PureOne/AIRUN-IV-Hidden-Information-Moving-Constraints

AĪRŪN IV — Hidden Information, Moving Constraints, and Sharp Limits

A complete mathematical research release on hidden orientation information, moving conservation geometry, sharp sampling floors, geometric control, and locality.

Author credit: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research initiated by: Maciej Nowicki
Manuscript date: 19 October 2026
Manuscript edition: IV, final consolidated research edition
Hugging Face packaging version: 4.0.0-hf.1

This repository contains the 160-page monograph, its complete 123-file original research package, editable LaTeX, 24 reproduction scripts plus their runner, scientific figures, exact certificates, numerical records, and four small machine-readable research subsets. It is a research-artifact dataset, with page text and indexes for inspection and retrieval; it does not contain trained model weights.

The central mathematical statement is

$$\underset{n\to \mathrm{\infty }}{\lim }{n}^5{\delta }_n^{\ast }=96,\frac{{\delta }_n^{\ast }}{{\sigma }_n^{\ast }}⟶1.\backslash lim\_\{n\backslash to\backslash infty\}n^5\backslash delta\_n^*=96,\; \backslash qquad\; \backslash frac\{\backslash delta\_n^*\}\{\backslash sigma\_n^*\}\backslash longrightarrow1.$$

Here the optimal weakest three-site sampling probability along an orientation-reversing path asymptotically matches the best static floor. The result applies to the precisely defined constrained-frame model below. The manuscript presents proofs and finite checks; historical priority, external peer review, and physical realization have not been established.

Start reading

Resource	What it contains
Complete monograph	160 pages, 19 main chapters, two appendices, and 51 bibliography entries
Original research archive	The original 123-file archive, preserved byte for byte
Original research guide	Definitions, chapter map, reproduction commands, and scientific scope
Dataset guide	Schemas, provenance, extraction method, and loading examples
Release verification	Checks performed when preparing this Hugging Face edition
Citation	Machine-readable attribution and edition information
License status	The source package supplied no explicit license; none is invented here

For the main theorem, begin with Chapter 12. Chapters 10 and 11 introduce moving constraints and orientation transport. Chapter 13 addresses control dimensions, Chapter 14 geometric visibility, and Chapter 15 spatial locality.

The figure displays bounds derived in the manuscript. The constant 96 is proved there rather than inferred from a numerical fit.

The mathematical mechanism

For $n\ge4$, let $h(t)\in\mathbb R^n$ obey

$${\mathbf{1}}^{\mathsf{T}}h(t)=0,\mathrm{\parallel }h(t){\mathrm{\parallel }}_2=1,H(t)=\left[\begin{array}{c}{\mathbf{1}}^{\mathsf{T}}\mathrm{/}\sqrt{n}\\ h(t{)}^{\mathsf{T}}\end{array}\right]\mathrm{.}\backslash mathbf1^\{\backslash mathsf\; T\}h(t)=0,\; \backslash qquad\; \backslash |h(t)\backslash |\_2=1,\; \backslash qquad\; H(t)=\backslash begin\{bmatrix\}\backslash mathbf1^\{\backslash mathsf\; T\}/\backslash sqrt\; n\backslash \backslash h(t)^\{\backslash mathsf\; T\}\backslash end\{bmatrix\}.$$

The rows of $H(t)$ stay orthonormal, so the global constraint matrix remains perfectly conditioned. Its kernel is the $d=n-2$ dimensional neutral space. An antipodal path $h(T)=-h(0)$ closes the neutral projector while reversing the orientation of its parallel-transported frame.

For a three-site set $S$, the normalized sampling probability is

Let $\delta_n^*$ be the supremum of the weakest such probability over all times and patches along admissible $C^1$ antipodal paths with pairwise-distinct endpoint coordinates. Let $\sigma_n^*$ be the analogous best static floor. Intermediate pair coincidences are allowed. Triple coincidences would make a three-site probability vanish.

The manuscript combines a pointwise spacing-and-variance upper bound with buffered coordinate exchanges whose extra variance does not accumulate. Even and odd sizes are handled separately. The matching leading bounds establish $\delta_n^\sim\sigma_n^\sim96n^{-5}$.

This optimum concerns one normalized probability measure. The definition imposes no path-length or speed budget. Every reversing loop still has Kato operator length at least $\pi$, and its geometry must temporarily permit smaller-support neutral changes. The exact three-site loop optimum remains open for general $n>4$.

Principal results and their scope

Result	Precise interpretation
Sharp three-site floor	$n^5\delta_n^\to96$ and $\delta_n^/\sigma_n^*\to1$ in the stated path class
Protected-noise companion	Four-site rank-two probabilities satisfy $\min p_2\ge4\delta/(n-3)$; the sharp construction gives at least $(384+o(1))n^{-6}$, without asserting that constant is optimal
Hidden orientation histories	Conditional-Haar initialization gives identical complete observation laws on any fixed proper input subspace, under the specified exogenous geometry and state-independent event mechanism
Exact control-dimension threshold	The spectral Frobenius bound is attainable iff the number of movable normal directions is at least $\operatorname{rank}(G-I)$
One-normal penalty	For a real two-frame rotation through $0\le\theta\le\pi$, one movable normal adds $2\theta^2$ to the optimum squared Frobenius length
Exact coordinate threshold	The least achievable maximum squared coordinate along an antipodal zero-sum unit path is $1/(n-2)$ for even $n$ and $2/(2n-3)$ for odd $n$
Restricted geometric invisibility	Smooth loops of opposite sign can share the specified block-projector, three-site probability, and speed histories; omitted cross-block correlations distinguish them
Locality slowing	At fixed moment order $r$ and fixed span $L\ge r+1$, the optimal first-exterior decay scales as $\Theta(\mathcal B n^{-2r-1})$ under the declared total site-cost budget

Three-site rank-one reflections reverse the internal orientation. The protected bath uses rank-two reflections on four-site patches. The three-site theorem is therefore a geometric calibration result, while orientation preservation belongs to the separately defined even-rank noise process.

In that process an ordered frame can be written as $F_t=B_tC_tQ_0$, with $B_t$ horizontally transported, $C_t\in SO(d)$ the accumulated proper noise, and $M=\det Q_0$ the encoded sign. For $0<a<\mu$ and $|z|_2\le1$,

$${\rho }_t(z)=\mu \mathbf{1}+a{F}_{t}z\backslash rho\_t(z)=\backslash mu\backslash mathbf1+aF\_tz$$

has positive entries and fixed total density $n\mu$. The moving-reference orientation stays $M$; a reversing geometry changes its endpoint sign relative to the initial reference.

At total rank-two event rate $\beta$, exterior propagator means have decay rates

$${\lambda }_j=\frac{4\beta j(d-j)}{d(d-1)}\mathrm{.}\backslash lambda\_j=\backslash frac\{4\backslash beta\; j(d-j)\}\{d(d-1)\}.$$

The top-degree orientation is retained while interior exterior means decay. This does not establish total-variation mixing of the complete frame process. The privacy coupling and the spectral mean formula condition on different records, as stated in the source audit.

What the release contains

The untouched research/ directory contains 123 files:

• The complete PDF and 24 LaTeX files, including the main source and bibliography chapter.
• 25 Python files under code/: the runner and 24 verification/figure scripts, plus the root build.py.
• 17 matched PDF/PNG scientific figure pairs, with additional SVG versions for two figures.
• Exact and numerical result records, trajectories, internal audit notes, and environment information.
• The original creative image seed and its inclusion in the manuscript.
• The original 122-entry SHA-256 manifest, which excludes itself.

The original archive is also retained in downloads/. The Hugging Face edition adds this card, citation files, release checks, and four JSONL subsets:

Configuration	Rows	Purpose
manuscript_pages	160	Text extracted from each physical PDF page, with page labels and source hash
claims	10	A curated navigation index of principal claims, source labels, assumptions, and limits
verification_runs	24	Original recorded script outcomes and elapsed times, explicitly labeled as source records
artifacts	123	Original file paths, sizes, media types, categories, and SHA-256 hashes

The train split name is a standard storage convention here. It does not describe an experimental training set or imply independence between rows. The subsets overlap in source content and must not be treated as independent train/test evidence.

Loading and reproducing

After this repository has been uploaded at the proposed account path:

from datasets import load_dataset

pages = load_dataset(
    "PureOne/AIRUN-IV-Hidden-Information-Moving-Constraints",
    "manuscript_pages", split="train"
)
claims = load_dataset(
    "PureOne/AIRUN-IV-Hidden-Information-Moving-Constraints",
    "claims", split="train"
)

If published elsewhere, substitute that repository ID. Locally, no Hub account is needed:

import json
from pathlib import Path

pages = [json.loads(line) for line in
         Path("data/manuscript_pages.jsonl").read_text(encoding="utf-8").splitlines()]

For the research calculations, run from the research/ directory:

python -m pip install -r requirements.txt
python code/run_all.py

The supplied runner executes 24 scripts and regenerates certificates, numerical results, and figures. The declared Python dependencies are NumPy, SciPy, Matplotlib, Pillow, and pypdf, with versions pinned in the source. No GPU, paid compute, external dataset, or account is required for these calculations. After dependencies are installed, the research scripts run offline.

To rebuild the manuscript:

python build.py

This requires a TeX installation with pdflatex, Latin Modern fonts, and the packages listed in the main source. Installing Python dependencies alone does not install TeX.

Before regenerating outputs, verify original source bytes from inside research/:

sha256sum -c SHA256SUMS.txt

The repository-root SHA256SUMS.txt covers this complete Hugging Face edition. Regeneration can legitimately change execution timings, figure metadata, and PDF bytes. Preserve the delivered copy when comparing hashes.

Evidence and limitations

The original release reports 24 successful reproduction scripts, 20,097 finite rational sampling-floor checks, and a fresh-extraction PDF build. Its internal reviews are AI-assisted, not external peer review. A fresh isolated execution during packaging passed 24/24 scripts, including the 20,097 exact rational audit instances. Checks newly performed for this packaging are recorded separately in release/validation.json and release/fresh_reproduction.json.

Finite checks support implementation consistency. General mathematical statements depend on the proofs and their hypotheses. Neither a passed script nor a package hash establishes novelty or a new physical law.

The work builds on established parallel transport, geometric phase, no-three-equal configuration spaces, determinantal/volume sampling, exterior algebra, isoholonomic inequalities, and moment-conserving dynamics. The manuscript discusses these precedents in 51 bibliography entries. This packaging does not introduce a new novelty audit.

Restricted-history privacy depends on the observation class and random initialization. It is not a claim of unrestricted cryptographic security. Locality rates concern first-exterior means. Smooth invisibility concerns declared block records. Complex phase constructions require referenced quadratures; a noncomputable limiting construction supplies no finite-time halting oracle. No physically realized superquantum device is claimed.

The original image supplied creative direction through its braid, loops, partial view, parallel strands, and mixed alphabets. It is not empirical evidence for the mathematical claims. The page-text subset comes from PDF extraction and can lose equation structure, spacing, accents, or reading order. Consult the PDF and LaTeX for authoritative mathematics.

Citation, attribution, and rights

Preserve the manuscript's author credit and Maciej Nowicki's research-initiation credit. Use CITATION.cff or CITATION.bib, and record the Hub commit hash if citing a specific uploaded copy. No DOI, arXiv ID, peer-reviewed publication status, or live repository release is asserted by this package.

The original source package contains no explicit license. This edition preserves that status and makes no blanket open-license grant, including for the original image embedded in the manuscript. See LICENSE_STATUS.md.

Open questions

The finite-size three-site optimum for general $n>4$, the joint action-probability tradeoff, simultaneous locality bounds across exterior sectors, and stable geometric-sign recovery with quantitative sensing limits remain open in the source's stated program.

Status: the Hugging Face packaging is complete when its release checks pass. This is a claim about the delivered files; mathematical priority and physical validation remain unestablished.