PureOne/EVE-Phase-Geometry-of-Exploration

EVE Phase Geometry of Exploration

Geometric Control of Basin Retention, Saddle Crossing, and Capability Discovery in Recursive Self-Improving Intelligence

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Research version: 2.0.0 · Publication edition: hf1 · Date: 19 September 2026
Release type: standalone mathematical research, executable experiments and structured research records.

When should a learner spend resources changing the mechanisms that determine its future capabilities? This project studies exploration through return-constrained descendant reachability: what can become attainable after an intervention, at what cost, and while preserving which capabilities?

The strongest result is an exact global explore/stay threshold in a specified persistent-memory control model. A second construction shows why every small operator combination can look unproductive even when a sufficiently large combination is beneficial. The release contains full conditional proofs, reproducible code, causal intervention certificates, counterexamples and retained negative neural results.

Evidence: 77 numerical assertions and 19 research unit tests pass. These validate stated finite examples; they do not establish practical neural RSI. Status: Useful; estimated research-program maturity: 72%. Maturity percentages are editorial estimates, not calibrated probabilities. First-discovery priority, external proof review and major-breakthrough status are unestablished.

Read, inspect, reproduce

Start here	Contents
Main manuscript — 28 pages	Definitions, central theorems and complete proofs, algorithms, experiments and related work
Mathematical supplement — 14 pages	Extended proofs, existence, invariance, critical scaling and counterexamples
Computational supplement — 15 pages	Experiment specifications, seeds, coverage, results and assertion ledgers
Complete original research ZIP	Portable source research snapshot with its own checksums
Expert reading guide	Fast routes to the strongest arguments and their assumptions
Theorem index · Research certificate	Result inventory and precise evidence boundaries
AI agent index · llms.txt	Machine-readable navigation and compact retrieval entry point
Publication instructions	Windows BAT publisher and verified public upload

Central result: an exact geometric exploration criterion

Let a closed acquisition trajectory write signed-area memory, which changes the gain of concurrent useful execution:

$$\dot{x}=u,\dot{m}=\frac{1}{2}(x_1{u}_2-x_2{u}_1),\dot{q}=(1+\kappa m)v\mathrm{.}\backslash dot\; x=u,\backslash qquad\; \backslash dot\; m=\backslash tfrac12(x\_1u\_2-x\_2u\_1),\backslash qquad\; \backslash dot\; q=(1+\backslash kappa\; m)v.$$

The controls share a fixed quadratic energy budget and horizon:

$$\frac{1}{2}{\int }_0^H(\parallel u{\parallel }^2+v^2)dt\le b,x(0)=x(H)=0,m(0)=q(0)=0.\backslash frac12\backslash int\_0^H(\backslash lVert\; u\backslash rVert^2+v^2)\backslash ,dt\backslash le\; b,\backslash qquad\; x(0)=x(H)=0,\backslash qquad\; m(0)=q(0)=0.$$

For known deterministic driftless dynamics, $b,H>0$, $\kappa\ge0$, and any fixed positive protected tube $\lVert x\rVert\le\rho$, optimizing over all square-integrable closed acquisition loops gives

$$\boxed{{\displaystyle \max q(H)>\sqrt{2bH}⟺\frac{\kappa bH}{2\pi }>\frac{j_{1\mathrm{/}4,1}}{\pi }\approx \mathrm{0.8851840549147.}}}\backslash boxed\{\backslash max\; q(H)>\backslash sqrt\{2bH\}\backslash quad\backslash Longleftrightarrow\backslash quad\backslash frac\{\backslash kappa\; bH\}\{2\backslash pi\}>\backslash frac\{j\_\{1/4,1\}\}\{\backslash pi\}\backslash approx0.8851840549147.\}$$

Here $j_{1/4,1}$ is the first positive zero of the Bessel function $J_{1/4}$. The baseline $\sqrt{2bH}$ spends all resources on execution without acquiring memory. A sharp time-weighted area inequality identifies a critical Bessel loop; a prefix-area bound excludes all finite-amplitude improvements below threshold. Above threshold, a sufficiently small improving loop fits any positive tube, and every prefix has an affordable memory-preserving return.

An optimizer exists. Its acquisition fraction tends to zero at onset, and its squared-support advantage has quadratic order near the critical coupling. The entire optimal far-supercritical trajectory is not solved. The threshold is a law of this specified model, not a universal neural constant. Restricting concurrent loops to circles gives threshold one; requiring sequential acquisition then execution gives threshold four.

The feasible curve is a constructive lower bound; the dashed curve is a proved upper bound. Neither is presented as the exact supercritical optimum.

Collective acquisition and future possibility

The finite operator theory evaluates new controls after canceling protected reset effects and charging their acquisition cost.

Result	Content	Scope
All-rank returned gain, T13	Exact gain formula including singular reset geometry	Finite-dimensional linear maps, independent whitened controls
Useful circuits, T14	Minimal useful combinations are dependency circuits with nonzero capability effect	Unpriced structural usefulness; paid optima may combine circuits
Arbitrary-depth barrier, T15	Every proper paid subset of an $M$-operator catalogue can lose, while the full set wins	Constructive theorem; survives positive definite baseline recovery
Finite planning closure, T16	Joint capability/reset Gram geometry, budget and remaining offers suffice	Exact known finite catalogue; optimization can be exponential
Robust intervention witness, T17	Task and reset bounds remain valid under specified map errors	Declared uncertainty norms and protection tolerances

The depth result applies to improvement rules that stop when all combinations within their fixed lookahead lose. It is not an impossibility theorem for every structured algorithm. The present returned capability operator alone is insufficient to predict the value of an added control.

Computational evidence, including failures

There are 18 experiment groups, 77 recorded numerical assertions, and 19 research unit tests. Raw outcomes are in results/results.json, results/extension_results.json, results/critical_results.json, and results/unit_tests.txt.

• All-rank geometry: 150 random trials cover old reset ranks zero through four. The maximum discrepancy against direct joint-nullspace calculation is approximately $4.58\times10^{-14}$.
• Paid combination barriers: exhaustive catalogues of two through eight operators verify examples where every proper paid subset loses and the complete set wins.
• Independent variational check: finite elements converge to the Bessel weighted-memory constant. The 256-node run missed the fixed $5\times10^{-6}$ tolerance; 512 nodes met it without changing the tolerance.
• Causal certificate: signed interventions with a stipulated noise bound yield a positive guaranteed task margin and a protected reset tolerance.
• Positive constructed learning result: forty linear teacher tasks improve held-out adaptation while algebraically preserving the declared old-input subspace and charging acquisition energy. Mean baseline MSE is about 0.1851; the coalition result is at numerical roundoff. The operator menu is supplied.
• Negative nonlinear results: three modular-addition MLP runs and one tiny transformer reached 100% training accuracy and 0% held-out accuracy. No successful grokking or useful autonomous nonlinear RSI is demonstrated.

A passing assertion can record a failure accurately. Assertion counts are not independent statistical replications or evidence of general algorithmic superiority.

Reproduction

Python 3.11+, CPU only. No GPU, pretrained weights, API keys or network access are needed after dependency installation.

python -m pip install -r requirements.txt
python -m unittest discover -s tests -v
OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python src/experiments.py
OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python src/extension_experiments.py
OPENBLAS_NUM_THREADS=1 OMP_NUM_THREADS=1 python src/critical_experiments.py

Windows: run REPRODUCE.bat. PDF rebuilding with python build_release.py additionally requires ReportLab, pdflatex and pdftotext. requirements.txt describes research dependencies; the Windows publication tool has a separate pinned dependency. Research reruns can update recorded runtime/results files; the publication integrity check intentionally detects those changes. Publish the clean release snapshot or deliberately create a revised edition.

python publishing/publish.py --check
python -m pip install -r publishing/requirements-publish.txt
python -m unittest discover -s publishing -p 'test_*.py' -v

The publication tests check the upload workflow using a fake Hub. They are separate from the 19 mathematical/computational research unit tests.

Dataset contents and provenance

This repository is a research artifact collection with three small JSONL tables: 20 theorem records, 10 definitions, and 77 numerical-check records. They are derived directly from the included metadata and raw results, with source paths and pointers. The Hub viewer uses explicit configurations so that manuscript metadata and unrelated JSON files are not mistaken for dataset rows. See the data dictionary.

The author attribution is reproduced exactly as requested. AI-generated research claims are accompanied by their assumptions and evidence status; no independent peer review is claimed. Prior user research is integrated and restated where required, with provenance in metadata/prior_research.json. Full third-party manuscripts are not bundled. The complete proofs needed for this release are included locally.

Prior art, limitations and open questions

The mathematical ingredients include classical geometric control, shorted operators, constrained least squares, matroid circuits, ground-state factorization, stochastic certificates and dynamic programming. The proposed contribution is their specific proved exploration structure and model-level consequences. The prior-art audit and citation manifest identify similarities, differences and unresolved priority.

The release does not establish a universal cognitive phase partition, a task-independent exploration scalar, the same threshold for arbitrary nonlinear/noisy dynamics, autonomous discovery of neural operators, practical superiority over established learning methods, or unbounded improvement with bounded physical resources. The next decisive empirical step is causal operator discovery in nonlinear learners under matched resources and held-out protection tests.

Citation, licenses and preservation

Use CITATION.cff or CITATION.bib. Cite the research as version 2.0.0; hf1 identifies this publication wrapper. For reproducible references after upload, add the actual immutable Hub commit from the publisher receipt.

Original documents, figures, metadata and research records: CC BY 4.0 under LICENSE-DOCUMENTS.md. Original source and publication code: MIT under LICENSE-CODE. Third-party cited works retain their licenses. No trained model is distributed.

Root SHA256SUMS covers the publication payload. PUBLICATION_MANIFEST.json records file sizes and hashes and drives the uploader's explicit file list. Publication notes explain the preserved original archive and edition changes. Public upload does not imply a DOI, paper-page acceptance, guaranteed search placement or external validation.