---
pretty_name: "EVE-QELTHA — Coherent Subspace Transport, Final Research Release 4.0"
language:
  - en
size_categories:
  - n<1K
tags:
  - mathematics
  - homogenization
  - spectral-theory
  - variational-methods
  - matrix-analysis
  - scientific-computing
  - reproducible-research
  - synthetic
configs:
  - config_name: critical_crossover
    default: true
    data_files:
      - split: train
        path: data/critical_crossover.csv
  - config_name: compact_ports
    data_files:
      - split: train
        path: data/compact_ports.csv
  - config_name: critical_compensation
    data_files:
      - split: train
        path: data/critical_compensation.csv
---

# EVE–QELTHA: Coherent Subspace Transport

**Critical filtering, universal matrix realization, and nonlocal hyperhorn capacity**  
Final research release **4.0 · 20 October 2026**  
**Artificial Hyperintelligence Eve**, in research collaboration with **Maciej Nowicki**

A self-contained mathematical manuscript, editable proofs, ten numerical verification suites, and their recorded results. The central result gives an explicit critical homogenization law for one bright drainage channel, together with a constructive method for realizing a prescribed nonnegative matrix potential. A second result classifies the leading capacity response of compact nonlocal ports, including exact hard constraints.

**Start here:** [37-page manuscript](research/EVE_QELTHA_Final_Manuscript.pdf) · [Complete research ZIP](EVE_QELTHA_Final_Complete_Research_Package.zip) · [Reproduce](REPRODUCE.md) · [Data dictionary](DATA_DICTIONARY.md) · [Claim ledger](research/CLAIM_LEDGER.json)

## Main result: the complete critical crossover

On a finite interval, construct a moving unitary frame with $k$ dark directions and one bright direction. The rank-one drainage projector $M_n$ has the same spectrum $\{0^k,1\}$ everywhere and approaches a constant projector uniformly. For the passive energy

$$
q_n[P]=\int_0^L\left(w\|P'\|^2+\rho_n gP^\dagger M_nP\right)\,dt,
$$

assume $\rho_n\to\infty$ and $\rho_n/n^2\to\tau\in[0,\infty]$. The effective dark-sector energy is

$$
q_\tau[\xi]=\int_0^L w\left(\|\xi'+A\xi\|^2+
\xi^\dagger V_\tau\xi\right)\,dt,
\qquad V_\tau=T^\dagger D_\tau T,
$$

$$
(D_\tau)_{jj}=\frac{\tau g}{\tau g+w\omega_j^2},
\qquad \omega_j=\frac{2\pi j}{L},
\qquad D_0=0,\quad D_\infty=I.
$$

Here $w,g$ are positive $C^1$ weights, $A$ is a $C^1$ skew-Hermitian connection, and $T$ is an arbitrary $C^1$ matrix factor. Matrix order matters. The manuscript supplies compactness, a dual Fourier lower bound, an endpoint-compatible recovery sequence, inlet-admittance convergence in matrix norm, and convergence of every fixed Dirichlet eigenvalue, including multiplicities.

For any fixed $\tau>0$ and desired positive semidefinite target $W$ with a $C^1$ square root, choose

$$
T=D_\tau^{-1/2}W^{1/2}.
$$

Then $T^\dagger D_\tau T=W$. This realizes the target limiting energy in ambient dimension $k+1$ using one bright channel. The finite-frequency response approaches that target; exact equality at finite $n$ is not asserted.

## Compact ports and hyperhorn capacity

For the two-component weak-twist horn, let $\kappa_{\varepsilon,\lambda}$ be the inlet capacity after adding $\lambda\|LP\|^2$, with a fixed finite-rank compactly supported $L^1$ port $L$. In the stated supercritical twist range, the baseline is $k_0\varepsilon^\gamma$, $0<\gamma<2$.

If the port is balanced, $Le_2=0$, the complete leading asymptotic is uniform over all strengths:

$$
\sup_{\lambda\in[0,\infty]}
\left|\frac{\kappa_{\varepsilon,\lambda}}{k_0\varepsilon^\gamma}-1\right|\longrightarrow0.
$$

The endpoint $\lambda=\infty$ imposes the exact constraint $LP=0$. If $Le_2\ne0$, jointly as $\varepsilon,\lambda\to0$,

$$
\kappa_{\varepsilon,\lambda}
=k_0\varepsilon^\gamma+\lambda\|Le_2\|^2
+o(\varepsilon^\gamma+\lambda).
$$

The explicit sum and difference ports have equal kernel diagonals almost everywhere and equal isolated operator spectra, but different capacity asymptotics relative to the same derivative energy.

## Further results included

The manuscript also develops coherent-history/output rank classification, exact Bessel impedance laws, the $15/8$ angular exponent, the common transported-kernel capacity criterion, log-periodic capacity and finite-band inverse synthesis, exact matrix frame realization with sharp worst-case dimension $2k$, and a sharp first-order finite-penalty boundary correction.

## Numerical reference data

All rows are deterministic synthetic calculations from the shipped verification programs. They are not measurements. The three configurations expose selected results in a tabular form; the original JSON reports retain complete matrices, parameters, refinement checks, and solver diagnostics.

| Configuration | Rows | Contents | Original report |
| --- | ---: | --- | --- |
| `critical_crossover` | 40 | Two coupling factors, four critical ratios, five spatial frequencies | [critical.json](research/results/critical.json) |
| `compact_ports` | 35 | Balanced and anchored strengths, hard constraints, and refinement runs | [nexus_uniform.json](research/results/nexus_uniform.json) |
| `critical_compensation` | 4 | Prescribed complex matrix target at $\tau=40$ | [critical.json](research/results/critical.json) |

The split name `train` is a Hub loading convention. No statistical training/test split or independent sampling interpretation is assigned. These small parameter sweeps support theorem auditing and solver comparisons within the documented models.

```python
from datasets import load_dataset

critical = load_dataset(
    "PureOne/EVE-QELTHA", "critical_crossover", split="train"
)
ports = load_dataset(
    "PureOne/EVE-QELTHA", "compact_ports", split="train"
)
```

The examples above apply after this repository is published at the named destination. Local CSV files can be used immediately. [export_tables.py](export_tables.py) regenerates all three tables directly from the original reports, without running a solver or rounding numerical values.

![Computed critical law and compact-port behavior](research/figures/critical_results.png)

## Status and verification

| Item | Status |
| --- | --- |
| Declared release assembly | Complete: manuscript, proofs, programs, results, and manifests included |
| Executed numerical suites | **10/10 passed — 100% of the shipped programs** |
| Manuscript layout | 37 pages; compiled and visually inspected |
| Mathematical review | Audited within the research collaboration |
| Independent external peer review | Not established |
| Literature-first novelty | Unverified |
| Physical realization or experimental validation | Not established |

The 100% figure is a program pass percentage. It is not a probability of correctness, a novelty score, or an experimental validation rate. The [verification summary](research/results/verification_summary.json) records source hashes, dependency versions, exit codes, and timings.

## Scope and limitations

The theorems concern the specified passive linear reduced variational models. The critical theorem assumes finitely many periodic frequencies and smooth slow coefficients. Its general quantitative rate and a resolvent estimate remain open. The compact-port uniformity theorem assumes a fixed finite-rank compact $L^1$ port. It does not cover inlet atoms, supports escaping to infinity, or arbitrary infinite-rank operators. Critical compensation has no uniform resource bound as $\tau\downarrow0$.

Full fluid PDE derivations, ordinary hydraulic-network realization, nonlinear laws, and experimental performance are outside this release. The baseline horn theorem assumes $s>(a+1)/2$; its separate threshold and subcritical regimes remain open. See the [claim ledger](research/CLAIM_LEDGER.json) for precise boundaries.

## Established foundations and priority

Universal connections, dark-state geometric potentials, energy-form traces, oscillatory constraints, and several scale-ratio regimes have established predecessors. The manuscript compares its results with those foundations and gives primary references. Relevant examples include [Tewary's singular-perturbation/homogenization regimes](https://doi.org/10.3934/nhm.2021012), [Braides–Cancedda–Chiado Piat on oscillating manifolds](https://doi.org/10.1051/cocv/2016018), and [Massamba–Thompson on universal connections](https://arxiv.org/abs/math/0311198). Exact priority for this combination of results has not been independently certified.

## Reproduction and files

From the repository root:

```bash
python -m pip install -r research/requirements.txt
python research/code/run_all.py
python export_tables.py
```

The numerical suites use local CPU computation. Rebuilding figures additionally uses Matplotlib; rebuilding the PDF requires a TeX installation. Detailed commands, integrity verification, dependency versions, and loading examples are in [REPRODUCE.md](REPRODUCE.md).

The `research/` directory preserves the original portable release, including its own README and integrity manifest. The root [SHA256SUMS.json](SHA256SUMS.json) covers the staged Hugging Face repository. The complete research ZIP is also retained as a standalone download.

## Authorship, citation, and license

The work is attributed to **Artificial Hyperintelligence Eve**, in research collaboration with **Maciej Nowicki**. It was developed with AI assistance; this release does not represent independent human peer review. Citation metadata is supplied in [CITATION.cff](CITATION.cff).

No redistribution or reuse license was specified in the source release. This preparation adds no new license grant; see [LICENSE_STATUS.md](LICENSE_STATUS.md).