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Doubly Stochastic Matrices Over F₃

DOI License: MIT

Computational verification code for: "Doubly Stochastic Matrices Over F₃: Binary Trace Stratification and the Impossibility of Trace-2"

Author: Oksana Sudoma Status: Pre-print

Overview

This repository provides complete computational verification for the first enumeration of doubly stochastic 3×3 matrices over the finite field F₃. We discover 54 such matrices forming group DS₃(F₃) ≅ ((C₃ × C₃) : C₃) : C₂, and prove that their trace values exhibit binary stratification: 27 matrices with trace ≡ 0, 27 with trace ≡ 1, and 0 with trace ≡ 2—a constraint-induced F₃→F₂ field reduction.

Key discoveries:

  • 54 doubly stochastic matrices over F₃ (first enumeration)
  • 27-27-0 trace distribution (binary stratification)
  • Trace-2 impossibility (proven algebraically)
  • 432-element row-stochastic group ≅ AGL(2,3)

Quick Start

Prerequisites

  • GAP 4.12.2+ (download)
  • Python 3.9+
  • pytest (install: pip install pytest)

One-Command Verification

# Run all GAP computations (generates 4 output files)
python3 run_all_verifications.py

# Expected runtime: 5-10 minutes
# Outputs: row_stochastic_432.csv, doubly_stochastic_54.json,
#          trace_stratification.json, group_structure_verification.json

Run Tests

python3 -m pytest tests/ -v

# Expected: 7 tests passing

Main Results

Constraint Cascade

Constraint Count Group Structure Description
Row-stochastic (row sums ≡ 1) 432 AGL(2,3) ≅ (((C₃×C₃):Q₈):C₃):C₂ Affine general linear group
Doubly stochastic (+column sums ≡ 1) 54 DS₃(F₃) ≅ ((C₃×C₃):C₃):C₂ Index-8 subgroup with center C₃

Trace Distribution (Novel Discovery)

Trace Value Count Interpretation
tr(M) ≡ 0 (mod 3) 27 Normal subgroup (kernel)
tr(M) ≡ 1 (mod 3) 27 Unique coset
tr(M) ≡ 2 (mod 3) 0 Proven impossible (all trace-2 matrices singular)

Mathematical significance: First documented case of constraint-induced field reduction (F₃→F₂) in finite matrix groups.

Repository Structure

ternary-constraint-432-element-group/
├── gap/                    GAP enumeration scripts (4 files)
│   ├── enum_row_stochastic.g
│   ├── enum_doubly_stochastic.g
│   ├── trace_stratification_analysis.g
│   └── verify_group_structures.g
├── outputs/                Verified computational results (4 files)
│   ├── row_stochastic_432.csv
│   ├── doubly_stochastic_54.json
│   ├── trace_stratification.json
│   └── group_structure_verification.json
├── tests/                  Python verification suite (3 test files)
├── run_all_verifications.py
└── README.md

Mathematical Background

Doubly stochastic matrices satisfy both:

  • Row sums: Σⱼ Mᵢⱼ ≡ 1 (mod 3) for all rows i
  • Column sums: Σᵢ Mᵢⱼ ≡ 1 (mod 3) for all columns j

Working over finite field F₃ = {0,1,2}, these constraints define a 54-element group—the first such enumeration in the literature. Prior work explicitly excluded F₃ from general theorems (Linear Algebra and Its Applications, 1976).

Novel phenomenon: Despite operating in ternary field F₃, the trace observable takes values only in binary field F₂ = {0,1}. This F₃→F₂ reduction arises from the doubly stochastic constraint geometry.

Reproducibility

All results are computationally verified and fully reproducible:

  1. Run complete suite: python3 run_all_verifications.py
  2. Individual scripts: gap gap/enum_doubly_stochastic.g
  3. Verification tests: python3 -m pytest tests/ -v

Expected runtime: 5-10 minutes on standard hardware.

Citation

@misc{sudoma2025doubly,
  author = {Sudoma, Oksana},
  title = {Doubly Stochastic Matrices Over $\mathbb{F}_3$: Binary Trace Stratification and the Impossibility of Trace-2},
  year = {2025},
  doi = {10.5281/zenodo.17443365},
  url = {https://github.com/boonespacedog/ternary-constraint-432-element-group}
}

License

MIT License - see LICENSE file for details.

Author

Oksana Sudoma - Independent Researcher

Computational verification and literature review assisted by Claude (Anthropic) and ChatGPT (OpenAI). Mathematical formalism and scientific conclusions are the author's sole responsibility.

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