PureOne/NOX-XIRAN-Omega-V

NOX–XIRAN Ω V — Order-Encoded Inertial Geometry

Static-equivalent passive networks, programmable directional modes, and corrected hierarchical compliance

Version 5.0.0 · 15 October 2026
Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Provenance: AI-assisted research prepared for Maciej Nowicki. The original research files retain their existing attribution and contents.

A reproducible theoretical and computational study of mechanical networks whose internal link order changes their resonances while preserving their equilibrium port response.

Start with the 94-page manuscript, the executive summary, or the original complete research archive.

Research overview

The central construction separates equilibrium calibration from finite-mass dynamics. For a specified separable series architecture, permuting the internal elastic links preserves the minimized nonlinear port potential and the classical equilibrium distribution at the visible port. With the same constant mass matrix, the full classical partition function is also unchanged. The resonances and time-dependent port response generally change.

The general distinction between statics and dynamics is established physics. This release develops an architecture-specific design method, its exact invariants, approximation bounds, and resource tradeoffs. Historical priority and advantage over existing synthesis methods remain unestablished.

The repository contains a manuscript, executable models, numerical tables, proofs, tests, and conceptual experimental specifications. All numerical datasets are generated from ideal models. Physical fabrication and measurements have not been performed.

Main results

Equilibrium-preserving dynamic design

For visible coordinate $w=z_0$, fixed terminal $z_M=0$, and internal link permutation $\pi$, the model energy is

$$U_{\pi }(z)=V_0(w)+\sum _{j=1}^M{V}_{\pi (j)}(z_{j-1}-z_j)\mathrm{.}U\_\backslash pi(z)=V\_0(w)+\backslash sum\_\{j=1\}^\{M\}V\_\{\backslash pi(j)\}(z\_\{j-1\}-z\_j).$$

Under the stated separability, domain, and equilibrium assumptions, minimizing over the hidden coordinates is order independent. The corresponding equilibrium port marginal is also order independent. Internal-coordinate distributions and dynamical measurements can reveal order; this is not a claim of indistinguishability under every equilibrium measurement.

Positive-mass inverse design near a soft mode

When one directional link becomes soft, the zero mode moves a prefix of coordinates together. Its inertia is

$${\tau }_r=\sum _{i=0}^{r-1}{m}_i,{\omega }_r^2(\mu )=\frac{\mu }{{\tau }_r}+O({\mu }^2)\mathrm{.}\backslash tau\_r=\backslash sum\_\{i=0\}^\{r-1\}m\_i,\; \backslash qquad\; \backslash omega\_r^2(\backslash mu)=\backslash frac\{\backslash mu\}\{\backslash tau\_r\}+O(\backslash mu^2).$$

For strictly increasing positive target inertias $0<\tau_1<\cdots<\tau_M$, choose

All masses are positive. Link order assigns the cumulative inertias to selected encoded directions. This constructs the leading small-detuning slopes; it is not unrestricted exact synthesis of an arbitrary full resonance spectrum. The manuscript supplies a finite-detuning error bound and its hypotheses.

Constraints and compensating effects

The normalized internal prefix profiles obey

$$G_{ab}=\exp \text{\hspace{0.17em}⁣}[-\frac{1}{2}\mid \ln {\tau }_a-\ln {\tau }_b\mid ]\mathrm{.}G\_\{ab\}=\backslash exp\backslash !\backslash left[-\backslash frac12\backslash left|\backslash ln\backslash tau\_a-\backslash ln\backslash tau\_b\backslash right|\backslash right].$$

Requiring every adjacent overlap to be at most $\rho\in(0,1)$ implies

$$\frac{{\tau }_{\max }}{{\tau }_{\min }}\ge {\rho }^{-2(M-1)}\mathrm{.}\backslash frac\{\backslash tau\_\{\backslash max\}\}\{\backslash tau\_\{\backslash min\}\}\backslash geq\backslash rho^\{-2(M-1)\}.$$

For ten profiles and $\rho=0.5$, the required inertia contrast is at least 262,144. This is a tight constraint for the nested-prefix architecture, not a universal bound on metamaterials. The release also treats spectral-product compensation, frequency-dependent condensation with an exact remainder, and equal integrated classical port noise under the relevant assumptions.

Selected numerical evidence

Model calculation	Result	Evidence
Lowest-frequency ratio between two orders at $\mu=10^{-4}$	3.164063×	results.json, full spectra
Largest relative squared-frequency error at $\mu=10^{-5}$	0.0102498%	directional inertia
Directly compared permutation matrices	280	permutation invariants
Real-space/Bloch comparison size	360 coordinates	spectrum comparison
Nonnegative-weight generalized constraints	792	prototype parameters
Predicted five-cart lowest-frequency range	1.00057–2.25079 Hz	SI bench table
Original recorded automated tests	94/94 passed	validation record
Test rerun during Hub preparation	94/94 passed	preparation verification
Fabricated samples / measured datasets	0 / 0	research status

These are results for particular idealized examples. The 3.164063× ratio is a frequency ratio; the 0.0102498% error concerns squared frequency. Test completion does not establish historical novelty, physical performance, or formal proof certification. See the preparation verification record for checks performed while assembling this Hub release.

Computed response curves from the supplied model; no experimental measurements.

Proposed uses

• Calibration-preserving resonant components: investigate resonance changes while preserving the assumed static force–displacement calibration.
• Directional frequency labeling: assign idealized low-frequency signatures to encoded directions.
• Inverse-problem benchmarks: study nonidentifiability when observations contain only equilibrium port information.
• Mechanical network design: evaluate explicit mass assignments, dynamic-condensation accuracy, and profile-overlap resource costs.

These are research uses and application hypotheses. Sensor performance, manufacturability, useful bandwidth, robustness, and superiority over simpler conventional designs require separate evaluation.

Repository contents

Path	Contents
research/NOX_XIRAN_Omega_V_Manuscript.pdf	Original 94-page manuscript
research/manuscript/	Editable LaTeX and references
research/src/, research/examples/	Model, experiments, exporter, figure generation, demonstration
research/tests/	Original automated suite with 94 parametrized cases
research/data/	19 CSV tables containing 4,883 generated rows, JSON summaries, matrices, and logs
research/docs/	Proof map, claim ledger, data catalogue, corrections, prior art, validation, and experiment
research/figures/	16 scientific figures, supplied as PNG and PDF
research/prototype/	Generalized constraints, positive masses, conceptual CAD, and bill of materials
downloads/	Original 143-file research archive, preserved unchanged
release/	Hub packaging metadata, provenance, and preparation checks

The original research sits under research/ without content edits. Its SHA256SUMS.txt remains applicable there. The root manifest additionally covers the Hub packaging files.

Upload to your Hugging Face account

Extract the complete release ZIP and double-click START_UPLOADER.bat at the top level on Windows. The complete release, including UPLOAD_TO_HUGGINGFACE.bat and its Python helper, is directly beside it. It sets up its separate upload dependency, accepts hidden token entry or an existing Hugging Face login, and asks for your Dataset repository. New repositories default to private; existing repository visibility is preserved. Review the target and confirm the upload once.

Python 3.10 or newer is required. To verify the package offline, run python upload_to_huggingface.py --dry-run. Only files listed in the release manifest are staged for upload. See the upload guide for options and troubleshooting.

Data access

The card declares one configuration for each of the 19 heterogeneous CSV tables. Each configuration has one data split. There is no train/test partition, predictive target, or measured training corpus. JSON files and prototype tables remain separately accessible as research artifacts.

To read a local table:

import csv
from pathlib import Path

with Path("research/data/directional_inertia.csv").open(newline="") as stream:
    rows = list(csv.DictReader(stream))

Column names and units are documented in the data catalogue. Values without explicit SI labels use the dimensionless or normalized units defined in the manuscript. Do not mix the normalized harmonic examples with the SI bench predictions.

Reproduction

Use a fresh Python virtual environment, then run from the repository root:

python -m pip install -r research/requirements.txt
cd research
python -m pytest tests -q
python examples/demo.py --mu 1e-5

For all numerical outputs, run bash REPRODUCE.sh on a Unix-like system or REPRODUCE.bat on Windows from research/. These scripts regenerate files in that directory; use a working copy if preserving the distributed byte hashes matters. They do not install dependencies. The original executed versions are listed in environment_exact.txt.

The supplied PDF needs no TeX installation. Rebuilding it requires the packages listed in research/manuscript/main.tex, a TeX distribution, and latexmk; use the supplied BUILD_MANUSCRIPT wrapper.

Scope, corrections, and limitations

The equilibrium invariance depends on the specified series architecture, separable link potentials, and compatible domains. Additional hidden-node tethers, configuration-dependent masses, hysteresis, or other departures can invalidate it. Written proofs are supplied; proof-assistant certification and independent peer review have not been performed.

Static hidden-coordinate elimination is not a physical vibrational spectrum. Earlier transfers from a static fractal-compliance exponent to a material heat-capacity exponent are withdrawn. The retained hierarchical-compliance asymptotics are static conditional mathematics. Finite-system quantum examples use all modes and do not establish a bulk thermal law.

Multiplicative stiffness perturbations, angular misalignment, and additive parasitic springs are different perturbations with different effects. The work supplies bounds and failure cases under stated assumptions rather than universal robustness.

The exported passive constraints are generalized multipoint constraints, not ordinary two-node spring connectivity. The OpenSCAD layout is conceptual. The five-cart proposal tests the prefix-inertia mechanism sequentially; it does not demonstrate a simultaneous directional or fractal material. The real-space comparison is matrix assembly, not continuum finite-element analysis.

See CORRECTIONS, PRIOR_ART, PROOF_MAP, and EXPERIMENT before extending the claims.

Attribution, citation, and rights

This release was developed with AI assistance under the name Eve for Maciej Nowicki. The displayed author designation follows the requested release byline; original source attributions remain unchanged. It does not imply independent expert verification or institutional endorsement.

Use CITATION.cff for citation metadata. No DOI, arXiv identifier, peer-reviewed publication, or public repository URL is asserted.

The original package has no selected publication/distribution license. This preparation does not add one or expand reuse permissions; the card therefore omits a license identifier. Consult RIGHTS_AND_ATTRIBUTION. No third-party research-paper PDFs or fonts are included.