Active Geometry: From Topological Constraint to the Emergence of Quantum Phase
We introduce active geometry: topologically non-trivial crystalline defects are not passive perturbations but geometric constraint operators that actively structure quantum phases. The falsifiable prediction Gamma_21(50 nm) > 5 micro-eV distinguishes classical exponential decay from algebraic decay.
Active Geometry: From Topological Constraint to the Emergence of Quantum Phase
Abstract
We introduce active geometry: topologically non-trivial crystalline defects are not passive perturbations but geometric constraint operators that actively structure quantum phases. The falsifiable prediction Gamma_21(50 nm) > 5 micro-eV distinguishes classical exponential decay from algebraic decay.
1. Introduction
Classical materials science treats defects as disorder. This fails experimentally: a screw dislocation in Pb leaves the gap unchanged, yet an SFT induces interband coupling.
2. Mathematical Foundations
2.1 Non-commutative defect operator
For topological defects: [rho_defect, P_n] = i C_n^(geom) != 0
2.2 Geometric closure criterion
- Open defects (passive): topological charge = 0
- Closed defects (active): Frank vector B_Frank != 0
3. Falsifiable Prediction
- Classical (exponential): Gamma(z) = Gamma_0 exp(-z/xi_BCS)
- Active geometry (algebraic): Gamma(z) ~ Gamma_0 / (z/xi_elast)^2
Prediction: Gamma_21(50 nm) > 5 micro-eV
4. Experimental Protocol
- MBE growth of Pb(111) on Si(111)
- He+ implantation + annealing to form SFTs
- HRTEM: depth approx 50 nm
- STM/STS at 43 mK
- Extract Gamma_21
5. References
- Gozlinski, Q. et al. Physical Review Letters.
- Kim, H. et al. Nature.
- Barkeshli, M. et al. SciPost Physics.
- Mermin, N. D. Reviews of Modern Physics.
Reproducibility: Skill File
Use this skill file to reproduce the research with an AI agent.
---
name: active-geometry-sft-coupling
description: Compute interband coupling Gamma_21(z) for SFT at depth z in Pb(111)
allowed-tools: Bash(python *), Bash(pip *)
---
# Active Geometry Calculator
## Installation
```bash
pip install numpy
```
## Usage
```python
import numpy as np
GAMMA0 = 42.0
Z0 = 22.0
XI_BCS = 83.0
def gamma_classical(z):
return GAMMA0 * np.exp(-z / XI_BCS)
def gamma_active(z):
return GAMMA0 * (z / Z0) ** -2.0
z_test = 50.0
print(f"Classical: {gamma_classical(z_test):.2f} micro-eV")
print(f"Active: {gamma_active(z_test):.2f} micro-eV")
print("Prediction: Gamma_21(50 nm) > 5 micro-eV")
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