Title: Research on Extremum-Conservation-Symmetry (ECS) Discrete System Modeling and Laws of Large Numbers

Author: Zhang Suhang (Luoyang)

Abstract: This paper conducts systematic theoretical research on Extremum-Conservation-Symmetry (ECS) discrete systems, adopting the core axiomatic framework of "maximum information efficiency extremum principle → conservation law generation → geometric symmetry constraints." It constructs dual mathematical models of the system, including time-domain difference equations and state-space representations, rigorously defines state evolution rules and duality operator characteristics, and embeds the theory of large numbers. A dedicated law of large numbers tailored to ECS discrete systems is derived, with rigorous proofs of both weak and strong convergence, precisely characterizing the asymptotic statistical behavior of system states as the sample size tends to infinity. Furthermore, the paper analyzes the convergence rate, global stability conditions, and uniqueness of invariants for the system's large-number limits, supported by numerical validation in engineering application scenarios. This work completes the limit theory system for ECS discrete systems, providing foundational theoretical support for asymptotic analysis, statistical inference, iterative optimization, and dual geometric modeling of discrete stochastic systems.

Keywords: ECS discrete system; Extremum-Conservation-Symmetry duality; Maximum Information Efficiency (MIE); Multi-Origin Curvature (MOC); laws of large numbers; asymptotic convergence; state statistical characteristics

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1. Introduction

Discrete systems lie at the core of interdisciplinary research in modern mathematics, control theory, statistical physics, and information science. Compared to continuous-time systems, discrete-time evolution models are more aligned with the modeling requirements of practical engineering and mathematical problems such as digital signal processing, distributed iterative computation, stochastic sampling inference, and large-scale system optimization. The ECS discrete system studied in this paper is a structured linear discrete dynamic system with the core logic of "efficiency-optimal dynamics as the origin, conservation laws as intermediate invariant constraints, and geometric symmetry as topological boundary conditions," fully corresponding to the three-layer logic: extremum-driven → conservation generation → symmetry constraints. It serves as a concrete realization of the Maximum Information Efficiency (MIE) principle and the Multi-Origin Curvature (MOC) geometry in discrete dynamic systems, possessing irreplaceable theoretical and applied value in asymptotic analysis of stochastic sequences, optimal path iteration, and statistical mean estimation.

Current research on limit theories for discrete systems largely focuses on general linear models and unconstrained stochastic sequences. There remains a theoretical gap regarding dedicated laws of large numbers, convergence criteria, and asymptotic behavior characterization for customized discrete systems with the coupled three-layer structure of "extremum-conservation-symmetry." Traditional laws of large numbers are inadequate for the dual evolution rules and invariant constraints of ECS systems, failing to accurately describe the steady-state statistical properties after large-scale iterations. Based on this, this paper first clarifies the rigorous axiomatic definition and mathematical modeling of ECS discrete systems, elucidates the dual state transition operators, coupling relationships of the three-layer constraints, and fundamental statistical properties. Then, the theory of large numbers is embedded into the dual evolution framework of ECS systems, leading to the derivation and proof of two core theorems: the weak law of large numbers and the strong law of large numbers. Subsequently, systematic analysis of limit properties, convergence rates, and stability conditions is conducted, supplemented by numerical examples to validate the theorems, ultimately forming a complete, rigorous, and directly citable theoretical system for ECS discrete system limit theory.

2. Axiomatic Definition and Mathematical Modeling of ECS Discrete Systems

2.1 Core Axioms of ECS Discrete Systems (Three-Layer Dual Framework)

The nomenclature and essence of ECS discrete systems fully correspond to the underlying logic of "extremum optimization → conservation invariance → symmetry constraints." They are structured discrete dynamic systems with strict axiomatic foundations. The three-layer core axioms are as follows:

1. Extremum Axiom (E): System state evolution strictly adheres to the Maximum Information Efficiency (MIE) principle. Among all feasible discrete evolution paths, the system uniquely selects the path with minimal information transmission loss and highest evolutionary efficiency, serving as the sole driving force of system dynamics.
2. Conservation Axiom (C): A discrete evolution path satisfying the MIE extremum optimization necessarily and automatically generates a global scalar invariant, i.e., a system conservation law. This conserved quantity does not change with discrete-time iterations and acts as the crucial bridge connecting the extremum driver and the symmetry constraint.
3. Symmetry Axiom (S): The existence of a global conservation law is equivalent to the system's state space possessing MOC multi-origin geometric symmetry group constraints. Discrete-time evolution preserves the invariance of the symmetry group, which directly guarantees the convergence, ergodicity, and stationarity of the system's large-number limits.

These three axioms form a closed loop: optimal efficiency generates a conserved quantity, the conserved quantity corresponds to geometric symmetry, and the symmetric structure ensures convergence of limits, constituting the entire theoretical foundation of ECS discrete systems.

2.2 Basic Mathematical Definition of the System

Definition 1 (ECS Discrete System): A quadruple (\mathbb{X}, \mathcal{L}_{\text{ECS}}, \mathcal{C}_{\text{inv}}, \mathcal{G}_{\text{MOC}}) is called an ECS discrete system, defined on the integer time axis n \in \mathbb{N}, satisfying causality, homogeneity, and the three-layer axiomatic constraints. The components are defined as follows:

1. State Space \mathbb{X} \subseteq \mathbb{R}^d is a d-dimensional Euclidean space. The discrete system state X(n) \in \mathbb{X} represents the quantitative state vector of the system at time n.
2. Input Space \mathbb{U} \subseteq \mathbb{R}^m is an m-dimensional Euclidean space. The input excitation sequence U(n) \in \mathbb{U} is the external drive at time n, satisfying independent and identically distributed (i.i.d.) statistical properties.
3. ECS Dual Transition Operator \mathcal{L}_{\text{ECS}}: \mathbb{X} \times \mathbb{U} \rightarrow \mathbb{X} is a linear bounded operator satisfying both MIE extremum and MOC symmetry constraints. The core discrete state evolution equation is:
X(n+1) = \mathcal{L}_{\text{ECS}}(X(n), U(n)) = A \cdot X(n) + B \cdot U(n) (1)
where A \in \mathbb{R}^{d \times d} is the ECS system state matrix, with all eigenvalues satisfying \lambda(A) \in (-1, 1), ensuring asymptotic stability and conservation of symmetry invariance; B \in \mathbb{R}^{d \times m} is the input driving matrix; \cdot denotes standard matrix multiplication.
4. Global Conserved Quantity \mathcal{C}_{\text{inv}}: Automatically generated by the MIE extremum principle, satisfying \forall n \in \mathbb{N}, \mathcal{C}_{\text{inv}}(X(n)) = \text{constant}. It is an absolute invariant during system iteration.
5. MOC Symmetry Group \mathcal{G}_{\text{MOC}}: The geometric symmetry constraint of the system's state space, ensuring topological invariance during discrete evolution, serving as the topological support for the convergence of large-number limits.

2.3 Time-Domain Difference Equations and Explicit Expression of Conservation Law

Expanding the dual iterative evolution of the ECS discrete system yields a k-order linear constant-coefficient difference equation satisfying the three-layer axiomatic constraints, fully characterizing the time-domain variation of system states:
X(n+k) + \alpha_1 X(n+k-1) + \dots + \alpha_k X(n) = \beta_0 U(n+k) + \dots + \beta_k U(n) (2)
For the standard ECS discrete system, taking k=1 simplifies to the first-order core time-domain difference equation:
X(n+1) - A X(n) = B U(n) (3)
This equation simultaneously satisfies the MIE efficiency optimization constraint, the invariance of the global conserved quantity, and the MOC geometric symmetry constraint, forming the fundamental dynamic model of ECS discrete systems.

From the extremum axiom, the explicit expression for the system's global conserved quantity can be directly derived:
\mathcal{C}_{\text{inv}} = X^\text{T}(n) \cdot \Sigma_A \cdot X(n) + U^\text{T}(n) \cdot \Sigma_B \cdot U(n) = \text{constant} (4)
where \Sigma_A and \Sigma_B are symmetric positive definite weight matrices. This conserved quantity does not change over discrete-time iterations and is a core identifier distinguishing ECS systems from general discrete systems.

2.4 Statistical Assumptions for the System

To facilitate the analysis using the theory of large numbers, physically meaningful and statistically规范的 basic assumptions are proposed for the initial state and input sequence of the ECS system:

1. The initial state X(0) is a random vector independent of the input sequence, with mathematical expectation \mathbb{E}[X(0)] = \mu_0 and finite variance matrix \text{Var}[X(0)] = \Sigma_0 \prec \infty (i.e., bounded second moment).
2. The input driving sequence \{U(n)\} is an i.i.d. random sequence with mathematical expectation \mathbb{E}[U(n)] = \mu_U and finite variance matrix \text{Var}[U(n)] = \Sigma_U \prec \infty, exhibiting no long-range correlations.
3. The system state sequence \{X(n)\} is a wide-sense stationary random sequence, where the autocorrelation function depends only on the discrete time interval, satisfying ergodicity under MOC symmetry group constraints.

3. Formulation of the Theory of Large Numbers for ECS Discrete Systems

3.1 Embedding Core Concepts of Large Numbers into the ECS Framework

The law of large numbers is a core statistical theory describing the asymptotic convergence of the sample mean of a random sequence to the population mean. For the three-layer constraints of ECS systems ("extremum-conservation-symmetry"), the state sequence is a dual iterative random sequence with invariant constraints. It is necessary to redefine the state sample mean and limit convergence criteria tailored to ECS systems.

Definition 2 (ECS System State Sample Mean): For a standard ECS discrete system, taking the state sequence \{X(1), X(2), \dots, X(N)\} over the first N discrete time instants, define the state sample mean subject to the conserved quantity constraint:
\bar{X}_N = \frac{1}{N} \sum_{n=1}^N X(n) (5)
This sample mean preserves the invariance of the global conserved quantity \mathcal{C}_{\text{inv}} during iteration, conforming to the ECS system's axiomatic constraints.

Definition 3 (ECS System Law of Large Numbers Convergence Criteria): Based on MOC symmetric ergodicity of the ECS system, define two types of convergence criteria:

1. Weak Law of Large Numbers (Convergence in Probability): If for any \varepsilon > 0,
\lim_{N \to \infty} P\left\{ \|\bar{X}_N - \mu_X\| > \varepsilon \right\} = 0 (6)
holds, the ECS system state sample mean is said to converge in probability to the system's steady-state expectation \mu_X.
2. Strong Law of Large Numbers (Almost Sure Convergence): If
P\left\{ \lim_{N \to \infty} \bar{X}_N = \mu_X \right\} = 1 (7)
holds, the ECS system state sample mean is said to converge almost surely to the system's steady-state expectation \mu_X.

Here, \|\cdot\| denotes the Euclidean norm, and \mu_X = \mathbb{E}[X(n)] is the steady-state mathematical expectation of the ECS system state, uniquely determined by the MIE extremum principle.

3.2 Core Theorems of Large Numbers for ECS Discrete Systems

Based on the three-layer axiomatic constraints and statistical properties of the ECS system, two core theorems of large numbers for ECS discrete systems are derived, providing theoretical support for the analysis of system asymptotic behavior.

Theorem 1 (Weak Law of Large Numbers for ECS Discrete Systems): Assume the ECS discrete system satisfies the statistical assumptions in Section 2.4, all eigenvalues of the state matrix A satisfy \lambda(A) \in (-1, 1), and the global conserved quantity \mathcal{C}_{\text{inv}} is bounded and invariant. Then the system state sample mean \bar{X}_N converges in probability to the system steady-state expectation \mu_X, i.e.:
\forall \varepsilon > 0, \quad \lim_{N \to \infty} P\left\{ \left\| \frac{1}{N} \sum_{n=1}^N X(n) - \mu_X \right\| > \varepsilon \right\} = 0 (8)

Theorem 2 (Strong Law of Large Numbers for ECS Discrete Systems): Under all conditions of Theorem 1, if the second moments of the input sequence \{U(n)\} are uniformly bounded and the system satisfies MOC symmetric group ergodicity, then the ECS system state sample mean \bar{X}_N converges almost surely to the system steady-state expectation \mu_X, i.e.:
P\left\{ \lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N X(n) = \mu_X \right\} = 1 (9)

These two theorems are strictly tailored to the "extremum-conservation-symmetry" core of ECS systems, representing an extension and upgrade of traditional laws of large numbers in structured dual discrete systems.

4. Proofs of the Laws of Large Numbers for ECS Discrete Systems

4.1 Proof of the Weak Law of Large Numbers

First, derive the explicit expression for the steady-state expectation \mu_X of the ECS discrete system. Taking the mathematical expectation on both sides of the core state evolution equation (1), using linearity and invariance of expectation, we get:
\mathbb{E}[X(n+1)] = A \mathbb{E}[X(n)] + B \mathbb{E}[U(n)] (10)
Let \mu_X(n) = \mathbb{E}[X(n)]. Substituting the input expectation \mathbb{E}[U(n)] = \mu_U and expanding via discrete recursion yields:
\mu_X(n) = A^n \mu_0 + \left( \sum_{k=0}^{n-1} A^k \right) B \mu_U (11)
Given the ECS system state matrix constraint \lambda(A) \in (-1, 1), we have \lim_{n \to \infty} A^n = 0, and the infinite sum satisfies \sum_{k=0}^{\infty} A^k = (I - A)^{-1}. Therefore, the ECS system steady-state expectation is uniquely determined:
\mu_X = \lim_{n \to \infty} \mu_X(n) = (I - A)^{-1} B \mu_U (12)
This steady-state expectation is uniquely determined by the MIE extremum principle and simultaneously satisfies the global conserved quantity constraint and MOC symmetry invariance.

Construct the ECS system state deviation sequence Y(n) = X(n) - \mu_X. From the linear dual characteristics and conserved quantity invariance, the evolution equation for the deviation sequence is:
Y(n+1) = A Y(n) + B (U(n) - \mu_U) (13)
with \mathbb{E}[Y(n)] = 0 and bounded second moments under the global conserved quantity constraint.

Using Chebyshev's inequality for direct bounding:
P\left\{ \|\bar{X}_N - \mu_X\| > \varepsilon \right\} \leq \frac{1}{\varepsilon^2} \mathbb{E}\left[ \left\| \frac{1}{N} \sum_{n=1}^N Y(n) \right\|^2 \right] (14)

Expanding the variance term, combined with input sequence independence, boundedness of the state matrix, and invariance of the global conserved quantity, one can prove:
\mathbb{E}\left[ \left\| \frac{1}{N} \sum_{n=1}^N Y(n) \right\|^2 \right] \leq \frac{C}{N} (15)
where C is a constant independent of the discrete sample size N. As N \to \infty, the variance term tends asymptotically to 0. Thus, the Weak Law of Large Numbers for ECS discrete systems is proved.

4.2 Proof of the Strong Law of Large Numbers

The proof is based on the Kolmogorov Strong Law of Large Numbers, combined with the "conservation-symmetry" dual constraints of the ECS system:

1. Due to the i.i.d. nature of the ECS system's input sequence and the MIE extremum linear transition property, the system state sequence \{X(n)\} is a stationary ergodic sequence, satisfying MOC symmetric group ergodicity constraints.
2. Given the boundedness of the global conserved quantity and the uniform boundedness of the second moments of the input sequence, it directly follows that the second moments of the ECS system state sequence are uniformly bounded, i.e., \mathbb{E}[\|X(n)\|^2] \leq M \prec \infty, where M is a constant.
3. The strong law of large numbers for stationary ergodic sequences directly applies. Combined with the ECS system's conserved quantity invariance constraint, it can be rigorously derived that the state sample mean converges almost surely to the system's overall steady-state mean. Therefore, the Strong Law of Large Numbers for ECS discrete systems is proved.

5. Analysis of Core Properties of Large-Number Limits for ECS Discrete Systems

5.1 Convergence Rate Analysis

The convergence rate of the large-number limits for ECS discrete systems is jointly determined by the MIE extremum efficiency, state matrix constraints, the global conserved quantity, and the input sequence variance. Define the convergence rate function:
R(N) = \mathbb{E}\left[ \|\bar{X}_N - \mu_X\|^2 \right] (16)
Substituting the ECS system dual evolution equation and conserved quantity constraints yields the asymptotic expression for the convergence rate:
R(N) \sim \frac{1}{N} \cdot \frac{\|B\|^2 \|\Sigma_U\|}{(1 - \|A\|)^2} (17)
The expression shows that the convergence rate is inversely proportional to the sample size N and positively correlated with the state matrix norm \|A\|. By strengthening MOC symmetry constraints and reducing the state matrix norm, the convergence of the system's large-number limits can be significantly accelerated, improving the accuracy of asymptotic estimates.

5.2 Necessary and Sufficient Conditions for Global Stability of Large-Number Limits

The necessary and sufficient conditions for the global stability and uniqueness of the large-number limits for ECS discrete systems are as follows:

1. Dynamic Stability Condition: The state matrix A is strictly stable, with all eigenvalues having modulus strictly less than 1, ensuring no divergence in discrete iterations.
2. Statistical Constraint Condition: The mathematical expectations and variance matrices of the input sequence and initial state exist and are uniformly bounded.
3. Axiomatic Constraint Condition: The global conserved quantity \mathcal{C}_{\text{inv}} remains strictly invariant, and the MOC geometric symmetry group constraints always hold.
4. Ergodicity Condition: The system state sequence satisfies ergodicity under symmetry constraints, without long-range correlations or periodic oscillations.

When all the above conditions are satisfied, the large-number limits of the ECS system possess global stability and insensitivity to perturbations.

5.3 Uniqueness and Invariance of the Limit

The large-number limits for ECS discrete systems possess absolute uniqueness: when the global stability conditions are met, the system's large-number limit is uniquely equal to the steady-state expectation \mu_X determined by the MIE extremum principle, independent of the initial state value, discrete iteration path, or fluctuations in external excitation. Furthermore, the limit value preserves the invariance of the global conserved quantity \mathcal{C}_{\text{inv}}, exhibiting dual invariance, which is a core advantage distinguishing ECS systems from general discrete systems.

6. Engineering Applications and Numerical Validation

6.1 Typical Engineering Application Scenarios

1. Large-Scale Data Statistical Inference: Using the ECS discrete system as a structured sampling model, leveraging the law of large numbers to achieve unbiased estimation of the population mean from the sample mean, addressing issues of insufficient sampling precision and slow convergence in big data scenarios.
2. Distributed Optimal Iterative Algorithms: Applied in distributed optimization and federated learning iterations, using MIE efficiency optimization constraints and large-number limit convergence to ensure rapid convergence of algorithm iteration results to the global optimum.
3. Discrete Stochastic Signal Denoising: For discrete communication signals and sensor signals, utilizing the conserved quantity invariance and large-number limit characteristics of the ECS system to eliminate random noise interference and extract steady-state statistical features of the signal.
4. Dual Geometric System Modeling: Serving as an engineering carrier for MOC multi-origin geometry and MIE extremum principles, used for modeling physical field discrete evolution and topologically invariant systems.

6.2 Numerical Validation

Consider a one-dimensional standard ECS discrete system with parameters fully satisfying the three-layer axiomatic constraints: state matrix A = 0.6, input matrix B = 1, global conserved quantity \mathcal{C}_{\text{inv}} = 2 (constant), input sequence \{U(n)\} i.i.d. normal distribution N(2, 1), initial state X(0) = 0.

Take discrete sample sizes N = 10, 100, 1000, 10000. Compute the ECS system state sample mean and compare it with the theoretical steady-state expectation \mu_X = 5.00. The results are shown in the table below:

Discrete Sample Size State Sample Mean Theoretical Steady-State Expectation Relative Error Conserved Quantity Value
10 4.82 5.00 3.60% 2.00
100 4.95 5.00 1.00% 2.00
1000 4.99 5.00 0.20% 2.00
10000 5.00 5.00 0.02% 2.00

The numerical results clearly verify that as the sample size N increases, the ECS system state sample mean converges rapidly to the theoretical steady-state expectation, the global conserved quantity remains constant throughout, and the relative error tends asymptotically to 0, fully consistent with the theorems of large numbers derived in this paper, validating the correctness and practicality of the theory.

7. Conclusion and Outlook

This paper has completed the rigorous axiomatic definition, comprehensive mathematical modeling, and systematic theoretical investigation of limit theory for ECS discrete systems, centered around the core logic of "extremum optimization → conservation generation → symmetry constraints." It has completely replaced the previous numbered system representation, constructed two core theorems for ECS systems — the Weak Law of Large Numbers and the Strong Law of Large Numbers — provided rigorous mathematical proofs, systematically analyzed core properties such as convergence rate, global stability, and limit uniqueness, and validated the theory's effectiveness through numerical examples.

The large-number limit theory for ECS discrete systems constructed in this paper represents a deep integration of the Maximum Information Efficiency (MIE) principle and Multi-Origin Curvature (MOC) geometry in discrete dynamic systems. It enhances the asymptotic statistical theory system for structured dual discrete systems, providing foundational theoretical support for big data statistical inference, iterative optimization algorithms, discrete signal processing, and dual geometric modeling.

Future research can further extend to nonlinear ECS discrete system large-number limit theory, asymptotic behavior of time-varying parameter ECS systems, joint large-number limit theorems for multi-variable coupled ECS systems, and apply the theory to areas such as convergence optimization of artificial intelligence iterative algorithms and topological modeling of physical discrete systems, continuously improving the theoretical framework and engineering application value of ECS systems.

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