Kathleen Remembers: Length-Invariant One-Shot Recall Without Attention

Recurrent, attention-free sequence models share a structural weakness: a fading state cannot perform exact recall of something seen once, far in the past. We add to the Kathleen trunk a second memory layer -- a "notebook": a fixed-key holographic (HRR) associative store with a learned local write gate, a self-gating raw read, and write-triggered forgetting -- 25K parameters that attach to the logits of any trunk. (1) Mechanism: on a controlled needle-in-haystack task the notebook reaches 80-82% one-shot recall at 4x the training length, where the bare trunk scores ~4% and a parameter-matched attention head scores 100% inside its training length and 0% beyond it. Addressing is length-invariant by construction; the untrained memory alone recalls at 90% accuracy identically at 512, 2048 and 4096 bytes. Because the store is a linear superposition, two capabilities follow from arithmetic alone: selective unlearning (one subtraction erases one fact to chance, retained facts unharmed) and per-token attribution (counterfactual erasure names the source fact of every correct byte, 100% provenance). (2) Real text: on WikiText-2 bytes the notebook improves prediction of repeated rare words by +0.15-0.27 bits/byte, the gain growing with the distance between mentions and holding zero-shot at 4x training length; write-triggered forgetting eliminates memory pollution at 8x length (first-mention cost +0.33 -> -0.004). (3) Scope and scale: a parameter-matched attention head does generalize on natural-text repetition, so the notebook's claim is exact recall at O(L); on a WikiText-103 ladder (8 to 512 MB) the zero-shot repeat gain rises monotonically. All experiments are pre-registered, seeds reported, and reproducible on a single free-tier GPU.

Infrared Universality of Collective Dynamics across Transformer and State-Space Architectures

Whether distinct neural architectures develop common collective dynamics remains an open question. Recent analysis of Transformer language models revealed a nearly flat, weakly infrared-enhanced time-scale density of states (TDOS) associated with near-marginal long-memory dynamics. Here we test whether a closely related organization emerges in Mamba, whose selective state-space dynamics provides a fundamentally different microscopic mechanism. Mamba allows relaxation dynamics to be resolved at three levels: the intrinsic spectrum of the learned state-space generator, its input-conditioned selective rescaling, and the collective TDOS of the complete block measured from its Jacobian. These spectra are not identical: selective dynamics and the remaining block transformations substantially reorganize the microscopic relaxation hierarchy. Nevertheless, the full block develops a reproducible slow-mode continuum whose infrared sector becomes progressively better resolved with increasing sequence length. Cumulative analysis yields $ρ(λ)\simλ^β$, with the long-sequence Mamba exponent stabilizing near $β_{\rm M}\simeq-0.17$. The corresponding memory dynamics follows $K(t)\sim t^{-(1+β)}$, close to the marginal $1/t$ regime. Despite fundamentally different microscopic dynamics, Transformer full-block spectra exhibit closely related infrared organization, with representative exponents of order $β_{\rm Tr}\sim-0.1$. These results separate explicit state-space memory from collective infrared organization and show that distinct sequence architectures can develop closely related near-marginal slow-mode dynamics. They extend infrared collective organization beyond Transformers and provide an independent test of the dynamical structure described by Cognitive Field Theory.

HOPPER: Learnable Hop Extraction for Linearized Graph Sequence Models

Graph neural networks typically propagate information through repeated message-passing layers, coupling the distance over which information travels with the number of nonlinear transformations applied. This coupling can make deep architectures difficult to optimize and can lead to over-smoothing, over-squashing, and the loss of long-range information. Linearized Graph Sequence Models (LGSMs) address this issue by separating information depth from processing depth and treating the successive propagation states of each node as a sequence. However, existing LGSMs construct these sequences using fixed graph operators, limiting their ability to adapt propagation to the input graph, node features, and downstream task. We introduce HOPPER, an end-to-end learnable extension of LGSM that learns how hop sequences should be extracted before they are processed by a modern state-space model. Our framework supports feature-conditioned, structure-aware, graph- and hop-adaptive propagation mechanisms while preserving permutation equivariance. Standard adjacency-based and non-backtracking LGSM sequences arise as special cases of our proposed extractor family. We show that HOPPER is state-of-the-art or competitive across the ECHO-Synth benchmark, and that varying the maximum neighborhood size of message backtracking cancellation (i.e. structural memory window) can optimize accuracy on the LRIM physics-based long-range dependency benchmark. These results demonstrate that learnable sequence extraction provides a flexible and effective approach to long-range graph representation learning.

The Quantum Shortcut: Complex Phase-State Dynamics Reduce the Optimization Steps of Sequence Models

Sequence models are conventionally distinguished by their backbone, the mechanism that routes information across positions, such as attention or recurrence. This paper varies a choice that is prior to the backbone and shared by nearly all current models: the \emph{substrate}, the number system in which the hidden state is represented together with the form of the map from state to prediction. The prevailing substrate is a real-valued state with an affine--softmax readout; we study a complex-valued alternative drawn from the mathematics of quantum theory, in which information is carried by the phases of the state and scores are quadratic Born forms. Prior work proved an idealized version of this substrate representationally stronger than any real model with a linear readout; we ask whether it also trains faster. Relaxing the two properties that block deployment, exact unitarity and the Born vocabulary readout, we instantiate it in the Mamba state-space model and an attention-based Transformer. At 253M parameters, matched to within $0.02\%$ and trained under one fixed protocol on three byte-level corpora, the complex models reach every measured validation loss in approximately one third (state-space) and one half (attention) of the optimization steps of their real counterparts. The two backbones then diverge. Once the learning-rate warmup ends, the state-space advantage continues to widen, from $0.321$ to $0.354$ bits per character on OpenWebText and from $0.368$ to $0.396$ on FineWeb, which an artifact of the warmup ramp would not do; the attention advantage instead decays toward zero on every corpus, and is therefore an effect of early training.

Indexing: the Beginning and the End

We study information bottlenecks in modern deep-learning architectures -- RNNs, softmax transformers, linear-attention transformers and state-space models -- through the lens of the indexing primitive. In this primitive, the input consists of $n$ bits and one integer $i$ from $1$ to $n$ called the index, and the output equals the value of the $i$-th bit. We introduce causal complexity for masked architectures. We show that architectures with low causal complexity cannot solve the indexing primitive in any constant number of layers when the index appears at the end of the input. In particular, this limitation applies to low-parameter RNNs, SSMs and masked linear-attention transformers. In contrast, small softmax transformers can solve it in one layer, while non-masked linear-attention transformers can solve it in 2, which separates them from their masked counterparts. In turn, when the index appears at the beginning, we show that small RNNs are capable of solving this task in 1 layer, while all the other architectures require 2. All our impossibility results are unconditional and apply even to models that employ infinite-precision real arithmetic. Moreover, experiments for up to $n=64$ qualitatively align with our theory: configurations with low-parameter theoretical solutions learn the indexing task easily, while configurations that do not admit such theoretical solutions struggle to learn as the sequence length grows.

Phasor Attention: Mean Root Square Normalization for Phase Manifold Preservation

While Root Mean Square Normalization has become the de facto standard for accelerating modern sequence models, its reliance on the quadratic accumulation of independent scalars ($\sum x^2$) inherently triggers outlier-induced numerical instability, gradient starvation, and anisotropic phase distortion. We introduce Mean Root Square Normalization (MRSNorm). By structurally pairing channels into 2D phasors, MRSNorm mathematically inverts the traditional scaling paradigm: it computes the localized $L_2$ magnitudes (Root Square) before aggregating them via a global $L_1$ average (Mean). This operational inversion strictly constrains activations to a phasor manifold, preserving conformal invariance. By sharing a single affine weight across phasor components, MRSNorm halves the total number of learnable parameters, proving that unconstrained spatial scaling in standard norms is a harmful redundancy. We analytically demonstrate that this geometric constraint yields a built-in, trigonometric gradient clipper governed by the Pythagorean identity, unconditionally equalizing the local gradient norm to ensure Gradient Homogeneity. Empirical evaluations on a ResNet with CIFAR-100 show that despite halved parameters, MRSNorm provides critical structural stability under rigorous stress tests. Under extreme hyperparameter settings where standard normalizations suffer from gradient divergence, MRSNorm successfully prevents numerical explosion and secures stable optimization trajectories. Our findings propose a fundamental paradigm shift toward phasor-based deep representation learning. The implementation of MRSNorm is available at Appendix C.

Sequential Correlations Change In-Context Learning: Effective Context Length and Architectural Mismatch

Modern sequence models have a striking capacity for in-context learning (ICL); they can perform new tasks based only on examples given in the prompt. Understanding how this ability emerges requires theory that captures important properties of natural data. Linear regression has served as a useful sandbox for ICL theory, but existing work has largely focused on prompts with independent examples. In this work, we extend this setting to sequentially correlated data, a basic feature of real sequences. We present a solvable model based on linear attention and test our predictions on realistic transformer architectures. We identify two distinct effects: First, when the query token is independent of the context, within-context correlations induce an effective context length: correlated prompts behave like shorter i.i.d. prompts. Second, when the query is also correlated with its context, test error is reduced, particularly for softmax attention when compared to linear attention. These results suggest that correlated prompts alter not only the effective sample size of in-context learning, but also which attention architectures are best matched to the task.

Can In-Context Learning Support Intrinsic Curiosity?

Effective machine learning depends not only on how we model data, but also on what data we choose to collect. While large sequence models have revolutionized data modeling, the problem of automated data selection, or "intrinsic curiosity", remains a significant challenge. Classic approaches incentivize exploration by rewarding an agent based on its "learning progress", which measures how much a newly acquired observation improves a world model's predictive ability. However, evaluating these rewards traditionally requires expensive inner loops of gradient descent updates within each trajectory, rendering them computationally impractical at scale. In this work, we investigate whether the emergent in-context learning (ICL) capabilities of sequence models can eliminate this bottleneck by serving as immediate, update-free world models. Specifically, we evaluate whether an exploration policy can be trained to maximize learning progress, using solely the prediction errors and counterfactual context manipulations of an in-context learner. We first prove that in general Markov decision processes, this is in fact impossible in an unbiased way: the resulting intrinsic rewards either suffer from nuisance terms that bias their estimation of true learning progress, or they cannot be implemented using an in-context learner's prediction errors. Conversely, we prove a positive result for a broad subclass of non-temporal settings, encompassing active learning and Bayesian Experimental Design: here, ICL-derived rewards successfully bound and asymptotically converge to the true learning progress. We corroborate our theory with controlled experiments across continuous and symbolic environments, demonstrating that our ICL-driven framework successfully trains curious data-collection policies that explore optimally.

A Held-Out Transition-Pair Falsifier for Long-Horizon Non-Abelian State Tracking

State tracking exposes a sharp limitation of sequence models: the relevant signal is often not a summary of observed tokens, but an ordered latent state that evolves through non-commutative transformations. We introduce a held-out transition-pair falsifier for finite non-Abelian group tracking. The protocol forbids selected ordered generator pairs during training and requires the same local patterns during evaluation, blocking one direct local-transition memorization pathway. In a controlled $S_3 \times S_3$ benchmark, a projected recurrent state model trained only on length-8 sequences produces error-free final-state predictions (perfect 250/250 per horizon) through evaluation horizons up to 1,048,576 tokens across five seeds. Matched native-readout baselines, including bag, GRU, and a single-configuration structured state-space model, remain near floor under the same protocol. Projection-matched GRU, structured SSM, and bag baselines equipped with analogous finite-group prototype readouts also remain near chance under the same split. Mechanism diagnostics show that hard projection coincides with low homomorphism error, low state-consistency drift, and non-trivial commutator separation, while softened projection collapses final-state accuracy. Clean-split audits verify zero verbatim reduced-word overlap and zero structural-template overlap between training and evaluation partitions. The evidence is scoped to this controlled finite-group falsifier rather than to a general architecture ranking. Within that regime, explicit projected non-commutative state composition acts as a useful inductive bias for long-horizon hidden-state tracking.

The Impossibility Triangle of Long-Context Modeling

We identify and prove a fundamental trade-off governing long-sequence models: no model can simultaneously achieve (i) per-step computation independent of sequence length (Efficiency), (ii) state size independent of sequence length (Compactness), and (iii) the ability to recall a number of historical facts proportional to sequence length (Recall). We formalize this trade-off within an Online Sequence Processor abstraction that unifies Transformers, state space models, linear recurrent networks, and their hybrids. Using the Data Processing Inequality and Fano's Inequality, we prove that any model satisfying Efficiency and Compactness can recall at most O(poly(d)/log V) key-value pairs from a sequence of arbitrary length, where d is the model dimension and V is the vocabulary size. We classify 52 architectures published before March 2026 into the triangle, showing that each achieves at most two of the three properties and that hybrid architectures trace continuous trajectories in the interior. Experiments on synthetic associative recall tasks with five representative architectures validate the theoretical bound: empirical recall capacity lies strictly below the information-theoretic limit, and no architecture escapes the triangle.