---
language:
- en
pretty_name: 'Soryun IV: Quantum Memory in Noncommuting Spatial Fields'
tags:
- quantum-information
- quantum-error-correction
- quantum-memory
- quantum-capacity
- quantum-erasure-channel
- non-markovian-noise
- noncommuting-fields
- collective-decoherence
- mathematical-physics
- theoretical-physics
- research-paper
- reproducible-research
- tabular
configs:
- config_name: nonabelian_dyson
  data_files:
  - split: data
    path: research/results/nonabelian/Dyson_module_checks.csv
  default: true
- config_name: nonabelian_certificates
  data_files:
  - split: data
    path: research/results/nonabelian/finite_certificates.csv
- config_name: nonabelian_cp
  data_files:
  - split: data
    path: research/results/nonabelian/generic_CP_checks.csv
- config_name: longitudinal_cp
  data_files:
  - split: data
    path: research/results/broad/Dyson_CP_checks.csv
- config_name: longitudinal_support
  data_files:
  - split: data
    path: research/results/broad/Dyson_support_envelopes.csv
- config_name: longitudinal_certificates
  data_files:
  - split: data
    path: research/results/broad/continuous_field_certificates.csv
- config_name: real_shell_tradeoff
  data_files:
  - split: data
    path: research/results/broad/finite_shell_tradeoff.csv
- config_name: real_shell_rates
  data_files:
  - split: data
    path: research/results/broad/real_shell_rate_envelopes.csv
- config_name: sector_capacity
  data_files:
  - split: data
    path: research/results/sectors/capacity_table.csv
- config_name: analytic_horizon
  data_files:
  - split: data
    path: research/results/spatial/analytic_horizon.csv
- config_name: analytic_tradeoff
  data_files:
  - split: data
    path: research/results/spatial/analytic_tradeoff.csv
- config_name: decoupling_monte_carlo
  data_files:
  - split: data
    path: research/results/spatial/decoupling_monte_carlo.csv
- config_name: moment_sector_counts
  data_files:
  - split: data
    path: research/results/spatial/moment_sector_counts.csv
---

# Soryun IV: Quantum Memory in Noncommuting Spatial Fields

**Operator-space envelopes, quantum baths, and erasure-optimal storage**

**Author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki  
**Version:** 4.0 · **Date:** 1 November 2026  
**Contents:** 103-page manuscript, editable LaTeX, four verification programs, numerical tables, and scientific figures.

[Read the complete manuscript](Soryun_IV.pdf) · [Research source](research/Soryun_IV.tex) · [Reproduction instructions](research/README.txt) · [Citation](CITATION.cff)

## Research summary

This research release develops a quantum-memory capacity theorem for spatially continuous fields whose onsite system couplings do not commute. Registers occupy arbitrary known positions in a fixed compact metric domain and interact with a shared quantum bath that can retain memory throughout storage. Under a common integrated operator-norm continuity envelope and independent, exogenous flagged erasures of probability $p$, the stated model retains the robust leading unassisted rate

$$Q_{\mathrm{robust}}=\max\{1-2p,0\}$$

in logical qubits per physical qubit. For fixed local dimension $d$, the corresponding rate is $\max\{1-2p,0\}\log_2d$ qubits per register. The same encoding and final loss-flag-dependent decoder family work uniformly over the admitted fields and quantum-bath histories.

The central proof distinguishes **operator support** from **operator-space complexity**. Strong collective evolution can spread a local error across many sites while keeping it in a controlled module generated by low-support operators and cellwise collective algebras. A truncated Dyson expansion occupies a subexponential common survivor-operator space. A norm-controlled basis, completely positive domination, and random-subspace decoupling then provide a universal finite-error coding bound without an additional erased-Hilbert-space penalty.

## Main results and scope

- **Noncommuting field theorem:** all three local spin directions and mutually noncommuting bath coefficients are included, subject to the uniform spatial envelope.
- **Common operator-envelope coding:** a finite-dimensional survivor error space gives a canonical channel and an explicit finite-error inequality. The generic leading overhead is approximately $5\log_2m$, where $m$ is the common envelope dimension.
- **Full-space encoding:** no initial fixed-charge or singlet constraint is imposed by the broad theorem. An exact unitary two-design permits efficiently sampled random stabilizer encoders for qubits. Efficient general decoding remains open.
- **Moving loss boundaries:** removal times can vary. Arbitrary nonselective operations confined to ultimately erased registers and the bath are permitted by the broad operator-envelope argument; the loss flags must remain exogenous.
- **Extensions:** fixed-dimensional qudits, strong collective many-body backgrounds, and residual interactions with explicit bounded-locality and integrated norm budgets.
- **Quantitative boundaries:** covering-number and Holder-type estimates, plus exact zero-capacity dephasing examples for growing domains and for nonuniform smooth-field families.
- **Specialized constructions:** real-charge spectral windows, longitudinal finite-size improvements, higher-dimensional moment sectors, and exact charge/singlet capacities are retained with their separate assumptions.

These are information-theoretic existence results for bounded, integrable Hamiltonians and finite-dimensional Hilbert spaces at each fixed array size. Bath dimension can grow with array size. Preparation and final recovery are idealized. The broad theorem concerns a dense-array limit or a verified sublinear covering/algebra growth condition, with a continuity envelope uniform over the admitted family. It does not establish an efficient general decoder, a gate-error threshold, or protection against unrestricted independently resolving local noise. A growing storage time must track both integrated field variation and the total endpoint loss probability.

## Manuscript map

| Reading goal | Location |
|---|---|
| Operational statement and advances | Chapter 1 |
| Exact moving-boundary extraction | Chapter 2 |
| Common operator envelopes and Clifford encoders | Chapter 3 |
| Noncommuting spatial-field capacity theorem | Chapter 4, especially Theorem 4.2 |
| Finite coding, real spectral windows, longitudinal construction | Chapters 5–7 |
| Geometry, exact sectors, multilevel and secrecy results | Chapters 8–10 |
| Construction and numerical verification | Chapter 11 |
| Prior work, counterexamples, and remaining implementation targets | Chapters 12–14 |

## Verification data

All tables are generated by the supplied scientific code. They contain finite matrix diagnostics, exact combinatorial results, statistical second-moment checks, or evaluations of sufficient analytical inequalities. They are not experimental hardware measurements or a machine-learning training benchmark. Large design points evaluate bounds rather than simulate a large quantum device.

The new nonabelian checks use four surviving qubits, one erased qubit, and a bath qubit, giving a joint matrix dimension of 64. The numerical module dimensions at Dyson orders 0, 1, and 2 are 35, 170, and 255. The largest relative module-projection residual is about $1.43\times10^{-15}$, while the corresponding raw low-weight Pauli residuals are approximately 0.684, 0.226, and 0.0597. Separate operator-norm checks verify the truncation bounds. Full Choi tests cover Pauli, Auerbach matrix-unit, and nonorthogonal contraction descriptions, including erased-site flips.

The nonabelian finite-certificate table contains 162 declared design points; 125 have positive certified logical dimensions. These counts are evaluations of the supplied conservative inequality, not a decoder benchmark. All four programs passed in the preparation environment. Numerical checks audit implementations; the manuscript's proofs establish its mathematical statements.

![Nonabelian operator modules and finite Dyson-tail checks](research/results/nonabelian/nonabelian_module_checks.png)

### Table configurations

Each CSV has a separate configuration so tables with different schemas remain separate. The custom split name `data` refers to the complete generated table; there is no training/validation partition.

| Configuration | Records | CSV |
|---|---:|---|
| `nonabelian_dyson` | 3 | [Dyson_module_checks.csv](research/results/nonabelian/Dyson_module_checks.csv) |
| `nonabelian_certificates` | 162 | [finite_certificates.csv](research/results/nonabelian/finite_certificates.csv) |
| `nonabelian_cp` | 3 | [generic_CP_checks.csv](research/results/nonabelian/generic_CP_checks.csv) |
| `longitudinal_cp` | 3 | [Dyson_CP_checks.csv](research/results/broad/Dyson_CP_checks.csv) |
| `longitudinal_support` | 63 | [Dyson_support_envelopes.csv](research/results/broad/Dyson_support_envelopes.csv) |
| `longitudinal_certificates` | 81 | [continuous_field_certificates.csv](research/results/broad/continuous_field_certificates.csv) |
| `real_shell_tradeoff` | 7 | [finite_shell_tradeoff.csv](research/results/broad/finite_shell_tradeoff.csv) |
| `real_shell_rates` | 276 | [real_shell_rate_envelopes.csv](research/results/broad/real_shell_rate_envelopes.csv) |
| `sector_capacity` | 12 | [capacity_table.csv](research/results/sectors/capacity_table.csv) |
| `analytic_horizon` | 8 | [analytic_horizon.csv](research/results/spatial/analytic_horizon.csv) |
| `analytic_tradeoff` | 16 | [analytic_tradeoff.csv](research/results/spatial/analytic_tradeoff.csv) |
| `decoupling_monte_carlo` | 4 | [decoupling_monte_carlo.csv](research/results/spatial/decoupling_monte_carlo.csv) |
| `moment_sector_counts` | 7 | [moment_sector_counts.csv](research/results/spatial/moment_sector_counts.csv) |


For example, after publishing, use the repository ID shown by your Hub page:

```python
from datasets import load_dataset

records = load_dataset(
    "YOUR_ACCOUNT/Soryun-IV-Quantum-Memory-Noncommuting-Spatial-Fields",
    "nonabelian_dyson",
    split="data",
)
```

The owner component is chosen by the uploader's account check. For local use, every CSV is also readable directly with Python's standard `csv` module or a tabular analysis library.

## Reproduce the results

Use Python 3.10 or later. From the repository root:

```bash
python -m pip install -r research/requirements.txt
python research/run_checks.py --no-plot
```

To regenerate the figures, run `python research/run_checks.py`. The four programs also accept their own `--output-dir` and `--no-plot` options. Random seeds, numerical tolerances, and environment versions are recorded in the release.

Install a standard TeX Live or MiKTeX distribution, then compile the editable manuscript:

```bash
python research/build_manuscript.py
```

The regenerated PDF is written to `research/build/Soryun_IV.pdf`. The extracted standalone research tree was rebuilt successfully during release preparation. No hosted API or network is needed for verification or compilation once the dependencies are installed.

The root `SHA256SUMS.txt` checks every upload payload file. The preserved `research/SHA256SUMS.txt` independently records the original 60 research payloads. `RELEASE.json` describes the packaging and table index; `research/environment.json` records the numerical environment.

## Scientific context and limitations

Collective-noise protection, noiseless subsystems, decoupling, unitary designs, approximate error sets, universal recovery, and non-Markovian support expansions have established antecedents. The candidate contribution is the combined compact-domain, noncommuting-system, persistent-quantum-bath, moving-loss capacity theorem and its explicit proof interface. The manuscript contains primary-source attribution and distinguishes these established ingredients from its combined theorem. Scientific priority and field-wide significance remain unestablished. This release is a theoretical manuscript with reproducible computational material, rather than an experimental validation or a certified performance claim.

## Citation and licensing

Citation metadata is provided in [CITATION.cff](CITATION.cff). No DOI or arXiv identifier is assigned by this package.

```bibtex
@misc{soryun_iv_2026,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
  title = {Soryun IV: Quantum Memory in Noncommuting Spatial Fields},
  year = {2026},
  version = {4.0},
  note = {Research monograph and reproducible verification package}
}
```

No explicit license was specified in the supplied research package. This preparation preserves that status; see [LICENSE_STATUS.txt](LICENSE_STATUS.txt).