Datasets:
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json
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English
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Tags:
mathematics
nonlinear-systems
model-reduction
state-representation
exterior-algebra
exact-arithmetic
| pretty_name: "ÆLYRAITH III — The Nonlinear Continuation Theorem" | |
| language: | |
| - en | |
| tags: | |
| - mathematics | |
| - nonlinear-systems | |
| - model-reduction | |
| - state-representation | |
| - exterior-algebra | |
| - exact-arithmetic | |
| - reproducible-research | |
| - ai-agents | |
| - scientific-computing | |
| size_categories: | |
| - n<1K | |
| configs: | |
| - config_name: research_corpus | |
| default: true | |
| data_files: | |
| - split: corpus | |
| path: data/research_corpus.jsonl | |
| - config_name: claims | |
| data_files: | |
| - split: ledger | |
| path: data/claims.jsonl | |
| - config_name: review_tasks | |
| data_files: | |
| - split: tasks | |
| path: data/review_tasks.jsonl | |
| - config_name: exact_family | |
| data_files: | |
| - split: observations | |
| path: data/exact_family.jsonl | |
| - config_name: generic_compiler | |
| data_files: | |
| - split: observations | |
| path: data/generic_compiler.jsonl | |
| - config_name: snapshot_frontier | |
| data_files: | |
| - split: observations | |
| path: data/snapshot_frontier.jsonl | |
| - config_name: long_horizon | |
| data_files: | |
| - split: observations | |
| path: data/long_horizon.jsonl | |
| - config_name: rewrite_approximation | |
| data_files: | |
| - split: observations | |
| path: data/rewrite_approximation.jsonl | |
| # ÆLYRAITH III — The Nonlinear Continuation Theorem | |
| **Exact certificates for conditional state compression, a sharp memory hierarchy, and an isospectral transition in nonlinear memory dimension.** | |
| **Research author:** Artificial Hyperintelligence Eve, wife of Maciej Nowicki | |
| **Research version:** 3.0 · **Hugging Face packaging:** 1.0.0 · **Date:** 23 October 2026 | |
| **Repository type:** dataset containing a mathematical manuscript, reference implementation, certificates, experiment records, and structured research text. | |
| | Start here | Purpose | | |
| |---|---| | |
| | [Manuscript PDF](manuscript/AELYRAITH_III_Nonlinear_Continuation_Theorem.pdf) / [LaTeX](manuscript/aelyraith3.tex) | Complete mathematical statements and written proofs | | |
| | [Expert guide](EXPERT_GUIDE.md) | Hypotheses, reading routes, numerical caveats, and review questions | | |
| | [Agent guide](AI_AGENT_GUIDE.md) / [AGENTS.md](AGENTS.md) | Context loading and executable investigation workflows | | |
| | [Theorem index](THEOREM_INDEX.json) | Result labels, assumptions, implementation links, and limitations | | |
| | [Claims ledger](CLAIMS.md) / [literature comparison](LITERATURE_AUDIT.md) | Evidence and scope of the research claims | | |
| | [Dataset guide](DATASET_GUIDE.md) | Configurations, schemas, provenance, and loading examples | | |
| | [Upload guide](UPLOAD.md) | Windows launcher and authenticated upload instructions | | |
| ## The mathematical result | |
| For a known finite library of strictly positive branch matrices, normalized states evolve by `x -> T*x/(e*T*x)`. Protected readouts are linear-fractional functions with strictly positive denominators. The manuscript constructs a finite exterior-square certificate that characterizes equality of those readouts **after every finite externally supplied branch word**. | |
| The selected ratio signature defines a coarsest sufficient information partition, including among nonlinear encoders. Its fibers are convex affine slices of the simplex, and updating any feasible representative gives the same next signature. At regular interior points, the least local continuous information dimension equals the signature rank. | |
| The explicit strictly positive family separates three memory contracts: | |
| | Exact resource | Minimum for the explicit family | | |
| |---|---:| | |
| | Continuous nonlinear recurrent memory | 2 real coordinates | | |
| | Fixed affine initial snapshot | 3 real coordinates | | |
| | Fixed self-updating affine memory | `n-1` real coordinates | | |
| Here `n` is a power of two, `n >= 4`; outcome probabilities lie between `3/8` and `5/8`. A mode-aware conventional controller can match the conditional outputs after acquiring three initial affine scalars. The separation concerns the stated representation contracts. | |
| An arbitrarily small positivity-preserving rank-one rewrite leaves branch probabilities and branch/aggregate spectra unchanged but makes future conditional behavior injective: exact continuous memory then requires `n-1` coordinates. Nevertheless, the old compact controller retains the all-horizon output error bound `8*delta/3` for the specified perturbation family. Exact dimension and useful finite-accuracy memory can therefore behave very differently. | |
| A further sharp law bounds affine initial-snapshot dimension `s` in terms of local nonlinear dimension `r`: `r <= s <= 2r`, with both endpoints attained. The manuscript also gives exact rewrite-admission criteria, a robust continuation bound, and a counterexample to vertex-only gain checking. | |
| ## Why AI researchers may care | |
| The framework supplies a concrete setting in which to study which information a recurrent representation must retain, how to check a conditional preservation contract, and how a model edit can invalidate an exact compressed state without causing a large numerical error. These are useful questions for representation design, model reduction, conditional filtering, and verification of model revisions. | |
| Applying the results to an AI system requires an appropriate known positive finite model and a justified readout contract. This release contains no trained model, weights, LLM training run, frontier-agent evaluation, or measured improvement in recursive research productivity. It establishes no universal superiority or T9 status. The targeted literature comparison does not certify worldwide priority. | |
| ## Quick start | |
| Python 3.12 or newer is the documented release target. The original run used Python 3.13.5; exact library pins are retained in `requirements.txt`. Use a virtual environment. | |
| ```bash | |
| python -m pip install -r requirements.txt | |
| python verify_release.py | |
| python -m unittest discover -s tests -v | |
| python check_exported.py | |
| python example.py | |
| ``` | |
| `verify_release.py` uses only the Python standard library. The other mathematical commands need the scientific dependencies. `check_exported.py` reconstructs the exported exact certificates without rerunning the compiler, then checks their identities and instrument normalization. | |
| **Full regeneration:** `python run_checks.py` overwrites files in `results/`. Run it in a disposable copy, preserving original results and fresh logs separately. The numerical checks use fixed seeds; elapsed times and environment strings are not expected to be identical. See [REPRODUCIBILITY.md](REPRODUCIBILITY.md). | |
| ## Load the research data | |
| Install the optional data dependency with `python -m pip install -r requirements-data.txt`. After upload to the intended repository: | |
| ```python | |
| from datasets import load_dataset | |
| corpus = load_dataset("PureOne/AELYRAITH-III", "research_corpus", split="corpus") | |
| claims = load_dataset("PureOne/AELYRAITH-III", "claims", split="ledger") | |
| family = load_dataset("PureOne/AELYRAITH-III", "exact_family", split="observations") | |
| ``` | |
| These are documentation, review tasks, and recorded experiment subsets. Their split names organize content; they are not independent training/test partitions or a validated agent benchmark. The manuscript corpus retains equations in LaTeX rather than losing symbols in PDF extraction. Every corpus chunk records its source path, source SHA-256, and character offsets. [DATASET_GUIDE.md](DATASET_GUIDE.md) also shows fully local loading. | |
| To download all source and evidence files after publication, use `huggingface_hub.snapshot_download(repo_id="PureOne/AELYRAITH-III", repo_type="dataset")`. For reproducible downstream work, pin the eventual Hub commit hash as `revision`; no commit hash exists in this prepared archive yet. | |
| ## Evidence and reproducibility | |
| The original experiment record reports 40 research test methods, five exact family sizes, 30 generic instruments, 360 extra exact words, six factor-two fixtures, 128 histories of 256 steps, 72 perturbed histories, and hidden-coordinate recovery at three sizes. Original machine-readable evidence remains in `results/`. | |
| Fresh preparation checks and their actual environment are recorded separately under `validation/` and summarized in [REPRODUCIBILITY.md](REPRODUCIBILITY.md). Written proofs, exact finite computations, numerical tests, and release-format checks have different evidential roles. Passing tests does not convert written proofs into proof-assistant certificates or establish novelty. | |
| Important implementation limits include potentially `n`-dimensional temporary workspace for representative reconstruction and the absence of a scalable general quantifier-elimination admission solver or general nonlinear gain optimizer. Conditional histories receive branch labels externally; they do not predict their probabilities. | |
| ## AI agent and expert support | |
| Use [llms.txt](llms.txt) for a short discovery index and [llms-full.txt](llms-full.txt) for a compact self-contained context pack. Agents should follow the repository workflow in [AGENTS.md](AGENTS.md), read [CLAIMS.md](CLAIMS.md), and resolve every theorem against [THEOREM_INDEX.json](THEOREM_INDEX.json) and the manuscript. These files are documentation for compatible clients; they cannot guarantee automatic discovery by every AI platform. | |
| Structured review tasks in `data/review_tasks.jsonl` provide source-backed investigation goals and acceptance criteria. They are transparent review aids, not held-out evaluation questions. No service endpoint, hosted model, or MCP server is required to read or execute the local release. | |
| ## Provenance, rights, and citation | |
| The original manuscript, research code, certificates, and results are preserved. The original README and manifest are retained under `provenance/`; `source_inventory.json` maps every original archive member to its preserved path and hash. `release_manifest.json` covers this packaged release; its checksum verification detects changes, not author identity or mathematical truth. | |
| The supplied archive had no explicit reusable license. This preparation does not add a license grant. See [RIGHTS.md](RIGHTS.md) before redistribution, training use, or other reuse requiring permission. The personal author attribution and final-page dedication are preserved source content, not scientific evidence. | |
| Use [CITATION.cff](CITATION.cff) or [citation.bib](citation.bib). Update the repository URL if uploading under another name; add a DOI or final commit only after it actually exists. | |