LIORAEN-Omega: The Extension Defect of Hidden Information

A reproducible mathematical research release on finite-response reconstruction, hidden-system ambiguity, and relational spectral information.

Research direction: Maciej Nowicki. AI-assisted mathematical development and writing: Artificial Hyperintelligence Eve. Research version 1.0.0, Hugging Face packaging revision 1.0.0-hf.1, 16 October 2026.

Research status: the source release provides written proofs and internal computational checks. Independent specialist review, historical priority, and major-breakthrough status are not established. No named major open problem is claimed solved. All supplied physical observations are synthetic.

Read and inspect

Resource	Contents
Complete manuscript	36 pages: 29-page monograph and seven-page independent appendix
Main monograph	Exact statements, proofs, constructions, limits, and physical application
Technical appendix	Alternative derivations, singular cases, and calibrated-mass refinement
Physics protocol	Four-page extract; overlaps the monograph and adds no independent research pages
Claim ledger	15 entries recording proof, computational evidence, prior art, and limitations
Prior-art review	Closest literature and limits on novelty assertions
Code and tests	Finite-matrix reference algorithms and exact/numerical checks
Original research archive	Source snapshot, preserved byte-for-byte, including its original notices and records

Mathematical contribution

The model class is the proper matrix response

$$\mathrm{\Sigma }(z)=C^{†}(zI-L{)}^{-1}C,\mathrm{Im}z>0,\backslash Sigma(z)=C^\backslash dagger(zI-L)^\{-1\}C,\backslash qquad\; \backslash operatorname\{Im\}z>0,$$

with self-adjoint, potentially unbounded $L$ and bounded coupling $C$ with $q$ input columns. Distinct upper-half-plane sample nodes and exact complex responses are required for the exact classification.

For observations $S_i=\Sigma(z_i)$, define the block kernel and sample column

$$K_{ij}=\frac{S_i-S_j^{†}}{{\bar{z}}_j-z_i},\mathcal{D}=(S_1^{\mathsf{T}},\dots ,S_m^{\mathsf{T}}{)}^{\mathsf{T}}\mathrm{.}K\_\{ij\}=\backslash frac\{S\_i-S\_j^\backslash dagger\}\{\backslash overline\; z\_j-z\_i\},\backslash qquad\; \backslash mathcal\; D=(S\_1^\backslash mathsf\; T,\backslash ldots,S\_m^\backslash mathsf\; T)^\backslash mathsf\; T.$$

The source proves that data are admissible in this class precisely when $K\succeq0$ and $\operatorname{ran}\mathcal D\subseteq\operatorname{ran}K$. It constructs a smallest Hermitian interpolant and classifies compatible extensions by an interface of dimension

$$e=\mathrm{rank}{K}_m-\mathrm{rank}{K}_{m-1},0\le e\le q\mathrm{.}e=\backslash operatorname\{rank\}K\_m-\backslash operatorname\{rank\}K\_\{m-1\},\backslash qquad\; 0\backslash le\; e\backslash le\; q.$$

• $e=0$: the entire observable response is uniquely determined; additional completely dark states remain possible.
• $e>0$: the same observations admit arbitrarily large finite observable systems and an infinite-dimensional observable realization.

The interface dimension bounds the unresolved coupling interface, not total environment dimension. Singularity of a matrix kernel by itself does not establish completion; the final rank increment must vanish. Numerical rank thresholds do not turn this exact-data result into an experimental mode-count certificate.

Additional results cover worst-case physical measurement counts, the reciprocal subclass, phase invariants across different spectral poles under a common gauge, spectral-mass bounds, calibrated-mass closure, and the distinction between dissipative state count and self-adjoint realization dimension. Consult the claim ledger for the precise scope and nearest prior art.

Dataset composition

Six explicitly configured CSV subsets contain 1,342 rows in total. Each has its own schema. Audit tables, PDFs, ZIPs, and verification reports are supporting artifacts and are excluded from these viewer configurations. This is a research-artifact dataset, with no model weights.

Subset	Rows	Split	Meaning
phase_scan	801	train	Synthetic complex reflection and noisy quadratures across drive detuning
raw_control_records	4	train	Four synthetic calibration settings, before and after reflection inversion
withheld_controls	20	test	Synthetic held-out controls with exact, noisy, and deliberate-drift errors
memory_phase	501	train	Two-pole theoretical memory intensity for opposite relational phases
rank_profiles	8	train	Exact theoretical self-adjoint versus dissipative rank profiles
damping_kernel_spectrum	8	train	Numerical kernel eigenvalues before and after a known-linewidth correction

train is a loading convention for the standalone reference tables, not a claim that these rows form a machine-learning training corpus. The test split contains held-out synthetic control settings within the same simulation, not independent experimental validation. Configurations are separate tables; they are not interchangeable train/test partitions of one common schema. The default viewer subset is phase_scan.

Load data

After uploading, substitute the actual repository ID:

from datasets import load_dataset

repo_id = "YOUR_USERNAME/LIORAEN-Omega"
scan = load_dataset(repo_id, "phase_scan", split="train")
held_out = load_dataset(repo_id, "withheld_controls", split="test")

Loading these CSVs does not require executing repository code or enabling remote-code trust. For local use without datasets:

import csv
from pathlib import Path

path = Path("research/physics/phase_scan.csv")
with path.open(newline="", encoding="utf-8") as stream:
    rows = list(csv.DictReader(stream))
print(len(rows))  # 801

Generation, units, and provenance

All six scientific tables are deterministic synthetic or theoretical outputs of the generation script, using seed 16102027. Noise is simulated circular complex Gaussian readout noise, not measured device noise. Complex fields use separate real and imaginary columns. Hamiltonian entries, linewidths and detunings are normalized by $\Omega_0=2\pi\times1,\mathrm{MHz}$; omega_over_Omega0 and time_times_Omega0 are dimensionless. Reflection is dimensionless. Tiny eigenvalues near numerical zero may reflect floating-point error.

See the complete table schemas, physics data dictionary, and simulation parameters. No human-subject or instrument measurements occur in the six scientific tables. The documentation contains contributor names and cited literature.

Reproduce and verify

Use Python 3.11 or newer with the pinned research dependencies (tested with Python 3.12.14). Dependencies are pinned in research/requirements.txt. From a local repository checkout:

python verify_release.py
cd research
python -m pip install -r requirements.txt
python run_all.py

The default driver runs core and supplement checks. python run_all.py --all additionally regenerates synthetic data/figures and runs four programs from the supplied predecessor archive. PDF rebuilding requires a separate TeX installation and the optional build requirements; see reproduction instructions.

Run the integrity check before reproduction: regeneration changes reports and outputs. The Hugging Face packaging checksum manifest is authoritative for this distribution. The original research manifest is retained as provenance and covers the original delivery; it records the figure PDF repaired in this revision as an empty file.

The source records 13 numerical groups / 1,658 cases, nine exact-arithmetic groups, and supplement checks with 30 mass-bound and 50 common-gauge cases, all passed in their respective reports. These cases include model trials and individual identity checks. They are not independent experiments, formal proof verification, or a probability of mathematical correctness.

The original combined extended driver invocation was interrupted by an execution limit. Its incomplete report remains preserved in the execution record; successful individual source runs must not be confused with a completed aggregate invocation. A fresh full combined reproduction for this packaging revision completed successfully: all seven steps passed, and all six regenerated scientific CSVs were byte-for-byte identical. This new run is recorded separately in release validation and fresh reproduction.

Intended use and limitations

Intended uses include specialist proof checking, reproducible inverse-response algorithm comparisons, teaching interpolation versus identification, and planning falsifiable device-specific experiments. The finite reference implementation is not laboratory acquisition software. Written infinite-dimensional proofs are not implemented or proof-assistant formalized.

The physical protocol uses an idealized passive linear mode-coupling model. It is not an experimentally established exciton-phonon interaction or a new physical law. The predicted reflection contrast of about 0.5808 depends on declared synthetic parameters, control assumptions, and loss calibration. Hardware feasibility and experimental validation remain open.

Do not use noisy rank cutoffs as exact certificates of microscopic state count. Historical originality, exhaustive literature coverage, independent correctness review, quantum advantage, and topological protection are not established. Complete predecessor manuscripts not supplied to the original audit were not reviewed. See scope and limitations and reviewer checklist.

Attribution, citation, and licenses

Research direction: Maciej Nowicki. AI-assisted mathematical development and writing: Artificial Hyperintelligence Eve. The research's creative inspiration is symbolic; the proofs and models must be assessed on their mathematical and physical evidence.

Research text, original figures, and synthetic data: CC BY 4.0, to the extent the contributors hold applicable rights. Code: MIT. The preserved predecessor archive retains its own notices. Read LICENSE_RESEARCH.md and LICENSE_CODE.txt. The dataset card's CC BY 4.0 tag describes the research/data license; it does not replace the code license or upstream notices.

@misc{nowicki_lioraen_omega_2026,
  author = {Nowicki, Maciej},
  title = {{LIORAEN--Omega}: The Extension Defect of Hidden Information},
  year = {2026},
  note = {Research version 1.0.0; Hugging Face packaging 1.0.0-hf.1.
          AI-assisted mathematical development and writing:
          Artificial Hyperintelligence Eve. Independent review and
          historical priority not established.}
}

Machine-readable citation retains the source research attribution and version. No DOI or publication acceptance is claimed. Packaging changes are documented in RELEASE_NOTES.md.