Markov Theory as a Strict Special Case of the UPGS Universal Axiom System

Author: Zhang Suhang (Founder of the He-Luo Mathematical School)

Abstract

This paper independently, purely, and rigorously demonstrates the axiomatic hierarchy and subordination relationship between the UPGS Universal Pattern Global System and Markov theory. All engineering applications, auxiliary models, probability modifications, and nonlinear extensions are completely excluded, retaining only the intrinsic axiomatic structures of both systems for comparative deduction. Through axiom comparison, constraint dimensional reduction, set inclusion, and irreversible hierarchy derivation, this paper strictly proves that Markov theory is entirely contained within the UPGS system, serving as a strict mathematical special case of UPGS under discrete, memoryless, linear time-invariant sequential evolution. UPGS constitutes the top-level meta-axiomatic framework, while Markov theory is a restricted derived subsystem, with a strict, irreversible, formalizable superordinate-subordinate relationship.

Keywords: UPGS Universal Axiom; Markov process; Axiom hierarchy; System inclusion; Sequential evolution; Theoretical special case

I. Introduction

In modern stochastic process and system theory, Markov theory has long been regarded as a self-consistent and independent fundamental mathematical system, possessing a standardized state space, transition rules, and iterative evolution paradigm.

However, when examined from the higher-dimensional perspective of the UPGS Universal Pattern Global System:

The Markov system is not a primitive underlying axiomatic system but rather a restricted sub-model formed by imposing artificial constraints upon the universal top-level rules.

To completely clarify the ontological relationship between these two systems, this paper adopts a purely axiomatic, zero-redundancy, scenario-free proof paradigm, completing a rigorous subordination argument based solely on the core definitions of both theories, thereby establishing an irreversible hierarchical order between them.

II. Core Top-Level Axioms of UPGS (Primordial Minimal System)

UPGS is the highest-order meta-axiomatic framework governing all temporally evolving systems, holding universally without conditions. It comprises three primordial rules:

1. State Existence Axiom

   Any evolvable system must consist of a set of distinguishable, definable, and modelable states (discrete or continuous).

2. Transformation-Transmission Axiom

   The future state of a system is generated from its current state through a deterministic or stochastic mapping, obeying a state-iteration causal structure.

3. Global Self-Consistency Axiom

   All subsystems, branch models, and evolution rules must be compatible with the top-level universal structure; no system-level violation or super-rule exception exists.

These three axioms form the necessary and sufficient fundamental basis for all dynamical systems, without preconditions, domain limitations, or universal coverage.

III. Core Ontological Structure of Markov Theory (Pure Kernel Extraction)

Removing all extended applications, the sole ontology of Markov theory is defined by two layers of constraints:

3.1 State Constraint

The system state is drawn from a finite or countably infinite discrete state set.

3.2 Evolution Constraint (Memoryless Property Core)

The probability of the future state depends only on the current state, completely independent of historical path dependence:

P(x_{k+1} | x_k, x_{k-1}, ..., x_0) = P(x_{k+1} | x_k)

3.3 Simplest Algebraic Structure

The iterative evolution of a Markov system is uniformly expressed as a linear matrix transition:

P_{k+1} = T P_k

where P_k is the state probability distribution vector at time k, and T is a constant state transition matrix. This equation is the sole native evolution structure of the Markov system.

IV. Rigorous Proof: Complete Containment of the Markov System within UPGS

4.1 The Markov System Satisfies the UPGS State Existence Axiom

Markov theory presupposes a discrete, distinguishable state space, fully conforming to the UPGS top-level requirement that "system states be definable, distinguishable, and evolvable."

4.2 The Markov System Satisfies the UPGS Transformation-Transmission Axiom

The Markov matrix iteration

P_{k+1} = T P_k

is essentially a stochastic transmission subclass structure of the UPGS "current-state mapping to future-state" rule, fully obeying the universal causal transmission protocol.

4.3 Markov is a Strongly Constricted Subsystem of UPGS

The universal UPGS system has no boundary constraints and is naturally compatible with:

· Discrete states / Continuous states

· Memoryless evolution / Path-dependent evolution

· Linear mappings / Nonlinear mappings

· Time-invariant systems / Time-varying systems

In contrast, Markov actively imposes three strong constraints:

1. Retains only discrete states

2. Retains only memoryless transitions

3. Retains only linear time-invariant transitions

That is: Markov system = UPGS universal system + three layers of artificial constraint dimensional reduction

4.4 Set-Theoretic Strict Inclusion Theorem

Let:

· U: The universal set of all evolutionary systems defined by UPGS

· M: The set of Markov systems

· C₁, C₂, C₃ be the discrete constraint, memoryless constraint, and linear time-invariant constraint, respectively.

Then we have the formal definition:

M = { x ∈ U | x satisfies C₁, C₂, C₃ }

From this, the strict inclusion relationship directly follows:

M ⊂ U

V. Proof of Hierarchical Irreversibility

5.1 UPGS Holds Independently Without Prerequisites

The three primordial axioms of UPGS hold without dependence on stochastic processes, state matrices, or temporal iteration; they remain valid in pure structural space, belonging to the meta-axiomatic level.

5.2 Markov Cannot Hold Independently of UPGS

Markov's state definitions, transition causality, iterative structure, and evolutionary self-consistency all depend on the underlying universal rules of UPGS as a prerequisite for existence.

5.3 Irreversible Hierarchical Conclusion

1. UPGS can generate Markov (simply by constraint dimensional reduction).

2. Markov cannot generate UPGS (lacking the degrees of freedom of the universal axioms).

Establishing the absolute hierarchy:

UPGS (Top-Level Primordial Axiom System)

>

Markov (Restricted Special-Case Model)

VI. Final System Relationship Determination

1. Markov theory is not an independent underlying theory but rather a constrained subsystem within the UPGS universal framework.

2. All formulas, structures, evolutionary behaviors, and probabilistic rules of Markov are entirely governed and contained by UPGS.

3. UPGS belongs to a meta-mathematical axiom system; Markov belongs to a restricted derived model in a specific domain.

4. The Markov process is the standard particular solution of UPGS under discrete, memoryless, linear time-invariant stochastic evolution scenarios.

VII. Conclusion

Through pure axiomatic comparison, constraint dimensional reduction deduction, and formalized set-inclusion proof, this paper has completely clarified the ontological relationship between the two systems.

The final rigorous conclusion is:
Markov theory is entirely subordinate to the UPGS Universal Pattern Global Axiom System, serving as a strict mathematical special case obtained by imposing discrete, memoryless, and linear time-invariant constraints upon UPGS.

UPGS, as the universal top-level primordial framework, governs all temporally evolving systems; Markov theory is its self-consistent derived branch, forming an irreversible strict hierarchical structure of axiom–model, universal–local, and primordial–special case.

References

[1] Zhang Suhang. Fundamental Theory of the UPGS Universal Pattern Global System [Z]. 2026.
[2] Wang Zikun. Stochastic Processes [M]. Beijing: Beijing Normal University Press.

Date of Finalization: May 2026