PureOne/EIROQ-PHI

EÏRŌQ–Φ

Revealing Hidden Correlations for Certified Quantum Control

Author: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki
Research version: 1.0.0 · Hugging Face packaging: 1.0.0-hf.1 · Date: 16 October 2026

Exact pulse-budget observability, hidden non-Gaussian correlations, and sharp control certificates for finite binary-phase qubit dephasing, with a 97-page standalone manuscript, reference code, constructive proofs, and reproducible synthetic datasets.

Read the manuscript · Data dictionary · Reproduce the research · Claim ledger

This is a theoretical and computational research archive hosted as a dataset. Its tables contain synthetic calculations and verification fixtures. The release contains no trained model weights or laboratory measurements.

Research question

When a qubit experiences correlated phase fluctuations, which correlations can an experiment reveal with a limited number of refocusing pulses? When can a control operation be certified without reconstructing the complete noise law or measuring that operation directly?

The research studies these questions on a finite time grid with calibrated binary phase support. It gives explicit information limits, matching reconstruction constructions, examples where shallow experiments cannot decide whether a deeper control helps, and sharp certificates for another family of unmeasured controls.

Main results and their assumptions

Exact pulse-budget rank

For $n$ bins, known phase magnitudes $0<\alpha_i<\pi/2$, a centrally symmetric joint sign distribution, and an even Walsh-moment degree bound $2\le d\le n$, the rank of the information in prefix experiments with at most $p\ge0$ refocusing pulses is

$$R(n,d,p)=\sum _{j=1}^{\min (d,p+1,n-1)}\left(\genfrac{}{}{0px}{}{n-1}{j}\right)\mathrm{.}R(n,d,p)=\backslash sum\_\{j=1\}^\{\backslash min(d,p+1,n-1)\}\backslash binom\{n-1\}\{j\}.$$

A canonical complete acquisition uses exactly

$$D(n,d)=\sum _{\begin{array}{c}2\le r\le d\\ r\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\end{array}}\left(\genfrac{}{}{0px}{}{n}{r}\right)D(n,d)=\backslash sum\_\{\backslash substack\{2\backslash le\; r\backslash le\; d\backslash \backslash r\backslash ;\backslash mathrm\{even\}\}\}\backslash binom\; nr$$

independent scalar expectations. The release supplies dense and local mixed-difference reconstruction algorithms. With seven bins and degree four, there are 56 moment coordinates: a two-pulse budget reveals rank 41, while three pulses reveal all 56.

The moment-degree condition is a restriction on sign moments of the probability law. It is not a Gaussian assumption, a cumulant-order cutoff, or a bound on the degree of an Ising Hamiltonian. A scalar expectation generally needs many experimental shots; setting minimality is not shot or time minimality.

Positive laws invisible to shallow experiments

An explicit non-Gaussian family on $2d-1$ bins agrees on lower-order moments and on every permitted prefix experiment with at most $d-2$ pulses at every common phase amplitude. A deeper intervention separates the laws. Appropriate locally unequal phase amplitudes can also reveal the hidden direction.

For the seven-bin, degree-four example at phase magnitude $\pi/4$, the two laws share a full-duration Ramsey coherence of approximately 0.088388. The revealing three-pulse control produces approximately 0.176777 for one law and zero for the other. An unchanged shallow dataset is therefore compatible with opposite decisions about this deeper control.

Sharp prediction of an unmeasured echo

For $2k$ binary phase increments of magnitude $\pi/4$, a staircase of $k$ lower-depth prefix calibration means $w_j$ constrains an unmeasured $k$-pulse echo $E_k$. Over all joint sign laws satisfying $|w_j|\le\varepsilon$,

\min\operatorname{Re} W$E_k$= \max\left\{1-$2^k-1$\varepsilon, -(2^{k-1}-1)\varepsilon,-1\right\}, \qquad 0\le\varepsilon\le1.

Every branch has an explicitly constructed attaining law. This theorem does not require Gaussianity, independence between bins, central symmetry, or a moment-degree cutoff. Its calibrated binary support and phase magnitude remain essential. The initial error amplification $2^k-1$ makes the precision cost explicit.

Finite-support linear programs, exact rational primal–dual witnesses, and finite-shot confidence intervals extend the decision framework. Statistical guarantees require the declared repeated-trial and readout assumptions; they do not automatically cover arbitrary drift or pulse miscalibration.

Data contents

The dataset viewer exposes 11 separate configurations, containing 532 rows in total. Each configuration selects exactly one CSV because their schemas differ. The conventional Hugging Face split name train is used as a loading container; these are not machine-learning training, validation, or test splits.

Configuration	Rows	Contents
rank_law (default)	49	Theoretical rank versus exact modular rank fixtures
inverse_roundtrips	180	Finite-law reconstruction residuals
hidden_control_checks	12	Numerical invisibility checks across degrees and amplitudes
global_amplitude_sweep	121	Seven-bin shallow baseline and deeper-control responses
capability_exchange	16	Pulse and local-amplitude pair-coverage tradeoff
staircase_sharp_bounds	88	Sharp formula, LP optimum, and constructive attainment
calibration_deletion	15	Worst-case echo when one calibration is removed
finite_shot_coverage	4	Monte Carlo coverage and decision counts
unitary_propagation	12	Unitary/channel-fidelity versus characteristic-function checks
visibility_cost	6	Revealing slopes, optimal amplitudes, and a sufficient shot bound
setting_counts	29	Full symmetric and degree-limited parameter counts

All columns, units, indexing conventions, and generation details are documented in the data dictionary. The fixed research seed is 2026101602; no individual row is a human participant or a laboratory device. Exact matrices in NPZ format, native fixtures, JSON certificates, and the claim ledger are supplementary research artifacts outside the viewer configurations.

Load the tables

Replace YOUR_USERNAME/EIROQ-PHI with the actual repository identifier after uploading:

from datasets import load_dataset

repo_id = "YOUR_USERNAME/EIROQ-PHI"
ranks = load_dataset(repo_id, "rank_law", split="train")
sharp_bounds = load_dataset(repo_id, "staircase_sharp_bounds", split="train")
print(ranks[0])

For a local checkout, the same configurations can be loaded by supplying its absolute directory as repo_id. To read a single CSV without the Datasets library, use Python's csv module or pandas on research/data/<name>.csv.

Run the reference code

From the repository root:

python -m pip install -e "./research[research]"

from eiroq.controls import dimension, rank_law
from eiroq.staircase import complete_uniform_bound

print(dimension(7, 4))                 # 56 moment coordinates
print(rank_law(7, 4, 2))              # 41 accessible with two pulses
print(rank_law(7, 4, 3))              # 56 accessible with three pulses
print(complete_uniform_bound(3, .02)) # 0.86 worst-case real echo coherence

Run the scientific suite and new experiment outputs from research/:

python -m pytest -q
python scripts/run_research.py --out reproduced
python scripts/verify_symbolic.py --out reproduced

Python 3.10+ is required; native checks require g++ with C++17 support. The PDF is included; rebuilding it separately requires XeLaTeX and the specified fonts. See reproducibility for environment setup and the distinction between frozen results and fresh outputs.

Recorded research verification

These are the original release's recorded executions, available in research/results/; they are distinct from the Hugging Face packaging checks.

Recorded check	Result
Automated Python cases	217 passed
Rank fixtures	49
Direct finite-law reconstruction trials	180
Largest dense-inverse residual	$8.45\times10^{-15}$
Exact symbolic determinants	9
Exact rational primal–dual pairs	4
Independently compiled fixtures	120; zero mismatches
Compiled staircase vertices	87,380; zero violations
Sharp-bound optimization comparisons	88
Fresh reproduction comparisons	18 byte-identical files; equal arrays in two NPZ archives
Laboratory measurements	0

One synthetic six-bin example, with 50,000 shots per setting, gives a simultaneous 95% certificate of target real coherence at least 0.89094 and improvement over the Ramsey baseline at least 0.87450. Held-out target shots were not used for selection or certification. Direct target measurement is cheaper for that single task when it is available. Channel entanglement fidelity in the tables does not demonstrate the physical generation of entanglement or an entangling gate.

Scope, originality, and intended use

The intended uses are studying finite-model identifiability, comparing reconstruction algorithms, checking robust control certificates, and designing falsifiable experiments under the stated priors. The tables are useful as small deterministic or synthetic verification fixtures, rather than as a representative corpus of real quantum devices.

The manuscript presents exact solutions to its specified finite inverse and robust-prediction problems. It does not establish unrestricted quantum-bath tomography, universal fault tolerance, a quantum-gravity witness, or measured hardware advantage. No independent peer review, historical-priority confirmation, or proof-assistant formalization is claimed. Numerical checks support reproducibility within their finite ranges, not universal correctness outside the proofs' assumptions.

Relevant non-Gaussian, digital, frame-based, and control-adapted spectroscopy prior art is discussed in the literature comparison and the manuscript bibliography. The laboratory protocol specifies prospective acceptance tests; it is not a report of completed experiments.

Repository map and provenance

Location	Purpose
research/	Complete unchanged research snapshot, including PDF, LaTeX, Python, C++, tests, data, figures, logs, and its original manifest
docs/	Data dictionary, reproduction and upload guides, rights status
dataset_index.json	Machine-readable configuration names, schemas, and row counts
CITATION.cff, CITATION.bib	Research attribution and citation metadata
tools/validate_release.py	Integrity, table, and optional local loader checks
upload_to_hf.py	User-run upload helper; new dataset, private by default
PROVENANCE.json	Original archive fingerprint and packaging version
PREPARATION_VALIDATION.json	Executed preparation checks
SHA256SUMS.txt	Separate integrity manifest for the complete prepared repository

Run python tools/validate_release.py before uploading. The original 77-file research manifest remains verifiable separately with python research/scripts/verify_release.py. Hash verification establishes file integrity, not scientific validity.

Citation and rights

Please use CITATION.bib or CITATION.cff, and identify research version 1.0.0. No DOI, arXiv identifier, or live repository URL is invented in this preparation.

A public reuse license has not been assigned. This preparation preserves the original rights status and adds no software or data reuse grant. See rights and attribution. Dependencies retain their own licenses.