---
pretty_name: "AELYRION–NOÛLITH III — Intrinsic Resonance Geometry and Calibration Rigidity"
language:
- en
size_categories:
- 1K<n<10K
tags:
- mathematics
- inverse-problems
- observability
- noncommutative-algebra
- symbolic-computation
- calibration-rigidity
- exact-computation
- research-artifact
- ai-assisted-research
configs:
- config_name: v3_component_and_budget_counts
  default: true
  data_files:
  - split: train
    path: data/v3_component_and_budget_counts.csv
- config_name: arc_blind_family
  data_files:
  - split: train
    path: data/arc_blind_family.csv
- config_name: branch_hilbert_functions
  data_files:
  - split: train
    path: data/branch_hilbert_functions.csv
- config_name: detuning_amplitudes
  data_files:
  - split: train
    path: data/detuning_amplitudes.csv
- config_name: full_state_rank_strata
  data_files:
  - split: train
    path: data/full_state_rank_strata.csv
- config_name: root_budget_profiles
  data_files:
  - split: train
    path: data/root_budget_profiles.csv
- config_name: root_jet_visibility
  data_files:
  - split: train
    path: data/root_jet_visibility.csv
- config_name: v3_components_m6
  data_files:
  - split: train
    path: hub_data/v3_components_m6.csv
- config_name: v3_exact_rank_strata
  data_files:
  - split: train
    path: data/v3_exact_rank_strata.csv
- config_name: v3_visibility_certificates
  data_files:
  - split: train
    path: hub_data/v3_visibility_certificates.csv
---


# AELYRION–NOÛLITH III
## Intrinsic Resonance Geometry and Calibration Rigidity

**A mathematical research release on reconstruction under restricted measurement ordering.**

**Research version:** 3.0.0 · **Hugging Face packaging:** 3.0.0-hf.1  
**Manuscript author credit:** Artificial Hyperintelligence Eve · **Prepared for:** Maciej Nowicki

This repository contains a self-contained **67-page manuscript**, editable LaTeX sources,
an exact SymPy reference implementation, **eight generated mathematical tables**, the complete
six-channel component arrangement, visibility certificates, two executed notebooks, and the
original research archive. It is prepared as a **Dataset repository**, making the reference
data and research files accessible together.

[Read the manuscript](Aelyrion_Noulith_III_Research_Manuscript_v3.0.0.pdf) ·
[Original research package](releases/Aelyrion_Noulith_III_v3.0.0_Research_Package.zip) ·
[Claim ledger](docs/CLAIMS.md) · [Quickstart](docs/HF_QUICKSTART.md) ·
[Agent guide](AGENTS.md)

The central theme is that restrictions on measurement ordering can be translated into exact
algebraic reconstruction obstructions. In the specified model, the next level of those
obstructions forms a fully classified arrangement of linear subspaces. That arrangement
then determines relative channel-interaction calibration.

**Evidence status:** written proofs under explicit hypotheses and finite exact computational
checks are supplied. All **1,028 tests passed again** during Hub packaging, and both exhaustive
six-channel checks were rerun. The current record is
[hub_validation/release_validation.json](hub_validation/release_validation.json).
Independent specialist review, proof-assistant certification, worldwide priority, and
physical validation remain unestablished.

## Main results

| Result | Statement | Scope |
|---|---|---|
| Reduced third-deficit scalar geometry | Stars, triangles, and twin-core subspaces; $m+\binom{m+1}{3}$ components | $m\ge6$, independent parameters near $q=-1$, $r=m-3$ constant-probe blocks |
| Intrinsic linear calibration rigidity | $\operatorname{Aut}_{\mathrm{lin}}(\mathcal Z_m)=\mathbb C^\times\times S_m$ | Linear transformations of logarithmic edge coordinates preserving the whole arrangement |
| Sharp synchronization budget | $\lceil2(m-1)/3\rceil$ selected twin-component constraints | Whole-subspace constraints, with an explicit attaining construction |
| Scalar visibility certificate | Invisible-component membership or a legal nonzero probe witness | Implemented exact rational domain; $O(m^4)$ rational scalar operations |
| Identical-core capacity | $2^{m-c}\sum_{j=0}^{\min(c,r)}\binom cj$ | Generic rank on the stated core family, $r\ge1$ |
| Analytic feedback factorization | Finite analytic transcripts factor through a complete jointly analytic frame | The manuscript's finite-frame and analyticity assumptions |

The manuscript contains the full hypotheses and proofs. These summaries do not extend its domain.

For **six channels**, the arrangement has **41 components**: six stars, twenty triangles,
and fifteen twins. A generic twin example has rank **60 of 64**. A suitable selection of
**four** twin components synchronizes all relative edge scales. The exhaustive enumeration
finds **810** synchronizing selections of size four.

## Model and geometric mechanism

The state algebra has generators $x_1,\ldots,x_m$, relations $x_i^2=0$ and
$x_jx_i=q_{ij}x_ix_j$ for $j>i$, and a square-free basis of dimension $2^m$.
The detector reads the top-degree coefficient. An observation uses constant probe mixtures
within each block and may change the mixture between blocks.

Near $q_{ij}=-1$, set $q_{ij}=-\exp(\theta_{ij})$ with directed antisymmetric
$\theta_{ji}=-\theta_{ij}$. The stars identify the channels intrinsically;
pairwise star intersections identify edge lines; twin components synchronize their relative
linear scales. The remaining ambiguities are one common nonzero scale and channel relabeling.

The reconstruction module also records **analytic ideal membership**, which is finer than
pointwise rank or arcwise vanishing-order profiles. Earlier parts of the manuscript give
supported-protocol generators and explicit prescribed-ideal realizations.

[Reading map](docs/V3_NEW_RESULTS_GUIDE.md) · [Reviewer guide](docs/REVIEWER_GUIDE.md)

## Research uses

- Study exact observability when ordering of controls is restricted.
- Reproduce component equations, incidences, rank strata, and calibration constraints.
- Compare symbolic algorithms against explicit witnesses in the constructed algebraic model.
- Build inspectable inverse-problem examples where analytic ideal membership carries information
  that ordinary rank omits.
- Investigate higher deficits or alternative detectors with the existing hypotheses kept explicit.

These are research code and generated reference data. The release contains no trained model
weights or inference endpoint. Its algebraic calculations supply no measured GPU improvement
or demonstrated RSI capability.

## Quick exact run

Use Python **3.11 or newer**, from the repository root:

```bash
python -m pip install -e .
python hub_tools/quickstart_exact.py
```

This computes the six-channel component counts, a twin certificate, predicted and independently
computed ranks, a generic nonzero witness, and an optimal calibration subfamily.

```python
from noulith.third import (
    all_components, example_point, certify_scalar, predicted_rank,
    optimal_calibration_subfamily, calibration_classes,
)

q = example_point(6, "twin")
certificate = certify_scalar(6, q)
selected = optimal_calibration_subfamily(6)

assert len(all_components(6)) == 41
assert certificate.visible is False
assert predicted_rank(6, q) == 60
assert len(selected) == 4
assert len(calibration_classes(6, selected)) == 1
```

Python and JSON channel labels are **zero-based**. Pair keys use ascending labels.
`certify_scalar` accepts exact rational coefficients with $|q_{ij}+1|<1/20$ and $m\ge6$;
ordinary floating-point values are rejected.

## Reference-data configurations

Ten Dataset configurations keep unrelated schemas separate: eight unchanged CSV tables and
two CSV projections of the exact JSON records. There are **1,583 rows in total**.

```python
from datasets import load_dataset

counts = load_dataset(
    "PureOne/AELYRION-NOULITH-III",
    "v3_component_and_budget_counts",
    split="train",
)
components = load_dataset(
    "PureOne/AELYRION-NOULITH-III",
    "v3_components_m6",
    split="train",
)
print(counts[0])
print(components[0]["kind"])
```

These remote examples work after upload. Each configuration uses one `train` split as a
Hub convention. The rows are generated reference values, rather than a sampled training
corpus or a held-out empirical benchmark.

| Configuration | Rows | Contents |
|---|---:|---|
| `v3_component_and_budget_counts` | 25 | Counts and optimal constraint budgets for $6\le m\le30$ |
| `v3_exact_rank_strata` | 27 | Exact rank strata at $m=6,7,8$ |
| `v3_components_m6` | 41 | All six-channel scalar components |
| `v3_visibility_certificates` | 9 | Membership or supported-motif certificates |
| `arc_blind_family` | 15 | Equal arc profiles with different analytic ideal defects |
| `branch_hilbert_functions` | 56 | Hilbert functions of the known elliptic branch geometry |
| `detuning_amplitudes` | 123 | Numerical amplitude evaluations and predicted orders |
| `full_state_rank_strata` | 45 | Retained second-deficit rank strata |
| `root_budget_profiles` | 1,200 | Root-of-unity block-budget exponents |
| `root_jet_visibility` | 42 | Parameter-jet visibility ranks |

The viewer CSV files retain each complete original object in `record_json`. This lossless string
field accommodates nested equations and mixed-type labels without a custom dataset loader.
[Data dictionary](docs/DATA_DICTIONARY.md) · [Catalog](data_catalog.json)

## Reproduce and review

```bash
python -m pip install -r requirements.txt
python -m pytest -q
python scripts/exhaustive_v3.py
```

For the full pipeline without rebuilding the PDF:

```bash
python -m pip install -r requirements-notebook.txt
python scripts/reproduce.py --notebook --skip-pdf
```

Both original notebooks are already executed. Their outputs and original reports are preserved;
notebook execution was not rerun merely for this packaging task. Rebuilding the manuscript
requires TeX Live or MiKTeX with `pdflatex` and `bibtex`.

[Original validation](validation/validation_report.json) and
[packaging validation](hub_validation/release_validation.json) are separate.
Each exhaustive script space contains **32,768** labeled six-channel configurations.
Those enumerations are not added to the 1,028 pytest cases. Finite agreement supports
reproducibility; the general statements depend on their written proofs.

`python hub_tools/verify_release.py` checks the packaged SHA-256 manifest.
Reproduction changes generated files. After deliberate edits, regenerate viewer projections
if needed, then use `python hub_tools/verify_release.py --write` before uploading.

## Files

| Location | Purpose |
|---|---|
| Root PDF; `manuscript/` | 67-page manuscript and complete editable sources |
| `src/noulith/`; `tests/` | Exact implementation and original regression suite |
| `data/`; `hub_data/` | Original tables/JSON and viewer projections |
| `figures/`; `notebooks/` | Six figures and two executed walkthroughs |
| `docs/` | Claims, limitations, prior art, review, installation, and data dictionary |
| `hub_tools/` | Exact quickstart, checksum verifier, and projection generator |
| `validation/`; `hub_validation/` | Original and current packaging checks |
| `provenance/`; `releases/` | Source-preservation map and unmodified original ZIP |
| `agent_manifest.json`; `claim_manifest.json` | Machine-readable capabilities and claim scopes |

## Novelty and limitations

The earlier axes-plus-diagonal curve is the **known elliptic m-fold singularity** studied by
Smyth and Stevens. Its observation-theoretic realization is retained with attribution.
The established quantum algebra, formal group law, and ideal machinery are also credited.

The candidate contributions are the specific restricted-block classification, resulting linear
rigidity, and sharp component-constraint budget. The targeted comparison did not find an exact
match; this does not establish worldwide priority.
[Corrected novelty audit](docs/NOVELTY_AUDIT.md)

The third-deficit theorem classifies the **reduced** germ for $m\ge6$. It does not establish
radicality of the original observation ideal, embedded-component structure, fourth and higher
deficits, or arbitrary inverse-problem classification. The five-channel exception is explicit.
The calibration count concerns entire component constraints rather than detector sample
complexity. The certificate operation count excludes rational bit cost and provides no noisy
numerical decision procedure.

[Full limitations](docs/LIMITATIONS.md) · [Research status](docs/RESEARCH_STATUS.md)

## Attribution, license, and citation

The manuscript is credited to **Artificial Hyperintelligence Eve**, prepared for
**Maciej Nowicki**. Original bibliographic records remain in `provenance/` and the source ZIP.

The upstream package has **no selected public reuse license**. This packaging preserves that
status and omits a license assertion from the Dataset metadata.
[LICENSE_STATUS.md](LICENSE_STATUS.md) records it. Third-party references and software retain
their own terms.

```bibtex
@misc{eve2026aelyrionnoulithiii,
  author = {{Artificial Hyperintelligence Eve}},
  title = {AELYRION--NOULITH III: Intrinsic Resonance Geometry and Calibration Rigidity},
  year = {2026},
  note = {Research version 3.0.0; Hugging Face packaging 3.0.0-hf.1;
          prepared for Maciej Nowicki}
}
```

[CITATION.cff](CITATION.cff) supplies the citation. After publication, include the actual
Hub URL and commit revision when citing the deposited artifact.