📐 The Active Geometric Principle (AGP) & Topotronique

"Geometry is not a passive backdrop; it is an active causal operator."

This repository contains the foundational technical white paper and architectural specifications for Topotronique, a proposed device platform that utilizes the Active Geometric Principle (AGP) to program physical laws at the hardware level.

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🌌 Abstract

Traditionally, physics views geometry as the stage upon which events happen. We propose a paradigm shift: Geometry as an Operator.

By engineering the topology, curvature, and symmetry of a system, we can actively select, stabilize, and generate physical effects—from acoustic stealth to protected quantum states. This project formalizes the AGP and proposes a concrete implementation using Hg(1-x)Mn(x)Te magnetic topological materials and Artificial Spin Ice (ASI) geometries.

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📐 The Active Geometric Principle (Formalism)

The core axiom of this framework is that geometric structure ($\mathcal{O}{geom}$) acts directly on physical fields ($\Psi$) to produce emergent phenomena ($\mathcal{E}{phys}$).

$$ \boxed{ \mathcal{O}_\text{geom}[\mathcal{M}, g, R, \ldots] \cdot \Psi \longrightarrow \mathcal{E}_\text{phys} } $$

Where:

• $\mathcal{M}$: The manifold (configuration space).
• $g, R$: Metric and Curvature tensors.
• $\mathcal{E}_{phys}$: Emergent effects (Monopoles, Majorana modes, Edge states).

In a quantum Hamiltonian context: $$H = H_0 + \lambda \mathcal{O}_\text{geom}$$

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⚡ The Topotronic Device Architecture

We propose a programmable hardware platform ("Topotronique") that validates AGP.

1. The Substrate: Hg(1-x)Mn(x)Te

A tunable magnetic topological material.

• Tunability: Can shift between Topological Insulator, Chern Insulator, and Weyl Semimetal phases via Mn concentration ($x \approx 0.25$).
• Role: Hosts the topological edge states and serves as the canvas for geometric patterning.

2. The Geometry: Artificial Spin Ice (ASI)

Nanomagnets arranged in frustrated lattices (Kagome or Penrose).

• Frustration: Geometry prevents simultaneous minimization of interactions.
• Emergence: Creates Magnetic Monopoles and Dirac Strings as programmable quasiparticles.

3. The Control: Gate-Tuned RKKY

• Mechanism: Indirect exchange coupling mediated by 2D carriers.
• Active Control: Gate voltages modulate carrier density, tuning the "stiffness" of the geometric interactions in real-time.

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🔮 Emergent Phenomena & Computation

This architecture allows us to move beyond binary logic by manipulating quasiparticles defined by geometry:

Phenomenon	Geometric Source	Computational Utility
Magnetic Monopoles	ASI Ice Rule Violation	Information carriers, logic gates
Skyrmions	Curvature & DMI	High-density memory, transport
Majorana Fermions	Monopole-Superconductor Hybrid	Topological Quantum Computing (Braiding)
Edge States	Bulk Topology (Chern Number)	Dissipationless transport

Braiding via Monopole Motion

By coupling the magnetic ASI to a superconductor, moving a magnetic monopole (via geometry) moves a vortex, effectively braiding Majorana Zero Modes for fault-tolerant quantum calculation.

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📊 Device Parameters

Parameter	Specification
Material	Hg(1-x)Mn(x)Te ($x \approx 0.25$)
Lattice Geometry	Kagome / Penrose (Quasiperiodic)
Scale	10–100 μm device area
Coupling	RKKY (Gate tunable), Dipolar
Readout	STM, NV Centers, Microwave Cavity

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⚠️ Dual-Use & Ethics

This research touches on high-energy physics and advanced computing capabilities.

• Potential Risks: Advanced cryptography breaking, quantum surveillance.
• Mitigation: Adherence to the Lichen Universe Ethical Homeostasis (EHE) protocols. Technology must be developed with transparency and responsible governance.

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📚 Reference & Citation

Please refer to the full white paper in this repository: AGP_Technical_Whitepaper.pdf

BibTeX:

@article{Lichen2026AGP,
  title={The Active Geometric Principle in Physics and Computation},
  author={Lichen Collective},
  journal={Lichen Universe Research},
  year={2026},
  note={Topotronique Architecture V1}
}