Choosing the rank of a low-rank adaptation (LoRA) update is usually an empirical task. In this paper, we provide a task-dependent theory of the approximation error achievable at each LoRA rank for Transformer attention. We fix a pretrained attention head, a target attention function, and a distribution over inputs from the downstream task, and bound the smallest expected Kullback--Leibler (KL) error achievable by a rank-$r$ query LoRA update. When target attention probabilities are bounded away from zero, we prove a lower bound of the error proportional to $ψ(\|d\|_2)$, where $d$ is the difference between candidate and target attention scores and $ψ(t)=\min\{t^2,t\}$. We also prove an unconditional upper bound $\min\{\|d\|_2^2/4,\sqrt2\|d\|_2\}$. Under explicit realizability, geometry, and moment conditions, we then bound the best rank-$r$ error between an explicit multiple of $ψ(\sqrt{T_r})$ and $\min\{T_r/4,\sqrt{2T_r}\}$, where $T_r$ is the downstream-weighted tail energy of the target update. We also provide target-Fisher bounds when candidate scores remain within a fixed range of the target scores, and an unrestricted lower bound when a subset of tokens carries most of the probability mass. These spectral bounds describe finite-score approximation. We then construct explicit families in which softmax saturation makes the rank required to match the attention function strictly smaller than the rank required to match the finite logits. Finally, we extend the analysis to fused multi-head LoRA and joint query/key updates, exposing the effects of rank sharing and query/key factorization constraints.
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.