EIDRION: A Shared-Center Theory of Functional Memory and All-Future Retention in Continual Learning
Year: 2026
Type: article
Abstract: EIDRION develops a conditional theory of continual learning in which historical prediction contracts are maintained through a shared operator center, nested protected subspaces, and independently calibrated functional bounds. Its central construction permits historical reference predictors to be discarded after calibration while preserving a statistical retention contract covering every future admitted update. The distinctive contribution is the connection between algebraic memory invariants, cumulative innovation energy, a bounded nonlinear residual, and one fixed exceptional set for future prediction changes. This creates a precise alternative to retaining a separate teacher network for every acquisition or treating stability solely through local gradient constraints. The theoretical mechanism Fix an external feature map φ and a finite output-class universe. The learner maintains a shared center M_t and an expanding protected feature span U_t. Its predictor combines the center readout with a trainable nonlinear residual: F_t(x) = M_t*φ(x) + p_t(x)u_t(x), where p_t(x) is the distance of φ(x) from U_t, and the residual satisfies a pointwise norm bound ‖u_t(x)‖ ≤ R_t. Each center innovation lies orthogonal to the previously protected span, and its directions are incorporated into the enlarged span. These conditions produce two linked invariants: old protected projections remain unchanged, and the squared distance between an earlier and later center equals the difference of their squared Hilbert–Schmidt norms. Consequently, deviation from an earlier center is bounded by: ‖F_t(x) − M_s*φ(x)‖ ≤ [√(‖M_t‖² − ‖M_s‖²) + R_t]p_s(x). This is a finite-update functional bound. It does not depend on a small-step Jacobian approximation. All raw neural layers may undergo arbitrary finite changes within the declared residual envelope; the external geometry defining that envelope remains fixed. What makes the synthesis distinctive At acquisition, a signed margin-to-power score measures how much admissible functional change can occur before the actual reference prediction changes. The score explicitly accounts for lossy consolidation into the shared center: an inaccurate or mismatched center cannot obtain protection simply by being called a memory. An independent sample calibrates a tolerance threshold for this fixed score. Strict admission conditions then place every future disagreement set inside the same exceptional set. Under the stated assumptions, this bounds the probability that an input drawn from the acquisition distribution ever experiences a change of reference prediction, throughout any countable sequence of admitted states. The confidence event covers future adaptive choices of center innovations, protected geometry, residual networks, and admissible radii. There is no additional confidence penalty for each gradient step or candidate update. Each newly acquired contract still requires valid evidence and its own confidence allocation. The resulting memory consists of a shared center, the complete protected span, fixed features, and certificate metadata. Historical reference networks need not remain stored after legitimate calibration. Their removal does not eliminate the costs of protected rank, evidence acquisition, feature storage, numerical control, or restricted plasticity. Additional theoretical contributions The manuscript develops several consequences and boundaries of this construction: Minimal protected-span growth: an orthogonal innovation requires its matrix rank in additional protected directions; centered logits bound that rank by the number of classes minus one. A functional response analysis: the stated smooth bounded residual preserves the rank of the centered raw-network Jacobian at positive residual power and radius, without guaranteeing successful optimization. Query-complete memory: preserving today’s center is weaker than preserving the information needed for a declared family of future queries. A supplied counterexample shows how center-preserving basis deletion can destroy tomorrow’s retention guarantee. Evidence–capacity duality: a separate linear-Gaussian reference model connects recoverability-aware evidence allocation with query-weighted residual modes and minimum-energy response geometry. Numerical defect accounting: explicit terms distinguish the exact-real invariants from leakage, imperfect nesting, center reconstruction defects, and forward arithmetic error. Orthogonal projection, Schur complements, Gaussian conditioning, minimum-energy control, and statistical tolerance bounds are established ingredients. The proposed contribution concerns their particular shared-center, prefix-energy, and discarded-reference synthesis. Historical priority for that exact synthesis remains unverified. Implementation and falsification Three PyTorch mechanisms implement distinct contracts: nested operator memory, immutable decision regions, and fresh checkpoint renewal. Their guarantees are deliberately separated. Reference retention can preserve acquisition errors; fresh renewal certifies accepted checkpoints and does not automatically imply an all-future ever-error bound. The source capture contains 260 locked MNIST/Fashion-MNIST runs and 96 structural runs. All 356 planned runs completed. Fresh preparation checks passed all 39 grouped fixtures and reconstructed 1,561 aggregate/interval values from bundled saved audit records. The standard experiments expose a substantial limitation: certified policies do not achieve the best mean accuracy on either tested image dataset. Strong simple prototype controls outperform them. The scalar energy certificate can become conservative enough to block useful acquisition, while stricter renewal can reject competent proposals. These negative regimes are retained as evidence identifying where better directional certificates, valid evidence reuse, and justified representation migration are needed. Scope and research significance The principal theorem requires fixed features and output classes, valid nested invariants, independent calibration, unchanged declared distributions, and appropriate numerical error control. The implementation provides float64 diagnostics rather than a globally verified interval enclosure. Modern-model scalability and general superiority remain unestablished. For researchers in continual learning, learning theory, kernel methods, statistical certification, and neural architecture design, EIDRION provides a concrete framework for investigating whether useful long-term plasticity can be governed by compact functional contracts. Its potential lies in making the retention–plasticity–evidence tradeoff explicit and mathematically auditable. Release contents and completeness: scientific version 1.0.0 includes a 58-page manuscript, executable source, theoretical supplements, positive and negative results, and reproduction guidance. The distributed main package has 100% coverage of its 3,197 manifest-listed main files. Four large raw-evidence companion archives are identified but absent; complete historical prediction/checkpoint auditing requires those companions. A second complete training reproduction was not performed.
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221062306