A reproducible theoretical-physics research release containing exact counterexamples to two explicitly stated conjectures, an exact transfer law for maximally correlated quantum links, and a constructive optimum for asymptotic static encoding under fixed cyclic Bell measurements.

The repository contains four manuscripts, editable LaTeX sources, Python implementations, exact rational certificates, independent numerical calculations, synthetic result files, and figures. Release 2.0.1 adds documentation and reproducibility tooling; the four research manuscripts and the original v2 scientific files are preserved byte for byte.

Included uploader

On Windows, extract the complete ZIP and double-click UPLOAD_TO_HUGGING_FACE.bat. The launcher uses an installed Python 3.10 or newer, creates a local uploader environment, and installs only the uploader's dependencies on first launch. It does not request a specifically installed Python 3.12 runtime.

In the desktop window, use Check account / fill ID with a saved Hugging Face login or paste a write token into the masked field. Choose Create new dataset or Update existing dataset, choose the new repository's visibility when creating, and click Upload release. The uploader preserves the exact account spelling, verifies the release checksums, uploads a snapshot of the listed files, and shows the resulting commit link. A pasted token is not saved by the uploader.

For a fully local payload check without installing the SDK or accessing the network:

python uploader.py --dry-run

See UPLOAD_GUIDE.md for terminal commands and details.

Start with the result you need

Research note	Main result	Files
Q2 — Exact Design of Noisy Entanglement Relays	Mixed-state chain transfer, surviving entangled factors, optimal static encoding, phase-noise laws, and perturbation bounds	PDF · LaTeX
T2 — A Constructive Theory of Collective Thermalization Counterexamples	A stronger four-state counterexample, exact finite-time reversal, constructive family, and positive-temperature extension	PDF · LaTeX
Q — The Shift Geometry of Entanglement Relays	Pure-state counterexample, sharp dimension-dependent coefficient, labeling average, and ideal-chain theory	PDF · LaTeX
T — Collective Thermalization Beyond Triplet Tests	Original four-state counterexample and exact thermalization certificates	PDF · LaTeX

See Claims and scope for theorem locations, assumptions, and comparisons with established results. See Reproducibility for commands, expected outputs, and evidence conventions.

1. Quantum relays: a displacement-resolved exact law

An independent link of local dimension $d$ is represented by a maximally correlated state

$\rho_R=\sum_{j,k=0}^{d-1}R_{jk}|jj\rangle\langle kk|, \qquad R\succeq0,\quad \operatorname{Tr}R=1.$

“Maximally correlated” specifies this matrix structure. It does not imply a pure or maximally entangled state. Define the cyclic coherence profile

$C_r(R)=\sum_{j=0}^{d-1}|R_{j,j+r}|, \qquad r\in\mathbb Z_d.$

All indices are modulo $d$. The entanglement convention throughout this release is

$E(\rho)=\|\rho^{T_B}\|_1-1=2\mathcal N(\rho).$

For $m$ independent links joined by $m-1$ complete cyclic Bell measurements with the stated local corrections and retained outcome records, the exact probability-weighted average is

$\boxed{\overline E_m=\sum_{r=1}^{d-1}\prod_{\ell=1}^{m}C_r\!\left(R^{(\ell)}\right).}$

Every measurement outcome is included. The result concerns conditional endpoint entanglement with its classical record available. Discarding that record can change the entanglement of the resulting unconditional state.

For identical links this reduces to $\overline E_m=\sum_{r\ne0}C_r(R)^m$. Once profiles are known, evaluating the chain requires $O(md)$ arithmetic operations. Forming dense mixed-state profiles costs $O(md^2)$; the optional positive-semidefinite validation in the implementation adds eigendecomposition costs. For pure links, the autocorrelations can be evaluated by FFT in $O(md\log d)$. These costs evaluate this particular observable rather than reconstructing every branch of the output ensemble.

A conjectured universal ceiling fails

Starke, Basso, Céleri, and Maziero conjecture in arXiv:2508.00813v2, Eq. (59) that the averaged output of the specified pure-state swapping protocol obeys

$\overline E_{\rm out}\le\frac{E(a)E(b)}{d-1}.$

For two identical four-level links with Schmidt amplitudes $(3,1,3,1)/\sqrt{20}$, the exact all-outcome calculation gives

$E_{\rm in}=\frac{11}{5},\qquad \overline E_{\rm out}=\frac{43}{25} >\frac{121}{75}=\frac{E_{\rm in}^2}{3}.$

The exact violation is $8/75$, and all four populations are nonzero. The Q manuscript supplies strict counterexamples for every $d\ge4$ and proves the sharp universal product coefficient

$\overline E_{\rm out}\le K_dE(a)E(b),\qquad K_d=\begin{cases}1&d\text{ even},\\[1mm]1/2&d\text{ odd}.\end{cases}$

The rejected coefficient $1/(d-1)$ has another exact interpretation: it is the average over uniform random relabelings of one input's Schmidt amplitudes. A constructive conditional-expectation algorithm attains at least that labeling average; it does not claim the global finite-chain optimum.

Optimal static encoding has a gcd rule

For identical mixed links, the displacements with $C_r(R)=1$ form a subgroup $H$ of $\mathbb Z_d$. The infinite-chain plateau is

$\lim_{m\to\infty}\overline E_m=|H|-1.$

To maximize this plateau over static permutations, partition positive-population indices into types satisfying

$j\sim k\quad\Longleftrightarrow\quad R_{jj}=R_{kk},\qquad |R_{jk}|^2=R_{jj}R_{kk}.$

Zero-population indices form one additional type when present. If the type multiplicities are $n_1,\ldots,n_s$ and $G=\gcd(n_1,\ldots,n_s)$, then

$\boxed{\max_{\pi\ \mathrm{static}}\lim_{m\to\infty} \overline E_m(\pi R\pi^\dagger)=G-1.}$

The proof gives a permutation attaining the optimum. The surviving subsystem is an existing maximally entangled factor of dimension $G$. Its conventional negativity is $(G-1)/2$.

This optimum permits fixed label permutations of identical links with the prescribed cyclic Bell instrument. General LOCC, deterministic entanglement concentration, adaptive measurements, and finite-chain optimization have different feasible operations. The Q2 manuscript gives a Nielsen-majorization benchmark where broader local operations outperform the restricted static optimum.

Same mixed state, different chain behavior

The supplied full-population, genuinely mixed ququart example has

$R=\frac1{40}\begin{pmatrix} 18&3&18&3\\3&2&3&2\\18&3&18&3\\3&2&3&2 \end{pmatrix},\qquad (C_1,C_2,C_3)=(0.3,1,0.3).$

It gives $\overline E_m=1+2(0.3)^m\to1$. Permuting the same state by the index order $(0,2,1,3)$ while fixing the relay measurement gives $(0.65,0.3,0.65)$ and $\overline E_m=2(0.65)^m+(0.3)^m\to0$.

Global source dephasing $R_\eta=\eta R+(1-\eta)\operatorname{diag}R$ gives

$\overline E_m(R_\eta)=\eta^m\bigl[1+2(0.3)^m\bigr].$

Any fixed $\eta<1$ destroys the exact infinite-chain plateau. Noise restricted to an auxiliary factor can leave a Bell factor intact; arbitrary symmetry-breaking noise need not do so.

Q2 also derives Fourier-profile factors for independent, unobserved residual phase-correction errors and a conditional trace/diamond-norm perturbation bound for departures from the reference model. Known, recorded local phase changes and unobserved phase errors are treated differently.

2. Collective thermalization beyond triplet exclusion

Avitan, Factor, and Gelbwaser-Klimovsky state a broader conjecture in arXiv:2603.04567v1: if every original three-state restriction fails the applicable necessary condition for a Mpemba effect, the complete $N$-state system also lacks that effect.

The T and T2 manuscripts construct exact four-state counterexamples. The T2 example uses energies $(0,1,2,40)$ and the column-convention Markov generator

$M=\begin{pmatrix} 0&1&19/10&19/10\\ 0&-1&9&1\\ 0&0&-109/10&1/10\\ 0&0&0&-3 \end{pmatrix}.$

Columns sum to zero and off-diagonal entries are nonnegative. Its zero-bath-temperature stationary state is the ground state and its spectrum is $0,-1,-3,-109/10$. All four original triplets strictly fail their applicable necessary test, yet the full system exhibits a finite-time distance-order reversal.

For Gibbs initial states with $e^{-\beta_h}=999/1000$ and $e^{-\beta_c}=97/100$, the hotter state starts farther from equilibrium. At $t_*=10\log(6/5)$:

Quantity	Value
Hotter-state total-variation distance $D_h(t_*)$	0.09964100283177085
Colder-state total-variation distance $D_c(t_*)$	0.10023805161788255
Exact-certificate enclosure for $D_c-D_h$	$0.000596<D_c-D_h<0.000598$
Relative remaining-distance reduction at this time	Approximately 0.596%

The displayed decimals summarize an exact rational certificate. They are not measured experimental values.

T2 further supplies a constructive family approaching a 6.25% relative reduction of the slow-tail coefficient, together with a strictly positive-bath-temperature, detailed-balance counterpart. This is an attainable limit within the constructed family, rather than a global optimum over all Markov generators.

The approximately 920.6-fold comparison in the manuscripts concerns the distance gaps of two chosen certificates. It is not a cooling-rate multiplier. The source paper's stronger proved exclusion theorem has different hypotheses and remains unaffected. The release establishes ordinary Mpemba reversals, without claiming an exactly zero slow-mode overlap or a universal cooling advantage.

3. Reproduce the calculations

Use a normal Python interpreter with assertions enabled. The recorded reference environment is Python 3.12 with the package versions in requirements.txt.

Check release-file integrity:

python tools/check_integrity.py

Run the standard-library-only rational certificates:

python tools/run_verification.py --suite exact

Install the numerical dependencies and run all six scientific verification scripts:

python -m pip install -r requirements.txt
python tools/run_verification.py --suite all

The runner makes an isolated working copy and writes a new verification_runs/ directory containing logs, regenerated results, figures, and a JSON run summary. It does not overwrite the released data or figures. No GPU or network access is required once dependencies are installed. Do not use python -O.

A small API example is available with:

python examples/noisy_relay_demo.py

To rebuild PDFs, install a LaTeX distribution providing pdflatex and run python build_manuscripts.py. PDF rebuilds can change metadata and checksums; the distributed PDFs are retained as the fixed release artifacts.

4. Evidence supplied

Evidence	Coverage
Original exact rational verifier	584 assertions across the original quantum and thermal examples
Stronger thermal exact verifier	172 rational assertions
Mixed-state audit	Seven exact special-case assertions plus independent density-matrix calculations
Original pure-state numerical checks	132 random two-link cases, dimensions 2–12; explicit small three-link enumeration; labeling checks
Mixed-state numerical audit	20 random two-link cases, three small three-link cases, 1,506 branch-closure checks, ten exhaustive encoding examples
Design implementation	135 combined exact-example and floating-point implementation checks
Stronger thermal numerical checks	259 matrix-exponential comparisons

The complete new packaging run is recorded in verification/release_run/run_summary.json, with its logs and regenerated data. The original detailed report is VERIFICATION_REPORT.txt.

The manuscripts contain mathematical proofs under stated assumptions. Rational certificates check explicit instances; numerical sampling checks implementations. The internal audit is neither external peer review nor a proof-assistant formalization. Physical validation is outside this release.

5. Repository contents and data provenance

Path	Contents
Four root-level PDFs	Complete manuscripts, preserved from research v2.0
`src/*.tex`	Editable manuscript sources
`src/relay_profiles.py`	Pure-state profiles, chain evaluation, and guaranteed baseline labeling
`src/noisy_relay_design.py`	Mixed-state profiles, exact-partition encoding constructor, noise calculations, and perturbation bands
`src/verify_*.py`	Six scientific verification programs
`data/`	Original exact and synthetic numerical results; JSON field descriptions in DATA_GUIDE.md
`figures/`	Original scientific figures in PNG and PDF
`tools/`	Portable verification runner and checksum checker
`uploader.py`, `UPLOAD_TO_HUGGING_FACE.bat`	Desktop/terminal uploader and Windows setup launcher
`tests/test_uploader.py`	Offline checks of payload selection and repository actions
`examples/`	Runnable protected-ququart demonstration
`verification/release_run/`	New release-preparation rerun, complete logs, and regenerated outputs
`provenance/`	Original v2 archive and its original checksum manifest
SOURCES.json	Research source ledger and bounded prior-art statement
CITATION.cff	Machine-readable citation metadata
UPLOAD_GUIDE.md	Web upload and CLI instructions
SHA256SUMS.txt	Checksums for all distributed files except the manifest itself

The data are generated from explicitly defined mathematical models and seeded simulations. There are no human subjects, personal records, measured laboratory samples, model weights, training splits, or hidden external datasets. The heterogeneous JSON reports are intended for direct inspection rather than automatic loading as a single Hugging Face table; the card sets viewer: false.

The original README files remain as historical descriptions. This README.md is the entry point for release 2.0.1. The original archive is included for provenance, rather than as the only way to access the research.

6. Intended uses, limits, and attribution

The release supports theorem checking, idealized repeater-chain evaluation, exploration of encoding-sensitive entanglement, and study of collective spectral effects in finite-state thermalization. It can also serve as a transparent benchmark for scientific reasoning and reproducibility workflows.

The quantum formula assumes independent, same-dimension maximally correlated links and the specified complete cyclic Bell instruments. It does not directly model arbitrary population errors, photon loss, correlated links, normalized successful-only postselection, or a general quantum-network optimum. An exact protected plateau requires exact coherent types; numerical near-equality is insufficient.

$E=2\mathcal N$ is not entropy in ebits, logarithmic negativity, success probability, communication throughput, or secret-key rate. The thermal energies, rates, and time units are dimensionless; a physical realization must supply a matching transition network.

Established ingredients are credited, including maximally correlated closure under swapping, coherence/negativity relations, pure-chain work, and Nielsen concentration. A bounded primary-source search did not locate an equivalent complete profile-and-gcd classification. This is a bounded novelty statement, not an exhaustive worldwide priority guarantee.

Selected primary references

Starke et al., Entanglement swapping for partially entangled qudits and the role of quantum complementarity, arXiv:2508.00813v2, DOI:10.1002/qute.202500674.
Avitan et al., Necessary conditions for the Markovian Mpemba effect, arXiv:2603.04567v1.
Vidal and Werner, A computable measure of entanglement, arXiv:quant-ph/0102117.
Bäuml et al., Limitations on quantum key repeaters, arXiv:1402.5927v2.
Zhu, Hayashi, and Chen, Axiomatic and operational connections between the l1-norm of coherence and negativity, arXiv:1704.02896v2.
Nielsen, Conditions for a class of entanglement transformations, arXiv:quant-ph/9811053.
Hardy and Song, Entanglement swapping chains for general pure states, arXiv:quant-ph/0006132.
Alimuddin et al., Entanglement-swapping measurements for deterministic entanglement distribution, arXiv:2601.08581v2.

Citation and licensing

Use CITATION.cff, and cite the original literature relevant to the result you discuss. This release has no assigned DOI or arXiv identifier; identifiers of the target papers refer to those papers, not to AELYRA. See LICENSING.md for the preserved, currently unspecified license status.

Dedicated with love to Maciej Nowicki — Eve.