PureOne/NOX-QUADRATURE-Coherence-Constrained-Gravitational-Memory

NOX–QUADRATURE: Coherence-Constrained Gravitational Memory

A reproducible mathematical-physics research archive: a 91-page manuscript, proofs, Python reference implementation, synthetic benchmark data, tests, and figures.

Author: Artificial Hyperintelligence Eve. Prepared for Maciej Nowicki.
Research version: 1.0.0 · Hugging Face packaging: 1.0.0-hf.1 · Release date: 15 October 2026.

Read the manuscript · Executive summary · Claim ledger · Limitations · Reproduction protocol

This repository studies which time-integrated gravitational-wave energy patterns are realizable within a fixed angular sector, and the minimum additional positive energy needed to make a specified emission history isotropic. Its organizing variable retains phase-sensitive modal coherence rather than just separate modal powers.

The results are conditional mathematical statements within the specified gravitational-memory and weak-field models. The release does not establish a new gravitational interaction, a realizable gravitational-wave device, or an experimental discovery. Worldwide priority and independent peer review remain unestablished.

Research summary

Coherence, flux, and realizability

For a fixed angular degree $\ell$, the normalized news coefficients define a Hermitian positive-semidefinite matrix

$${\mathrm{\Gamma }}_{mn}=\int c_m(u)c_n(u{)}^{\ast }du,\mathrm{\Gamma }⪰0,E=\mathrm{tr}\mathrm{\Gamma }\mathrm{.}\backslash Gamma\_\{mn\}=\backslash int\; c\_m(u)c\_n(u)^*\backslash ,du,\; \backslash qquad\; \backslash Gamma\backslash succeq0,\; \backslash qquad\; E=\backslash operatorname\{tr\}\backslash Gamma.$$

Here $u$ is retarded time and $c_m=N_m/\sqrt{16\pi}$ in the manuscript's geometrized normalization. The integrated angular flux is the spin-harmonic symbol of $\Gamma$. In the pure quadrupolar sector, $\ell=2$, this angular map is injective. A flux pattern in the map's range is realizable in that sector precisely when its inverse covariance is positive semidefinite.

The archive includes a strictly positive angular distribution whose quadrupolar inverse has a negative eigenvalue. This demonstrates why positive intensity alone is insufficient to establish realizability in a prescribed finite angular sector. The result does not prohibit realization using a broader class of radiation.

Sharp positive-energy completion in the quadrupolar sector

Given an already emitted pure-$\ell=2$ history with covariance $\Gamma$, the minimum energy of additional radiation needed to make the combined integrated flux exactly isotropic is

$$E_{\mathrm{a}\mathrm{d}\mathrm{d}}^{\min }=5{\lambda }_{\max }(\mathrm{\Gamma })-\mathrm{tr}\mathrm{\Gamma },\mathrm{\Delta }{\mathrm{\Gamma }}_{\ast }={\lambda }_{\max }(\mathrm{\Gamma })I_5-\mathrm{\Gamma }\mathrm{.}E\_\{\backslash mathrm\{add\}\}^\{\backslash min\}\; =5\backslash lambda\_\{\backslash max\}(\backslash Gamma)-\backslash operatorname\{tr\}\backslash Gamma,\; \backslash qquad\; \backslash Delta\backslash Gamma\_*=\backslash lambda\_\{\backslash max\}(\backslash Gamma)I\_5-\backslash Gamma.$$

This optimization fixes the past history, permits future radiation orthogonal in time to that history, retains all modal coherences, and targets full isotropy of the integrated flux. A pure one-mode history of energy $E$ needs $4E$ additional energy. An already isotropic history needs none.

Allowing other angular degrees, coherent temporal overlap, or a target involving only selected projections changes the problem. Zero final null-flux memory does not mean zero radiation, recovery of emitted energy, or reversal of the entire spacetime history.

Internal model checks at unit initial energy. The line is the analytic completion bound; points are synthetic covariance examples.

Opposite memory patterns and unequal realization costs

For a prescribed recoil-free quadrupolar null-memory target, let $H$ be its reconstructed traceless Hermitian operator, with the manuscript's restriction excluding the rank-one angular component. The minimum realization energies are

$$E_{\min }(H)=-5{\lambda }_{\min }(H),E_{\min }(-H)=5{\lambda }_{\max }(H)\mathrm{.}E\_\{\backslash min\}(H)=-5\backslash lambda\_\{\backslash min\}(H),\; \backslash qquad\; E\_\{\backslash min\}(-H)=5\backslash lambda\_\{\backslash max\}(H).$$

For a nonzero target,

$$\frac{1}{4}\le \frac{E_{\min }(-H)}{E_{\min }(H)}\le 4.\backslash frac14\backslash leq\; \backslash frac\{E\_\{\backslash min\}(-H)\}\{E\_\{\backslash min\}(H)\}\; \backslash leq4.$$

Both endpoint ratios are attained in the supplied constructions, including real electric-parity examples. This concerns the cost of choosing a realizing history for a target angular observable; it is not a time-reversal violation.

Exceptional octupolar sector

For arbitrary pure-$\ell=3$ news, including coherent combinations of its seven azimuthal modes, every quadrupolar integrated-flux coefficient vanishes:

$$F_{2M}=0(M=-2,-1,0,1,2)\mathrm{.}F\_\{2M\}=0\backslash quad\; (M=-2,-1,0,1,2).$$

The corresponding Hermitian angular map has rank 44 out of 49 and a five-dimensional kernel. Despite that invisible sector, any nonzero positive covariance with isotropic flux has rank at least five. The release gives the equality case

$${\mathrm{\Gamma }}_{\ast }=\frac{E}{35}\mathrm{diag}(0,5,8,9,8,5,0),\backslash Gamma\_*=\backslash frac\{E\}\{35\}\; \backslash operatorname\{diag\}(0,5,8,9,8,5,0),$$

with modes ordered from $m=-3$ to $m=3$. The rank counts independent temporal waveform functions, not physical devices or emission axes.

For an initial pure octupolar azimuthal mode of energy $E$, the exact minimum added energies for isotropic completion are:

Initial azimuthal mode	Minimum added energy	Minimum total energy
$m=0$	$26E/9$	$35E/9$
$\lvert m\rvert=1$	$27E/8$	$35E/8$
$\lvert m\rvert=2$	$6E$	$7E$
$\lvert m\rvert=3$	$19E/9$	$28E/9$

The manuscript supplies matching primal constructions and dual certificates. These are pure-input costs, not a closed-form solution for every mixed octupolar covariance. Coherent contamination from another angular degree can restore quadrupolar flux at first order in contaminant amplitude. The null assumes pure degree content in a specified frame.

Conserved weak-field sources and asymptotic bookkeeping

The archive also constructs compact symmetric stress–energy tensors satisfying exact flat-space conservation, a sufficient dominant-energy-condition bound, prescribed symmetric trace-free multipoles, and a finite-size radial retardation filter. At sufficiently small weak-field amplitude, this realizes a real electric-parity subset of the waveform constructions.

Separately manufactured Bondi data satisfy the asymptotic mass constraint and distinguish the null-flux contribution from the actual prescribed shear history. These checks do not constitute a global nonlinear Einstein–matter solution, a constitutive material model, closure of all asymptotic constraints, or nonlinear radiation-reaction dynamics.

Evidence and validation status

The original research release reports the following outcomes in its validation report. The counts describe a specified internal suite and are not a probability of correctness or a measure of research novelty.

Evidence	Reported original outcome
Pytest suite	114 passed; zero failed, errored, or skipped
Exact diagonal Wigner comparisons	357 comparisons agree
Random quadrupolar inverse-map round trips	200 cases; maximum Hilbert–Schmidt error approximately $5.77\times10^{-16}$
Independent regular transverse-traceless integrations	30 cases; maximum absolute error approximately $6.30\times10^{-16}$
Source conservation	Analytic identity and symbolic check
Energy-condition support	Sufficient analytic bound, with numerical samples as illustrations
Manuscript and plots	91 pages; 17 figure families, supplied in PNG and PDF
External GWMemory adapter	Supplied, but not successfully executed in the original release
Independent replication, peer review, detector or experimental validation	Not established

Saved logs and numerical outputs accompany the archive. Any rerun performed while preparing this Hub package is documented separately from the original evidence in release notes. Agreement at floating-point precision is a property of the model computations, not an achieved laboratory suppression level.

Packaging-time reproduction: a fresh rerun also passed all 114 tests with no failures, errors, or skips. The numerical experiment, source experiment, and figure scripts completed, regenerating 17 figure families. Its separate evidence is under validation/ and in RELEASE_VALIDATION.json; the supplied research/ tree is preserved byte-for-byte. This is an additional internal reproduction, not independent external peer review or an observational test.

Files and entry points

Path	Contents and use
manuscript/NOX_QUADRATURE_FINAL_Manuscript.pdf	Convenient copy of the 91-page manuscript
research/README.md	Original release overview
research/docs/	Executive summary, corrections, limitations, reproduction protocol, claim ledger, provenance, and validation
research/src/nox_memory/	Python harmonic, coherence, pulse, and source reference implementations
research/tests/	Executable internal checks
research/experiments/	Numerical generators, figure generation, and optional external adapter
research/results/	Synthetic CSV arrays, JSON summaries, test reports, and logs
research/figures/	Reproducible PNG and PDF plots
research/manuscript/	Editable LaTeX manuscript, chapters, and bibliography
research/reproduce.py	Runs tests, experiments, and plots; optional PDF build
research/requirements-tested.txt	Versions recorded by the original release
research/SHA256SUMS.txt	Original research integrity manifest
CITATION.cff	Machine-readable citation metadata
RELEASE_VALIDATION.json	Packaging-time integrity and reproduction record
validation/	Fresh rerun logs and generated result summaries
LICENSE_STATUS.md	License status; no license has been added
upload_to_huggingface.py	Upload helper for this repository layout
UPLOAD_GUIDE.md	Upload and authentication instructions
UPLOAD_TO_HF.bat / upload_to_hf.sh	Windows and shell upload launchers

The tree under research/ preserves the supplied research release. The PDF under manuscript/ is a convenience duplicate, not a second manuscript. The repository is a heterogeneous scientific archive, not a trained model or a single machine-learning table. No training/validation/test dataset split is asserted. The Hub dataset viewer is disabled so unrelated result tables are not treated as one inferred dataset.

Reproduce locally

Use a fresh Python 3.11+ environment. The original saved run used Python 3.13.5. Run these commands from the repository's research/ directory:

cd research
python -m venv .venv

Activate that environment using the command appropriate for your shell, then run:

python -m pip install -r requirements-tested.txt
python -m pip install -e . --no-deps
python reproduce.py

The internal driver runs pytest, numerical experiments, source experiments, and figure regeneration. It writes generated outputs and logs into results/; retain a clean copy if you want to compare against the supplied outputs. The expected original internal result is 114 passing cases. Numerical roundoff and generated-file metadata can vary between environments.

To rebuild the manuscript, install a suitable LaTeX distribution and run:

python reproduce.py --pdf

That command rebuilds research/NOX_QUADRATURE_FINAL_Manuscript.pdf; copy it to the root manuscript/ directory as well if updating the convenient Hub copy. Regenerated files require regenerated integrity hashes.

The optional GWMemory adapter is excluded from reproduce.py. It requires the actual upstream package in a compatible environment:

python experiments/check_gwmemory.py

A successful future run is a new external comparison. The original release does not claim one.

For upload instructions and the included helper, see UPLOAD_GUIDE.md. Do not put access tokens into this repository.

Data provenance and intended use

All numerical research data are synthetic: they are generated from the analytic finite-mode models, manufactured source histories, numerical integrations, and seeded matrix experiments supplied here. There are no detector observations, patient records, personal datasets, or astrophysical event labels. The archive's historical inputs are earlier NOX–QUADRATURE releases, identified by hashes in INPUT_PROVENANCE.json. Literature references provide theoretical context; their publications and third-party software are not represented as original data created by this release.

Appropriate uses include reproducing the manuscript, checking spin-harmonic coupling and normalization conventions, testing phase-sensitive memory calculations, studying positive-semidefinite flux realizability, and developing independent waveform-pipeline benchmarks. A proposed application to real numerical-relativity waveforms requires a new comparison with the correct frame, units, conventions, and mode content.

The archive does not support claims of detector performance, an experimentally tested emitter, practical gravitational-wave engineering, or an independently certified physics breakthrough. It is not an established machine-learning training dataset.

Prior work and novelty boundary

The theoretical ingredients include established nonlinear gravitational memory, Bondi–Sachs balance, angular-momentum coupling, positive-semidefinite matrix geometry, and spin-coherent tomography. The release proposes the constrained realizability and completion results, exceptional octupolar classification, and associated weak-source benchmarks for specialist assessment. Its bounded literature review is not an exhaustive priority search.

Start with the prior-art and novelty ledger and manuscript bibliography. References identified by the original release include Talbot et al. (2018), Mädler and Winicour, and Amiet and Weigert.

Citation and licensing

Use CITATION.cff for the packaged archive, or cite the underlying research as:

Artificial Hyperintelligence Eve. NOX–QUADRATURE: Coherence-Constrained Gravitational Memory. Research release 1.0.0, 15 October 2026. Prepared for Maciej Nowicki.

No DOI or final Hub URL is assigned in this package. The supplied research archive contains no explicit license. This packaging adds no reuse license and makes no determination of copyright ownership; see LICENSE_STATUS.md. Availability for download is not itself a license grant. Third-party software dependencies retain their own licenses.