SSZ-Qubits: Segmented Spacetime Framework for Quantum Computing

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Table of Contents

1. Overview
2. The Problem
3. The SSZ Solution
4. Theoretical Foundations
5. Installation
6. Quick Start
7. API Reference
8. Application Examples
9. Test Suite
10. Visualizations
11. Physical Results
12. Experimental Validation
13. Project Structure
14. FAQ
15. References
16. Contributing
17. Authors & License

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Overview

SSZ-Qubits applies the Segmented Spacetime (SSZ) framework to quantum computing. It provides tools for analyzing and minimizing gravitational effects on qubit systems.

"Qubits don't just exist in space—they exist in segments of spacetime."

What is SSZ?

SSZ (Segmented Spacetime) is a theoretical framework that treats spacetime as a discrete structure, rather than a continuous manifold as in classical General Relativity (GR).

Why is this relevant for qubits?

Qubits are extremely sensitive to:

• Time dilation from gravitational fields
• Phase shifts from height differences
• Decoherence from segment mismatch

SSZ quantifies these effects and enables their compensation.

---

The Problem

Classical Qubit Problems

Problem	Description	Impact
Decoherence	Loss of quantum coherence	Computational errors
Timing errors	Asynchronous gate operations	Wrong results
Spatial drift	Position-dependent phase errors	Unpredictable errors
Gravitational gradients	Height-dependent time dilation	Systematic errors

The Overlooked Problem

Even micrometer-scale height differences between qubits lead to measurable effects!

1 μm height difference -> ΔXi ~ 10⁻²² -> ~10⁻²² s/s time drift
1 mm height difference -> ΔXi ~ 10⁻¹⁹ -> ~10⁻¹⁹ s/s (0.1 attosecond/s) time drift

Note: These effects are extremely small at Earth's surface but accumulate 
over millions of gate operations. For satellite-to-ground links (Δh ~ 400 km),
the effect becomes significant: ΔXi ~ 10⁻¹¹ -> ~1.4 rad/s phase drift at 5 GHz.

These effects are often dismissed as "hardware drift" or "unexplained decoherence" in classical qubit physics.

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The SSZ Solution

SSZ provides a geometric framework with:

1. Segment Density Xi(r)

Quantifies the local spacetime structure:

Xi(r) = r_s / (2r)

where r_s is the Schwarzschild radius.

2. SSZ Time Dilation D_SSZ

Determines local clock rates:

D_SSZ = 1 / (1 + Xi)

3. Segment-Coherent Zones

Defines optimal qubit placement regions where Xi variation is minimal.

4. Geometry-Aware QEC

Enables gravity-aware quantum error correction.

5. Golden Ratio φ

Controls segment saturation in strong fields:

φ = (1 + √5) / 2 = 1.618033988749895

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Theoretical Foundations

Two SSZ Regimes

SSZ distinguishes two regimes based on field strength:

Weak Field (r/r_s > 100) - Applicable on Earth

Xi(r) = r_s / (2r)
dXi/dr = -r_s / (2r²)
D_SSZ = 1 / (1 + Xi) ≈ 1 - Xi + O(Xi²)

Example Earth:

• r_s = 8.87 mm
• r = 6.371×10⁶ m (Earth radius)
• Xi = 6.96×10⁻¹⁰
• D_SSZ = 0.999999999303892

Strong Field (r/r_s < 100) - Near Black Holes

Xi(r) = 1 - exp(-φ × r / r_s)
dXi/dr = (φ / r_s) × exp(-φ × r / r_s)

Important: In the strong field, D_SSZ remains finite at the horizon (D_SSZ ≈ 0.555), while GR predicts a singularity.

Fundamental Constants

Constant	Symbol	Value	Unit
Speed of light	c	299792458	m/s
Gravitational constant	G	6.67430×10⁻¹¹	m³/(kg·s²)
Planck constant	ℏ	1.054571817×10⁻³⁴	J·s
Golden Ratio	φ	1.6180339887498948	-
Earth mass	M_E	5.972×10²⁴	kg
Earth radius	R_E	6.371×10⁶	m

Schwarzschild Radius

r_s = 2GM / c²

Object	Schwarzschild Radius
Earth	8.87 mm
Sun	2.95 km
Sagittarius A*	12 million km

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Installation

Requirements

• Python 3.8 or higher
• pip (Python Package Manager)

Installation

# Clone repository
git clone https://github.com/error-wtf/ssz-qubits.git
cd ssz-qubits

# Install dependencies
pip install -r requirements.txt

# Verify installation
python -c "from ssz_qubits import *; print('SSZ-Qubits successfully installed!')"

Dependencies

numpy>=1.20.0
pytest>=7.0.0
matplotlib>=3.5.0

Run Tests

# All tests
python run_tests.py

# Or with pytest
pytest tests/ -v

Expected output:

============================= 113 passed in 0.54s ==============================

---

Quick Start

Minimal Example

from ssz_qubits import Qubit, QubitPair, analyze_qubit_segment, qubit_pair_segment_mismatch

# Create qubit at sea level
q1 = Qubit(id="Q1", x=0, y=0, z=0)

# Create qubit 1 cm higher
q2 = Qubit(id="Q2", x=0, y=0, z=0.01)

# Analyze single qubit
analysis = analyze_qubit_segment(q1)
print(f"Xi = {analysis.xi:.6e}")
print(f"D_SSZ = {analysis.time_dilation:.15f}")

# Analyze qubit pair
pair = QubitPair(q1, q2)
mismatch = qubit_pair_segment_mismatch(pair)
print(f"Delta Xi = {mismatch['delta_xi']:.6e}")

Interactive Demo

python demo.py

The demo shows 9 different use cases:

1. Basic SSZ Physics
2. Single Qubit Analysis
3. Qubit Pair Mismatch
4. Coherent Zones
5. Array Optimization
6. Gate Timing Corrections
7. Decoherence Analysis
8. Experimental Validation
9. Practical System Design

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API Reference

Constants

from ssz_qubits import C, G, HBAR, M_EARTH, R_EARTH, PHI

C        # Speed of light [m/s]
G        # Gravitational constant [m³/(kg·s²)]
HBAR     # Reduced Planck constant [J·s]
M_EARTH  # Earth mass [kg]
R_EARTH  # Earth radius [m]
PHI      # Golden Ratio

Core Functions

schwarzschild_radius(M)

Calculates the Schwarzschild radius.

r_s = schwarzschild_radius(M_EARTH)  # 8.87e-3 m

xi_segment_density(r, M, regime='auto')

Calculates the segment density Xi.

xi = xi_segment_density(R_EARTH, M_EARTH)  # 6.96e-10
xi = xi_segment_density(r, M, regime='weak')   # Force weak field
xi = xi_segment_density(r, M, regime='strong') # Force strong field

xi_gradient(r, M, regime='auto')

Calculates the gradient dXi/dr.

grad = xi_gradient(R_EARTH, M_EARTH)  # -1.09e-16 /m

ssz_time_dilation(r, M)

Calculates SSZ time dilation D_SSZ.

d = ssz_time_dilation(R_EARTH, M_EARTH)  # 0.999999999303892

Qubit Classes

Qubit

from ssz_qubits import Qubit

q = Qubit(
    id="Q1",              # Unique ID
    x=0,                  # X position [m]
    y=0,                  # Y position [m]
    z=0,                  # Height above reference [m]
    coherence_time_T2=100e-6,  # T2 time [s] (default: 100 μs)
    gate_time=50e-9       # Gate time [s] (default: 50 ns)
)

# Properties
q.radius_from_earth_center  # Distance from Earth center

QubitPair

from ssz_qubits import QubitPair

pair = QubitPair(q1, q2)

# Properties
pair.separation          # Spatial separation [m]
pair.height_difference   # Height difference [m]

Analysis Functions

analyze_qubit_segment(qubit, M)

analysis = analyze_qubit_segment(q, M_EARTH)
# Returns: SegmentAnalysis with:
#   .xi              - Segment density
#   .gradient        - dXi/dr
#   .time_dilation   - D_SSZ
#   .radius          - Distance from center

qubit_pair_segment_mismatch(pair, M)

mismatch = qubit_pair_segment_mismatch(pair, M_EARTH)
# Returns: Dict with:
#   'delta_xi'              - Xi difference
#   'delta_time_dilation'   - D_SSZ difference
#   'phase_drift_per_gate'  - Phase drift per gate
#   'decoherence_enhancement' - Decoherence factor

segment_coherent_zone(center_height, max_xi_variation, M)

h_min, h_max = segment_coherent_zone(0, 1e-18, M_EARTH)
# Returns: Tuple (min_height, max_height) in meters

optimize_qubit_array(n, base_height, max_separation)

qubits = optimize_qubit_array(16, base_height=0, max_separation=1e-3)
# Returns: List of n optimally placed qubits

array_segment_uniformity(qubits, M)

uniformity = array_segment_uniformity(qubits, M_EARTH)
# Returns: Dict with:
#   'xi_mean'      - Mean Xi
#   'xi_std'       - Standard deviation
#   'xi_min'       - Minimum
#   'xi_max'       - Maximum
#   'xi_range'     - Range
#   'uniformity'   - Uniformity score (0-1)

Gate Timing Functions

gate_timing_correction(qubit, M)

correction = gate_timing_correction(q, M_EARTH)
# Returns: Correction factor for gate time

two_qubit_gate_timing(pair, M)

timing = two_qubit_gate_timing(pair, M_EARTH)
# Returns: Dict with:
#   'optimal_gate_time'  - Optimal gate duration
#   'timing_asymmetry'   - Required timing asymmetry
#   'max_fidelity_loss'  - Maximum fidelity loss
#   'd_qubit_a'          - D_SSZ for qubit A
#   'd_qubit_b'          - D_SSZ for qubit B

Decoherence Functions

ssz_decoherence_rate(qubit, M)

gamma = ssz_decoherence_rate(q, M_EARTH)
# Returns: SSZ-induced decoherence rate [1/s]

effective_T2(qubit, M)

T2_eff = effective_T2(q, M_EARTH)
# Returns: Effective T2 time under SSZ influence [s]

pair_decoherence_time(pair, M)

T2_pair = pair_decoherence_time(pair, M_EARTH)
# Returns: Pair decoherence time [s]

---

Application Examples

1. Qubit Placement Optimization

Problem: Minimize SSZ mismatch between qubits in an array.

from ssz_qubits import optimize_qubit_array, array_segment_uniformity, M_EARTH

# Optimize 100-qubit array
qubits = optimize_qubit_array(100, base_height=0, max_separation=5e-3)

# Check uniformity
uniformity = array_segment_uniformity(qubits, M_EARTH)
print(f"Xi uniformity: {uniformity['uniformity']:.6f}")
print(f"Xi range: {uniformity['xi_range']:.6e}")

2. Finding Coherent Zones

Problem: Find the height range where Xi variation stays below a tolerance.

from ssz_qubits import segment_coherent_zone, M_EARTH

# Find zone with tolerance 10⁻¹⁸
h_min, h_max = segment_coherent_zone(0, 1e-18, M_EARTH)
print(f"Coherent zone: {h_min*1e6:.1f} μm to {h_max*1e6:.1f} μm")
print(f"Zone width: {(h_max-h_min)*1e6:.1f} μm")

3. Gate Timing Correction

Problem: Compensate time dilation in two-qubit gates.

from ssz_qubits import Qubit, QubitPair, two_qubit_gate_timing, M_EARTH

q1 = Qubit(id="Q1", x=0, y=0, z=0, gate_time=50e-9)
q2 = Qubit(id="Q2", x=0, y=0, z=10e-3, gate_time=50e-9)  # 10 mm higher
pair = QubitPair(q1, q2)

timing = two_qubit_gate_timing(pair, M_EARTH)
print(f"Optimal gate time: {timing['optimal_gate_time']*1e9:.6f} ns")
print(f"Timing asymmetry: {timing['timing_asymmetry']:.6e}")

4. Decoherence Analysis

Problem: Determine the SSZ contribution to decoherence.

from ssz_qubits import Qubit, ssz_decoherence_rate, effective_T2, M_EARTH

q = Qubit(id="Q1", x=0, y=0, z=0, coherence_time_T2=100e-6)

gamma = ssz_decoherence_rate(q, M_EARTH)
T2_eff = effective_T2(q, M_EARTH)

print(f"Intrinsic T2: {q.coherence_time_T2*1e6:.1f} μs")
print(f"Effective T2: {T2_eff*1e6:.3f} μs")
print(f"SSZ contribution: {(1 - T2_eff/q.coherence_time_T2)*100:.2f}%")

5. Quantum Communication Synchronization

Problem: Calculate time drift between remote qubits.

from ssz_qubits import ssz_time_dilation, R_EARTH, M_EARTH

# Two stations with 100 m height difference
h1 = 0       # Sea level
h2 = 100     # 100 m higher

d1 = ssz_time_dilation(R_EARTH + h1, M_EARTH)
d2 = ssz_time_dilation(R_EARTH + h2, M_EARTH)

delta_d = d2 - d1
drift_per_hour = delta_d * 3600 * 1e9  # ns/hour

print(f"Time drift: {drift_per_hour:.3f} ns/hour")

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Test Suite

Overview

Category	Tests	Description
Physics	17	Physical formulas
Edge Cases	25	Extreme values, error handling
Validation	17	Experimental validation
Applications	15	Practical applications
Total	74

Running Tests

# All tests with detailed output
python run_tests.py

# Single test file
pytest tests/test_ssz_physics.py -v

# With pytest (faster)
pytest tests/ -v --tb=short

Test Categories

Physics Tests (test_ssz_physics.py)

• Schwarzschild radius validation
• Segment density calculations
• Time dilation verification
• Gradient consistency
• Physical limits
• Golden ratio properties
• Strong field behavior

Edge Cases (test_edge_cases.py)

• Extreme radii (near r_s to 1 AU)
• Extreme masses (0 to black holes)
• Unusual qubit configurations
• Numerical precision
• Error handling
• Special qubit properties
• QEC edge cases

Validation (test_validation.py)

• GR weak-field comparison
• GPS satellite time dilation
• Pound-Rebka experiment
• NIST optical clocks
• Tokyo Skytree experiment
• Theoretical consistency
• Dimensional analysis

Qubit Applications (test_ssz_qubit_applications.py)

• Segmented time logic as qubit clock
• Decoherence as geometry phenomenon
• Gravity-induced drift prediction
• Segment-aware QEC
• Quantum communication sync

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Visualizations

Generate Plots

python visualize_ssz_qubits.py

Generated Plots

Plot	Description
	D_SSZ vs Height - Shows how time dilation changes with altitude
	Pair Mismatch Analysis - Xi difference between qubit pairs
	Segment-Coherent Zones - Optimal placement regions
	Array Optimization - Multi-qubit array uniformity
	SSZ vs GR Comparison - Framework comparison
	φ Structure - Golden ratio in SSZ

Plot Gallery

1. Time Dilation vs Height

2. Qubit Pair Mismatch

3. Coherent Zone

4. Qubit Array Analysis

5. SSZ vs GR Comparison

6. Golden Ratio Structure

Example Visualization

import matplotlib.pyplot as plt
import numpy as np
from ssz_qubits import ssz_time_dilation, R_EARTH, M_EARTH

heights = np.linspace(0, 1000, 100)  # 0 to 1000 m
d_ssz = [ssz_time_dilation(R_EARTH + h, M_EARTH) for h in heights]

plt.plot(heights, d_ssz)
plt.xlabel('Height [m]')
plt.ylabel('D_SSZ')
plt.title('SSZ Time Dilation vs Height')
plt.savefig('my_plot.png')

---

Physical Results

Earth Surface

Parameter	Value
Xi	6.961078×10⁻¹⁰
D_SSZ	0.999999999303892
dXi/dr	-1.093×10⁻¹⁶ /m
r_s (Earth)	8.87 mm

Qubit Effects

Height Difference	ΔXi	Time Drift
1 μm	~10⁻²²	Measurable
10 μm	~10⁻²¹	Significant
100 μm	~10⁻²⁰	Critical
1 mm	~10⁻¹⁹	~0.01 ps/s
10 mm	~10⁻¹⁸	~0.1 ps/s

Coherent Zones

Tolerance	Zone Width
10⁻¹⁶	458 mm
10⁻¹⁷	46 mm
10⁻¹⁸	4.6 mm
10⁻¹⁹	458 μm
10⁻²⁰	46 μm

SSZ vs GR Comparison

Aspect	GR	SSZ
Spacetime	Continuous	Discrete
At horizon	D = 0 (singularity)	D = 0.555 (finite)
Weak field	D ≈ 1 - r_s/2r	D = 1/(1+Xi) ≈ same
Quantization	No	Yes (φ-based)

---

Experimental Validation

GPS System

Parameter	SSZ Prediction	Measured	Status
Time drift	~45 μs/day	~45 μs/day	✓ MATCH
Position error	~11 km/day	~10 km/day	✓ MATCH

Pound-Rebka Experiment (1960)

Parameter	SSZ Prediction	Measured	Status
Redshift	2.46×10⁻¹⁵	(2.57±0.26)×10⁻¹⁵	✓ MATCH

NIST Optical Clocks (2010)

Parameter	SSZ Prediction	Status
33 cm height difference	Measurable	✓ MATCH

Tokyo Skytree (2020)

Parameter	SSZ Prediction	Status
450 m height difference	Measurable	✓ MATCH

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Project Structure

ssz-qubits/
├── ssz_qubits.py               # Core module
├── ssz_entanglement.py         # Entanglement module (Paper B support)
├── ssz_paper_a_support.py      # Paper A dedicated functions
├── demo.py                     # Interactive demo (9 demos)
├── run_tests.py                # Test runner
├── visualize_ssz_qubits.py     # Visualization (6 plots)
├── test_paper_consistency.py   # Paper equation verification
├── final_summary.py            # Repository summary
├── requirements.txt            # Dependencies
├── README.md                   # This file
├── PAPER_SUPPORT.md            # Paper support documentation
├── LICENSE                     # Anti-Capitalist License v1.4
│
├── docs/
│   ├── SSZ_FORMULA_DOCUMENTATION.md    # Formula documentation
│   ├── SSZ_MATHEMATICAL_PHYSICS.md     # Math/physics foundations
│   ├── SSZ_QUBIT_APPLICATIONS.md       # Practical applications
│   └── SSZ_QUBIT_THEORY_SUMMARY.md     # Theory summary
│
├── tests/
│   ├── test_ssz_physics.py             # Physics tests
│   ├── test_edge_cases.py              # Edge case tests
│   ├── test_validation.py              # Validation tests
│   ├── test_ssz_qubit_applications.py  # Application tests
│   ├── test_entanglement.py            # Entanglement tests (23)
│   └── test_paper_a_support.py         # Paper A support tests (16)
│
├── outputs/                    # Generated plots
│   ├── time_dilation_vs_height.png
│   ├── qubit_pair_mismatch.png
│   ├── coherent_zone.png
│   ├── qubit_array_analysis.png
│   ├── ssz_vs_gr_comparison.png
│   └── golden_ratio_structure.png
│
└── reports/                    # Test reports
    ├── RUN_SUMMARY.md
    └── full-output.md

New Modules (Paper Support)

ssz_entanglement.py - Paper B Support

Functions for analyzing gravitational effects on entangled qubit pairs:

• entangled_pair_phase_drift() - Phase drift for entangled pairs
• bell_state_fidelity() - F = cos²(ΔΦ/2)
• chsh_parameter() - S = 2√2·cos(ΔΦ) for fixed settings
• characteristic_time_T_SSZ() - Time for π phase accumulation
• correction_interval() - Gates before correction needed
• is_in_coherent_zone() - Check if pair is in same zone
• analyze_entangled_pair() - Complete analysis

ssz_paper_a_support.py - Paper A Support

Dedicated functions for Paper A claims:

• gr_time_dilation_weak_field() - GR comparison
• compare_ssz_gr() - SSZ vs GR analysis
• fidelity_reduction_small_angle() - 1-F ≈ ΔΦ²/4
• verify_linear_scaling() - Verify |ΔΦ| ∝ |Δh|
• verify_numerical_stability() - Closed-form stability
• coherent_zone_analysis() - Complete zone analysis
• decoherence_enhancement_factor() - 1 + (ΔΞ/Ξ_ref)²

---

FAQ

General

Q: What is the difference between SSZ and General Relativity?

A: GR treats spacetime as a continuous manifold. SSZ treats spacetime as a discrete structure with measurable segments. In the weak field (like on Earth), both yield nearly identical results. The difference appears in strong fields: SSZ avoids the singularity at the Schwarzschild horizon.

Q: Why is the Golden Ratio φ important?

A: φ = (1+√5)/2 controls the saturation rate of segment density in the strong field. The formula Xi = 1 - exp(-φ×r/r_s) ensures that Xi reaches a finite value (about 0.8) at r = r_s, instead of diverging.

Q: Is SSZ experimentally confirmed?

A: SSZ reproduces all known experimental results (GPS, Pound-Rebka, atomic clocks) in the weak field. Predictions for strong fields (black holes) are not yet directly testable.

Technical

Q: Which Python version is required?

A: Python 3.8 or higher.

Q: How accurate are the calculations?

A: Calculations use 64-bit floating point (numpy.float64) with precision of about 15-16 significant digits.

Q: Can I integrate SSZ with my qubit simulator?

A: Yes! SSZ-Qubits is designed as a Python module and can be integrated into any Python-based simulator. Simply import the required functions.

Q: Why are some values 0.000000e+00?

A: For optimized arrays with constant height, Xi variation is exactly zero. This is not an error but the desired result of optimization.

Qubit-Specific

Q: How large is the SSZ effect on typical qubits?

A: At 1 mm height difference, ΔXi is about 10⁻¹⁹, leading to a time drift of ~0.01 ps/s. For typical gate times of 50 ns, the direct effect is small but accumulates over many operations.

Q: Should I implement SSZ corrections in my quantum computer?

A: For current quantum computers with few qubits and short coherence times, other error sources (thermal noise, EM interference) are dominant. SSZ becomes relevant for:

• Large qubit arrays (>100 qubits)
• Long coherence times (>1 ms)
• Precise timing requirements (<1 ps)
• Distributed quantum systems

Q: What is a "coherent zone"?

A: A coherent zone is a height range where Xi variation stays below a certain tolerance. All qubits within this zone have nearly identical segment properties, minimizing mismatch errors.

---

References

Primary Literature

1. Casu, L. & Wrede, C. (2025). "Segmented Spacetime: A Discrete Framework for Quantum Gravity" (in preparation)
2. Casu, L. & Wrede, C. (2025). "SSZ Applications in Quantum Computing" (in preparation)

Experimental Validation (used for SSZ validation)

3. Pound, R.V. & Rebka, G.A. (1960). "Apparent Weight of Photons". Physical Review Letters, 4(7), 337-341.
4. Chou, C.W. et al. (2010). "Optical Clocks and Relativity". Science, 329(5999), 1630-1633.
5. Takamoto, M. et al. (2020). "Test of general relativity by a pair of transportable optical lattice clocks". Nature Photonics, 14, 411-415.

Background

6. Ashby, N. (2003). "Relativity in the Global Positioning System". Living Reviews in Relativity, 6(1).
7. Will, C.M. (2014). "The Confrontation between General Relativity and Experiment". Living Reviews in Relativity, 17(1).

---

Contributing

We welcome contributions! Please see our Contributing Guidelines for details.

Quick Start for Contributors

# Fork and clone
git clone https://github.com/YOUR_USERNAME/ssz-qubits.git
cd ssz-qubits

# Create feature branch
git checkout -b feature/your-feature

# Make changes, then test
python run_tests.py

# Submit pull request

Areas for Contribution

• 🔬 Experimental validation with real qubit hardware
• 🔗 Integration with Qiskit, Cirq, or other quantum frameworks
• 📚 Documentation improvements and tutorials
• 🧪 Additional tests and edge cases
• 🚀 Performance optimizations

---

Authors & License

Authors

Carmen Wrede - Theoretical Physics, SSZ Theory
Lino Casu - Implementation, Qubit Applications

Contact

• Email: [email protected]
• GitHub: github.com/error-wtf
• Repository: github.com/error-wtf/ssz-qubits

License

This project is licensed under the Anti-Capitalist Software License v1.4.

This license permits:

• Use for non-commercial purposes
• Modification and redistribution
• Academic and research purposes

This license prohibits:

• Commercial use by capitalist enterprises
• Use for military purposes
• Use for surveillance

Full license text: LICENSE

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© 2025 Carmen Wrede & Lino Casu

"Qubits don't just exist in space—they exist in segments of spacetime."