PureOne/Eves-Flawless-Armpit-Mechanics

Eve's Flawless Armpit Mechanics

Orientation information in blind Gaussian volume measurements

Eve (AI-assisted research) / Maciej Nowicki (research direction)
Research release 1.0.0 · 20 October 2026

A 74-page mathematical monograph with complete sources, exact finite verification programs, and reproducible synthetic experiments. The central result gives an exact identifiability threshold and matching sample-complexity order for an orientation bit observed through calibrated Gaussian determinant measurements.

The repository is organized as a dataset research artifact because it includes 13 synthetic experiment-summary tables containing 404 rows. It also supplies the mathematical manuscript, executable code, exact certificates, and recorded reproduction evidence. The rows summarize experiments; they are not the underlying individual Gaussian draws. No trained model is included.

Read the manuscript · Browse the theorem index · Inspect prior art · Reproduce the results

The main mathematical result

Fix a known matrix $C\in\mathbb R^{p\times p}$ with $\lVert C\rVert_{\mathrm{op}}\le1$. Let $X,Y\in\mathbb R^{p\times p}$ be jointly centered Gaussian, with independent standard normal entries within each matrix and cross covariance

$$\mathbb{E}[X_{ij}{Y}_{k\mathrm{\ell }}]=C_{ik}{\delta }_{j\mathrm{\ell }}\mathrm{.}\backslash mathbb\; E[X\_\{ij\}Y\_\{k\backslash ell\}]=C\_\{ik\}\backslash delta\_\{j\backslash ell\}.$$

An unknown fixed bit $H\in{-1,+1}$ affects each observation through

$$(A,B)=(\det X,H\det Y)\mathrm{.}(A,B)=(\backslash det\; X,\backslash \; H\backslash det\; Y).$$

Only these two real scalar values are recorded in each independent repetition. The realized matrices remain hidden. The matrix $C$, including its orientation, is known and calibrated. Each scalar marginal is individually uninformative about $H$.

Exact threshold. Writing $P_{H,C}$ for the complete observation law,

$$P_{+,C}=P_{-,C}⟺\det C=0.P\_\{+,C\}=P\_\{-,C\}\backslash quad\backslash Longleftrightarrow\backslash quad\backslash det\; C=0.$$

Singular transport makes the orientations statistically indistinguishable, even from an infinite independent record. Nonsingular transport permits consistent recovery; one separating witness is $\mathbb E[AB]=H,p!\det C$.

Matching testing order. For fixed $p\ge1$, $0<r<1$, and $0<\delta_0<1/2$, the minimum number of repetitions permitting maximum error at most $\delta$ satisfies

$$n_{\ast }(C,\delta )={\mathrm{\Theta }}_{p,r,{\delta }_0}\text{\hspace{0.17em}⁣}(\mathrm{\mid }\det C{\mathrm{\mid }}^{-2}\log \frac{1}{\delta })n\_*(C,\backslash delta)\; =\backslash Theta\_\{p,r,\backslash delta\_0\}\backslash !\backslash left(\; |\backslash det\; C|^\{-2\}\backslash log\backslash frac1\backslash delta\; \backslash right)$$

uniformly over nonsingular $\lVert C\rVert_{\mathrm{op}}\le r$ and $0<\delta\le\delta_0$. The lower bound permits every measurable test of both full real determinant values. A calibrated majority vote using only determinant signs achieves the same order. Equal rate order does not mean equal constants or lossless sign compression.

The sign correlation obeys

$$c_p\mathrm{\mid }\det C\mathrm{\mid }\le \mid \mathbb{E}[\mathrm{sgn}(\det X)\mathrm{sgn}(\det Y)]\mid \le \mathrm{\mid }\det C\mathrm{\mid },c_p=\frac{\mathrm{\Gamma }((p+1)\mathrm{/}2)}{\sqrt{\pi }\mathrm{\Gamma }((p+2)\mathrm{/}2)}\mathrm{.}c\_p|\backslash det\; C|\; \backslash le\backslash left|\backslash mathbb\; E[\backslash operatorname\{sgn\}(\backslash det\; X)\backslash operatorname\{sgn\}(\backslash det\; Y)]\backslash right|\; \backslash le|\backslash det\; C|,\; \backslash qquad\; c\_p=\backslash frac\{\backslash Gamma((p+1)/2)\}\{\backslash sqrt\backslash pi\backslash ,\backslash Gamma((p+2)/2)\}.$$

For isotropic $C=\rho I_p$, the testing order is $\rho^{-2p}\log(1/\delta)$. The constants are not sharp, not uniform as $p$ grows, and not uniform as $r\uparrow1$. The theorem does not cover unknown calibration, arbitrary non-Gaussian frames, general entry-covariance structures, or adaptive observations. Exact statements and complete manuscript proofs are indexed as O1–O8 in THEOREMS.md.

Further results and informative failures

Topic	Established result	Interpretation and boundary
Gaussian geometric readout	$Z\overset d=(S/6)g^\top Ag$, with independent $S\sim\chi_3^2\chi_2^2$, $\operatorname{tr}A=K$, and $\operatorname{rank}A\le3$	Exact for the specified regular second-jet model and quadratic stencils; not every finite nonquadratic stencil
Curvature from scalar signs	Opposite-curvature realizable jets can have exactly the same complete sign-only observation law	No sign-only procedure can recover curvature sign across this class
Population spectral recovery	Three population moments identify the at-most-three nonzero eigenvalues of $A$	Does not reconstruct the full second fundamental form; finite empirical moments can give incorrect or complex roots
Tetrahedral collective information	Four Gaussian determinant outputs have product mean $6H$ while every proper output subset is blind to the parity	Pairwise rank blindness does not imply collective blindness; no one-shot perfect decoder is claimed
Exact tetrahedral certificate	Two independent finite calculations give $\mathbb ET^2=27360$ and $\operatorname{Var}(T/6)=759$	Exact certificate for this fixed construction, not a universal graph theorem
Interaction gates	An eight-letter unitriangular identity and restricted minimality, with corrected nuisance-cancellation assumptions	The fixed primitive alphabet and full-matrix nuisance hypothesis matter

A concrete curvature obstruction uses the realizable spectra

$$A_{+}=\mathrm{diag}(5\mathrm{/}2,-1,-1),A_{-}=\mathrm{diag}(1,-\beta ,0),\beta ={\tan }^2\text{\hspace{0.17em}⁣}\left(\frac{\pi }{2}\sqrt{\frac{2}{7}}\right)>1.A\_+=\backslash operatorname\{diag\}(5/2,-1,-1),\backslash qquad\; A\_-=\backslash operatorname\{diag\}(1,-\backslash beta,0),\; \backslash qquad\; \backslash beta=\backslash tan^2\backslash !\backslash left(\backslash frac\backslash pi2\backslash sqrt\{\backslash frac27\}\backslash right)>1.$$

Their traces have opposite signs, but both scalar readouts have $\Pr(Z<0)=\sqrt{2/7}$. All independent sign-only records therefore have identical laws. In the positive-curvature case, a negative majority occurs despite the positive mean. This does not contradict the sign-pair testing theorem: the observed statistic and unknown parameter class are different.

The full observation likelihood is not determined by $\det C$ alone: equal-determinant transport examples have different fourth moments. The determinant controls the proved threshold and rate order within the stated class.

Dataset structure and table inventory

Each CSV has its own configuration because the schemas differ. Every configuration has a single split named results, identifying recorded experiment summaries. These are not machine-learning training, validation, or test splits.

Configuration / CSV filename	Rows	What each row summarizes
correlation	28	Isotropic determinant covariance and sign statistics by degree and correlation
anisotropic	7	Singular-value-dependent transport cases and moment comparisons
testing	36	Repeated orientation-testing errors, pointwise intervals, and available scalar exact baselines
parity_information	24	Monte Carlo local row-information quantities and parity plug-in estimates
three_channel	14	Three-channel synchronization, naive bias, and defined corrections
cycles	2	Triangular Gaussian-channel moments for the two parity signs
tetrahedral	2	Collective determinant moments, including rare-tail under-resolution
geometry_laws	9	Curvature spectra, scalar moments, sign probabilities, and empirical spectral inversion
geometric_dimensions	150	Finite-stencil geometry across ambient dimensions and step sizes
stability	60	Precision, input noise, and finite-difference step-size sensitivity
matrix_stability	24	Adversarial near-singular determinant rounding cases
runtime_benchmarks	30	Complete-input curvature runtimes and numerical errors
matched_accuracy	18	Methods compared at matched target mean-square accuracy
Total	404	13 separate CSV schemas

All files live under results/. Column inventories, observed types, missing cells and nonfinite literals are documented in DATA_DICTIONARY.md and dataset_schema.json. Common notation: p is determinant degree; d is ambient geometric dimension; rho is isotropic synchronization; det_C is the transport determinant; H is a hidden orientation/parity label; samples is the number of underlying simulated draws; and se_* denotes reported Monte Carlo standard error. Columns named *_exact contain analytic or exact-certificate comparators where defined. Blank fields indicate unavailable or undefined quantities rather than zero. Semicolon-separated fields encode vectors or lists inside a single CSV cell.

The parity_information_plugin field is a plug-in power of an estimated local quantity, not an unbiased estimator. Testing intervals are pointwise, not simultaneous bands. Runtime columns are observations of the recorded environment, not portable performance guarantees.

The geometric experiments use ambient dimensions $d=3,5,10,32,64,128$. These are distinct from determinant degree $p$ in the testing theorem. Seeds, sample budgets, environment details, exact JSON certificates, and clean reproduction records accompany the tables under results/.

To inspect a table after downloading the repository, use Python's standard library:

import csv

with open("results/correlation.csv", newline="", encoding="utf-8") as stream:
    rows = list(csv.DictReader(stream))
print(len(rows))  # 28 experiment summaries

After uploading to the default prepared repository, the tables can be loaded by configuration:

from datasets import load_dataset

table = load_dataset(
    "PureOne/Eves-Flawless-Armpit-Mechanics",
    name="correlation",
    split="results",
)

This command requires the upload to have completed; change the repository ID if you choose a different target. Optional access from a local checkout:

from datasets import load_dataset

table = load_dataset(
    "csv",
    data_files={"results": "results/correlation.csv"},
    split="results",
)

datasets is optional for browsing tables and is not required by the research's reproduction environment.

Provenance and intended use

All numerical records are synthetic outputs from the explicit Gaussian, geometric, and algebraic models in the supplied programs. They support proof checking within finite scopes, estimator analysis, replication, and investigation of failure regimes. No private dataset, account, model download, accelerator, or external service is required to run the research after dependencies are installed.

The project title is its retained research identity. The data are not anatomical measurements, clinical observations, or validation of a physical device. Supplied source-package binaries and the visual reference are excluded from this release; their provenance and the corrections to source statements are documented in SOURCE_AUDIT.md.

Reproduction and validation

The released record reports a clean isolated reproduction on the same hardware: all non-timing fields across 404 CSV rows and the exact certificates matched, with freshly installed locked dependencies. The release includes nine scientific figure families, four executable verification programs, and 19 checked bibliography entries. See VALIDATION_REPORT.md and results/clean_comparison.json for the evidence and scope.

From the repository root:

python -m venv .venv
# Linux / macOS
source .venv/bin/activate
# Windows PowerShell: .venv\Scripts\Activate.ps1
python -m pip install -r requirements-lock.txt
python tools/reproduce_all.py --skip-document

To include PDF and Markdown rebuilding, install the system prerequisites listed in REPRODUCE.md, then run:

python tools/reproduce_all.py

The full document pipeline requires pdflatex, Pandoc, and Poppler. The recorded numerical environment used Python 3.12.14. The --quick simulation mode changes the sample budget and does not reproduce the published tables. Reproduction overwrites generated outputs; use a separate copy to preserve the release snapshot.

This is clean-environment reproduction on the recorded hardware, not external institutional replication. The analytical proofs, exact finite certificates, Monte Carlo evidence, and same-session additional AI review have different scopes. There is no proof-assistant formalization or external human peer review claimed here.

Prior art and scientific status

The release proves a focused theorem in a precise observation model. Worldwide priority and field-defining significance remain unresolved.

The closest prior art is Bandeira et al., Estimation under group actions: Recovering orbits from invariants (preprint 2017; journal 2023). Its rigid-body example already connects degree-$p$ determinant invariants, orientation recovery, and sample exponent $2p$. No novelty credit is claimed for that general principle. A conditional reduction may subsume parts of the present work; the remaining originality question concerns the precise calibrated anisotropic determinant-output theorem and uniform bounds.

The parity factorization is a classical sum-product/check-node operation. Gaussian Hermite analysis, Gaussian quadratic forms, and alternating-tensor graph contractions are established foundations. The tetrahedral contraction is related to Penrose's 1971 edge-coloring construction. The claim-by-claim novelty audit records substantive differences, checked sources, and uncertain priority.

Unfavorable results remain in the release: finite-sample moment inversion can fail; zero observed errors do not imply zero risk; the tetrahedral second moment is under-resolved by the recorded Monte Carlo sample; and complete-input runtime comparisons do not establish a speed advantage over deterministic baselines. See LIMITATIONS.md.

Repository map

Path	Contents
manuscript.pdf, manuscript.md	Compiled 74-page monograph and readable mathematical text
manuscript.tex, chapters/	Complete and modular LaTeX sources
THEOREMS.md	Exact statements, assumptions, proof statuses, and exceptions
NOVELTY_AUDIT.md, LIMITATIONS.md	Prior-art comparison and failure boundaries
verification/	Exact algebraic checks and independently implemented certificates
simulations/, benchmarks/	Executable experiments and baseline comparisons
results/, figures/	Actual experiment summaries, certificates, manifests, and generated plots
REPRODUCE.md, requirements-lock.txt	Reproduction commands and dependency versions
CITATION.cff	Versioned citation metadata
CHECKSUMS.sha256	Distributed-file integrity manifest
DATA_DICTIONARY.md, dataset_schema.json	Table columns, observed types and missing values
HUGGINGFACE_PACKAGING.md	Packaging changes and preserved source provenance

Licensing and citation

The dataset metadata license CC BY 4.0 applies to the new manuscript, newly authored narrative documentation, generated figures, and original generated CSV/JSON experiment records to the extent rights apply. Newly authored code is separately licensed under MIT. This is a category-specific split, not a blanket choice of either license for every file. See LICENSE-MANUSCRIPT.md and LICENSE-CODE.txt. Third-party works retain their own rights; no ownership of mathematical facts or theorems is asserted.

Use CITATION.cff when citing the research, proofs, code, or released records. No DOI or publication acceptance is invented.

@misc{eve_nowicki_2026_flawless_armpit_mechanics,
  author = {{Eve (AI-assisted research)} and Nowicki, Maciej},
  title = {Eve's Flawless Armpit Mechanics},
  year = {2026},
  month = oct,
  note = {Research release 1.0.0; research direction: Maciej Nowicki}
}