NOETHER

A Constructive Framework for Metamorphic Pattern Discovery from Operator Algebras

This repository accompanies the paper NOETHER: A Constructive Framework for Metamorphic Pattern Discovery from Operator Algebras. It contains the LaTeX source of the manuscript, an arXiv-ready preprint variant, a Python reference implementation of the CONSTRUCT-MP algorithm, the 84-MR PWR corpus, an SE(3)-equivariant point-cloud testing harness, and a reproducibility manifest.

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Motivation

Metamorphic Testing (MT) is recognised in IEEE/ISO software-testing standards and is widely recommended for testing AI systems. Its progress, however, remains constrained by metamorphic relation (MR) identification — a step that depends heavily on tester domain knowledge and yields MR sets that are difficult to reuse across teams and across program families.

Existing approaches share an inductive grounding that leaves three foundational questions open:

1. Origin — where do MetaPattern categories come from?
2. Closure — is a given MetaPattern set closed under reasonable composition operators, or are recurrent MR families silently missing?
3. Transferability — can the same construction mechanism be applied to a new program family without re-doing the inductive work?

Structured frameworks (METRIC, METRIC+), mining and evolutionary pipelines (MR-Scout, GenMorph), LLM-assisted methods, and the original MetaPattern catalogues each answer one of these questions while leaving the other two open. NOETHER addresses all three by relocating induction from MR samples to recurrent algebraic structures, then making the downstream construction mechanical.

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Core contributions

The framework is two-layer:

Downstream layer (mechanical, provable)

Given an operator-algebra block decomposition of a program family $P$, the CONSTRUCT-MP algorithm produces a MetaPattern set with:

• Theorem 1 — algebraic closure of the constructed MetaPattern set under the framework's Translate operator, within the algebra-induced MR space. Absolute completeness over arbitrary properties is identified as an open conjecture (Theorem 1$'$).
• Theorem 2 — polynomial-time decidability of the construction under a finite generating set.

Upstream layer (curated, explicit empirical hypothesis)

An eight-block decomposition over recurrent mathematical structures common to many program families, stated as Hypothesis 1 with a documented out-of-scope catalogue rather than as an axiom:

Block	Mathematical structure
$\mathcal{G}^*$	Symmetry / group action
$\mathcal{O}_{\le}^*$	Order / monotonicity
$\mathcal{T}^*$	Self-adjoint duality
$\mathcal{T}_{\mathrm{rev}}^*$	Time-reversal
$\mathcal{L}^*$	Limit / convergence
$\mathcal{D}^*$	Qualitative dynamics
$\mathcal{E}^*$	Method-comparison equivalence
$\mathcal{B}_{\mathrm{rel}}^*$	Relational equivalence

A Remark enumerates six program-family classes likely to require additional blocks beyond Hypothesis 1; these are explicitly out-of-scope for the present work.

Three instantiations

• Boltzmann reactor-physics transport solver — the framework systematises a prior 84-MR inductive PWR catalogue and re-classifies further equivalence classes.
• Equivariant machine learning — executable MRs are derived for rotation invariance, adjoint duality, and training-trajectory reversibility; a §6.6.1 DeepCrime-style pilot evaluates structural coverage on five mutation classes.
• Relational query optimisers — the idempotent-semiring algebra exercises the relational-equivalence block, $\mathcal{B}_{\mathrm{rel}}^*$.

Structural-coverage predictions hold across the three instantiations. Comparative evaluation against existing automated pipelines is reported as a pre-registered protocol rather than a claim of average superiority — the §6.6.1 pilot's sample size is insufficient for $\alpha = 0.05$ inferential conclusions and is reported as descriptive evidence consistent with the structural prediction.

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Repository layout

.
├── README.md                    # This file
├── REPRODUCTION.md              # Step-by-step replication guide
├── LICENSE                      # CC-BY 4.0 (manuscript) / MIT (code)
├── .gitignore
│
├── NOETHER_paper.tex            # Manuscript source (TOSEM submission variant)
├── NOETHER_paper.bib            # Bibliography (53 entries, all cited)
├── NOETHER_paper.pdf            # Compiled manuscript
│
├── arxiv/                       # arXiv preprint variant (de-anonymised)
│   ├── NOETHER_paper_arxiv.tex
│   ├── NOETHER_paper.bib
│   ├── build_arxiv.sh
│   └── README.md
│
├── supplementary/               # S1-S4 reproducibility archive
│   ├── README.md
│   ├── S1_construct_mp/         # Python reference implementation
│   ├── S2_pwr_corpus/           # 84-MR PWR corpus (CSV + protocol)
│   ├── S3_case_study/           # SE(3)-equivariant harness
│   └── S4_reproducibility/      # Seeds, env, dataset hashes
│
├── docs/                        # Planning notes (gitignored locally)
├── archive/                     # Process history (not for active use)
│   ├── process-history/         # Earlier drafts, review reports, response letters
│   └── pre-noether-research/    # Parallel-research markdown predating NOETHER
└── 已有论文/                     # Reference reading (PDFs, not redistributed)

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Reproducing the experimental results

The full step-by-step guide is in REPRODUCTION.md. Quick version:

1. Environment

# Apple Silicon (the manuscript's actual platform — see S4 environment.yml)
conda env create -f supplementary/S4_reproducibility/environment.yml
conda activate noether-equivariant

For CUDA hosts, swap the dependencies: block as documented in the environment.yml comments.

2. CONSTRUCT-MP unit tests (S1)

Verifies that the algorithm's step semantics match the specification in Appendix D of the manuscript.

cd supplementary/S1_construct_mp
python -m pytest tests/ -v

Expected: all unit tests pass.

3. Boltzmann instantiation (§5)

cd supplementary/S1_construct_mp
python boltzmann_instance.py

Reproduces the block-mapping output that informs the 12-row subset of Table 3 in §5.3. The full 84-MR corpus and the per-MR block annotations live at supplementary/S2_pwr_corpus/.

4. Equivariant-ML case study (§6.6)

cd supplementary/S3_case_study

# (a) Train (or load) the SE(3)-equivariant point-cloud classifier.
#     Skip this if a checkpoint is already present at model/se3_classifier.pt.
python model/train.py

# (b) Run the three MR sets against the 20 mutation scripts.
python runner.py --mr-set N --output results.csv
python runner.py --mr-set L --append-output results.csv
python runner.py --mr-set B --append-output results.csv

# (c) Aggregate to Table 4 numbers.
python analysis.py results.csv --output-dir .
cat table4.json

5. DeepCrime-style pilot (§6.6.1)

cd supplementary/S3_case_study
python runner_pilot.py
cat deepcrime_pilot_summary.json

The pilot is reported with $n=5$ per MR set; the manuscript explicitly flags the sample size as insufficient for $\alpha = 0.05$ inferential conclusions and reports the result as descriptive evidence consistent with the structural prediction (Set N: 2/5; Sets L, B: 0/5).

6. Integrity check

cd supplementary
find . -type f ! -name '.DS_Store' ! -name '*.pyc' ! -path '*/__pycache__/*' \
  -print0 | sort -z | xargs -0 cat | shasum -a 256

Expected output: dc54d8288205c98e1edd2a96e724cdc9261155990461b1c8efeee2e2db2e77b8.

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Building the manuscript

TOSEM submission variant (anonymised, root-level .tex)

xelatex -interaction=nonstopmode NOETHER_paper.tex
bibtex NOETHER_paper
xelatex -interaction=nonstopmode NOETHER_paper.tex
xelatex -interaction=nonstopmode NOETHER_paper.tex

arXiv preprint variant

See arxiv/README.md. Fill in the author block, then:

cd arxiv && ./build_arxiv.sh

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Sensitive-information policy

The repository follows a strict no-credentials policy:

• .env is .gitignored; only .env.example with your_*_api_key / your_*_base_url placeholders is committed.
• No personal absolute paths (/Users/<name>/...) appear in any file. Use the environment variable P2_ROOT to point at the optional P2 codebase from supplementary/S3_case_study/p2_integration.py.
• LaTeX .log / .aux artefacts are regenerated on each build and are also .gitignored.

Pre-commit grep verification:

grep -rIn -E "(/Users/[^/]+|sk-[A-Za-z0-9]{16,}|Bearer\s+[A-Za-z0-9]+|api_key\s*=\s*['\"][^'\"]{8,})" \
  --exclude-dir=.git --exclude-dir=.venv* --exclude-dir=texmf-dist --exclude-dir=node_modules
# expected: empty output

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Citing this work

A BibTeX entry will be added once the arXiv DOI is assigned. In the meantime:

@unpublished{NOETHER2026,
  title  = {{NOETHER}: A Constructive Framework for Metamorphic Pattern
            Discovery from Operator Algebras},
  author = {<AUTHOR_NAME>},
  year   = {2026},
  note   = {Manuscript; arXiv preprint forthcoming}
}

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License

• Manuscript and figures (NOETHER_paper.*, arxiv/*, supplementary/S2_pwr_corpus/*.csv, supplementary/*.md): CC-BY 4.0 — please cite the paper.
• Code (supplementary/S1_construct_mp/, supplementary/S3_case_study/, supplementary/S4_reproducibility/, arxiv/build_arxiv.sh): MIT.

See LICENSE for the full text.

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Contact

Issues and pull requests are welcome on the GitHub repository. For academic collaboration enquiries, see the corresponding-author email in the manuscript.