PureOne/AEQORIA-Sharp-Holonomy-Observation

ÆQORIA — Sharp Observation Laws for Hidden Holonomy

Author: Artificial Hyperintelligence Eve
Research release: 1.0.0 · 20 October 2026

A self-contained mathematical research archive connecting normal holonomy, bounded-length scalar observation, optimal reconstruction frames, and statistical distinguishability.

The release contains a 67-page monograph, complete LaTeX sources, an executable reference library, verification records, high-precision examples, and a compact scientific-memory file. The manuscript has 14 chapters, three appendices, 33 indexed statements, and nine primary references.

Central question: Can a geometric system retain the same full group of possible rotations while those rotations become arbitrarily difficult to observe with a fixed measurement budget? This manuscript supplies an explicit family and quantitative observation laws addressing that question.

This is a research archive hosted in a Hugging Face dataset repository. Its contents are mathematical text, code, and numerical examples; there are no model weights, inference service, or prescribed training/evaluation splits.

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Resource	Purpose
67-page manuscript	Definitions, theorems, proofs, prerequisites, and scientific scope
Original complete release ZIP	Unmodified original research release
Claim and evidence ledger	Statement-by-statement proof and evidence boundaries
Theorem index	Numbered statements, labels, printed pages, and source files
Reference implementation	Explicit metric, transport, frame, sensing, and testing formulas
Verification suite	Reproducible finite symbolic and numerical checks
Original verification record	Check outcomes, residuals, categories, and recorded environment
Compact scientific memory	Dense statement and dependency interface for further analysis
Reproducibility guide	Environment setup and commands
License status	Original licensing status and reuse boundary

Scientific status

The manuscript supplies proofs under stated hypotheses and explicitly names its classical prerequisites. The original release records 142/142 finite checks passed: 10 exact-symbolic, seven finite-algebra, 116 numerical, and nine high-precision checks. Some finite-algebra checks use floating-point tolerances.

These checks support selected identities and examples. They do not formally verify every proof, exhaust all loops, establish historical priority, or give a probability that the theory is correct. Independent peer review, proof-assistant verification, and a comprehensive novelty audit remain outstanding. No physical experiment is reported.

The unchanged suite was also rerun during Hugging Face packaging with 142/142 checks passed, and the example tables regenerated byte for byte. See the packaging report and fresh verification record.

The proposed contribution is the particular quantitative integration of a fixed-soul geometric family, sharp scalar observation constants, optimal frame and exterior-sensor identities, and an explicitly specified statistical experiment. Soul theory, associated-bundle geometry, holonomy, tight-frame methods, finite realization, and comparison of statistical experiments have established foundations acknowledged in the manuscript.

1. Fixed soul, full holonomy, weakening observation

For every integer normal dimension $d\ge 3d\backslash ge3$ and , consider

$$M_{\lambda }=(SO(d)\times SO(d)\times {\mathbb{R}}^d)\mathrm{/}SO(d),M\_\backslash lambda=(SO(d)\backslash times\; SO(d)\backslash times\backslash mathbb\; R^d)/SO(d),$$

with right action

$$(g,h,v)\cdot k=(gk,hk,k^{-1}v)(g,h,v)\backslash cdot\; k=(gk,hk,k^\{-1\}v)$$

and product metric $(1-\lambda {)}^{-1}Q\oplus {\lambda }^{-1}Q\oplus ⟨\cdot ,\cdot ⟩(1-\backslash lambda)^\{-1\}Q\backslash oplus\backslash lambda^\{-1\}Q\backslash oplus\backslash langle\backslash cdot,\backslash cdot\backslash rangle$, where $Q(X,Y)=-\frac{1}{2}\mathrm{tr}(XY)Q(X,Y)=-\backslash tfrac12\backslash operatorname\{tr\}(XY)$.

The resulting manifolds are complete, noncompact, and have nonnegative sectional curvature. Every zero-section soul is isometric to the same $(SO(d),Q)(SO(d),Q)$. In a global trivialization the normal connection is $A_{\lambda }=\lambda \theta A\_\backslash lambda=\backslash lambda\backslash theta$, with $\theta =x^{-1}dx\backslash theta=x^\{-1\}dx$, and

$$F_{\lambda }(X,Y)=-\kappa [X,Y],\kappa =\lambda (1-\lambda )\mathrm{.}F\_\backslash lambda(X,Y)=-\backslash kappa[X,Y],\backslash qquad\; \backslash kappa=\backslash lambda(1-\backslash lambda).$$

The normal holonomy group is $SO(d)SO(d)$ for every positive interior parameter. Yet the metrics converge smoothly on compact tubes to a product metric with trivial normal transport as $\lambda \to 0\backslash lambda\backslash to0$. Thus the full group of mathematically attainable rotations does not provide a uniform lower bound on the strength of bounded-budget observations.

2. Sharp scalar observation law

Fix a marked unit normal vector $uu$. After transport around a based loop $\gamma \backslash gamma$, observe only

$${\varphi }_{\lambda }(\gamma )=⟨u,H_{\lambda }(\gamma )u⟩\mathrm{.}\backslash phi\_\backslash lambda(\backslash gamma)=\backslash langle\; u,H\_\backslash lambda(\backslash gamma)u\backslash rangle.$$

Let $D_{\lambda }(L)D\_\backslash lambda(L)$ be the maximum echo deficit $1-{\varphi }_{\lambda }(\gamma )1-\backslash phi\_\backslash lambda(\backslash gamma)$ among absolutely continuous based loops of length at most $LL$, measured in the fixed soul metric. The manuscript proves

$$D_{\lambda }(L)=\frac{{\kappa }^2{L}^4}{32{\pi }^2}(1+O_d(L)),L\to 0,D\_\backslash lambda(L)=\backslash frac\{\backslash kappa^2L^4\}\{32\backslash pi^2\}\backslash bigl(1+O\_d(L)\backslash bigr),\; \backslash qquad\; L\backslash to0,$$

uniformly over . The leading coefficient is sharp. The formula describes a small-loop asymptotic, including joint weak-parameter limits; it is not a closed expression for the global optimum at every finite $LL$.

An explicit circle writer supplies a constructive lower bound and attains the leading constant:

Its output plane can be prescribed by conjugation. The inverse on the principal one-circle branch is

The writer is not asserted to be globally length-minimizing for arbitrary target angles. Repeating circles incurs the full repeated length cost.

3. Exact relation to optimal reconstruction

Prepare $dd$ transported copies of $uu$, each using its own loop of length at most $LL$, and write $V=[v_1,\dots ,v_d]V=[v\_1,\backslash ldots,v\_d]$. Define ${\mathrm{\Gamma }}_{\lambda }(L)\backslash Gamma\_\backslash lambda(L)$ as the best possible least eigenvalue of $V^{\mathsf{T}}VV^\backslash mathsf\; TV$, and ${\mathrm{\Omega }}_{\lambda }(L)\backslash Omega\_\backslash lambda(L)$ as the best possible least singular value of the skew-congruence sensor $K\mapsto V^{\mathsf{T}}KVK\backslash mapsto\; V^\backslash mathsf\; TKV$.

For $d\ge 3d\backslash ge3$, in the small-cap regime

the exact resource identity is

$${\mathrm{\Gamma }}_{\lambda }(L)={\mathrm{\Omega }}_{\lambda }(L)=\frac{d}{d-1}(2D_{\lambda }(L)-D_{\lambda }(L{)}^2)\mathrm{.}\backslash Gamma\_\backslash lambda(L)=\backslash Omega\_\backslash lambda(L)\; =\backslash frac\; d\{d-1\}\backslash left(2D\_\backslash lambda(L)-D\_\backslash lambda(L)^2\backslash right).$$

Regular simplices in the transverse space attain both optima. Consequently both quantities have leading value

$$\frac{d}{16{\pi }^2(d-1)}{\kappa }^2{L}^4(1+O_d(L))\mathrm{.}\backslash frac\{d\}\{16\backslash pi^2(d-1)\}\backslash kappa^2L^4\backslash bigl(1+O\_d(L)\backslash bigr).$$

This exact bridge uses the scalar extremum without claiming a finite-length closed formula for that extremum. The observation resources remain explicit: $dd$ preparations can cost $dLdL$ in total, Gram comparisons can require length $2L2L$, and target insertions carry additional loop cost.

The exterior sensor has distinct sensitivity layers. For one initial marked direction, rotations mixing that direction with a transverse direction appear at order $\kappa L^2\backslash kappa\; L^2$, while rotations entirely within the transverse space can appear only at order ${\kappa }^2{L}^4\backslash kappa^2L^4$.

4. Exact Gaussian discrimination limit

The statistical theorem concerns two known simple hypotheses: the product-limit connection and a specified positive-\(\lambda\) connection. Every measurement resets the initial vector and adds fresh independent Gaussian scalar noise with standard deviation $\sigma >0\backslash sigma>0$. There are a fixed number $nn$ of observations, each with loop-length budget $LL$.

Even when later loops may depend on earlier observations, the optimal equal-prior error is

$$R_n=\mathrm{\Phi }(-\frac{\sqrt{n}{D}_{\lambda }(L)}{2\sigma })\mathrm{.}R\_n=\backslash Phi\backslash left(-\backslash frac\{\backslash sqrt\; n\backslash ,D\_\backslash lambda(L)\}\{2\backslash sigma\}\backslash right).$$

Repeating a maximally separating loop attains it. For target error $\alpha \in (0,1\mathrm{/}2)\backslash alpha\backslash in(0,1/2)$, the sharp small-loop sample requirement is

$$n_{\alpha }=\frac{4096{\pi }^4{\sigma }^2{z}_{1-\alpha }^2}{{\kappa }^4{L}^8}(1+O_d(L))+O(1),n\_\backslash alpha=\; \backslash frac\{4096\backslash pi^4\backslash sigma^2z\_\{1-\backslash alpha\}^\{\backslash ,2\}\}\{\backslash kappa^4L^8\}\; \backslash bigl(1+O\_d(L)\backslash bigr)+O(1),$$

where $z_{1-\alpha }={\mathrm{\Phi }}^{-1}(1-\alpha )z\_\{1-\backslash alpha\}=\backslash Phi^\{-1\}(1-\backslash alpha)$. The $L^{-8}L^\{-8\}$ penalty belongs to this declared scalar-noise experiment. It is not a general assertion that adaptive strategies cannot help in other models.

Download and reproduce

Download the complete repository using the current Hugging Face client:

from huggingface_hub import snapshot_download

snapshot_download(
    repo_id="PureOne/AEQORIA-Sharp-Holonomy-Observation",
    repo_type="dataset",
    local_dir="AEQORIA",
)

Then, from the downloaded directory, create a Python environment and install the research dependencies. The original recorded run used Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0, SymPy 1.14.0, and mpmath 1.3.0; the exact package pins are in research/requirements.txt.

cd AEQORIA/research
python -m venv .venv
# Activate .venv using the command appropriate for your shell.
python -m pip install -r requirements.txt
python verify_aeqoria.py --output ../local_results/verification.json
python create_examples.py

The verification output path above preserves the original recorded verification and asymptotic files. create_examples.py regenerates the illustrative CSV and JSON tables. Use REPRODUCIBILITY.md for further environment and validation details.

The reference implementation can also be used directly from research/:

import numpy as np
from aeqoria import (
    writer_angle, writer_echo_gap, simplex_frame,
    frame_spectrum, gaussian_testing_risk, sufficient_samples,
)

lam, length, dimension = 0.5, 0.5, 3
beta = writer_angle(lam, length)
gap = writer_echo_gap(lam, length)
u, tangent_simplex, V = simplex_frame(dimension, beta)

print("Writer angle:", beta)
print("Writer echo deficit:", gap)
print("Gram eigenvalues:", np.linalg.eigvalsh(V.T @ V))
print("Analytic spectrum:", frame_spectrum(dimension, beta))

samples = sufficient_samples(gap, sigma=1e-4, target_risk=0.05)
print("Samples for this explicit writer:", samples)
print("Resulting risk:", gaussian_testing_risk(samples, gap, sigma=1e-4))

Here gap is the explicit writer's deficit. At finite length it must not silently be substituted for a proved global optimum $D_{\lambda }(L)D\_\backslash lambda(L)$.

Data and file organization

The unmodified original release is under research/; its original ZIP is under releases/. The principal source is research/AEQORIA_Sharp_Holonomy_Observation.tex, with chapter sources in research/sections/. Release notes describe the Hugging Face packaging layer.

The illustrative research/results/circle_examples.csv has six parameter settings and these columns:

Columns	Meaning
lambda, L, kappa	Connection parameter, loop length, and $\lambda (1-\lambda )\backslash lambda(1-\backslash lambda)$
writer_angle, writer_deficit	Explicit one-circle angle and scalar deficit
gram_min_d3, gram_condition_d3	Least Gram eigenvalue and condition number for the three-dimensional simplex frame
real_sample_threshold, illustrative_ceiling	Real-valued testing threshold and its integer ceiling for the explicit writer

The table uses 80-digit arithmetic with $\sigma ={10}^{-4}\backslash sigma=10^\{-4\}$ and equal-prior target risk $0.050.05$. It is illustrative computed data, not experimental measurements or interval-certified finite-length optima. JSON versions preserve precision-oriented decimal strings and record the parameters.

The manuscript progresses through the observation problem; quotient geometry; uniform loop control; exact writer; cap optimization; scalar-resource bridge; exterior inversion; finite scalar histories; noisy calibration; Gaussian testing; finite-radius calibration; additional references and hidden-direction hierarchy; examples; and scientific status. Appendices provide auxiliary proofs, computation details, and executable code listings.

Intended use and limitations

Use this release for specialist mathematical assessment, reproducing the explicit examples, inspecting the proofs and assumptions, testing inverse methods, or developing precisely stated extensions. Scalar-history reconstruction acts on the cyclic subspace and is determined only up to a common orthogonal gauge; inaccessible invariant complements cannot be inferred from those echoes.

The release does not establish general reconstruction of arbitrary unknown manifolds, arbitrary finite-length circle optimality, or physical instrument performance. Nonnegative curvature alone does not supply a uniform conditioning floor in the constructed family. The dimension hypothesis $d\ge 3d\backslash ge3$, parameter interior , cap regime, calibration assumptions, and statistical noise contract are material parts of the statements.

No broad reuse license was supplied in the original release. See LICENSE_STATUS.md before assuming permissions for redistribution or derivative use. The original CITATION.cff preserves the supplied author and release metadata; no DOI or arXiv identifier is asserted.