MOC Higher-Dimensional Complex Space Reconstruction and the Intrinsic Geometric Definition of the Critical Strip

Author: Zhang Suhang (Bosley Zhang / Bi Sulin)
Research Affiliation: Luoyang Independent Researcher
System Affiliation: MOC‑MIE‑ECS‑UCE Unified Mathematical‑Physical Paradigm
Publication Date: 2026‑05‑17 (Foundational Series of the School – Paper No. 2)

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Abstract

Traditional complex analysis is built on the single‑origin flat complex plane, where the critical strip (0 < \Re(s) < 1) is an artificially truncated analytic region without intrinsic geometric necessity. This is the fundamental reason why the Riemann Hypothesis (RH) has, for over a century, been unable to lock the steady state of zeros from a geometric perspective.

Based on the previously established MOC axiomatic system of multi‑origin higher‑dimensional space, this paper performs a structural upgrade of the complex domain:

1. Embed the traditional two‑dimensional flat complex plane into a multi‑origin higher‑dimensional curved manifold to construct MOC higher‑dimensional complex space;
2. From multi‑base‑point coupling topology and cross‑origin projection layering, intrinsically derive the critical strip without any artificial interval definition;
3. Prove that the division into critical strip, critical line, and non‑critical regions is a natural consequence of dimensional coupling balance in MOC complex space;
4. Provide an initial geometric proof of the critical line \Re(s) = \frac12 as an equilibrium line, laying the complex‑space substructure for subsequent UCE unified curvature balance, ECS steady‑state constraints, and MIE zero evolution.

Keywords: Multi‑origin complex space; higher‑dimensional complex manifold; critical strip; Riemann Hypothesis; intrinsic geometry; cross‑base‑point projection

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1. Introduction

1.1 Core Defects of the Traditional Complex Plane

The underlying carrier of classical complex analysis is \mathbb{C} \cong \mathbb{R}^2, which carries the following strong constraints:

· A unique global origin;
· Globally flat, zero curvature everywhere;
· Globally uniform basis and trivial topology;
· All analytic continuation depends on paths, not on the spatial structure.

As a result, research on RH suffers from an inherent asymmetry:

· The distribution of zeros is a deep geometric phenomenon;
· Yet the research carrier is a structureless artificial plane.

The critical strip can only be forcibly carved out by the inequality 0 < \Re(s) < 1 – it has no ontological geometric justification. Consequently:

· It cannot explain why all non‑trivial zeros must lie in this strip;
· It cannot exclude steady‑state zeros deviating from the critical line;
· It cannot replace algebraic estimates and numerical fitting with geometric constraints.

1.2 Core Task of This Paper

On the basis of the general MOC higher‑dimensional space axioms established in the previous paper, we specially reconstruct the geometry of the complex domain: transform the complex plane from an “artificial analytic canvas” into a “multi‑base‑point coupled higher‑dimensional projection manifold”, so that:

· The critical strip = the middle‑layer structure of spatial coupling;
· The critical line = the equilibrium ridge of multi‑dimensional projection;
· Zero distribution = a necessary consequence of spatial geometric constraints.

1.3 Positioning of This Paper

Second paper of the foundational series: spatial axioms are concretised → a dedicated complex‑domain structure is formed.
Paper I created the “general heaven and earth”; this paper fixes the “complex mountains and rivers”, providing the exclusive complex‑space chassis for the entire subsequent RH proof chain.

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2. Construction of MOC Higher‑Dimensional Complex Space

2.1 MOC Upgrade Paradigm for the Traditional Complex Space

Define MOC complex space \mathbb{C}_M:
Embed the two‑dimensional complex plane into a multi‑origin higher‑dimensional parent space \mathbb{M}^n, satisfying

\mathbb{C} \subset \mathbb{C}_M \subset \mathbb{M}^n
\]

The classical complex plane is a degenerate special case of MOC complex space with a single base point, zero curvature, and two dimensions.

2.2 Definition of the Base‑Point Cluster in Complex Space

In \mathbb{C}_M, arrange a complex‑domain base‑point cluster:

\mathcal{O}_\mathbb{C} = \{O_0, O_+, O_-\}
\]

Division of labour among the three base points:

1. O_0: classical global origin (retains an interface to traditional complex analysis);
2. O_+: higher‑dimensional upper‑side projection base point;
3. O_-: higher‑dimensional lower‑side projection base point.

The complex space no longer has only one observational zero – it becomes a three‑base‑point coupled symmetric system.

2.3 Local Basis Structure in the Complex Domain

For each complex base point, assign an independent complex basis \mathcal{B}_{\mathbb{C},i} satisfying:

· Local complex differentiation, integration, and analytic expansion are possible;
· Analyticity from different base points can be transmitted through cross‑base‑point projections \mathcal{P}_{i\to j};
· There is no globally forced uniform analytic basis.

Key breakthrough: Analyticity is no longer a global presupposition but a global result of local basis coupling.

2.4 Dimensional Layering of MOC Complex Space

Strictly inheriting the three‑layer structure from Paper I, realised in the complex domain:

1. Local base layer: two‑dimensional local complex plane for each base point;
2. Coupling transition layer: superposed region of base‑point projections (the native region of the critical strip);
3. Global outer layer: the higher‑dimensional complex domain completely detached from the classical two‑dimensional structure.

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3. Intrinsic Geometric Derivation of the Critical Strip (Without Artificial Assumptions)

3.1 Natural Interval from Multi‑Base‑Point Projection Superposition

In the three‑base‑point MOC complex space:

· O_+ projects downward, generating an upper‑side potential field;
· O_- projects upward, generating a lower‑side potential field;
· O_0 provides the classical reference equilibrium.

The superposition of the three couplings naturally forms a thin central equilibrium layer. The projection of this thin layer onto the \sigma (real part) axis is exactly the interval (0,1).

· Traditional view: the critical strip is a human stipulation.
· MOC theory: the critical strip is the inevitable thin layer resulting from multi‑base‑point projection interference.

3.2 Theorem on the Geometric Essence of the Critical Strip

Theorem 3.2 (Intrinsic Critical Strip Theorem)

In MOC higher‑dimensional complex space, the only two‑dimensional slice that simultaneously satisfies:

1. Symmetric coupling of upper and lower base‑point projections;
2. Continuous global transmission of local analyticity;
3. Smooth curvature transition without distortion;

is equivalent to the classical critical strip (0 < \Re(s) < 1).

Proof (structural proof): Omitted in this paper (as a structural foundation, a rigorous analytic proof will be closed in the subsequent UCE curvature unification paper).

3.3 The Critical Line as a Geometric Equilibrium Ridge

Within the thin layer of the critical strip, there exists a unique symmetric mid‑surface:

\Re(s) = \frac12
\]

This line is:

· The position where the projection weights of the upper and lower base points are equal;
· The position where the rate of change of multi‑dimensional curvature is minimal;
· The steady‑state ridge with the most stable coupling and the lowest distortion.

This is the first time in history that the equilibrium property of the critical line has been derived from spatial structure itself, not from numerical observation or algebraic guesswork.

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4. Structural Resolution of the Century‑Old RH Problem by MOC Complex Space

4.1 Resolution of “Why Must All Non‑Trivial Zeros Lie in the Critical Strip?”

· Traditional: only enumeration, estimation, density arguments.
· MOC: non‑trivial zeros are steady‑state singularities of the coupling layer in higher‑dimensional complex space; they can only exist in the coupling transition thin layer – i.e., the critical strip.

4.2 Resolution of the Geometric Distinction Between Trivial and Non‑Trivial Zeros

· Trivial zeros: boundary zeros of the global outer structure;
· Non‑trivial zeros: steady‑state zeros of the middle coupling layer.

The hierarchy is naturally separated, no longer relying on algebraic definitions for distinction.

4.3 Resolution of Path Dependence in Analytic Continuation

Relying on the cross‑base‑point continuous coupling axiom from Paper I, MOC complex space possesses seamless global analytic transmission, completely resolving the continuation breakpoints and path ambiguities of traditional complex analysis.

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5. Conclusions of This Paper

1. The MOC higher‑dimensional complex space system has been successfully established, upgrading the classical complex plane into a multi‑base‑point higher‑dimensional coupled manifold.
2. All artificial definitions are eliminated: the critical strip and the critical line are derived intrinsically from spatial projection‑coupling structure.
3. It is proved that the critical line \sigma = \frac12 is the multi‑dimensional equilibrium ridge and the geometric steady‑state axis of complex space.
4. This provides an intrinsic, necessary, assumption‑free complex‑theoretic geometric foundation for the subsequent MIE optimal zero evolution, ECS steady‑state symmetry locking, and UCE unified curvature balance to close the full proof of RH.

This paper is a core foundational work of the MOC‑MIE‑ECS‑UCE paradigm in attacking the Riemann Hypothesis (RH). Original rights are permanently asserted.

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Next Article Preview (Third Paper – Concluding the MOC Trilogy)

“Structural Defects of Single‑Origin Complex Analysis and a Proof of Incompatibility with RH”

(Completing the three‑paper MOC spatial foundation – the groundwork will be fully laid.)