Critical Windows of Complexity Control: When Transformers Decide to Reason or Memorize

Transformers trained on modular arithmetic exhibit sharp transitions between memorization, generalization, and collapse. We show that weight decay acts as a scalar empirical control parameter for these regimes, and introduce two cheap online diagnostics, mean pairwise attention-head cosine similarity and entropy standard deviation, that track training dynamics from attention activations alone and complement loss-landscape diagnostics at lower compute cost. Across eleven experimental conditions and three model scales (0.82M to 85M parameters), the weight-decay axis separates memorization, developmental grokking, and collapse. A near-transition logistic fit localizes the memorization-to-developmental boundary at $λ_c=0.0158$ (95% CI [0.0109, 0.0200], N=210); a power-law fit gives an empirical exponent $ν=0.757$ (CI [0.725, 0.799]). Reference exponents $ν=1/2$ and 3D Ising $ν\approx 0.63$ lie outside this empirical CI under our four-bin grid, so we report $ν$ as empirical and defer universality-class identification to denser finite-size-scaling work. A horizon-matched multi-task replication (n=280, four modular operations) preserves the weight-decay control pattern; a paired attention-head re-initialization experiment at $λ=0.05$ changes Phase-2 amplitude (Cohen's $d=-1.190$, n=10, $p_t=4.5 \times 10^{-3}$), while matched weight-norm clipping does not. Three cross-architecture probes (4L MLP, 4L LSTM, and 4L Mamba; each n=70) replicate the weight-decay-controlled transition with architecture-specific $λ_c$ values. Main diagnostic claims are scoped to modular arithmetic in small transformer attention models; the non-attention experiments are scope probes, and architecture-wide, language-model, and universality-class claims are out of scope.

While empirical scaling laws for LLM reasoning are well-documented, the theoretical mechanisms governing out-of-distribution (OOD) generalization remain elusive. We formalize reasoning via optimal transport, projecting discrete trajectories into a continuous metric space to quantify domain shifts using the Wasserstein-1 distance. Invoking Kantorovich duality, we bound OOD generalization via architectural Lipschitz continuity and functional approximation limits. This exposes two primary constraints. First, position-dependent attention (e.g., Absolute Positional Encoding) fails to preserve shift invariance, yielding an $Ω(1)$ Lipschitz constant and expected risk, whereas shift-invariant mechanisms (e.g., Rotary Embeddings) preserve equivariance and bound the error. Second, by mapping sequential backtracking to a Dyck-$k$ language, we establish a strict circuit depth lower bound for $\text{TC}^0$ Transformers. Scaling physical layer depth is necessary to avert representation collapse -- a constraint that scaling representation width cannot bypass due to irreducible approximation bounds in Barron spaces. Evaluations across 54 Transformer configurations on combinatorial search corroborate these bounds, demonstrating that generalization risk degrades monotonically with the Wasserstein domain shift.

Standard Transformers have a fixed computational depth, limiting their ability to generalize to tasks that require variable-depth reasoning. The usual remedy, Chain-of-Thought (CoT), spends tokens to reason, inflating the key--value cache and making latency grow with the step count, so memory becomes the limiting cost when reasoning is served over large query batches. We study a depth-recurrent Transformer that decouples computational depth from parameter count by iterating a shared-weight block, so that each added reasoning step costs flat memory and linear latency, with no token generation. Three ingredients keep the recurrence stable for 20+ thinking steps: a silent thinking objective that supervises only the final output, LayerScale initialization, and an identity-biased gate that opens a gradient highway across steps. We characterize it on three compositional domains with decreasing structural bias: graph reachability (adjacency masking), nested boolean logic (relative positioning), and unstructured relational text (no positional cue). We find a \emph{computational frontier}: accuracy climbs once the thinking-step count meets the task's complexity, reaching near-perfect performance on the two structured tasks and a lower plateau on unstructured text. How it climbs depends on the structural bias---abruptly from chance on the graph task, gradually on the other two. Depth recurrence extrapolates beyond the training range: it succeeds on the graph task where fixed-depth models barely extrapolate, and on the two sequence tasks comes within two points of fixed-depth Transformers that use $4$--$6.4\times$ more parameters. On the graph task, whose adjacency mask makes propagation depth verifiable, intermediate per-step supervision---a standard recipe for deep iterative models---consistently \emph{harms} this extrapolation. We release the code for reproducibility.