General Machine Learning: Theory for Learning Under Variable Regimes

We study learning under regime variation, where the learner, its memory state, and the evaluative conditions may evolve over time. This paper is a foundational and structural contribution: its goal is to define the core learning-theoretic objects required for such settings and to establish their first theorem-supporting consequences. The paper develops a regime-varying framework centered on admissible transport, protected-core preservation, and evaluator-aware learning evolution. It records the immediate closure consequences of admissibility, develops a structural obstruction argument for faithful fixed-ontology reduction in genuinely multi-regime settings, and introduces a protected-stability template together with explicit numerical and symbolic witnesses on controlled subclasses, including convex and deductive settings. It also establishes theorem-layer results on evaluator factorization, morphisms, composition, and partial kernel-level alignment across semantically commensurable layers. A worked two-regime example makes the admissibility certificate, protected evaluative core, and regime-variation cost explicit on a controlled subclass. The symbolic component is deliberately restricted in scope: the paper establishes a first kernel-level compatibility result together with a controlled monotonic deductive witness. The manuscript should therefore be read as introducing a structured learning-theoretic framework for regime-varying learning together with its first theorem-supporting layer, not as a complete quantitative theory of all learning systems.

💡 Research Summary

The paper introduces a novel theoretical framework called General Machine Learning (GML) to study learning processes that operate under variable regimes—situations where the data source, task definition, update mechanism, evaluator, and memory rules may change over time. Traditional learning theory (PAC, Bayesian PAC, online learning, universal induction) assumes a fixed evaluative frame; when regimes shift, the very identity of the learning problem can be lost. GML addresses this by imposing a Hadamard‑style well‑posedness requirement: admissible learning trajectories must exist, be identifiable at an appropriate equivalence level, and remain stable under small perturbations of experience, memory, or regime.

The core objects are (i) a protected‑core operator Φ that captures the evaluative component that must stay invariant across regime changes, (ii) an admissibility predicate Γ that certifies which regime transitions are semantically legitimate, and (iii) a protected equivalence relation ∼_Φ that declares two evaluators equivalent when they represent the same learning problem at the protected level. Together with a regime carrier set R, local state spaces S_r, memory spaces M_r, interfaces O_r, actions A_r, and local evaluators V_r, the framework formalizes a learning system as a tuple G = (R, {S_r}, {M_r}, {O_r}, {A_r}, {V_r}, T, Φ, Γ, ∼_Φ, U).

Four standing structural requirements are imposed: structural closure, dynamical stability, bounded statistical capacity, and evaluator invariance. Under these, the canonical definition (Definition 3.1) states that a system learns when its evaluative performance improves across active regimes while remaining admissible, coherent, structurally stable, and compatible with the protected core.

The paper’s first theorem‑supporting layer consists of several key results:

1. Strict Extension Theorem (Section 6.1) – Shows that classical fixed‑ontology learning is a degenerate boundary case of GML; when regime curvature is negligible, existing PAC and online results are recovered.

2. Protected‑Stability Template (Section 6.2) – Provides sufficient conditions for memory and evaluator cores to survive admissible regime transformations. It introduces a regime‑variation cost and proves non‑vacuity through both numerical (convex) and symbolic (deductive) witnesses.

3. Structural Obstruction Argument (Section 6.3) – Demonstrates that in genuinely multi‑regime settings a faithful reduction to a single ontology is impossible when protected cores and memory dependencies are regime‑specific, formalizing an “obstruction” to fixed‑ontology reduction.

4. Kernel‑Level Semantic Alignment (Section 7) – Establishes a partial kernel‑level alignment principle for layers that are semantically commensurable at the protected level, enabling limited functional isomorphisms across regimes.

A concrete two‑regime example (Section 8) illustrates these concepts. The first regime is a convex regression setting where admissibility certificates, protected cores, and variation costs are computed analytically. The second regime is a deductive logical inference setting; symbolic witnesses exhibit monotonicity of the protected core and validate the transport structure. Both examples confirm that the protected‑stability template is substantive and that the admissibility predicate can be satisfied in practice.

The manuscript also situates GML relative to existing learning theories (Section 9, Appendix A), arguing that GML subsumes them as special cases while offering a principled way to handle evaluator and memory evolution. Limitations are candidly discussed: the current work does not provide explicit sample‑complexity bounds, nor does it prescribe concrete algorithms for optimal regime transport. Future directions include deriving statistical capacity bounds for specific families of admissible transports, designing algorithms that respect protected‑core constraints, and extending kernel‑alignment theory to richer semantic layers.

In summary, the paper establishes a rigorous, structural foundation for learning under changing regimes, defines the minimal objects required for well‑posedness, proves several foundational theorems, and supplies concrete examples. It opens a new line of inquiry that shifts the focus from static performance improvement to the continuity and admissibility of the learning problem itself across dynamic environments.