Ǝlūmiraŋ: Infinite Hex on the full infinite plane and Non-dyadic Laver algebras in 4D: Real-Time Neural Weight Rearrangement Through Nonlinear Ray-Tracing Operators, a novel RSI Method
Year: 2026
Type: article
Abstract: Discovering, certifying, and reusing the geometry of learning: an executable framework for bounded recursive self-improvement research. Ǝlūmiraŋ investigates a concrete problem for recursive self-improvement (RSI): how can a system discover useful changes to its own learning coordinates, determine when those changes remain valid, and reuse the recovered structure across subsequent tasks? This release presents a mathematical architecture and working implementation that connect hidden-geometry discovery, exact global optimization certificates, reusable adaptation, nonlinear Hamiltonian updates, and a controller that checks which guarantee applies to each proposed change. The central result is an exact recognition algorithm for a regular family of polynomial objectives whose apparent nonconvexity conceals a positive-definite quadratic problem behind a nonlinear canonical transformation. Starting from expanded rational polynomial coefficients, the algorithm recovers a hidden Lagrangian subspace from the highest homogeneous term, constructs rational symplectic coordinates, reconstructs an integrable polynomial shear, and verifies the complete objective identity. Acceptance yields an explicit global minimizer, its exact objective value, and a positive weighted sum-of-squares certificate. For RSI researchers, the distinctive contribution is the connection between discovering an optimization representation, verifying its applicability, and retaining it as reusable structure for future adaptation. The learned object changes how subsequent weight updates are computed. The recognition theorem covers the stated regular family in arbitrary even dimension and polynomial degree 2m ≥ 4; the title’s four-dimensional setting remains an implemented case. Distinctive mathematical and architectural results Hidden nonlinear geometry recovered from the objective. The useful coordinate chart is recovered algorithmically from the polynomial. A complete identity check establishes the global guarantee for an accepted input. Exact recognition has polynomial complexity in the dense rational coefficient input length, with coefficient acquisition charged separately. One-gradient adaptation after geometry discovery. For new objectives sharing the recovered chart and latent metric, one fresh gradient identifies an unknown latent target and determines the exact global minimizer. This gives a precise mechanism for amortizing structural discovery across tasks. Shared-family membership is a substantive assumption or a separately checked identity; one gradient alone does not certify it. A complete response to physical linear perturbations of the quartic family. Cached geometry reduces every such tilt to a smaller quadratic problem, distinguishing a unique global minimum, a nonunique family of minima, and unboundedness. This identifies mathematical boundaries at which an acquired optimization structure changes behavior. Sharp conditioning and executable nonlinear update geometry. The work includes an attained optimality theorem for symplectic transformations realizing a supplied descent direction, an optimal scaling rule, and a sharp square-root conditioning bound for scaled Newton directions. A separate construction realizes a prescribed second-order local update curve through at most seven explicit, globally invertible Hamiltonian shear stages. Certificate-carrying changes to learning coordinates. The integrated controller installs a recovered chart while preserving current effective weights, proposes adaptation, and checks the full task identity before assigning a global certificate. When that identity fails, it uses bounded fallback steps with exact actual-objective acceptance. The architecture therefore distinguishes global solution certificates from local decrease checks. Explicit contracts for algebraic actions and graph changes. Non-dyadic Laver-related tables supply action labels and rewrite constraints, including guards against invalid self-distributive reasoning. The Infinite Hex component supplies finite boundary-connectivity certificates for local replacements relative to an unchanged exterior. These are separately verified components with defined preservation properties. Results and verification The release combines mathematical proofs, exact rational certificates, independently implemented checking code, numerical experiments, and negative controls. 18 exact hidden-coordinate cases recovered and certified, spanning dimensions 2, 4, 6, and 8 and polynomial degrees 4 and 6. 1,995 exact identity evaluations performed by the separately implemented certificate checker, alongside structural, integrability, positivity, minimum, and stationarity checks. 32 changing-target tasks solved exactly with one fresh gradient per task after discovery. 256 numerical changing-target tasks solved in one update each. The experiment records 64 setup gradients, 256 online gradients, and 512 objective-value gates. 24 nonlinear tanh-neuron training runs, with 137 total accepted Gauss–Newton updates, 118 direct geodesic-acceleration updates, and 115 symplectic-chart updates. 12 globally certified new-task minima in the integrated controller, with two altered identities detected and six accepted fallback steps. All 18 accepted within-task changes strictly decreased their respective actual objectives. Eight deliberately tampered certificates rejected, with additional counterexamples showing the limits of finite gradient observations and untrusted degree assumptions. 68,161,536 finite-algebra triples and 327,680 glued-graph cases checked in the retained algebraic and connectivity modules. The archived reproduction run passed all ten stages. Fresh release audits also rechecked the 18 global certificates, all eight tampering controls, and the controller’s certified minima and fallback decreases. Potential for RSI research The framework offers a concrete setting for investigating self-improvement through reusable learning geometry. A system can acquire an internal optimization representation, preserve its current effective model while installing that representation, and use the retained structure to compute later adaptations under explicit mathematical conditions. This suggests research directions in reusable optimizer memory, certification of local polynomial models for neural parameter blocks, adaptation under changing objectives, reversible coordinate transformations, and controllers that monitor the validity of their own optimization assumptions. The perturbation classification provides a model for detecting when previously useful structure remains stable, becomes degenerate, or ceases to define a bounded problem. The separation between the discovery engine and certificate checker also supports experiments in which a complex or heuristic proposal mechanism is paired with a smaller exact verifier. For broader objectives, the included fallback architecture provides a starting point for combining structural proposals with measured improvement. Extending these mechanisms to large neural systems and preserving capabilities across repeated changes remain substantive research opportunities. Interpretation of the results The demonstrated self-modification concerns optimization coordinates and weight-update mechanisms within defined tasks. An equivalent method supplied with the same cached polynomial geometry matches the one-gradient result, and exact direct geodesic acceleration also achieves one-update solutions on the structured numerical target tasks. In the nonlinear-neuron experiment, the symplectic chart uses slightly fewer accepted updates but takes more elapsed time. The contribution is therefore the connected discovery-certification-reuse architecture and its exact mathematical contracts. The release does not establish unrestricted neural-loss recognition, general optimizer superiority, large-model scaling, autonomous AGI, or open-ended recursive capability growth. Historical priority of the combined formulation remains unestablished; the manuscript identifies and credits its mathematical and algorithmic precedents. Release contents Revision 2.0 includes the self-contained 38-page manuscript, editable manuscript sources, executable discovery and adaptation modules, separate certificate checkers, synthetic datasets, archived benchmark traces, exact certificates, controller audits, rejection examples, scientific figures, citation metadata, and integrity records. The exact engines use Python’s standard library; numerical experiments use NumPy. A single reproduction entry point reruns the checks while preserving the archived evidence. PureOne/Elumirang · Datasets at Hugging Face Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225166273