Part One: Core Dualism — The World is Composed of Two Fundamental Modes

1.Two Working Media of the Brain

Electrical Signal

• Physical Characteristics: Linear, integrable, transmitted along a definite path
• Information Type: Shannon Information
• Pursued Goal: Certainty
• Nature: Consistent but Incomplete
  

Electromagnetic Wave (Magnetic Field)

• Physical Characteristics: Non-linear, non-integrable, diffusive, without a definite path
• Information Type: Gödel Information
• Pursued Goal: Meaning
• Nature: Complete but Inconsistent, carrying intuition and inspiration
  

2.Two Types of Information

Part Two: The Common Root of Difficult Problems

1.Unified Problem Perspective

These problems originate from the resonant nature of free particles.

• In mathematics: manifested as non-integrability
• In physics: manifested as difficulty in integrating macro and micro scales
• In computation: manifested as NP asymmetry
  

2.The History and Physical Significance of Maxwell’s Equations

• The first mathematical formula containing time irreversibility
• Revealed non-linear, non-integrable electromagnetic phenomena
• Served as the foundational source of inspiration for modern quantum mechanics and relativity

Part Three: Core Mechanisms — “Field,” “Resonance,” and “Simultaneity”

1.Field: The superposition of multiple particles’ resonance forms a non-integrable holistic process, namely “simultaneity.”

2.Simultaneity: The superposed expression of particles with degrees of freedom under infinite rules.

3.Resonance and Dissipative Structures

• Poincaré proved that in complex systems, the relationships among elements are unknown, and energy is manifested in the form of potential energy.
• Resonance is the infinite unfolding of simultaneity and forms the basis of dissipative structures.

4. Analogy — Bitcoin Mining: Multiple mining machines operate independently and in parallel, presenting non-integrable resonance similar to fields and magnetic fields in physics.

Part Four: The Way Forward — Mathematical Tools and Philosophical Thought

1. Topology

• Traditional calculus is suited for integrable systems
• Poincaré pioneered topology to study non-integrable phenomena
• Chen-Ning Yang’s gauge field theory and James Simons’ expansion of field theory foundations
  

2. Philosophical Thought — Gödel’s Incompleteness Theorem

• Defines the boundaries of formal systems
• Guides scientific exploration beyond these boundaries
• Develops into a “metaphilosophy of precision,” expanding scientific methodology

This article systematically elaborates the dualistic foundation of the world and its non-integrable unified theory, spanning the deep mechanisms of physics, information, the brain, and computation. It reveals the common roots of current scientific problems and points toward directions for future exploration.