 3.9 – 3.11, conditions (i) are performed in accordance with the union of graphs $H^{un}={\bigcup }_{i\in I}G(P{F}_i)$. In definitions 3.12 – 3.14, conditions (i) are realized in accordance with the intersection of graphs $H^{in}={\bigcap }_{i\in I}G(P{F}_i)$.

Definition 3.9.

A union of plafales $P{F}^{un}$ is defined by $H^{un}$ and the following condition holds:

(i) ${\tilde{\mathcal{V}}}_{\mathrm{common}}={\bigcup }_{\mu \in I}{\tilde{\mathcal{V}}}_{\mu }$ and ${\tilde{\mathcal{E}}}_{\mathrm{common}}={\bigcup }_{\nu \in I}{\tilde{\mathcal{E}}}_{\nu }$.

Definition 3.10.

A docking of plafales $P{F}^{doc}$ is defined by $H^{un}$ and the following condition holds:

(i) ${\tilde{\mathcal{V}}}_{\mathrm{common}}=\langle {\tilde{\mathcal{V}}}_{\eta },\dots ,{\tilde{\mathcal{V}}}_{\mu }\rangle ,\eta <\dots <\mu$ and ${\tilde{\mathcal{E}}}_{\mathrm{common}}=\langle {\tilde{\mathcal{E}}}_{\varrho },\dots ,{\tilde{\mathcal{E}}}_{\chi }\rangle ,\varrho <\dots <\chi$.

Definition 3.11.

A docking• of plafales $P{F}^{doc•}$ is defined by $H^{un}$ and the following condition holds:

(i) Condition (i) of definition 3.9 is performed for $1\le r_1\le r-1$ common vertices and for $1\le t_1\le t-1$ common edges; condition (i) of definition 3.10 is realized for $r_2=r-r_1$ common vertices6 and for $t_2=t-t_1$ common edges7.

Definition 3.12.

An intersection of plafales $P{F}^{in}$ is defined by $H^{in}$ and the following condition holds:

(i) ${\tilde{\mathcal{V}}}_{\mathrm{common}}={\bigcap }_{\mu \in I}{\tilde{\mathcal{V}}}_{\mu }$ and ${\tilde{\mathcal{E}}}_{\mathrm{common}}={\bigcap }_{\nu \in I}{\tilde{\mathcal{E}}}_{\nu }$.

Definition 3.13.

A merger of plafales $P{F}^m$ is defined by $H^{in}$ and the following condition holds:

(i) ${\tilde{\mathcal{V}}}_{\mathrm{common}}=\langle {\tilde{\mathcal{V}}}_{\eta },\dots ,{\tilde{\mathcal{V}}}_{\mu }\rangle ,\eta <\dots <\mu$ and ${\tilde{\mathcal{E}}}_{\mathrm{common}}=\langle {\tilde{\mathcal{E}}}_{\varrho },\dots ,{\tilde{\mathcal{E}}}_{\chi }\rangle ,\varrho <\dots <\chi$.

Definition 3.14.

A merger• of plafales $P{F}^{m•}$ is defined by $H^{in}$ and the following condition holds:

(i) Condition (i) of definition 3.12 is performed for $1\le r_1\le r-1$ common vertices and for $1\le t_1\le t-1$ common edges; condition (i) of definition 3.13 is realized for $r_2=r-r_1$ common vertices and for $t_2=t-t_1$ common edges.

Agreement8. In definitions 3.10, 3.13, in the case of the chain of operations ${\tilde{\mathcal{V}}}_{\mathrm{common}}=\langle {\underbrace{\langle {\tilde{\mathcal{V}}}_{\alpha },\dots ,{\tilde{\mathcal{V}}}_{\mu }\rangle ,\dots ,\langle {\tilde{\mathcal{V}}}_{\beta },\dots ,{\tilde{\mathcal{V}}}_{\zeta }\rangle }}_{k\ge 1},{\underbrace{{\tilde{\mathcal{V}}}_{\varsigma },\dots ,{\tilde{\mathcal{V}}}_{\xi }}}_{m\ge 0}\rangle$, $k+m=\left|I\right|,\alpha <\dots <\varsigma$ and taking into account $\left|I\right|!$ permutations for ${\tilde{\mathcal{V}}}_{\mathrm{common}}$, we get ${\tilde{\mathcal{V}}}_{\mathrm{common}}=\langle {\tilde{\mathcal{V}}}_{\alpha },\dots ,{\tilde{\mathcal{V}}}_{\varsigma }\rangle .$ For the same reason, this is performed for ${\tilde{\mathcal{E}}}_{\mathrm{common}}$.

On the other hand, definitions 3.9 – 3.14 can be formulated in the following equivalent form: in general, $S^k(G(PF))\stackrel{★=\{\cup ,\cap \}}{\to }S(G(PF))\stackrel{S(P{F}_{cam})}{\to }\diamond PF$.

Figure 6. Advanced operations.

Figure 6. Advanced operations.

Definition 3.15.

A product• of two plafales $P{F}^{prod•}=PF\times P{F}^{\prime }$ is defined by the graph product $\prod =G(PF)\times G(P{F}^{\prime })$ and the following condition holds:

(i) $\forall {\tilde{\mathcal{V}}}^{prod•}\in S({\tilde{\mathcal{V}}}^{prod•}):{\tilde{\mathcal{V}}}^{prod•}=\tilde{\mathcal{V}}\times {\tilde{\mathcal{V}}}^{\prime }$, $\forall {\tilde{\mathcal{E}}}^{prod}\in S({\tilde{\mathcal{E}}}^{prod}):{\tilde{\mathcal{E}}}^{prod}=\mathcal{U}$-standard.

(i) is realized in accordance with ∏. The mentioned ${\tilde{\mathcal{E}}}^{prod}$ can be different.

We stress that $P{F}^{prod}$ is a particular case of $P{F}^{prod•}$.

Definition 3.16.

A decomposition of plafales is a collection of maps such that $S(G(PF))\stackrel{H^d}{\to }S(G(PF))\stackrel{S^{\complement }(P{F}_{cam})}{\to }S(\diamond PF)$ and the following conditions hold:

(i) for k-vertices, which were located at the common vertice, of k-decomposited graphs we have: ${\tilde{\mathcal{V}}}_{\mathrm{common}}$ corresponds to one of the above vertices (k-variations) and for a collection of remaining $(k-1)$-vertices corresponds a collection of arbitrary $\mathcal{U}$-standard objects. By the same argument, this is performed for l-edges;

(ii) ${\tilde{\mathcal{V}}}_{\mathrm{no}\mathrm{common}}$ and ${\tilde{\mathcal{E}}}_{\mathrm{no}\mathrm{common}}$ remain unchanged;

where $H^d$ is an operation of the decomposition of graphs; here $S^{\complement }(P{F}_{cam})$ is a collection of $S(P{F}_{cam})$ and $S(\diamond PF)$ is a collection of $\diamond PF$ for $S(G(PF))$ respectively.

Remark 3.

The other types of operations will be discussed in a further paper.

4. Geo-Space

Let us introduce the following concept. A geo-space is a tuple $\langle {\mathbb{R}}^3,U\rangle$ and is denoted by ${\left|PF\right|}^U$, where U is a universe of $\mathcal{U}$-standard objects such that ${\mathbb{R}}^3\cap U=⌀$. The objects of the following type $\langle \mathrm{Object}\subset {\mathbb{R}}^3,\diamond \mathrm{Object}\subset U\rangle$, which will be represented in this section, are implemented in ${\left|PF\right|}^U$.

4.1. Geo-Plafal

Taking into account $PF=\langle G(PF),\diamond PF\rangle$, let us provide a definition of a geometric representation of $PF$ in ${\left|PF\right|}^U$.

Definition 4.1.

$\langle |G(PF)|,\diamond PF\rangle$ is said to be a geo-plafal and is denoted by $\left|PF\right|$, here $|G(PF)|$ is the geometric realization of $\mathcal{K}=G(PF)$ in ${\mathbb{R}}^3$ such that $\left|\mathcal{K}\right|\to \diamond PF$ in accordance with $\mathcal{K}\to \diamond PF$.

Let us give a comment to definition 4.1. We use the metric or coherent topologies.

On the other hand, $\left|PF\right|$ can be formulated by analogy with the concept of the plafal. For any 1-dimensional (topological) simplicial complex9$\left|\mathcal{K}\right|$ in ${\mathbb{R}}^3$ let there be a correspondence between each vertex and $|\tilde{\mathcal{V}}|=\mathcal{U}$-standard, and also between each edge and $|\tilde{\mathcal{E}}|=\mathcal{U}$-standard. The mentioned $|\tilde{\mathcal{V}}|$ and $|\tilde{\mathcal{E}}|$ can be the same for $V(|\mathcal{K}|)$ and for $E(|\mathcal{K}|)$. Thus we obtain an object, which is called a geo-plafal and is denoted by $\left|PF\right|$ or ${\left|PF\right|}_j^i$, where i is a number of vertices of $\left|\mathcal{K}\right|$, j is a number of edges of $\left|\mathcal{K}\right|$. In the general case, $\langle \left|v\right|,\left|\tilde{\mathcal{V}}\right|\rangle$ is called a vertex of the geo-plafal and is denoted by $\left|\mathcal{V}\right|$; $\langle \left|e\right|,\left|\tilde{\mathcal{E}}\right|\rangle$ is called an edge of the geo-plafal and is denoted by $\left|\mathcal{E}\right|$. By $V(|PF|)$ we denote a collection of vertices, by $E(|PF|)$ we denote a collection of edges. Between $\left|\mathcal{E}\right|$ and ${\left|\mathcal{V}\right|}_1,{\left|\mathcal{V}\right|}_2$, which are connected by $\left|\mathcal{E}\right|$, there is not necessarily a logical relation. If $\forall \left|\mathcal{V}\right|\in V(|PF|):\left|\mathcal{V}\right|=\langle \left|v\right|,\left|v\right|\rangle$ and $\forall \left|\mathcal{E}\right|\in E(|PF|):\left|\mathcal{E}\right|=\langle \left|e\right|,\left|e\right|\rangle$, then we say that $\left|PF\right|=\left|\mathcal{K}\right|$ is a trivial geo-plafal and write _*$\left|PF\right|$. In fact, $\left|PF\right|=\langle V(|PF|),E(|PF|)\rangle$, $\left|PF\right|=\langle \left|\mathcal{K}\right|,\diamond \left|\mathcal{K}\right|=\langle S(|\tilde{\mathcal{V}}|),S(|\tilde{\mathcal{E}}|)\rangle \rangle$, where $S(|\tilde{\mathcal{V}}|)$ is a collection of $|\tilde{\mathcal{V}}|$ and $S(|\tilde{\mathcal{E}}|)$ is a collection of $|\tilde{\mathcal{E}}|$. By $S(|PF|)$ we denote a collection of geo-plafales. By $S(|\mathcal{K}|)$ we denote a collection of $\left|\mathcal{K}\right|$ as supports of geo-plafales. The concepts of the labeled plafal and management systems are performed for $\left|PF\right|$.

Definition 4.2.

A geo-plafal morphism $\left|PF\right|\stackrel{{\Lambda }^{*}=\langle \vartheta ,o\rangle }{\to }{\left|PF\right|}^{\prime }$ is a tuple of maps’s collections such that

(1) $\left|\mathcal{K}\right|\stackrel{\vartheta }{\to }{\left|\mathcal{K}\right|}^{\prime }$, $\vartheta$ is a morphism between polyhedra;

(2) $\diamond \left|\mathcal{K}\right|\stackrel{o=\langle o^{*},o^{**}\rangle }{\to }\diamond {\left|\mathcal{K}\right|}^{\prime }$: $S(|\tilde{\mathcal{V}}|)\stackrel{o^{*}}{\to }S(|\tilde{\mathcal{V}}{|}^{\prime })$ and $S(|\tilde{\mathcal{E}}|)\stackrel{o^{**}}{\to }S(|\tilde{\mathcal{E}}{|}^{\prime })$.

Note that (2) is realized in accordance with (1).

Claim 4.3.

Geo-plafales and geo-plafal morphisms form the category Geo-Plafales, together with the componentwise compositions $\langle \vartheta ,o\rangle \circ \langle {\vartheta }^{\prime },o^{\prime }\rangle =\langle \vartheta \circ {\vartheta }^{\prime },o\circ o^{\prime }\rangle$ and identities $i{d}_{\left|PF\right|}=\langle i{d}_{\left|\mathcal{K}\right|},i{d}_{\diamond \left|\mathcal{K}\right|}\rangle$. The proof is left to the reader.

Definition 4.4.

A product of two geo-plafales ${\left|PF\right|}^{prod}=\left|PF\right|\times {\left|PF\right|}^{\prime }$ is defined by the product of polyhedra $\prod =\left|\mathcal{K}\right|\times {\left|\mathcal{K}\right|}^{\prime }$ and the following condition holds:

(i) $\forall |\tilde{\mathcal{V}}{|}^{prod}\in S(|\tilde{\mathcal{V}}{|}^{prod}):|\tilde{\mathcal{V}}{|}^{prod}=|\tilde{\mathcal{V}}|\times |\tilde{\mathcal{V}}{|}^{\prime }$, $\forall |\tilde{\mathcal{E}}{|}^{prod}\in S(|\tilde{\mathcal{E}}{|}^{prod}):|\tilde{\mathcal{E}}{|}^{prod}=\left|e\right|$.

Notice that (i) is realized in accordance with ∏.

Definition 4.5.

A coproduct of two geo-plafales ${\left|PF\right|}^{coprod}=\left|PF\right|+{\left|PF\right|}^{\prime }$ is defined by the disjoint union of polyhedra $\coprod =\left|\mathcal{K}\right|+{\left|\mathcal{K}\right|}^{\prime }$ and the following condition holds:

(i) in the general case, $\diamond \left|\mathcal{K}\right|+\diamond {\left|\mathcal{K}\right|}^{\prime }$ is a coproduct of categories.

Let us remark that (i) is performed in accordance with ∐.

Claim 4.6.

In Geo-plafales, there’re no initial and terminal objects. The (co)-product of any two geo-plafales does not always exist. Geo-plafales is not a (quasi)-topos. The proof is similar to the proofs in the section 3.

It is easily proved that $\mathbf{Plafales}\stackrel{\mathcal{G}=\langle \mathcal{L},\diamond \mathbf{F}\rangle }{\to }\mathbf{Geo}-\mathbf{Plafales}$ such that $\mathbf{SCpx}\stackrel{\mathcal{L}}{\to }\mathbf{Top}$ and $\diamond \mathbf{F}$ is a collection of identity functors, here SCpx is the category of abstract simplicial complexes, Top is the category of topological spaces. Note that advanced operations are performed for geo-plafales. The reader will have no difficulty in showing that $\left|PF\right|=\langle CW,\diamond PF\rangle$, where $CW$ is a 1-dimensional cellular complex.

4.2. Point

We begin with some notation for ${\mathbb{R}}^3$. $pl$ is a limit point; $S(pl)$ is a collection of limit points.

Definition 4.7.

$\langle pl,\diamond pl=\mathcal{U}-\mathrm{standard}\rangle$ is a limit point in ${\left|PF\right|}^U$ and is denoted by ${\left|PF\right|}^{pl}$ such that there is a correspondence between $pl$ and $\diamond pl$. $S(|PF{|}^{pl})$ is a collection of limit points in ${\left|PF\right|}^U$.

If $\diamond pl=pl$ (a singleton in $\mathbf{Top}$), then we say that ${\left|PF\right|}^{pl}=pl$ is a trivial point and write _*${\left|PF\right|}^{pl}$. The concept of the management system is realized for ${\left|PF\right|}^{pl}$.

Definition 4.8.

A morphism ${\left|PF\right|}_i^{pl}\stackrel{t_{pl}=\langle \varsigma ,\mathcal{R}\rangle }{\to }{\left|PF\right|}_j^{pl}$ is a tuple of maps’s collections such that

(1) ${(pl)}_i\stackrel{\varsigma }{\to }{(pl)}_j$;

(2) $\diamond {(pl)}_i\stackrel{\mathcal{R}}{\to }\diamond {(pl)}_j$.

Note that (2) is performed in accordance with (1).

Claim 4.9.

$S(|PF{|}^{pl})$ and morphisms $t_{pl}$ form the category PFpl.

Proof. 

$i{d}_{{\left|PF\right|}^{pl}}=\langle i{d}_{pl},i{d}_{\diamond pl}\rangle$. We get the following commutative diagrams:

□

Remark 4. The concept of ${\mathbb{R}}^3$ with isolated points and the concept of a flickering point, as a superposition of limit and isolated points, will be discussed elsewhere.

4.3. Manifold

Let us introduce the following notation for ${\mathbb{R}}^3$. M is a topological manifold; $S(M)$ is a collection of manifolds.

Definition 4.10.

$\langle M,\diamond M=\mathcal{U}-\mathrm{standard}\rangle$ is a manifold in ${\left|PF\right|}^U$ and is denoted by ${\left|PF\right|}^M$ such that there is a correspondence between M (as a whole object) and $\diamond M$. $S(|PF{|}^M)$ is a collection of manifolds in ${\left|PF\right|}^U$.

If $\diamond M=M$ (an object of $\mathbf{TopMfd}$), then we say that ${\left|PF\right|}^M=M$ is a trivial manifold and write _*${\left|PF\right|}^M$; here $\mathbf{TopMfd}$ is the category of topological manifolds. The concept of the management system is realized for ${\left|PF\right|}^M$.

Remark 5. The concept of (open) cover $\mathcal{J}$ of M (including the relationship with (pre)-sheaf) and a collection of $\mathcal{U}-\mathrm{standard}$ objects, which corresponds to $\mathcal{J}$, will be the object of another paper.

Definition 4.11.

A morphism ${\left|PF\right|}_i^M\stackrel{{\rm Y}=\langle \varrho ,\mathcal{D}\rangle }{\to }{\left|PF\right|}_j^M$ is a tuple of maps’s collections such that

(1) $M_i\stackrel{\varrho }{\to }{M}_j$, $\varrho$ is a morphism between topological manifolds;

(2) $\diamond M_i\stackrel{\mathcal{D}}{\to }\diamond M_j$.

Notice that (2) is realized in accordance with (1).

Claim 4.12.

$S(|PF{|}^M)$ and morphisms Υ form the category PFM. Omitted.

Definitions 4.4, 4.5 (in terms of the product manifold and the disjoint union of topological manifolds) and results of claim 4.6 are performed for $\mathbf{PFM}$.

4.4. Dynamical System

Let us summarize and use the results of [13,14]. $\langle |PF{|}^U,G,\Omega \rangle$ is a dynamical system and is denoted by _d${\left|PF\right|}^U$ such that $\mathcal{S}=\langle {\mathbb{R}}^3,\tau \rangle$10 undergoes homeomorphic transformations, here G is a topological group, $\Omega$ is a collection of identity functors, results of advanced operations and management systems of subsections 4.1 – 4.3.

Definition 4.13.

A homomorphism _d${\left|PF\right|}_i^U{\stackrel{\Psi =\langle \Phi ,\Delta \rangle }{\to }}_d{\left|PF\right|}_j^U$ is a tuple of maps’s collections such that

(1) ${\mathcal{S}}_i\stackrel{\Phi }{\to }{\mathcal{S}}_j$, $\Phi$ is a homomorphism between dynamical systems;

(2) $\Delta$ is a collection of maps.

Let us remark that (2) is realized in accordance with (1).

Figure 7. Dynamical system.

Figure 7. Dynamical system.

5. p = np (with Pre-Prepared Tape)

The proof of the equality of complexity classes P and NP will be given in a constructive way using the apparatus of the constructive theory of serendipity approximations (CTSA)11 [4,15–24].

Remark 6. The total volume of CTSA is about 1000 papers (they can be traced evolutionarily by the indicated references).

In the general case, by S we denote an instance of an NP-complete problem. Using the deterministic machine, C is a type of S with answer “Yes” after the checking, otherwise is $\overline{C}$. The implementation model will be described after theorem 5.1. We will operate with a plane in ${\left|PF\right|}^U$. By ${\left|PF\right|}_{pl}^U$ we denote this topological subspace.

Theorem 5.1.

Based on the theory of plafales, there exists an algorithm that allows to check all instances of an NP-complete problem12 in polynomial time.

Proof. 

1st stage. Computational template (CT). Let us consider the following labeled geo-plafal: ${\left|PF\right|}_{N_0}^{N_N}=\langle \left|G(PF)\right|=\mathrm{SFE},\diamond PF\rangle$, where SFE is a serendipity finite element (square: $\left|x\right|\le 1,\left|y\right|\le 1$) on N vertices (nodes)13. Specified vertices ${\left|v\right|}_i$ are evenly spaced on the boundary of SFE ($\left|x\right|=1,\left|y\right|=1$) by the rule

$$\mathcal{G}=\left\{\begin{array}{c}i\in \{4·l+1\};-1\le x<1,y=-1,\hfill \\ i\in \{4·l+2\};x=1,-1\le y<1,\hfill \\ i\in \{4·l+3\};-1<x\le 1,y=1,\hfill \\ i\in \{4·(l+1)\};x=-1,-1<y\le 1,\hfill \\ l\in {\mathbb{Z}}_{+}\cup \left\{0\right\}.\hfill \end{array}\right.$$

For $\diamond PF$ we have the following ⁱ$\left|\mathcal{V}\right|={\langle \left|v\right|}_i,|\tilde{\mathcal{V}}{|}_i=\langle S_i,{\mathcal{H}}_i={\zeta }_i(t)·C_i+{\theta }_i(t)·{\overline{C}}_i\rangle \rangle$, here ${\mathcal{H}}_i$ is a linear combination such that ${\zeta }_i(t)+{\theta }_i(t)=1$, ${\zeta }_i(t),{\theta }_i(t)\in \mathbb{R}$, where t is time; and $\forall \left|\mathcal{E}\right|\in {\left|PF\right|}_{N_0}^{N_N}:\left|\mathcal{E}\right|=\langle \left|e\right|,\left|e\right|\rangle$.

2nd stage. Relationship between $S_i$ and basis function (BF)14 $L_i(x,y)$. Without loss of generality, let us consider the word $S_i$ over a given alphabet into a representation over the alphabet $\left\{\overline{0,9}\right\}$. Then to each $S_i$ corresponds a unique “weight” ${\chi }_i={\displaystyle \frac{S_i}{{\sum }_{i=1}^N{S}_i}}$; however ${\chi }_i={\displaystyle \frac{1}{G}}{\int \int }_D{L}_i(x,y)dxdy\ge 0$, G is the area of D ($\left|x\right|\le 1,\left|y\right|\le 1$), where N is a quantity of instances of an NP-complete problem. Hence to each ${\chi }_i$ corresponds a unique collection of the basis functions (UCBF) in the CTSA. A relationship between the word $S_i$ and UCBF is structurally shown15.

Figure 8. Computational template (in general).

Figure 8. Computational template (in general).

3rd stage. The analysis of the field function (FF)16. Let us configure the CT. Without loss of generality, ${\chi }_1<{\chi }_2<\dots <{\chi }_l<\dots$. So, ⁱ$\left|\mathcal{V}\right|={\langle \left|v\right|}_i({\chi }_i),|\tilde{\mathcal{V}}{|}_i\rangle$.

We will operate with the non-standard Dirichlet boundary value problem [4]: $f(x,y,t)={\sum }_{i=1}^N{\lambda }_i(t)·L_i(x,y)$ such that $(★)=\left\{\begin{array}{c}{g}_i={\displaystyle \frac{1}{S_i}},g_1>g_2>\dots >g_l>\dots ,\hfill \\ {\lambda }_i(t)={\zeta }_i(t)·(g_i-g_{i+1})+g_{i+1},\hfill \\ {\lambda }_i(t)\in (g_{i+1},g_i),t<T^{*},\hfill \\ {\lambda }_i(t)=\left[\begin{array}{c}{g}_i,t\ge T^{*},\hfill \\ g_{i+1},t\ge T^{*}.\hfill \end{array}\right.\hfill \end{array}\right.$ Let us comment. For C we get ${\lambda }_i(t)=g_i$, otherwise is ${\lambda }_i(t)=g_{i+1}$ (see Figure 9).

Taking into account the field function $f(x,y,t)={\sum }_{i=1}^N{\lambda }_i(t)·L_i(x,y)$ and $\mathcal{G}$, let us provide the claim, which is a generalization of the results of Ph. D. thesis [19].

Claim 5.2.

For SFE on N vertices the condition of invariant of the field function (stability of the field function with respect to the basis function) is the following

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i=5}^N{\lambda }_i(t)}{N-4}},N=4·k,\hfill \\ {\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i\in \{5,9,\dots ,4·l+1\}}{\lambda }_i(t)}{N-1}}+{\displaystyle \frac{{\sum }_{i\in \{4·l+2,4·l+3,4·(l+1),1\le l<N\}}{\lambda }_i(t)}{N-5}},N=4·k+1,\hfill \\ {\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i\in \{4·l+1,4·l+2,1\le l\le N\}}{\lambda }_i(t)}{N-2}}+{\displaystyle \frac{{\sum }_{i\in \{4·l+3,4·(l+1),1\le l<N\}}{\lambda }_i(t)}{N-6}},N=4·k+2,\hfill \\ {\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i\in \{4·l+1,4·l+2,4·l+3,1\le l\le N\}}{\lambda }_i(t)}{N-3}}+{\displaystyle \frac{{\sum }_{i\in \{8,12,\dots ,4·(l+1)\}}{\lambda }_i(t)}{N-7}},N=4·k+3.\hfill \end{array}\right.\end{array}$$

Proof. 

For SFE on 8 nodes the above condition is ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i=5}^8{\lambda }_i(t)}{4}}$ [4,19]. Let us consider the BF of SFE on N nodes in the following representation $L_i(x,y)=={\sum }_{k,l=0}^r{\sigma }_{kl}·x^k·y^l={\sum }_{k,l=0}^2{\mu }_{kl}·x^k·y^l·Q(x,y)$ such that $minr=\left\{\begin{array}{c}{\displaystyle \frac{N}{4}}if4\mid N,\hfill \\ {\displaystyle \frac{N}{4}}]+1if4\nmid N.\hfill \end{array}\right.$ Let us show the rule of arrangement of the number of nodes: $N\equiv b(\mathrm{mod}4)$ (see Figure 10). In the case of $b\in \{1,2,3\}$, b nodes are spaced on b sides of SFE.

$K_1(x,y),\dots ,K_8(x,y)$ is a collection of the basis functions of SFE on 8 nodes [4,17,22]. Therefore, $K_1(-1,-1)=1$, $K_2(1,-1)=1$, $K_3(1,1)=1$, $K_4(-1,1)=1$, $K_5(0,-1)=1,$$K_6(1,0)=1$, $K_7(0,1)=1$, $K_8(-1,0)=1$. Let us show one of the collections of the basis functions for SFE·8.

Let us introduce the following system.

$$\begin{array}{c}\hfill \mathcal{Y}=\left\{\begin{array}{c}{\mathcal{W}}_1=\underset{q_1=(N+{\displaystyle \frac{2·b^3}{3}}-4·b^2+{\displaystyle \frac{19·b}{3}}-4)/4}{\underbrace{{\lambda }_5(t)·L_5(x,y)+\dots +{\lambda }_{4·l+1}(t)·L_{4·l+1}(x,y)}}=K_5(x,y)·g_5=\hfill \\ =K_5(x,y)·(\underset{q_1=(N+{\displaystyle \frac{2·b^3}{3}}-4·b^2+{\displaystyle \frac{19·b}{3}}-4)/4}{\underbrace{{\lambda }_5(t)·Q_5(x,y)+\dots +{\lambda }_{4·l+1}(t)·Q_{4·l+1}(x,y)}}),\hfill \\ {\mathcal{W}}_2=\underset{q_2=(N-{\displaystyle \frac{4·b^3}{3}}+6·b^2-{\displaystyle \frac{17·b}{3}}-4)/4}{\underbrace{{\lambda }_6(t)·L_6(x,y)+\dots +{\lambda }_{4·l+2}(t)·L_{4·l+2}(x,y)}}=K_6(x,y)·g_6=\hfill \\ =K_6(x,y)·(\underset{q_2=(N-{\displaystyle \frac{4·b^3}{3}}+6·b^2-{\displaystyle \frac{17·b}{3}}-4)/4}{\underbrace{{\lambda }_6(t)·Q_6(x,y)+\dots +{\lambda }_{4·l+2}(t)·Q_{4·l+2}(x,y)}}),\hfill \\ {\mathcal{W}}_3=\underset{q_3=(N+{\displaystyle \frac{2·b^3}{3}}-2·b^2+{\displaystyle \frac{b}{3}}-4)/4}{\underbrace{{\lambda }_7(t)·L_7(x,y)+\dots +{\lambda }_{4·l+3}(t)·L_{4·l+3}(x,y)}}=K_7(x,y)·g_7=\hfill \\ =K_7(x,y)·(\underset{q_3=(N+{\displaystyle \frac{2·b^3}{3}}-2·b^2+{\displaystyle \frac{b}{3}}-4)/4}{\underbrace{{\lambda }_7(t)·Q_7(x,y)+\dots +{\lambda }_{4·l+3}(t)·Q_{4·l+3}(x,y)}}),\hfill \\ {\mathcal{W}}_4=\underset{q_4=(N-b-4)/4}{\underbrace{{\lambda }_8(t)·L_8(x,y)+\dots +{\lambda }_{4·(l+1)}(t)·L_{4·(l+1)}(x,y)}}=K_8(x,y)·g_8=\hfill \\ =K_8(x,y)·(\underset{q_4=(N-b-4)/4)}{\underbrace{{\lambda }_8(t)·Q_8(x,y)+\dots +{\lambda }_{4·(l+1)}(t)·Q_{4·(l+1)}(x,y))}}.\hfill \end{array}\right.\end{array}$$

(6)

Using $\mathcal{Y}$, we have ${\sum }_{i=1}^N{\lambda }_i(t)·L_i(x,y)={\sum }_{1\le i\le 4}{\lambda }_i(t)·L_i(x,y)+{\sum }_{1\le i\le 4}{\mathcal{W}}_i=={\sum }_{1\le i\le 4}{\lambda }_i(t)·K_i(x,y)·Q_i(x,y)+{\sum }_{1\le i\le 4}{\mathcal{W}}_i$. Let us remark that $L_1(-1,-1)==K_1(-1,-1)$, $L_2(1,-1)=K_2(1,-1)$, $L_3(1,1)=K_3(1,1)$, $L_4(-1,1)=K_4(-1,1)$.

Taking into account the above condition ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i=5}^8{\lambda }_i(t)}{4}}$ for SFE·8 and the properties of the BF

$\left\{\begin{array}{c}{L}_1(x,y)+L_2(x,y)+L_5(x,y)+\dots +L_{4·l+1}(x,y)=1,\hfill \\ L_2(x,y)+L_3(x,y)+L_6(x,y)+\dots +L_{4·l+2}(x,y)=1,\hfill \\ L_3(x,y)+L_4(x,y)+L_7(x,y)+\dots +L_{4·l+3}(x,y)=1,\hfill \\ L_1(x,y)+L_4(x,y)+L_8(x,y)+\dots +L_{4·(l+1)}(x,y)=1,\hfill \end{array}\right.$ we get $\mathcal{I}=\left\{\begin{array}{c}{Q}_1(0,-1)+Q_2(0,-1)+Q_5(0,-1)+\dots +Q_{4·l+1}(0,-1)=1,\hfill \\ Q_2(1,0)+Q_3(1,0)+Q_6(1,0)+\dots +Q_{4·l+2}(1,0)=1,\hfill \\ Q_3(0,1)+Q_4(0,1)+Q_7(0,1)+\dots +Q_{4·l+3}(0,1)=1,\hfill \\ Q_1(-1,0)+Q_4(-1,0)+Q_8(-1,0)+\dots +Q_{4·(l+1)}(-1,0)=1,\hfill \end{array}\right.$ and in accordance with coordinates of $Q_j$ in $\mathcal{I}$ we have ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i=5}^8{g}_i}{4}}$. Our goal is that the choice of the basis functions is insignificant. So, the expected value of difference of two field functions with different collections of the basis functions is equal to zero. To do that, let us show for the first equation of $\mathcal{I}$. $Q_1(0,-1)++Q_2(0,-1)=0$ and $Q_5(0,-1)+\dots +Q_{4·l+1}(0,-1)=1$, then $Q_{4·j+1}(0,-1)={\displaystyle \frac{1}{q_1}}$, where $q_i$ is a quantity of nodes on the side of SFE (see $\mathcal{Y}$).

Let us show for $b=0$. $\forall j\in \mathbb{N}\setminus \left\{\overline{1,4}\right\}:Q_j(x,y)={\displaystyle \frac{4}{N-4}}$, here $(x,y)$ are coordinates in $\mathcal{I}$. Thus we obtain ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}={\displaystyle \frac{{\sum }_{i=5}^N{\lambda }_i(t)}{N-4}}$. □

Remark 7.

There is an additional condition of symmetry of boundary values in some nodes for SFE·12 with n-parametric interpolation ($n>13$) and SFE·16 [19].

Further, we will use the results of Stepanets’ school17. Let us provide the results of [26]. Let $\mathcal{M}$ be a set of all sequences $m={\left\{m_k\right\}}_{k=1}^{\infty }$ of nonnegative numbers such that $\left|m\right|:={\sum }_{k=1}^{\infty }{m}_k\le 1$. Let r be a positive number and ${\mathcal{A}}_r$ is a set of all nonincreasing sequences $\alpha ={\left\{{\alpha }_k\right\}}_{k=1}^{\infty }$ of positive numbers such that ${lim}_{k\to \infty }{\alpha }_k=0$. In the case of $r\in (0,1)$, we have the following condition: ${\sum }_{k=1}^{\infty }{\alpha }_k^{1/(1-r)}<\infty$. Denote by ${\gamma }_n=\{k_1,k_2,\dots ,k_n\}$ a collection of n different integers $n=1,2,\dots ,$ and for any $m\in \mathcal{M}$ and $\alpha \in {\mathcal{A}}_r$, set

$${\mathcal{E}}_n(m)={\mathcal{E}}_n(\alpha ,r,m):=\sum _{k=1}^{\infty }{\alpha }_k·m_k^r-\underset{{\gamma }_n}{sup}\sum _{k\in {\gamma }_n}{\alpha }_k·m_k^r,$$

and

$${\mathcal{E}}_n={\mathcal{E}}_n(\alpha ,r):=\underset{m\in \mathcal{M}}{sup}{\mathcal{E}}_n(m)=\underset{m\in \mathcal{M}}{sup}{\mathcal{E}}_n(\alpha ,r,m).$$

(i) Let $\alpha \in {\mathcal{A}}_r$ and $r\ge 1$. Then $\forall n\in \mathbb{N}$$\exists s^{*}>n$ such that

$${\mathcal{E}}_n(\alpha ,r)=(s^{*}-n)·{\left(\sum _{k=1}^{s^{*}}{\alpha }_k^{-1/r}\right)}^{-r}.$$

$s^{*}$ is determined by

$$\underset{s>n}{sup}(s-n)·{\left(\sum _{k=1}^s{\alpha }_k^{-1/r}\right)}^{-r}=(s^{*}-n)·{\left(\sum _{k=1}^{s^{*}}{\alpha }_k^{-1/r}\right)}^{-r}.$$

The exact upper bound is realized by the sequence $m^{*}={\left\{m_k^{*}\right\}}_{k=1}^{\infty }$ from $\mathcal{M}$ such that

$$m_k^{*}=\left\{\begin{array}{c}{({\alpha }_k^{1/r}·\sum _{i=1}^{s^{*}}{\alpha }_i^{-1/r})}^{-1},k\in [1,s^{*}],\hfill \\ 0,k>s^{*}.\hfill \end{array}\right.$$

(ii) Let $\alpha \in {\mathcal{A}}_r$ and $r\in (0,1)$. Then $\forall n\in \mathbb{N}$ is performed the following equality

$${\mathcal{E}}_n={\mathcal{E}}_n(\alpha ,r)={\left({(s^{*}-n)}^{1/(1-r)}·{\left(\sum _{k=1}^{s^{*}}{\alpha }_k^{-1/r}\right)}^{r/(r-1)}+\sum _{k=s^{*}+1}^{\infty }{\alpha }_k^{1/(1-r)}\right)}^{1-r},$$

here $s^{*}>n$ is the biggest natural number such that

$$s-n\le {\alpha }_s^{1/r}\sum _{k=1}^s{\alpha }_k^{-1/r},\mathrm{for}\mathrm{all}s\in (n,s^{*}].$$

The exact upper bound is realized by the sequence $m^{*}={\left\{m_k^{*}\right\}}_{k=1}^{\infty }$ from $\mathcal{M}$ such that

$$m_k^{*}=\left\{\begin{array}{c}({\alpha }_k^{-1/r}·{(s^{*}-n)}^{1/(1-r)}·{\left(\sum _{i=1}^{s^{*}}{\alpha }_i^{-1/r}\right)}^{1/(r-1)}·{\mathcal{E}}_n^{1/(r-1)},k\in [1,s^{*}],\hfill \\ {\alpha }_k^{1/(1-r)}·{\mathcal{E}}_n^{1/(r-1)},k>s^{*}.\hfill \end{array}\right.$$

Note. Since solutions of these extremal problems may be of an independent interest, in the paper [26], the authors propose a new method of finding these solutions, which leads to the required result by essentially shorter and more transparent way.

Let us combine all of the above into the following provisions. It is easily shown that the conditions of [26] can be implemented for the field function $f(x,y,t)$. Indeed, ${\lambda }_i(t)>0,{\lambda }_i(t)\downarrow ,{lim}_{i\to \infty }{\lambda }_i(t)\to 0,L_i(x,y)=m_i^r$ and it is clear that ${\sum }_{i=1}^{\infty }{\lambda }_i^{1/(1-r)}(t)<\infty$ in the case of $r\in (0,1)$. This means that the results of [26] provide an effective tool for the analysis of the field function18. Notice that the above results give an opportunity to make the estimate for $\mathcal{N}\to \infty$ (word length).

Ideology of P = NP. Due to checking the first M instances (see 1st stage: configuration of nodes) with total computational complexity of $O(n^k)$, we can identify $(N-M)$ instances without checking them19, here M is polynomially bounded. Agreement. Without loss of generality it can be assumed that a functional type of ${\lambda }_i(t)=f_i(t)$ is not principled; all required $\{g_i,g_{i+1}\}$ are defined at time $t=T^{*}$. Note that this configuration is an example of the soft mathematical modeling [25].

Let us implement the results of claim 5.2. We can assume without loss of generality that ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(t)}{4}}·\varpi (N,t)={\displaystyle \frac{{\sum }_{i=5}^N{\lambda }_i(t)}{N-4}}$, where $\varpi (N,t)$ is a stabilization function.

Claim 5.3.

The value of the field function at the barycentre20 ${\displaystyle \frac{{\sum }_{i=1}^N{\lambda }_i(t)}{N}}$ equals $\kappa ·\varpi (N,t)·{\mathcal{E}}_n(\lambda (t),1)$, here $\kappa =const$, see (i).

Proof. 

It is easily shown that ${\displaystyle \frac{{\sum }_{i=1}^{s^{*}}{\lambda }_i(T^{*})}{s^{*}}}=\kappa ·\varpi (s^{*},T^{*})·{\mathcal{E}}_n(\lambda (T^{*}),1)$, where $s^{*}=M$. Taking into account ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i(T^{*})}{4}}·\varpi (N,T^{*})={\displaystyle \frac{{\sum }_{i=5}^N{\lambda }_i(T^{*})}{N-4}}$, by induction, we obtain the statement. □

Corollary 5.4.

We get $\varpi (N,T^{*})={\displaystyle \frac{4·{\sum }_{i=1}^4{\lambda }_i(T^{*})}{4·N·\kappa ·{\mathcal{E}}_n(\lambda (T^{*}),1)-(N-4)·{\sum }_{i=1}^4{\lambda }_i(T^{*})}}$, ${\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i^r(t)}{4}}\times \times {(\varpi (N,t))}^r={\displaystyle \frac{{\sum }_{i=5}^N{\lambda }_i^r(t)}{N-4}}$ and ${\displaystyle \frac{{\sum }_{i=1}^N{\lambda }_i^r(t)}{N}}$ equals $\upsilon ·{(\varpi (N,t))}^r·{\mathcal{E}}_n(\lambda (t),0<r<1)$, here $\upsilon =const$, see (ii). The proof is omitted.

Remark 8.

Claim 5.3 links two scientific schools: CTSA and Stepanets’ school.

Using claim 5.3, corollary 5.4 and the results of the Bogolyubov principle of the decay of correlations for an infinite three-dimensional systems [27] for field function (see Figure 10), we have the following system $\mathcal{U}$ for determination of ${\lambda }_i(t),$$i>s^{*}$.

$$\mathcal{U}=\left\{\begin{array}{c}{\displaystyle \frac{{\sum }_{i=1}^N{\lambda }_i(t)}{N}}=\kappa ·\varpi (N,t)·{\mathcal{E}}_n(\lambda (t),1),(1)\hfill \\ {\displaystyle \frac{{\sum }_{i=1}^4{\lambda }_i^r(t)}{4}}·{(\varpi (N,t))}^r={\displaystyle \frac{{\sum }_{i=5}^N{\lambda }_i^r(t)}{N-4}},(2)\hfill \\ {\displaystyle \frac{{\sum }_{i=1}^N{\lambda }_i^r(t)}{N}}=\upsilon ·{(\varpi (N,t))}^r·{\mathcal{E}}_n(\lambda (t),0<r<1),(3)\hfill \\ {\displaystyle \frac{d}{dt}}F(t)=-\mathcal{L}F(t)+\mathbf{a}{\mathcal{L}}^{\mathrm{int}}{F(t),F(t)|}_{t=0}=F(0).(4)\hfill \end{array}\right.$$

Let us comment. (4) is the initial-value problem of the BBGKY hierarchy21. Taking into account $s^{*}=M$, for operators $\mathcal{L}$ (which is defined by the Poisson bracket of free particles with boundary conditions on $\partial W_{s^{*}}$) and $\alpha {\mathcal{L}}^{\mathrm{int}}$ (in the case of $t>0$) we get

$${(\mathcal{L}F(t))}_{s^{*}}(x_1,\dots ,x_{s^{*}})={\mathcal{L}}_{s^{*}}{F}_{s^{*}}(t)=\sum _{i=1}^{s^{*}}{〈p_i,{\displaystyle \frac{\partial }{\partial q_i}}〉}_{\partial W_{s^{*}}}{F}_{s^{*}}(t,x_1,\dots ,x_{s^{*}}).$$

(5.1)

Each particle is characterized by the phase coordinates $(q_i,p_i)\equiv x_i\in {\mathbb{R}}^3\times {\mathbb{R}}^3,i\ge 1.$

$$\begin{array}{c}\hfill {(\mathbf{a}{\mathcal{L}}^{\mathrm{int}}F(t))}_{s^{*}}(x_1,\dots ,x_{s^{*}})=\\ \hfill ={\sigma }^2\sum _{i=1}^{s^{*}}{\int }_{{\mathbb{R}}^3\times {\mathbb{S}}_{+}^2}d{p}_{s^{*}+1}d\eta \langle \eta ,(p_i-p_{s^{*}+1})\rangle \times (F_{s^{*}+1}(t,x_1,\dots ,q_i,p_i^{*},\dots ,x_{s^{*}},\\ \hfill ,q_i-\sigma \eta ,p_{s^{*}+1}^{*})-F_{s^{*}+1}(t,x_1,\dots ,x_{s^{*}},q_i+\sigma \eta ,p_{s^{*}+1})),\end{array}$$

(5.2)

$\langle \eta ,(p_i-p_{s^{*}+1})\rangle ={\sum }_{\alpha =1}^3{\eta }^{\alpha }(p_i^{\alpha }-p_{s^{*}+1}^{\alpha }),{\mathbb{S}}_{+}^2=\{\eta \in {\mathbb{R}}^3\left|\right|\eta |=1,\langle \eta ,(p_i-p_{s^{*}+1})\rangle >0\},$ for impulses $p_i^{*},p_{s^{*}+1}^{*}$: $p_i^{*}=p_i-\eta \langle \eta ,(p_i-p_{s^{*}+1})\rangle ,p_{s^{*}+1}^{*}=p_{s^{*}+1}+\eta \langle \eta ,(p_i-p_{s^{*}+1})\rangle .$ Also for (4) we take into account s-particles correlation functions $G_{s^{*}}(t)$ [27], so

$$F_{\left|Y\right|}(t,Y)=\sum _{\mathrm{P}:Y={\bigcup }_i{X}_i}\prod _{X_i\subset \mathrm{P}}{G}_{|X_i|}(t,X_i),Y\equiv (x_1,\dots ,x_{s^{*}}).$$

(5.3)

The chaos condition for s-particles correlation functions is as follows [27]:

$$\begin{array}{c}\hfill G_{s^{*}}(t,x_1,\dots ,x_{s^{*}})=\\ \hfill =\sum _{n=0}^{\infty }{\displaystyle \frac{1}{n!}}{\int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3}d{x}_{s^{*}+1}\dots d{x}_{s^{*}+n}{\mathbf{U}}_{s^{*}+n}(t,1,\dots ,s^{*}+n)\prod _{i=1}^{s^{*}+n}{G}_1(0,x_i){\chi }_{{\Gamma }_{s^{*}+n}}.\end{array}$$

(5.4)

Note that (5.4) can be obtained on the basis of the solutions of Liouville’s equations.

Taking into account $\mathcal{U}$ and (5.1) – (5.4), $f(x,y,t)$ is a phase space: each z-axis at the node is a particle of unit mass and diameter $\rho >0$ such that $|q_i-q_j|\ge \rho$ and ${\lambda }_{1\le i}^{0<r<1}(t)\in (g_{i+1},g_i)$. This implies that, ${lim}_{t\to T^{*}-0}{\lambda }_i(t)\to \{g_i,g_{i+1}\}$ such that $r=\omega (t)$ (taking into account $t=T^{*}$) and the number of r is polynomially bounded.

Remark 9. For a metrizable space $\mathcal{S}=\langle {\mathbb{R}}^3,\tau \rangle$ the following condition holds: existence of the space of sequences of integrable translation-invariant functions [27].

The implementation model. 1. Deterministic Turing machine (DTM). Using the DTM with polynomial memory, we operate with pre-prepared tape ${\chi }_1<{\chi }_2<\dots <{\chi }_l<\dots$ (see 2nd stage). Total computational complexity of the analysis of the field function (see 3rd stage) is checking the first M instances with total computational complexity of $O(n^k)$ (see ideology of P = NP), checking ${\lambda }_{1\le i\le s^{*}}^{0<r<1}(t)$ with total computational complexity of $O(n^l)$ and analysis of $\mathcal{U}$ with total computational complexity of $O(n^n)$. Thus we have leveled the surplus of instances $(N-M)$ and P = NP is performed with pre-prepared tape. 2. Quantum Turing machine (QTM) and DTM. Using the QTM, we perform the above tape [28].

Proposition 5.5.

Using the QTM, theorem 5.1 is performed for BQP = NP.

Remark 10. For proposition 5.5: ${\mathcal{H}}_i$ is a vector in Hilbert space (see 1st stage).

5.1. Appendix

Let us give a strengthening of the results [26] in the case of $|m_k|\le 1,{\alpha }_k\ge 0$.

Let $\mathcal{M}$ be a set of all sequences $m={\left\{m_k\right\}}_{k=1}^{\infty }$ of real numbers such that

$$|m_k|\le 1,k=1,2,\dots ,(k\in \mathbb{N}),$$

(5.5)

and $\left|m\right|:={\sum }_{k=1}^{\infty }{m}_k\le 1$. Let also N be a fixed positive integer and ${\mathcal{A}}_N$ be a set of all nonincreasing sequences $\alpha ={\left\{{\alpha }_k\right\}}_{k=1}^{\infty }$ of nonnegative numbers such that ${\alpha }_k=0,k>s$.

Denote by ${\gamma }_n=\{k_1,k_2,\dots ,k_n\}$ a collection of n different integers $n=0,1,2,\dots ,$ and for any $m\in \mathcal{M}$ and $\alpha \in {\mathcal{A}}_N$, set

$${\mathcal{E}}_n(m)={\mathcal{E}}_n(\alpha ,m):=\sum _{k=1}^{\infty }{\alpha }_k·m_k-\underset{{\gamma }_n}{sup}\sum _{k\in {\gamma }_n}{\alpha }_k·m_k=\sum _{k=1}^N{\alpha }_k·m_k-\underset{{\gamma }_n}{sup}\sum _{k\in {\gamma }_n}{\alpha }_k·m_k$$

and

$${\mathcal{E}}_n={\mathcal{E}}_n(\alpha ):=\underset{m\in \mathcal{M}}{sup}{\mathcal{E}}_n(m)=\underset{m\in \mathcal{M}}{sup}{\mathcal{E}}_n(\alpha ,m).$$

(5.6)

Lemma 5.6.

Let $\alpha \in {\mathcal{A}}_N,N\in \mathbb{N}$. Then for any $n=0,1,\dots ,$$n<N$

$${\mathcal{E}}_n(\alpha )=\sum _{k=n+1}^N{\alpha }_k.$$

In this case, the exact upper bound on the right-hand side of the relation (5.6) is realized by the sequence $m^{*}={\left\{m_k^{*}\right\}}_{k=1}^{\infty }$ such that

$$m_k^{*}=\left\{\begin{array}{c}1,k\in [1,N],\hfill \\ -1,k\in [N+1,2·N-1],\hfill \\ 0,k>2·N-1.\hfill \end{array}\right.$$

(5.7)

Proof. 

Let ${\mathcal{M}}^{\prime }$ be a set of all sequences $m\in \mathcal{M}$ such that ${\alpha }_1·m_1\ge \dots \ge {\alpha }_N·m_N$. Then

$${\mathcal{E}}_n=\underset{m\in {\mathcal{M}}^{\prime }}{sup}{\mathcal{E}}_n(m).$$

Indeed, let $m\in \mathcal{M}$. We construct a sequence $m^{\prime }\in {\mathcal{M}}^{\prime }$ such that ${\mathcal{E}}_n(m)\le {\mathcal{E}}_n(m^{\prime })$.

Step 1. If ${max}_{k\in [1,N]}{\alpha }_k·m_k={\alpha }_{k_1}·m_{k_1},k_1>1$, then represent the number $m_{k_1}$ in the form $m_{k_1}={\overline{m}}_{k_1}+{\tilde{m}}_{k_1}$, where ${\alpha }_1·(m_1+{\overline{m}}_{k_1})={\alpha }_{k_1}·m_{k_1}$. Consider the sequence $m^{(1)}={\left\{m_k^{(1)}\right\}}_{k=1}^{\infty }$ such that

$$m_k^{(1)}=\left\{\begin{array}{c}{m}_1+{\overline{m}}_{k_1},k=1,\hfill \\ {\tilde{m}}_{k_1},k=k_1,\hfill \\ m_k,k\ne 1,k_1.\hfill \end{array}\right.$$

Since ${\alpha }_1\ge {\alpha }_{k_1}$, we have $m_1^{(1)}=m_1+{\overline{m}}_{k_1}\le m_{k_1}$, $|m^{(1)}|=|m|$, ${\mathcal{E}}_n(m^{(1)})\ge {\mathcal{E}}_n(m)$ and ${max}_{k\in [1,N]}{\alpha }_k·m_k^{(1)}={\alpha }_1·m_1^{(1)}$. If ${max}_{k\in [1,N]}{\alpha }_k·m_k={\alpha }_1·m_1$, then we set $m^{(1)}:=m$.

Step 2. If ${max}_{k\in [2,N]}{\alpha }_k·m_k^{(1)}={\alpha }_{k_1}·m_{k_2}^{(1)},k_2>2$, we represent the number $m_{k_2}$ in the form $m_{k_2}={\overline{m}}_{k_2}+{\tilde{m}}_{k_2}$, where ${\alpha }_2·(m_2+{\overline{m}}_{k_2})={\alpha }_{k_2}·m_{k_2}$. Consider the sequence $m^{(2)}={\left\{m_k^{(2)}\right\}}_{k=1}^{\infty }$ such that

$$m_k^{(2)}=\left\{\begin{array}{c}{m}_2^{(1)}+{\overline{m}}_{k_2},k=2,\hfill \\ {\tilde{m}}_{k_2}^{(1)},k=k_2,\hfill \\ m_k^{(1)},k\ne 2,k_2.\hfill \end{array}\right.$$

Similarly, we have $m_2^{(1)}=m_2+{\overline{m}}_{k_2}\le m_{k_2}$, $|m^{(2)}|=|m^{(1)}|=|m|$, ${\mathcal{E}}_n(m^{(2)})\ge {\mathcal{E}}_n(m^{(1)})\ge {\mathcal{E}}_n(m)$ and ${\alpha }_1·m_1^{(2)}\ge {max}_{k\in [2,N]}{\alpha }_k·m_k^{(2)}={\alpha }_2·m_2^{(2)}$. If ${max}_{k\in [2,N]}{\alpha }_k·m_k^{(2)}={\alpha }_2·m_2^{(1)}$, then we set $m^{(2)}:=m^{(1)}$.

Step N. Continuing this procedure, at Nth step we finally construct the sequence $m^{(N)}={\left\{m_k^{(N)}\right\}}_{k=1}^{\infty }\in \mathcal{M}$ such that $|m^{(N)}|=|m|$, ${\alpha }_1·m_1^{(N)}\ge {\alpha }_2·m_2^{(N)}\ge \dots \ge {\alpha }_N·m_N^{(N)}$ and ${\mathcal{E}}_n(m^{(N)})\ge {\mathcal{E}}_n(m^{(N-1)})\ge \dots \ge {\mathcal{E}}_n(m^{(1)})\ge {\mathcal{E}}_n(m)$. Setting $m^{\prime }:=m^{(N)}$ we see that it satisfies all the necessary relations.

By virtue of (5.5), for any sequence $m\in {\mathcal{M}}^{\prime }$ we have

$${\mathcal{E}}_n(m)=\sum _{k=n+1}^N{\alpha }_k·m_k\le \sum _{k=n+1}^N{\alpha }_k.$$

To finish the proof it is sufficient to make sure that the sequence $m^{*}={\left\{m_k^{*}\right\}}_{k=1}^{\infty }$ of the form (5.7) belongs to the set $\mathcal{M}$ and

$${\mathcal{E}}_n(m^{*})=\sum _{k=n+1}^N{\alpha }_k.$$

□

6. General Comments

Section 2.

1. Let us comment an informal moment for a visual perception. $PF$ is a “sandwich-structure”: $G(V,E)$ is a support and $\diamond PF$ is a superstructure over the support.

2. Labeled plafal. Generally, we use the index collection in an alternative form $V^{•}(PF)(E^{•}(PF))\stackrel{\{{\pi }_1,{\pi }_2\}}{\to }\left\{\overline{1,\eta }\right\}$ with due regard for $\varnothing :\varnothing \left\{\overline{1,\eta }\right\}$, here $\eta \in \{i,j\}$.

Section 4.

${\mathbb{R}}^3\cap U=⌀$ means that a universe of $\mathcal{U}$-standard objects is located outside of ${\mathbb{R}}^3$. Let us provide a visual perception by analogy in computer sciense. ${\mathbb{R}}^3$ is a directory and U is a collection of external files.

Section 5.

Historical background. Dual role of the basis functions of serendipity finite elements is proved by O. Zienkiewicz (1968). His group constructed a biquadratic basis (SFE·8), but Zienkiewicz’s paradox is formed: in some nodes exist negative loads (unnatural spectrum). In 1982 Khomchenko constructed a 13-parametric bicubic basis (SFE·12) with positive loads [20], [21], thus Zienkiewicz’s paradox was solved. Methods of constructing alternative (physically correct) bases are presented in [16].

Perspective. Plafal (*-plafal) complex and plafal (*-plafal) cell complex are generalizations of the plafal (*-plafal), and this will be the object of another paper. We will give a short description. For any abstract simplicial complex (n-dimensional cell complex) let there be a correspondence between each face (k-skeleton) and $\mathcal{U}$-standard (${\mathcal{U}}^{*}-\mathrm{standard}$ respectively).