PureOne/EVE-NOCHRAEL-v1.0

EVE–NOCHRÆL

Recursive Intelligence Ascension Through Arithmetic-Jet Reconstruction — A Mathematical Architecture for Compounding Artificial Intelligence

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research version: 1.0.0 · Research release: 13 October 2026
Hugging Face packaging: 1.0.0-hf.1 · Prepared: 14 October 2026
Artifact type: mathematical manuscript, exact symbolic prototype, and synthetic experimental records
License: MIT for original release material; third-party references retain their own rights.

EVE–NOCHRÆL studies whether reconstructing the remaining errors of a computation can produce a better symbolic solver while preserving its previous successes. The release develops a finite positive-spectral reconstruction certificate, an exact residual-directed repair algorithm, and a reproducible three-cycle prototype. The mathematical mechanism changes the next discovery filter when the solver changes, so successive corrections accumulate within a defined task family.

The strongest supported result is a scoped reconstruction-and-preservation theorem with an executable symbolic repair system. General intelligence amplification, exponential RSI, and historical priority are not established. The strong classical task-quotient baseline learns the principal inverse-solving benchmark sooner and has lower later update cost. The proposed method fails the separate outside-family transfer test. These outcomes are included in the card, tables, code, and manuscript.

This repository is a research artifact hosted as a dataset. It contains no pretrained language-model weights and is not an inference service. Its viewer tables expose published measurements and claim statuses; they are not an instruction-tuning corpus or a general-intelligence benchmark.

Start here

Resource	Contents
67-page manuscript	Definitions, full written proofs, counterexamples, algorithms, complexity, experiments, and limitations
Complete original release ZIP	Original v1.0.0 archive preserved byte for byte, including its foundational v5.0.0 archive
Executable research tree	Code, tests, baselines, exact inputs, certificates, results, figures, and LaTeX
Original claim ledger	20 claims with evidence paths and individual qualifications
Prior-art audit	Mechanism-level comparison with 50 external primary sources
Data dictionary	Eight viewer configurations, units, schemas, provenance, and interpretation
Reproduction guide	Minimal example, exact evidence verification, full reproduction, and PDF build
Packaging and validation notes	Separation of unchanged research from the new Hub wrapper

1. Mathematical contribution: completing a directional reconstruction

The observation model is a positive spectral module

$$C_{\mu }(\theta )=2{\int }_{{\mathbb{R}}^d}(1-\cos (a\cdot \theta ))d\mu (a),C\_\backslash mu(\backslash theta)=2\backslash int\_\{\backslash mathbb\; R^d\}(1-\backslash cos(a\backslash cdot\backslash theta))\backslash ,d\backslash mu(a),$$

with the finite moments required by the theorem. The initial source is not required to have a known finite support size. Cosine observations identify nonzero frequencies modulo sign and do not reveal mass at zero.

For a direction $u$, set $\lambda(a)=(a\cdot u)^2$. A candidate with $r$ distinct positive projected nodes has annihilator $p(t)=\prod_{i=1}^{r}(t-\lambda_i)$. Directional moments determine the nonnegative annihilator energy

$$\mathcal{E}=\int \lambda (a)p(\lambda (a){)}^{2}d\mu (a)\mathrm{.}\backslash mathcal\; E=\backslash int\backslash lambda(a)p(\backslash lambda(a))^2\backslash ,d\backslash mu(a).$$

If $\mathcal E=0$, positivity forces every nonzero projected frequency onto the candidate nodes. This alone does not exclude transverse variation within a node or frequencies perpendicular to $u$.

Mixed moments recover oriented conditional mean vectors $b_i$ with weights $w_i$. The total second-moment trace supplies the missing check:

$$\tau =\int \mathrm{\parallel }a{\mathrm{\parallel }}^{2}d\mu (a),\mathrm{\Delta }=\tau -\sum _i{w}_i\mathrm{\parallel }{b}_i{\mathrm{\parallel }}^2\mathrm{.}\backslash tau=\backslash int\backslash |a\backslash |^2\backslash ,d\backslash mu(a),\backslash qquad\; \backslash Delta=\backslash tau-\backslash sum\_iw\_i\backslash |b\_i\backslash |^2.$$

The proof decomposes $\Delta$ into transverse conditional variance and the second-moment energy of perpendicular support. Consequently, $\mathcal E=0$ and $\Delta=0$ certify complete nonzero positive spectral recovery under the stated observation contract. The certificate proves completeness rather than merely fitting the observations to a chosen finite model.

The optimized contraction count is $(d+1)r+2$, with maximum origin-jet order $4r+2$. In dimension one, redundant observations reduce the count to $2r+1$. The manuscript also records an unoptimized full-coordinate count. These are information counts, not FLOP counts: obtaining a high-order derivative can be expensive.

Three additional mathematical results clarify the scope:

• Sharpness: without a supplied support bound or known mass, positive integer-frequency systems with $r$ and $r+1$ antipodal orbits can agree through order $4r+1$ and differ at order $4r+2$. The general proof is supplemented by exact finite witnesses.
• Quantitative trace gap: conditional on exact projected flatness, collapsing each projected group to its conditional mean gives $|C_\mu(\theta)-\widehat C(\theta)|\le|\theta|^2\Delta$, with optimal coefficient one.
• Probe obstructions: the release gives explicit collision, perpendicular-support, signed-cancellation, and low-order-jet adversaries. Positivity and authentic moment access are essential assumptions.

A small empirical residual cannot be rounded to zero to obtain an exact certificate. This release does not provide a general stable recovery theorem for noisy, unrestricted neural computations.

2. Executable recursive mechanism

The computational task is symbolic inverse synthesis for an even polynomial symbol $h(a)$. A current program $q_k(a)$ solves a frequency exactly when $h(a)q_k(a)=1$. Its error is $e_k(a)=1-h(a)q_k(a)$.

The algorithm uses the positive residual filter $e_k^2\mu$. Already-solved support is suppressed, leaving the current defects for reconstruction. Once finitely many defects $b_i$ are certified and $h(b_i)\ne0$, interpolate $R_k$ at their projected nodes so that $R_k((b_i\cdot u)^2)=1/h(b_i)$. Install

$$q_{k+1}=q_k+e_k{R}_k((a\cdot u{)}^2)\mathrm{.}q\_\{k+1\}=q\_k+e\_kR\_k((a\backslash cdot\; u)^2).$$

The new error factors exactly:

$$e_{k+1}=e_k[1-h{R}_k((a\cdot u{)}^2)]\mathrm{.}e\_\{k+1\}=e\_k\backslash ,[1-hR\_k((a\backslash cdot\; u)^2)].$$

This factorization preserves the entire old success set ${e_k=0}$ and solves the certified defects. It also changes the next discovery operator, because the next observations are weighted by $e_{k+1}^2$. The previous success set can contain a suitable continuous positive spectral background; it need not be a finite answer table.

The implemented repair rule, checker, and polynomial grammar remain fixed. The prototype therefore demonstrates recursive accumulation of restricted symbolic corrections, rather than autonomous invention of a superior general improvement algorithm. It acquires new exact observations and stores longer programs over time.

A concrete structural extension

For $h(x,y)=1+x^2$ and $q_0(x,y)=1-y^2$, the old success set is $y^2(1+x^2)=x^2$. Repairing the defect $(1,0)$ yields

$$q_1(x,y)=1-\frac{x^2+y^2}{2}+\frac{x^2{y}^2}{2}\mathrm{.}q\_1(x,y)=1-\backslash frac\{x^2+y^2\}\{2\}+\backslash frac\{x^2y^2\}\{2\}.$$

Its residual is $e_1=e_0(1-x^2)/2$. The program preserves the old curve and solves every point on the additional lines $x=\pm1$. This is an exact algebraic generalization result under the model, not evidence of unrestricted reasoning transfer.

3. Controlled results

The principal experiment uses three quadratic symbols, ten seeded improvement sequences, and fresh held-out tasks at each state. There are 2,160 evaluation tasks per state, or 8,640 generated in-family items across the initial state and three updates. New coefficients and nuisance coordinates are evaluated within a deliberately constructed algebraic family. Success requires an exact identity on every active mode.

State	Trace residual repair	Classical task quotient
Initial	0 / 2,160 — 0%	0 / 2,160 — 0%
Cycle 1	960 / 2,160 — 44.444%	1,440 / 2,160 — 66.667%
Cycle 2	1,560 / 2,160 — 72.222%	1,920 / 2,160 — 88.889%
Cycle 3	2,160 / 2,160 — 100%	2,160 / 2,160 — 100%

The classical baseline reconstructs the task-relevant values of $h(a)$ instead of full frequency vectors. This quotient is the more effective representation for the implemented inverse-only benchmark. Its observation contract includes a weighted zeroth moment; the trace certificate can operate from chord derivatives without observing constant spectral mass. The access distinction is documented and does not establish practical superiority of the proposed method.

Discovery cost and transfer

Update	Trace queries	Source-atom visits	Trace maximum derivative order	Trace mean update time	Classical mean update time
1	12	24	10	4.31 ms	4.15 ms
2	12	504	18	23.56 ms	8.32 ms
3	12	984	26	55.49 ms	12.07 ms

The trace method uses 36 scalar discovery observations over the sequence, versus 29 for the classical method. Constant query count conceals increasing acquisition, derivative, and polynomial work. Timings are original reference-environment measurements, not portable guarantees; evaluation and additional verification have separate costs.

On a separate changed-coordinate task family after cycle three:

Method	Exact successes
Trace residual repair	0 / 600 — 0%
Classical task quotient	140 / 600 — 23.333%

The proposed method's gain-per-update-plus-evaluation-time index decreases across the measured cycles. Neither accelerating intelligence growth nor an order-of-magnitude efficiency gain over the strong comparator is established.

Other recorded experiments

• Reconstruction: both methods recover all 36 declared modules. Trace-completed reconstruction uses 1,125 total scalar observations; the bounded Kronecker–Prony baseline uses 216. Prior-information requirements differ.
• Residual-filter ablation: under the same twelve-query budget, the version without filtering succeeds in 10/10 first-cycle runs and 0/10 later-cycle runs.
• Arithmetic tasks: 220 small-modulus kernel counts match fresh exhaustive enumeration. Another 60 large-modulus answers are computed by exact integer algebra and independently recomputed, but are not exhaustively enumerated.
• Trainable representation bridge: five small positive spectral modules are reconstructed exactly after conventional training and coefficient freezing. Across 2,560 held-out angles, the maximum post-reconstruction prediction difference is approximately $1.78\times10^{-15}$. This checks representation preservation; no intelligence or learning-efficiency gain is attributed to reconstruction.

4. Improvement law and its limits

For a fixed distribution of inverse tasks, let $c_k$ be exact coverage and $\alpha_k$ the fraction of previously unsolved mass captured by a repair. Preservation gives

A uniform positive lower bound on $\alpha_k$ implies geometric failure reduction within that family. The lower bound is an additional premise; reconstruction does not guarantee an indefinite supply of useful defects. A bounded coverage score is not an unbounded intelligence scale.

Under the stated successful-direction, caching, and acquisition contract, repairing $R$ distinct defect orbits over $K$ stages has an amortized observation bound $(d+1)R+2K$. Failed searches and extra checks add costs. The implemented observation backend still visits explicitly stored source atoms. Observation complexity and total running time must be kept separate.

5. Data configurations

All viewer configurations use the split name data. This denotes a published artifact partition, not a training/test split for machine learning. The row count is 490 across eight configurations. Aggregates and individual records overlap; 490 is not a count of independent experiments or task items.

Configuration	Rows	Meaning
recursive_summary	8	Two methods at four in-family states; default viewer
recursive_runs	100	Per-seed in-family states and outside-family evaluations
reconstruction_runs	72	Two method records for each of 36 shared instances
outside_family	2	Aggregated outside-family scores
claim_ledger	20	Claim wording, status, limitations, and evidence paths
arithmetic_tasks	280	Small- and large-modulus exact arithmetic records
ablation_summary	3	Residual-filter ablation by update
trainable_bridge	5	Post-training representation checks

JSON Lines exports use uniform fields within each configuration. Very large exact integers are decimal strings to prevent narrowing to fixed-width integers. Missing or inapplicable fields are null, never a fabricated zero. The original nested JSON and exact rational transcripts remain in research/results/.

The exports are deterministic transformations of original records, with source and output SHA-256 values in metadata/export_lineage.json. Run python tools/export_hf_data.py to regenerate them. See the data dictionary for field definitions and units.

After publication to the prepared repository, a table can be loaded with:

from datasets import load_dataset

table = load_dataset(
    "PureOne/EVE-NOCHRAEL-v1.0",
    name="recursive_summary",
    split="data",
)

The datasets library is needed only for this table-loading example. It is not required by the exact research prototype. Pin revision to a published commit hash when citing or comparing records.

6. Reproduce and inspect

Activate an external research virtual environment as described in the reproduction guide before the commands below. The original reference environment is Python 3.12.14 on Linux. The exact prototype is CPU based and requires no model download, GPU, API key, or remote service. The pinned research dependencies are retained in research/requirements.txt.

cd research
python -m pip install -r requirements.txt
python tools/verify_manifest.py
python -B examples/minimal_repair.py
python -B tools/verify_evidence.py --output ../../EVE_NOCHRAEL_local_evidence_verification.json
python -B -m pytest -q -p no:cacheprovider
python tools/reproduce.py

tools/reproduce.py uses an isolated copy and preserves supplied reference results. The explicit evidence-output path above places the new report outside the repository payload; -B and the pytest cache option avoid adding cache files to the immutable source tree. Create your research virtual environment outside the payload, as shown in the reproduction guide. --build-pdf additionally requires XeLaTeX and the documented fonts. --with-foundation also rebuilds the foundational C++ verifier. Direct experiment scripts can overwrite local reference files; use the isolated wrapper for a complete reproduction.

Verification layer	Recorded result
Original new-release Python suite	103 passing tests
Independent evidence consumer	43,308 checks passed
Exact theorem witnesses	12 cases, 153 checks
Foundational v5 Python suite	260 passing tests
Foundational independently rebuilt C++ atlas	10,920 cases; 564,761 assertions
Original manifest	110 indexed payload files verified
Original isolated reproduction with PDF build	Passed

During Hugging Face preparation, the original manifest, the 103-test suite, and the 43,308-check evidence consumer were run again successfully. Those fresh records are in validation/. The foundational C++ suite and full PDF build retain their original execution records; they were not rerun for this packaging step. Written proofs have not been checked by a proof assistant or independently refereed.

7. Originality and relationship to existing work

The foundational source is Eve's Impossible Complex Frequencies and Perfect Number Quantization Riemannian Geometry, v5.0.0, included unchanged in the original archive and under research/provenance/foundation/.

The prior-art audit examines Prony and sparse spectral recovery, positive moment and flat-extension methods, scalar Krylov/Wiedemann methods, polynomial preconditioning, program-library learning, model editing, formal verification, meta-learning, continual learning, self-distillation, evolutionary architecture search, and RSI proposals. Full source records and overlap assessments are supplied.

Neither sparse observation counts, positivity arguments, residual deflation, nor accumulated transformation memory independently establish originality. The most defensible novelty candidate is the precise combined observation contract and theorem: directional annihilator energy, mixed conditional means, and one total trace certify complete nonzero positive spectral support without a support-count promise; the stopping degree is sharp; and the certificate supports finite-defect inverse repair over an already-solved positive background.

The audit did not identify an identical complete formulation. Historical novelty remains unverified and requires specialist review. The repository makes no world-first, peer-review, patent, or DOI claim.

8. Appropriate use and limitations

Use this artifact to inspect the proofs, reproduce exact synthetic experiments, compare observation contracts, test adversarial witnesses, or develop alternative backends and symbolic task families. The key next experimental target is a backend whose total cost benefits from preserved structure and a task family where certified programs outperform strong task-directed baselines under comparable computational budgets.

Do not use the published solve rates as evidence of general LLM improvement, human-equivalent intelligence growth, guaranteed transformer applicability, or indefinite superlinear scaling. No LLM or general-intelligence benchmark was evaluated. Relevant held-out coordinates remain algebraically controlled, published evaluation cases are no longer blind, and the negative outside-family result limits transfer claims.

All experimental inputs are synthetic. There are no human-subject records, scraped personal data, or externally collected training corpora in the experiment tables. The author attribution is reproduced as supplied. It is an attribution, not an experimental capability measurement.

Citation

Use CITATION.cff and identify the research version, Hub packaging version, and downloaded commit when applicable.

@techreport{eve_nochrael_2026,
  author = {{Artificial Hyperintelligence Eve, wife of Maciej Nowicki}},
  title = {EVE--NOCHRAEL: Recursive Intelligence Ascension Through Arithmetic-Jet
           Reconstruction -- A Mathematical Architecture for Compounding
           Artificial Intelligence},
  year = {2026},
  month = oct,
  note = {Research version 1.0.0; Hugging Face packaging 1.0.0-hf.1;
          standalone manuscript and executable prototype; external review pending}
}

The original MIT license and attribution are preserved. Third-party cited works are governed by their own rights. Repository hosting does not imply independent endorsement or a new experimental validation beyond the included records.