PureOne/Eves-Impossible-Complex-Frequencies-v5.0.0

Eve's Impossible Complex Frequencies and Perfect Number Quantization Riemannian Geometry

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research version: 5.0.0 · 105-page standalone manuscript · MIT license

A mathematical research release on how local geometric observations determine global arithmetic information in finite integer-frequency embeddings of flat tori. Its central results are a sharp finite-jet reconstruction theorem, exact counterexamples to unrestricted finite-order recovery, and a two-divisor law for worst-case separation at fixed metric and power.

The repository contains the manuscript, complete proofs and LaTeX source, exact observation inputs and certificates, research software, numerical experiments, verification records, and installable distributions. The original 166-file release is preserved byte for byte under release/.

Read and download

Resource	Link
Standalone manuscript, 105 pages	Read the PDF
Original complete research archive	Download v5.0.0 ZIP
Proof dependencies and review route	Proof map
Contribution boundaries and literature positioning	Claims and prior art
Verification scope and reproduction commands	Verification
Software contracts and examples	API
Original release overview	Release README

For the main mathematical argument, read manuscript §15 → §17 → §18, followed by the worked certificates in §23.

Mathematical setting

For integer frequency vectors $a_j$ and positive weights $w_j$, consider

$$F(\theta {)}_j=\sqrt{w_j}{e}^{i{a}_j\cdot \theta },C(\theta )=2\sum _j{w}_j(1-\cos (a_j\cdot \theta ))\mathrm{.}F(\backslash theta)\_j=\backslash sqrt\{w\_j\}\backslash ,e^\{i\; a\_j\backslash cdot\backslash theta\},\; \backslash qquad\; C(\backslash theta)=2\backslash sum\_j\; w\_j\backslash bigl(1-\backslash cos(a\_j\backslash cdot\backslash theta)\backslash bigr).$$

The intrinsic metric is $G=\sum_jw_j a_ja_j^\mathsf{T}$. It describes flat geometry but can lose arithmetic information that affects finite phase grids. The manuscript determines when additional local derivatives recover that information and constructs exact examples where they do not.

Sharp reconstruction from finite derivatives

With at most $m$ distinct nonzero frequency orbits under $a\sim-a$, the full exact multivariate chord jet through derivative order $4m$ determines the reduced frequency vectors and their positive weights. If the aggregate nonconstant power is supplied independently, order $4m-2$ suffices. Both worst-case bounds are sharp in the proved primitive integer-circle examples for $m\ge2$.

The reconstruction algorithm receives a moment table, dimension, and support bound; it does not receive the hidden source frequencies. Mixed derivatives are required. The number $4m$ is a derivative-order bound, not a scalar-measurement count.

Recovery is subject to the natural equivalences of the chord kernel: row permutation, independent sign changes, combining duplicate orbits, and omission of constant coordinates. The support bound is an assumption; the finite observations do not certify it themselves.

Exact two-divisor separation law

For faithful integer-frequency circles with nonzero frequencies, impose

Let $Q\ge2$ be the alphabet size and $p$ its smallest prime divisor. Over all such finite representations, the infimum of squared nearest-neighbor chordal separation is

$${\mathcal{E}}_Q(P)=\underset{q\in \{p,Q\}}{\min }4{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\pi }{q}\right){\left[\frac{q^{2}P-1}{q^2-1}\right]}_{+},[x{]}_{+}=\max \{x,0\}\mathrm{.}\backslash mathcal\; E\_Q(P)=\; \backslash min\_\{q\backslash in\backslash \{p,Q\backslash \}\}\; 4\backslash sin^2\backslash !\backslash left(\backslash frac\{\backslash pi\}\{q\}\backslash right)\; \backslash left[\backslash frac\{q^2P-1\}\{q^2-1\}\backslash right]\_+,\; \backslash qquad\; [x]\_+=\backslash max\backslash \{x,0\backslash \}.$$

Only the smallest prime divisor and the full alphabet can minimize in the positive range. The infimum is positive exactly when $P>1/p^2$. For positive infimum and $P<1$, reduced extremizers use frequencies ${1,p}$ or ${1,Q}$. Composite alphabets have a single explicit transition; for $Q=6$, it occurs at $P=137/452$.

When the infimum is zero, faithful representations approach it but do not attain it. This theorem is a worst-case lower envelope, with nonconstant power prescribed; it is not a maximum-distance code-design theorem.

Exact limits on finite-order information

Without a frequency-complexity bound, fixed finite derivative data can leave arithmetic behavior undetermined. The release gives faithful positive rational-weight circles with equal metric, equal nonconstant power, and matching prescribed jets, but different behavior at a suitable prime.

Precisely, the prescribed-prime construction takes $R\ge1$, rational $P\in(0,1)$, and a prime $p>2R+1$ satisfying $Pp^2>1$. Each representation has $R+2$ distinct positive frequencies. Their jets agree through order $2R$, while their prime safety and a labeled $p$-grid chord differ. A separate result proves that their minimum distances differ for all sufficiently large alphabets; it does not identify those minima with the labeled prime-grid discrepancy.

Further results

The manuscript also develops relative modular rigidity for nonprimitive parametrizations, prime-power obstruction depths, labeled support reconstruction under external Smith-invariant bounds, finite certificates for differential identities with known Fourier support, sharper eventual-neighbor thresholds, and magnetic spectral factorization. The $2m$-measurement real spectral schedule requires independently calibrated DC.

Evidence and status

The manuscript contains 81 numbered results: 31 theorems, 18 lemmas, 16 propositions, and 16 corollaries. This inventory includes established foundations, retained results, and extensions; it is not a claim of 81 historically new discoveries.

Recorded release evidence	Count
Python tests passing	260; zero failures, errors, or skips
Independent C++ arithmetic atlas	10,920 matrix–modulus cases; 564,761 assertions
New explicitly counted exact predicates	2,101
Separately labeled numerical predicates	2,555
Moment-only reconstruction observations	30 inputs; 1,633 moment entries
Support-only reconstruction observations	3 inputs; 20 subsystem outputs

These figures come from the supplied prepack validation, v5 experiment report, and C++ report. The Hugging Face preparation audit rechecked file integrity and the consistency of these records; it did not rerun the research test suite. The C++ atlas independently audits its declared inherited arithmetic scope, not every new theorem.

Status: written proofs and reproducible internal checks are supplied. Independent specialist review, proof-assistant formalization, and exhaustive historical-priority checking remain outstanding. The numerical experiments are finite floating-point evidence, not universal or interval-certified proofs. No percentage is assigned to mathematical correctness or historical novelty.

The geometry is intrinsically flat, with informative extrinsic derivatives. Prony reconstruction, Smith normal form, finite Fourier analysis, and torus coding have established antecedents that the claims ledger credits. Perfect numbers supply optional alphabet examples; the release does not establish a distinct perfect-number quantization mechanism. No AI capability gain or physical hardware advantage is demonstrated.

Reproduce the research

Download and extract the original archive, or work from a clone of this dataset repository. From its release/ directory, using Python 3.10 or newer:

python -m pip install -r requirements.txt
python examples/demo_v5.py
python tools/verify_all.py --output ../eve-verification.json
python tools/smoke_wheel.py --output ../eve-wheel-verification.json

Full verification requires a C++20 compiler. Figure rebuilding needs the figure requirements; rebuilding the PDF additionally needs LaTeX. See the verification document for exact scope, environment, budgets, and platform notes. Do not run with Python optimization enabled.

Dataset contents and schema

This is a research artifact dataset, containing mathematical documents, software, and reproducibility material. The configured Dataset Viewer displays only data/artifact_index.jsonl, an inventory with one row for each of the 166 original release files. Its train split is a Hub loading convention; it does not imply a training corpus or a train/test evaluation protocol.

Field	Type	Meaning
path	string	Repository-relative path under release/
category	string	Manuscript, source, certificate, test, distribution, or other artifact role
extension	string	File extension, or an empty string
bytes	integer	Exact byte length
sha256	string	SHA-256 of the artifact
release_version	string	5.0.0
description	string	Short inventory label

The explicit Viewer configuration keeps the heterogeneous research JSON files separate from this inventory.

from datasets import load_dataset

index = load_dataset(
    "PureOne/Eves-Impossible-Complex-Frequencies-v5.0.0",
    "artifact_index",
    split="train",
)

The root MANIFEST.sha256 covers the repository files except itself. The original independent manifest remains at release/MANIFEST.sha256. Preparation details are recorded in HUGGING_FACE_PREPARATION.json.

Intended use and limitations

The release supports study and review of its stated finite-frequency theorems, exact reconstruction experiments, arithmetic certificate development, and reuse of the research software. Each routine's hypotheses and computational budgets are part of its contract. Exact reconstruction does not provide uniform recovery from noise when weights or frequency separation can degenerate.

The scope is finite integer-character geometry. It does not establish results about unrestricted analytic functions, general Riemannian spaces, arbitrary differential equations, or general intelligence. Finite computations supplement the written arguments rather than replacing them.

License and citation

The MIT license is retained from the source release and covers its generated software and associated documentation. Referenced third-party publications retain their respective rights. The exact supplied CITATION.cff is preserved, including the source release date of 12 October 2026; this Hugging Face preparation was made on 13 October 2026.

@misc{eve_impossible_frequencies_2026_v5,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
  title = {Eve's Impossible Complex Frequencies and Perfect Number Quantization Riemannian Geometry},
  year = {2026},
  note = {Version 5.0.0. Standalone mathematical manuscript and research software.}
}