Learning must acquire new capabilities while preserving both prior responsibilities and the capacity to learn again. We introduce Generalized Residual Closure (GRC), a framework for learning as recursive closure of future-relevant discrepancies: closing a residual establishes the conditions for subsequent prediction, interaction, and learning. Under a complete representation-relation description at a fixed learner-world boundary, persistent internal learning has two primitive modes: Transformation within a representation and revision of the Representation itself. We establish a local tangent decomposition under regularity assumptions and a criterion for when representation revision is necessary. In an affine model, we derive a necessary-and-sufficient condition for stability-plasticity compatibility and the unique solution of a constrained quadratic update problem, which preserves registered old responsibilities while reducing residuals with an effective safe response. We prove that reconstructive semantic protection weakly enlarges the safe-response operator relative to preserving an exact historical realization. Dynamic sufficiency and future-closure viability extend representation adequacy from current prediction to lawful future updating and continued learning. Growth Learning expands the lawful closure domain or lowers optimal closure cost without regression of the registered capability-cost frontier; a conditional commit rule maintains this order. Restricted-sector recoveries and a conditional representation theorem connect the framework to optimization, machine learning, and control. Together, these results organize adaptation, representation revision, and reusable capability within a common account of continued learning.
Figures & tables
Figure 2: Stability is a geometry of lawful change. A line and a patch of a plane illustrate nested safe subspaces; the drawing is conceptual, not to scale. Reconstructive protection permits additional directions when historical realization constraints are unnecessary. Strict response gain still requires those directions to affect the residual through B .
Paradigm
Predeclared structure
Candidate GRC revision
Optimization
Coordinates and metric
Representation or metric
Deep learning
Differentiable family
Outer machine family
ResNet
Interface and additive form
Owner or correction form
LoRA/adapters ( Hu et al., 2022 )
Factorized edit family
Rank, family, or owner
RAG ( Lewis et al., 2020 )
Evidence-access boundary
Separate missing evidence from internal failure
RL/control
State/action abstraction
State representation
Table 1: Representative restricted choices and candidate GRC revisions. Rows describe conditional correspondences, not universal algorithmic equivalence or guaranteed improvement.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Mechanism
Correspondence and status
Choices that may reopen
Quadratic optimization
Exact stated quadratic-step algebra with frozen representation
Edit geometry, representation
Deep networks
Conditional containment of a declared differentiable family
Machine/program family
Residual blocks
Additive forward-map structure
Correction owner and form
Attention
Exact allocation for the stated entropic read objective
Read grammar and candidates
RNN/SSM/filtering
Conditional realization of update-sufficient state
Recurrent-state representation
Sparse experts
Structural ownership interpretation
Owner creation, merge, retirement
Appendix
Table 2: Status of the sector map. Reopened choices are candidate freedoms, not guarantees of discovery, strict improvement, or lower complete cost.
What should a machine learning model learn when data is missing during training? We look at the learning process from a dynamical systems perspective, cast data missingness as a structured loss of actuation that limits controllability of the parameter error dynamics, and ultimately derive adaptation mechanisms with Lyapunov stability characteristics that throttle model updates in ways that preserve learning coherence under partial, intermittent observability. Under recurrent excitation, our analysis provides ISS-type residual-to-state bounds with respect to a bounded closed-loop mismatch between the loss residual and the preconditioned update geometry. We evaluate the efficacy of our directional observability-aware adaptive learning approach on multimodal contexts, reinforcing its premise in promoting learning coherence and stability even in pathologically sparse domains and problems.
Dimitrios Pylorof, Humberto E. Garcia
U.S. Department of Energy Idaho National Laboratory
Memory consolidation determines both what a learner can do now and which changes remain implementable later. We develop a finite-model synthesis of operational state abstraction and optimal control under the stability-evidence-revision (SER) framework. ``Poincaré meets Bellman'' names two complementary roles: qualitative dynamics identifies reusable action-response structure, and dynamic programming prices acquisition, retention, reuse, merging, and forgetting. Recurrence enters separately through the timing and value of future demands. We distinguish active quotient merging from historical information erasure, characterize exact repair by zero-error functional coding and causal migration, and derive a Bellman recursion over the joint law of hidden state and complete deployed memory. A first-return model yields an explicit retention rule. Conditional results show how factor sharing avoids enumerating combinations and how independent informative observations improve identification, while leaving some zero-error evidence budgets unchanged. Finite enumerations verify the coding and retention calculations. The synthesis gives an exact benchmark for specified finite models, without claiming universal recurrence, bounded-memory open-ended learning, or tractable global planning.
Xin Li
Department of Computer Science University at Albany
Neural networks can acquire new capabilities while damaging existing ones, but what determines whether new learning persists remains unclear. We identify functional compatibility, the extent to which incoming learning can coexist with behaviour that must be preserved, as an experimentally manipulable causal determinant of persistence. From identical neural states, we vary compatibility while matching unrestricted learning opportunity and imposing a common retention requirement. Persistent learning increases with compatibility across independent directions, convolutional and transformer architectures, vision and text, and a ten-seed replication. Learning rules and retention constraints determine how much compatible opportunity is retained, whereas nonlinear geometry limits the matched intervention at larger update norms. Functional compatibility therefore reframes stability-plasticity from preventing forgetting to determining which new learning can coexist with existing function and persist.
Hossein Javidnia
School of Computing Dublin City University Dublin, Ireland