Training, learning and inference: unified dynamics of neural systems

Conventional neural networks strictly separate learning and inference because if parameters are updated during inference, outputs become unstable and even the inference function itself is not well defined [1, 2, 3]. This paper shows that DynamicGate MLP structurally permits learning inference concurrency [4, 5]. The key idea is to separate routing (gating) parameters from representation (prediction) parameters, so that the gate can be adapted online while inference stability is preserved, or weights can be selectively updated only within the inactive subspace [4, 5, 6, 7]. We mathematically formalize sufficient conditions for concurrency and show that even under asynchronous or partial updates, the inference output at each time step can always be interpreted as a forward computation of a valid model snapshot [8, 9, 10]. This suggests that DynamicGate MLP can serve as a practical foundation for online adaptive and on device learning systems [11, 12].

A learner's present predictions do not fully determine how it will learn next. We study the dynamics of hidden learning structure through two related operations. Fiber Regulation Theory (FRT) describes protected repair of existing function and the learning responses left available after repair. Fiber Acquisition Theory (FAT) describes the formation and renewal of trainable structure, distinguishing neutral installation, access to a learning signal, and additional functional directions. The two operations also constrain one another. A repair obligation can force a change in a learning channel, while acquired features can make new protected repairs feasible or reduce the minimum coefficient cost of existing ones. We characterize joint admissibility and response reachability, and give a finite supervised-learning construction in which the two operations together produce a favorable response absent from either operation alone. We implement protected head repair and fixed-capacity LoRA renewal in Qwen2.5-32B with AdamW, under replay and output distillation. Across two training trajectories differing in seed and within-task example order, the coordinated policy achieves higher teacher-forced Exact than either separate policy in all four comparisons. It also has the lowest macro NLL under both learning rules on the second trajectory. The implementations realize structure-conditioned self-regulation within specified information and action constraints: accessible state readouts and arrived evidence guide repairs and structural renewals, and ordinary learning continues from the modified state.

A learning system can occupy execution states that are indistinguishable under every declared present-behavior readout yet respond differently to future training. We formalize this through fiber fingerprints: controlled future-learning response laws restricted to present-behavior equivalence classes. Prefix-compatible finite probes induce a predictive quotient functor, a Nerode-type minimal recursively sufficient representation, and a canonical set-level predictive fiber without assuming smoothness, reversibility, finite rank, or a manifold. Under an explicit finite-dimensional Hilbert realization, response decomposes into visible, visible-mode-reuse, and irreducible-new sectors; a history-reachability bridge retains only distinctions generated by natural training histories. Conditional mechanism results then identify a graph-Hodge chronology decomposition, a regular switching class with root-mean-square scale $\sqrt{p}η^{3/2}$ and finite-scale corrections, and an exact Adam moment section whose immediate adaptive field is constant while common future gradients can reveal hidden moment differences. Frozen Transformer--LoRA--AdamW studies with Qwen2.5-7B and Mistral-7B-v0.3 support a local action backbone, longer-horizon first-return non-closure, and fresh visible-relative completion with output-range reuse and a low-rank irreducible sector. Stronger claims remain bounded by preregistered negative or mixed results: re-anchored transport is unresolved above its measurement floor; the strict finite-grid Hodge--$3/2$ conjunction is unmet despite prospective contraction; Qwen accessibility is not established in the frozen raw moment chart; and Mistral revelation is future-context dependent rather than bank invariant. Within these support-, scale-, metric-, and context-resolved boundaries, present behavior is not a sufficient statistic for declared future learning.